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[
"CONFORMAL CONTINUATIONS AND WORMHOLE INSTABILITY IN SCALAR-TENSOR GRAVITY",
"CONFORMAL CONTINUATIONS AND WORMHOLE INSTABILITY IN SCALAR-TENSOR GRAVITY",
"CONFORMAL CONTINUATIONS AND WORMHOLE INSTABILITY IN SCALAR-TENSOR GRAVITY",
"CONFORMAL CONTINUATIONS AND WORMHOLE INSTABILITY IN SCALAR-TENSOR GRAVITY"
]
| [
"K A Bronnikov \nInstitute of Gravitation and Cosmology\nCentre for Gravitation and Fundam. Metrology\nVNIIMS\n3-1 M. Ulyanovoy St119313MoscowRussia\n\nPFUR\n6 Miklukho-Maklaya St117198MoscowRussia\n",
"S V Grinyok ",
"\nBauman Moscow State Technical University\nMoscowRussia\n",
"K A Bronnikov \nInstitute of Gravitation and Cosmology\nCentre for Gravitation and Fundam. Metrology\nVNIIMS\n3-1 M. Ulyanovoy St119313MoscowRussia\n\nPFUR\n6 Miklukho-Maklaya St117198MoscowRussia\n",
"S V Grinyok ",
"\nBauman Moscow State Technical University\nMoscowRussia\n"
]
| [
"Institute of Gravitation and Cosmology\nCentre for Gravitation and Fundam. Metrology\nVNIIMS\n3-1 M. Ulyanovoy St119313MoscowRussia",
"PFUR\n6 Miklukho-Maklaya St117198MoscowRussia",
"Bauman Moscow State Technical University\nMoscowRussia",
"Institute of Gravitation and Cosmology\nCentre for Gravitation and Fundam. Metrology\nVNIIMS\n3-1 M. Ulyanovoy St119313MoscowRussia",
"PFUR\n6 Miklukho-Maklaya St117198MoscowRussia",
"Bauman Moscow State Technical University\nMoscowRussia"
]
| [
"Grav. & Cosmol",
"Grav. & Cosmol"
]
| We study the stability of static, spherically symmetric, traversable wormholes existing due to conformal continuations in a class of scalar-tensor theories with zero scalar field potential (so that Fisher's well-known scalar-vacuum solution holds in the Einstein conformal frame). Specific examples of such wormholes are those with nonminimally (e.g., conformally) coupled scalar fields. All boundary conditions for scalar and metric perturbations are taken into account. All such wormholes are shown to be unstable under spherically symmetric perturbations. The instability is proved analytically with the aid of the theory of self-adjoint operators in Hilbert space and is confirmed by a numerical computation. | null | [
"https://export.arxiv.org/pdf/gr-qc/0411063v1.pdf"
]
| 16,180,861 | gr-qc/0411063 | 0bede287f671fe3479e6922b6859e4fce88b346a |
CONFORMAL CONTINUATIONS AND WORMHOLE INSTABILITY IN SCALAR-TENSOR GRAVITY
2004
K A Bronnikov
Institute of Gravitation and Cosmology
Centre for Gravitation and Fundam. Metrology
VNIIMS
3-1 M. Ulyanovoy St119313MoscowRussia
PFUR
6 Miklukho-Maklaya St117198MoscowRussia
S V Grinyok
Bauman Moscow State Technical University
MoscowRussia
CONFORMAL CONTINUATIONS AND WORMHOLE INSTABILITY IN SCALAR-TENSOR GRAVITY
Grav. & Cosmol
102372004
We study the stability of static, spherically symmetric, traversable wormholes existing due to conformal continuations in a class of scalar-tensor theories with zero scalar field potential (so that Fisher's well-known scalar-vacuum solution holds in the Einstein conformal frame). Specific examples of such wormholes are those with nonminimally (e.g., conformally) coupled scalar fields. All boundary conditions for scalar and metric perturbations are taken into account. All such wormholes are shown to be unstable under spherically symmetric perturbations. The instability is proved analytically with the aid of the theory of self-adjoint operators in Hilbert space and is confirmed by a numerical computation.
Introduction
In our recent paper [1] we have considered spherically symmetric perturbations of wormhole solutions to the Einstein-massless scalar field equations which exist for scalar fields nonminimally coupled to gravity [2,3]. The equations of motion were reduced to a single wave equation for the scalar field perturbation which in this case comprises the only dynamical degree of freedom. An analysis of this wave equation leads to the conclusion that such wormholes are unstable under spherically symmetric (monopole) perturbations, and this instabilty is of catastrophic nature since the increment of perturbation growth has no upper bound.
In this paper we continue this study and extend it in two respects. First, we discuss more general background configurations, namely, static, spherically symmetric wormholes which appear in arbitrary scalar-tensor theories (STT) of gravity in which the effective gravitational constant can change its sign due to conformal continuation (CC) [4]. The investigation is, however, restricted to massless fields for which Fisher's well-known solution holds in the Einstein frame. Second, we examine the problem in more detail, including the behaviour of metric perturbations related to those of the scalar field. A physically meaningful metric perturbation of an initially regular configuration should be regular everywhere. This requirement turns out to impose an additional constraint on the scalar field perturbations, which makes the stability problem quite nontrivial. We finally prove that there exists at least a single growing mode of physically meaningful perturbations, i.e., such wormholes are indeed unstable. However, contrary to 1 e-mail: [email protected] 2 e-mail: [email protected] the conclusion of Ref. [1], the perturbation grows at a finite rate.
We thus find that gravitational instabilities, whose existence seems to be quite natural at surfaces where the gravitational coupling changes its sign (see, e.g., Ref. [5] for a discussion in a cosmological setting), still need much effort in their detailed study and even discovery.
The paper is organized as follows. Sec. 2 is a brief description of the background static configuration and its place among more general configurations of this kind, i.e., static, spherically symmetric wormhole solutions of a general class of scalar-tensor theories (STT) admitting conformal continuations (CCs) [4]. Sec. 3 discusses spherically symmetric perturbation equations and the corresponding gauge freedom. Sec. 4 is devoted to a stability investigation for wormholes, both analytical, using the theory of self-adjoint operators in Hilbert space, and numerical.
Conformal continuations and
wormhole solutions of scalar-tensor theories
STT in Jordan and Einstein pictures
Consider the general (Bergmann-Wagoner-Nordtvedt) class of STT, where gravity is characterized by the metric g µν and the scalar field φ; the action is
S = d 4 x √ g f (φ)R[g] + h(φ)g µν φ ,µ φ ,ν − 2U (φ) + 16πG L m .(1)
Here R[g] is the scalar curvature, f, h and U are certain functions of φ, varying from theory to theory, L m is the matter Lagrangian, and G is the gravitational constant, not necessarily coinciding with its Newtonian value. The action (1) is simplified by the well-known conformal mapping [6]
g µν = g µν /|f (φ)|,(2)
accompanied by the scalar field transformation φ → ψ such that
dψ dφ = ± |l(φ)| f (φ) , l(φ) def = f h + 3 2 df dφ 2 .(3)
In terms of g µν and ψ , the action for U = L m = 0 , the case of massless scalar-vacuum fields to be considered here, takes the form
S = d 4 x g(sign f ) R[g] + [sign l(φ)]g µν ψ ,µ ψ ν(4)
(up to a boundary term which does not affect the field equations). Here R[g] is the Ricci scalar obtained from g µν . The space-time M J = M[g] with the metric g µν is referred to as the Jordan conformal frame (or picture), generally regarded as the physical frame in STT; the Einstein conformal frame M E = M[g] with the field ψ then plays an auxiliary role (see, however, discussions of the physical meaning of various conformal frames in [7,8] and references therein). The action (4) corresponds to conventional general relativity if f > 0 , and the normal sign of scalar kinetic energy is obtained for l(φ) > 0 . Scalar fields in anomalous STT, in which l(φ) < 0 , lead to a kinetic term in (4) with a "wrong" sign, are called phantom scalar fields. Such fields (with different potentials) are sometimes invoked in modern cosmological studies to describe dark energy.
Exact static, spherically symmetric scalar-vacuum solutions of the theory (1) are well known [9,2]. Among them, wormhole solutions are generic in the case of phantom scalar fields [2]. Their stress-energy tensor T µν manifestly violates the null energy condition (NEC) T µν k µ k ν ≥ 0 , k µ k µ = 0 , such violation being a necessary condition for wormhole existence [10], therefore wormhole solutions in their presence would have been naturally expected. One can note that, according to such solutions, both space-times M J and M E have wormhole properties, i.e., represent regular static traversable bridges between two flat asymptotics. The stability of such configurations has also been proved [11,12] by a direct study of perturbation equations, though it seems quite strange for a field system with energy density unbounded from below. This question evidently deserves further investigation.
Our interest here will be in other wormhole solutions which appear in STT with l(φ) > 0 , in cases when the space-time manifold M E is mapped, according to (2), to only a part of M J ; this phenomenon was named conformal continuation (CC) [4,13]. The Jordan space-time M J is then globally regular, represents a wormhole, and it two non-intersecting parts map to two singular spacetimes M E and M E ′ .
Fisher's solution and its conformal continuations
The general static, spherically symmetric solution to the Einstein-scalar equations that follow from (4), was first found by Fisher [9] and was repeatedly rediscovered afterwards. Let us write it in the form suggested in [2], restricting ourselves to the "normal" case l > 0 :
ψ(u) = Cu + ψ 0 ,(5)ds 2 E = e 2γ(u) dt 2 − e 2α(u) du 2 − e 2β(u) dΩ 2 = e −2mu dt 2 − k 2 e 2mu sinh 2 (ku) k 2 du 2 sinh 2 (ku) + dΩ 2(6)
where the subscript "E" stands for the Einstein frame; dΩ 2 = dθ 2 + sin 2 θ dφ 2 is the linear element on a unit sphere; C (scalar charge), m > 0 (mass in geometric units), k > 0 and ψ 0 are integration constants, of which the first three are related by
k 2 = m 2 + 1 2 C 2 .(7)
Without loss of generality we put C > 0 and ψ 0 =0 . We are here using the harmonic radial coordinate u ∈ R + in M E [g], satisfying the coordinate condition α = 2β + γ . Another convenient form of the solution is obtained in isotropic coordinates: with y = tanh(ku/2), Eqs. (5), (6) are converted to
ψ(y) = C k ln 1 + y 1 − y ,(8)ds 2 E = A(y) dt 2 − k 2 (1 − y 2 ) 2 y 4 A(y) (dy 2 + y 2 dΩ 2 ), A(y) = 1 − y 1 + y 2m/k .(9)
The solution is asymptotically flat at u → 0 (y → 0 ), has no horizon when C = 0 and is singular at the centre (u → ∞, y → 1 − 0 , ψ → ∞). When the scalar field is "switched off" (C = 0 , k = m), the Schwarzschild solution is recovered.
A feature of importance is the invariance of (8), (9) under the inversion y → 1/y , noticed probably for the first time by Mitskievich [14]. Due to this invariance, the solution (8), (9) considered in the range y > 1 describes quite a similar configuration, but now y → ∞ is a flat asymptotic and y → 1 + 0 is a singular centre. An attempt to unify the two ranges of y , or, in other words, the two copies of Fisher's solution, is meaningless due to the singularity at y = 1 . We shall see that such a unification, leading to a wormhole, is achieved in M J [g] where the singularity is smoothed out (in case C = √ 6m) owing to the conformal factor. The corresponding Jordan-frame solution for any f (φ) and h(φ) such that l(φ) > 0 are obtained from (5), (6) using (2), (3). If the function f (φ) is everywhere finite, M J has the same basic properties as M E .
However, according to [4], there is a class of STT in which some solutions produce structures of M J drastically different from that of M E . Namely, let us use the φ field reparametrization freedom of the action (1) [φ = φ(φ new )] and fix the parametrization by putting in (1) h(φ) ≡ 1 . Then [4], if the function f (φ) has a simple zero at some φ = φ 0 , there is a subfamily of static, spherically symmetric solutions to the field equations admitting a CC. The latter means that a singular surface in M E , corresponding to φ = φ 0 , maps according to (2) to a regular surface S trans in M J . Then M can be continued in a regular manner through this surface, and the global properties of M J can be considerably richer than those of M E : in the new region one can possibly find, e.g., a horizon or another spatial infinity. The above result was obtained in [4] for STT (1) in arbitrary dimensions and with arbitrary potentials U (φ). It was also shown [4] that a wormhole was a generic type of a conformally continued Jordan-frame manifold. Before studying perturbations of such generic solutions (but with U (φ) ≡ 0 , so that we have Fisher's solution in the Einstein picture), we discuss a specific example which makes evident the relations between Einstein and Jordan quantities.
Example: wormholes with a conformally coupled scalar field
A particular example of a CC is given by a free massless conformally coupled scalar field in GR. The latter is obtained when we put in (1)
f (φ) = 1 − φ 2 /6, h(φ) ≡ 1. U (φ) = L m = 0. (10)
A transition sphere S trans , if any, corresponds to φ 2 =6 . The transformation (3) now takes the form
dψ dφ = 1 1 − φ 2 /6 .(11)
We assume that spatial infinity, where ψ → 0 , corresponds in the Jordan space-time M J to |φ| < √ 6 , where f (φ) > 0 , so that the gravitational coupling has its normal sign. Then (11) gives
ψ = √ 6 tanh −1 (φ/ √ 6) + ψ 0 , ψ 0 = const.(12)
Using (5) and (6), it is now easy to write the metric in the Jordan picture.
A CC through the sphere S trans (u = ∞, y = 1 , φ = √ 6), which is singular in M E , is obtained when the infinity of the conformal factor 1/f in (2) compensates the zero of both g tt and g θθ simultaneously. This happens when, in accord with (7),
k = 2m = 2C/ √ 6,(13)
which selects a special subfamily among all solutions. We will restrict the consideration to this subfamily.
In terms of the isotropic coordinate y , the solution in the Jordan picture has the form [2]
ds 2 = (1+yy 0 ) 2 1 − y 2 0 dt 2 (1+y) 2 − m 2 (1+y) 2 y 4 (dy 2 + y 2 dΩ 2 ) , φ(y) = √ 6 y + y 0 1 + yy 0 ,(14)
where y 0 = tanh(ψ 0 / √ 6). The range 0 < y < 1 , describing the whole manifold M E in the Fisher solution, corresponds to only a region M J ′ of the manifold M J of the solution (14). In all cases, y = 0 corresponds to a flat asymptotic, where φ → √ 6y 0 < √ 6 . The global properties of the solution depend on the sign of y 0 : a) y 0 < 0 . The solution is defined in the range 0 < y < 1/|y 0 |. At y = 1/|y 0 |, there is a naked attracting central singularity:
g tt → 0 , r 2 → 0 , φ → ∞. b) y 0 = 0 , φ = √ 6y , y ∈ R + .
In this case it is helpful to pass to the conventional radial coordinate r , substituting y = m/(r − m). The solution
ds 2 = (1 − m/r) 2 dt 2 − dr 2 (1 − m/r) 2 − r 2 dΩ 2 , φ = √ 6m/(r − m)(15)
is the well-known BH with a conformal scalar field [15,16]. The infinite value of φ at the horizon r = m does not make the metric singular since, as is easily verified, the energy-momentum tensor remains finite there. This solution has been shown to be unstable under radial perturbations [17].
c) y 0 > 0 . This is the wormhole solution discussed in [1,2] and, among other solutions, re-analyzed now. The solution is defined in the range y ∈ R + . At y → ∞, we find another flat spatial infinity, where φ → √ 6/y 0 , r 2 → ∞ and g tt tends to a finite limit.
The whole manifold M J can be represented as the union
M J = M J1 ∪ S trans ∪ M J2(16)
where the region M J 1 (y < 1 ) is, according to (2), in one-to-one correspondence with the whole manifold M E of Fisher's solution (5), (6). The "antigravitational" (f (φ) < 0 ) region M J 2 (y > 1 ) is in a similar correspondence with another "copy" of Fisher's solution,
M E ′ [g]
, where, instead of (12),
ψ = √ 6 coth −1 (φ/ √ 6) + ψ ′ 0 , ψ ′ 0 = const.(17)
The metric g µν of this second Einstein-frame manifold M E ′ should also be regularized by the factor 1/f on S trans , hence the integration constants in it should satisfy the condition (13). Moreover, one can verify that, to provide a smooth transition in the Jordan-frame metric g µν through S trans , all the constants k , h, C and ψ 0 should coincide in M E and M E ′ . The latter statement is proved using the coordinate y which is common on both sides of S trans . This example well illustrates the general properties of conformally continued solutions [4]. Namely, in the region beyond S trans , there can be a singularity due to l(φ) = 0 , as happens in the above case a) at y = 1/|y 0 | > 1 . If there is no such singularity, we obtain a wormhole. Case b), with a horizon, is exceptional, inherent only to the field (10) in four dimensions. Thus, for a more general action discussed in [1,18], with (18) with the coupling constant ξ > 0 , in case ξ > 1/6 all solutions exhibiting a CC describe wormholes, whereas for ξ < 1/6 everything depends on an integration constant similar to y 0 in the above example, and we may have either a wormhole or a naked singularity.
f (φ) = 1 − ξφ 2 , h(φ) ≡ 1. U (φ) = L m = 0,
The stability analysis developed below covers wormhole solutions obtained in the theory (1) under the conditions
h ≡ 1, U = L m = 0, l(φ) > 0,(19)
with an arbitrary function f (φ), having a simple zero at some φ = φ 0 . In other words, we consider massless scalar fields in a general non-phantom STT, for which wormhole solutions exist due to a CC.
Spherically symmetric perturbations and gauge freedom
Consider small spherically symmetric (monopole) perturbations of any static, spherically symmetric solution of the theory (1), (19). The only dynamical degree of freedom is evidently related to the scalar field due to the generalized Birkhoff theorem [19]: if we take a timeindependent scalar field, the equations of motion automatically lead to a static solution. We will use, for simplicity, the Einstein conformal frame, since the perturbation equations in M J , being equivalent to those in M E , look much more complicated, and it is even hard to decouple different components of the Einstein equations. However, the boundary conditions that select physically meaningful perturbation should be formulated for variables specified in M J and only then converted to Einstein-frame quantities.
In the Einstein picture, the equations of motion are the Einstein-scalar field equations due to (4)
∇ α ∇ α ψ = 0,(20)R ν µ = −ψ ,µ ψ ,ν .(21)
We now write the metric in M E in the form
ds 2 E = e 2γ dt 2 − e 2α du 2 − e 2β dΩ 2 ,(22)
where the functions α, β, γ as well as the scalar field ψ are split into a static background part and a small (linear) time-dependent perturbation:
α = α(u) + δα(u, t)
where u is a radial coordinate, and similarly for β , γ and ψ . Now, in addition to the freedom of choosing the radial coordinate u , we have an additional freedom of specifying the frame of reference in perturbed spacetime, called gauge freedom. The latter makes it possible to specify (by hand) some linear relation between the perturbations. Certain care is needed to ensure that the resulting perturbation will not be a "pure gauge", i.e., will not be removable by coordinate transformations.
Consider an infinitesimal coordinate transformation of the static metric (22) preserving its spherical symmetry, i.e.,
x µ new = x µ old + ζ µ , ζ µ = (η,ξ, 0, 0),(23)
where the time dependence of the perturbationsξ,η is separated:η = η(u) e Ωt andξ = ξ(u) e Ωt . To preserve the diagonal form of the metric, we should put Ωξ = η ′ e 2γ−2α (where α and γ are unperturbed), so that all perturbations are expressed in terms of η(u). For the metric functions and the scalar field φ we obtain (omitting the factor e Ωt )
δα = 1 Ω e 2γ−2α [η ′′ + η ′ (2γ ′ − α ′ )], δβ = β ′ Ω η ′ e 2γ−2α , δγ = 1 Ω [Ω 2 γ + γ ′ η ′ e 2γ−2α ], δφ = φ ′ Ω η ′ e 2γ−2α ,(24)
where the prime denotes d/du . Let there be a static configuration with β ′ = 0 and φ ′ = 0 . Then, if we have nontrivial time-dependent perturbations under the gauge condition δβ = 0 (or δφ = 0 ), Eqs. (24) immediately lead to η ′ = 0 , which means that our perturbation cannot be caused by a transformation like (23), i.e., is physical. The same is true for any gauge of the form f 1 δβ + f 2 δφ = 0 where f 1 and f 2 are any fixed functions of u , provided f 1 β ′ + f 2 φ ′ = 0 . The reason is that β and φ are scalars with respect to coordinate transformations of the 2-surfaces (x 0 , x 1 ). Thus, choosing such gauges, we can be sure that the perturbations to be studied will be physical. For other gauges, involving δα and/or δγ , an additional inspection will be required.
A more general approach to the problem of gauge in perturbation theory for spherically symmetric spacetimes can be found in Ref. [20]; though, in the present case, our explicit treatment seems more transparent.
Stability analysis
The problem
We have considered our set of linear perturbation equations using two different systems of analytical computation, Maple and Mathematica, which made it possible to compare the results and to avoid errors.
We use the Einstein conformal frame, in which the equations are much simpler, and the gauge δψ = 0 (25) which is manifestly physical (see Sec. 3) and, in addition, transforms to δφ = 0 in the Jordan frame. Moreover, according to Eq. (20), we have the following relation between the metric perturbations:
δα = 2δβ + δγ.(26)
Two independent components of the Einstein equations for perturbations in the gauge (25) may be written as
e 2γ R 0 1 = 2[δβ ′ − β ′ (δβ + δγ) − γ ′ δβ] = 0, e 2α R 2 2 = 2β ′′ (2δβ + δγ) − 2 e 2β+2γ δβ + e 4β δβ − δβ ′′ = 0(27)
Here primes denote derivatives with respect to u , the harmonic radial coordinate in the Einstein frame, α , β and γ describe the background configuration and satisfy the static field equations. We separate the variables using the substitution
δ(r, t) = δ(u) e Ωt ,(28)
where δ is a perturbation of any variable in our problem. After substitution of (28) into (27), δγ is expressed from the first equation, and then the second equation takes the form ( e Ωt is omitted)
δβ ′′ − Ω 2 δβ s 4 (u) + F (u) δβ ′ + G(u) δβ = 0,(29)
where s, F , G are functions of u obtained from the metric (6):
F (u) = −2β ′′ /β ′ , G(u) = −2β ′′ + 2β ′′ γ ′ /β ′ + 2 e 2β+2γ , s(u) = e β .(30)
A few words about the boundary conditions. At spatial infinity the choice is evident: δβ → 0 . At the transition sphere S trans δβ should be finite, as well as its first two derivatives in u . This is necessary for the metric perturbations in the Jordan picture to be finite and smooth at S trans , which is easily checked using the transformation (2), (3) and expressions in terms of the invariant length in the Jordan frame.
As usual, we perform a transition from (29) to a Schrödinger-like form of the perturbation equation:
d 2 y/dx 2 + [E − V (x)]y(x) = 0,(31)
where
x = 1 m s 2 du, δβ = y s exp − 1 2 F du ,(32)V (x) = 2(β xx /β x ) 2 − (β xxx + 2β xx γ x )/β x + 3β xx + 5β 2 x − 4β x γ x − 2m 2 e 2γ−2β ,(33)
where the subscript x denotes d/dx and E = −m 2 Ω 2 . The notations are chosen in such a way that the potential V (x) and the "energy" E are dimensionless. The asymptotic forms of V (x) are
V (x) ≈ 2/x 3 (x → ∞, spatial asymptotic), V (x) ≈ −1/(4x 2 ) (x → 0, the sphere S trans ). (34)
Thus we have a quadratic potential well at S trans , which is placed at x = 0 by choosing the proper value of the arbitrary constant in the definition of x in Eq. (32).
The boundary condition at spatial infinity (u → 0 , x → ∞) is y → 0 while the asymptotic form of any solution of (31) with E < 0 at large |x| is
y ≈ C 1 e mΩ|x| + C 2 e −mΩ|x| , C 1,2 = const. (35)
An admissible solution is the one with C 1 = 0 , with only a decaying exponential. At the other asymptotic, x → 0 , the condition that follows from the above continuity requirements reads y/ √ x < ∞ whereas the solution of (31) behaves as
y ≈ √ x(C 3 + C 4 ln x).(36)
It follows that we must select the solution with C 4 = 0 .
In other words, our problem is to find out whether there is a solution to the boundary-value problem for Eq. (31) such that y → 0 as x → ∞, y/ √ x < ∞ as x → 0 and E < 0 . In the remainder of the section we solve this problem.
Summary of the solution
We begin with a proof of the fact that the Hamiltonian operator H related to Eq. (31) is self-adjoint and is bounded from below. To this end, we use an auxiliary operator T which has the same singularity at S trans as H . The one-sided boundedness indicates that the real part of the increment Ω cannot be infinite. A further comparison of T and H shows that the continuous parts of their spectra coincide and lie in the non-negative part of the real number axis. So, if there are any solutions of our boundary value problem with E < 0 , they belong to a discrete spectrum.
To prove the existence of a solution with E < 0 we use the well-known fact from quantum mechanics (its more general form is called the minimax principle) that the lower bound µ 0 of the spectrum of an operator A is the infimum of the functional
(ψ, Aψ),(37)
where the parentheses denote the scalar product (defined a bit later), the infimum is taken on the set of functions ψ which lay in the definition domain of T , and the norm ψ = 1 . Thus the value (ψ, Aψ) for any specified function ψ is an upper estimate for µ 0 , and if it is negative, then µ 0 < 0 . Functions which may closely resemble the unknown function that realizes the infimum can give values of the functional (37) closest to µ 0 . We guess such a function, which shows that the ground state of H lies below zero. This function is a ground state of a certain operator which is similar to H but simpler.
The solution
Consider the auxiliary differential equation
− d 2 dx 2 y(x) − 1 4x 2 y(x) = Ey(x)(38)
and investigate the question of self-adjointness of the Schrödinger operator
T y(x) ≡ − d 2 dx 2 y(x) − 1 4x 2 y(x)(39)
on the subset D(T ) of real Hilbert space L 2 ([0, ∞)) such that, for y(x) ∈ D(T ), (a) our boundary conditions (BCs) hold (so that, e.g., |y|/ √ x < ∞ as x → 0 ) and (b) T y(x) ∈ L 2 . The space L 2 ([0, ∞)) is a Hilbert space with an inner (scalar) product defined as the Lebesgue integral
(ϕ, ψ) = ∞ 0 ϕ * ψ dx,(40)
where the star stands for complex conjugation. D(T ) is dense in L 2 since C ∞ 0 (0, ∞) ⊂ D(T ), where C ∞ 0 (0, ∞) is the the subset of functions in C ∞ (0, ∞) with a compact support separated from 0 . It is a dense subset in L 2 [21].
One can show that the operator (39) defined in this way is symmetric and therefore closable [21]. Obviously, the BCs of our Hilbert space are homogenous. The Schrödinger equation (38) related to the operator T has the solution
c 1 √ xK 0 ( √ −Ex) + c 2 √ xI 0 ( √ −Ex),(41)
where E is the "energy" corresponding to −m 2 Ω 2 of our problem, so to prove the instability we should show that there are "quantum states" with E < 0 ; K 0 and I 0 are the zero-order modified Bessel functions of the first kind. Neither of these functions, nor their any linear combination, satisfy our BCs. This means that the operator T − EI , E < 0 has a bounded inverse operator (T − EI) −1 with a definition domain dense in L 2 . The existence of a reverse operator follows from the well-known alternative: under given homogenous BCs, either the differential equation L[y] = g(x) has a uniquely defined solution y(x), or the homogeneous equation L[y] = 0 has a non-zero solution. In our case,
L[y] ≡ − d 2 dx 2 y(x) − 1 4x 2 y(x) − Ey(x).(42)
The boundedness and the density property of the definition domain of (T − EI) −1 in L 2 follow from studying the properties of solutions to the equation L[y] = g(x)
with nonzero g(x) ∈ L 2 . The existence of (T − EI) −1 , E < 0 means that the domain (−∞, 0) ⊂ ρ(T ), ρ(T ) being the resolvent set of T . Considering in a similar way Eq. (42) with E > 0 , one can show that [0, ∞) is a continuous spectrum.
Thus we have shown that T is a closed symmetric operator which contains real numbers in its resolvent set. It satisfies the conditions of the second corollary of Theorem X.1 in [21]: If the resolvent set of a closed symmetric operator contains at least one real number, then this operator is self-adjoint. The self-adjointness of this operator was also mentioned in passing in Ref. [22].
The proved properties of T make it possible to use the wealth of results obtained in the theory of selfadjoint operators. In particular, we use the following two theorems: Theorem 2 (Kato [22]). Let the conditions of Theorem 1 hold and A be bounded from below (or from above, or from both sides), then A + B is bounded from below (or from above, or from both sides), but not necessarily with the same bound (bounds).
Considering T as A in these theorems, we can rewrite (31) as
T y(x) + V (x)y(x) = Ey(x),(43)
where
V (x) = V (x) + 1/4x 2 .(44)
Since V is bounded (this is true since V (x) → 0 as x → ∞, and V (x) is bounded everywhere), the conditions of Theorem 1 are fulfilled, and the operator
H ≡ −d 2 y/dx 2 + V (x)y(x)(45)
connected with Eq. (31) is self-adjoint on D(T ). Using the spectral theorem for unbounded operators [21], we prove that T is non-negative and consequently is bounded from below. Therefore the operator H is bounded from below too (Theorem 2).
We now wish to show that the continuous spectrum of H coincides with the continuous spectrum of T . We use the following theorem:
Theorem 3 [21]. Let A be a self-adjoint operator and C a symmetric operator such that C(A n + i) −1 , n ∈ N is a compact operator. Then, if B = A + C is self-adjoint on D(A), σ ess (A) = σ ess (B). 3 The compactness of the operator V (T + i) −1 follows from its integral representation:
( V (T + i) −1 f )(x) = ∞ 0 dy K(x, y) f (y), K(x, y) = V (x)G(x, y),(46)
where G(x, y) is the Green function, core of the integral operator (T + i) −1 . As follows from the asymptotic properties of V (x) and G(x, y), K(x, y) ∈ L 2 ([0, ∞) × [0, ∞)). So, V (T + i) −1 is a Hilbert-Schmidt operator and hence is compact [21]. The conditions of Theorem 3 are fulfilled, and σ ess (H) = σ ess (T ) = [0, ∞). Since σ = σ ess ∪σ disc [21], the remaining part of σ(H) belongs to the discrete spectrum.
Eq. (29) may also be expressed in a non-Schrödinger form using the new variable w instead of u :
u = 1 2h ln 1/w + 1(47)
and further converted to a normal form:
16E + 11 + 32E 2w − 64 3(1 + 4w) + 1 4w 2 + 1 4(1 + w) 2 − 1 6(1 + w) − 32 (1 + 4w) 2 n(w) + d 2 n(w) dw 2 = 0, (48) where n(w) = δβ(w) exp − (2w − 1) dw 2w(1 + w)(1 + 4w)
.
We cannot solve this equation, but its truncated version d 2 n(w) dw 2 + 16E + 11 + 32E 2w + 1 4w 2 n(w) = 0 (50) has the solution:
n(w) = c 1 M (11 + 32E) 16 √ −E , 0, 8 √ −Ew + c 2 W (11 + 32E) 16 √ −E , 0, 8 √ −Ew ,(51)
where M and W are Whittaker functions. If the first argument, usually denoted as the index k , takes on the values k = 1/2, 3/2, . . . , n + 1/2 , n ∈ {0, N} , then M (k, 0, w) ≡ W (k, 0, w), and the corresponding quantity E has the values
E k = − 11 32 − k 2 8 1 − 1 + 11 2k 2 ,(52)
which satisfy the equation
(11 + 32E k ) 16 √ −E k = k.(53)
Performing the inverse coordinate transformation from the variables of (48) to the variables of (31), we see that the functions y corresponding to W (k, 0, 8
√ −E k w) belong to D(T ).
Hence E takes a discrete set of values in the interval [−3/8 + √ 23/32, 0), and they lie in σ disc of the Schrödinger operator related to Eq. (50). The latter operator has an infinite discrete spectrum with the limiting point 0 . Any other eigenvalue E gives a solution related to the essential spectrum, or the resolvent set of the latter operator. The solutions obtained make it possible to apply the minimax principle ( [21], theorem XIII.1). The theorem is applicable because H is self-adjoint and bounded from below.
We use the solution with k = 1/2 as a trial function to find an upper estimate of the lower bound of σ(H) :
µ ′ 0 = ψ(E 1/2 , x), Hψ(E 1/2 , x) ,(54)
where the parentheses denote the scalar product on L 2 , i.e., Lebesgue integration,
µ ′ 0 = (ψ(E 1/2 , x), H ′ + V ′ ψ(E 1/2 , x)) = E 1/2 + (ψ(E 1/2 , x), V ′ ψ(E 1/2 , x)) = E 1/2 + W (1/2, 0, 8 −E 1/2 w) 2 U (w)dw W (1/2, 0, 8 −E 1/2 w) 2 r(w)dw ,(55)
where U (w) = 64 3(1 + 4w) − 1 4(1 + w) 2 + 1 6(1 + w)
+ 32 (1 + 4w) 2 ,(56)
and r(w) = 16(1+1/w). Integration gives µ ′ 0 ≃ −0.039 . Since the essential spectrum begins from 0 , according to the minimax principle, µ ′ 0 is an upper estimate of the ground state which is thus located below zero.
Thus we have proved the existence of negative eigenvalues of the operator H under physically justified BCs, and therefore there are exponentially growing solutions (at least one) of our perturbation equation (29).
We have also solved our boundary-value problem (29) numerically, applying the Fortran procedure SLEIG and using the coordinate transformation (47). We found a single discrete eigenvalue at −0.048 , which fits our estimate of µ ′ 0 . The corresponding problem for Eq. (48) is not suitable for using SLEIG because Fortran does not "understand" the BC y/ √ x < ∞. Recall that we have been working in the Einstein frame, so that the coordinates used cover only half of the wormhole, so that we should use two copies of this patch and verify whether the metric perturbations remain really smooth at the transition sphere S trans . Computation of the corresponding metric perturbations (δβ , δγ ) at the point where f (φ) = 0 shows that their first derivatives in l (where l is the Gaussian radial coordinate in the Jordan picture, such that g ll = −1 ) are zero, while the second derivatives are finite, so that the linearized gravity equations are there meaningful and hold. We conclude that the metric perturbations found are physical, and the wormholes under consideration are unstable under small spherically symmetric perturbations. The decay rate depends on the value of m in (6), since E = −m 2 Ω 2 . The wormhole radius (understood, for simplicity, as √ −g 22 at the transition sphere rather than throat radius which is smaller but generically of the same order) is proportional to m and also depends on the integration constant in the solution of (3). Let us discuss the special case of conformal coupling. According to (14), the wormhole radius is r wh = 2m (1 + y 0 )/(1 − y 0 ) . If we assume y 0 ≪ 1 , then both r wh and the throat radius are approximately equal to 2m. The characteristic time of decay, τ = 1/Ω, is proportional to m (which has the dimension of length):
τ ≃ m/ √ µ 0 ≃ 5m.(57)
For a wormhole radius of the order of a typical stellar size ∼ 10 6 km, the time τ is a few seconds (slightly greater than the time needed for a light signal to cover the stellar diameter). If y 0 increases under fixed wormhole radius, then m decreases, so τ decreases too. We see that, for all wormholes with a fixed radius, τ ≤ 5m. Similar estimates can be obtained for other STT characterized by different f (φ).
Theorem 1 (
1Rellich [22]). Let A be a self-adjoint operator on D(A) and B a symmetric operator on D(B), so that D(B) ⊃ D(A) and Bψ ≤ a ψ + b Aψ , b < 1 . Then the operator A + B is self-adjoint and D(A + B) = D(A).
ψ(E 1/2 , x) is a normalized function obtained from W (1/2, 0, 8 −E 1/2 w) by the substitution which transforms Eq. (48) into (31). Explicit integration in x is impossible because we cannot represent w(x) in elementary functions, but we can integrate in w after necessary substitutions. It is convenient to represent H as a sum of two Schrödinger operators, where the first, H ′ , corresponds to Eq. (50) and the second, V ′ , corresponds to the remaining part of the operator. We obtain
We denote: σ disc = discrete spectrum, σcont = continuous spectrum, σess = essential spectrum[21]. In our case, σess consists of σcont and a possible limiting point of σ disc .
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| []
|
[
"arXiv:astro-ph/0410431v3 5 May 2005 Strong Lensing Probabilities in a Cosmological Model with a Running Primordial Power Spectrum",
"arXiv:astro-ph/0410431v3 5 May 2005 Strong Lensing Probabilities in a Cosmological Model with a Running Primordial Power Spectrum"
]
| [
"Tong-Jie Zhang [email protected] \nDepartment of Astronomy\nDepartment of Astronomy\nBeijing Normal University\n100875BeijingP.R.China\n",
"Zhi-Liang Yang [email protected] \nDepartment of Astronomy\nBeijing Normal University\n100875BeijingP.R.China\n",
"Xiang-Tao He [email protected] \nBeijing Normal University\n100875BeijingP.R.China\n"
]
| [
"Department of Astronomy\nDepartment of Astronomy\nBeijing Normal University\n100875BeijingP.R.China",
"Department of Astronomy\nBeijing Normal University\n100875BeijingP.R.China",
"Beijing Normal University\n100875BeijingP.R.China"
]
| []
| The combination of the first-year Wilkinson Microwave Anisotropy Probe (WMAP) data with other finer scale cosmic microwave background (CMB) experiments (CBI and ACBAR) and two structure formation measurements (2dF-GRS and Lyman α forest) suggest a ΛCDM cosmological model with a running spectral power index of primordial density fluctuations. Motivated by this new result on the index of primordial power spectrum, we present the first study on the predicted lensing probabilities of image separation in a spatially flat ΛCDM model with a running spectral index (RSI-ΛCDM model). It is shown that the RSI-ΛCDM model suppress the predicted lensing probabilities on small splitting angles of less than about 4 ′′ compared with that of standard power-law ΛCDM (PL-ΛCDM) model. | 10.1142/s0217732305016142 | [
"https://export.arxiv.org/pdf/astro-ph/0410431v3.pdf"
]
| 16,197,205 | astro-ph/0410431 | c8b91f9c48fe9f4a776383a9b051c9e4bff60a45 |
arXiv:astro-ph/0410431v3 5 May 2005 Strong Lensing Probabilities in a Cosmological Model with a Running Primordial Power Spectrum
Tong-Jie Zhang [email protected]
Department of Astronomy
Department of Astronomy
Beijing Normal University
100875BeijingP.R.China
Zhi-Liang Yang [email protected]
Department of Astronomy
Beijing Normal University
100875BeijingP.R.China
Xiang-Tao He [email protected]
Beijing Normal University
100875BeijingP.R.China
arXiv:astro-ph/0410431v3 5 May 2005 Strong Lensing Probabilities in a Cosmological Model with a Running Primordial Power Spectrum
Subject headings: cosmology:theory-dark matter-galaxies:halos-gravitational lensing-early universe-large-scale structure PACS numbers:9880Es9862Sb9535+d9880Cq
The combination of the first-year Wilkinson Microwave Anisotropy Probe (WMAP) data with other finer scale cosmic microwave background (CMB) experiments (CBI and ACBAR) and two structure formation measurements (2dF-GRS and Lyman α forest) suggest a ΛCDM cosmological model with a running spectral power index of primordial density fluctuations. Motivated by this new result on the index of primordial power spectrum, we present the first study on the predicted lensing probabilities of image separation in a spatially flat ΛCDM model with a running spectral index (RSI-ΛCDM model). It is shown that the RSI-ΛCDM model suppress the predicted lensing probabilities on small splitting angles of less than about 4 ′′ compared with that of standard power-law ΛCDM (PL-ΛCDM) model.
Introduction
Mapping the mass distribution of matter in the universe has been a major challenge for modern observational cosmology. One of the direct procedures to weigh matter in the universe is measuring its deflection of light by gravity. The statistics of gravitational lensing can provide us with a very powerful probe of the mass distribution of the Universe. It is well known that the Jodrell-Bank VLA Astrometric Survey (JVAS) and the Cosmic Lens All-Sky Survey (CLASS) (Browne & Myers 2000;Myers et al. 2003;Browne et al. 2003) have provided us with observations of strong lensing probabilities for small image separations ranging from 0.3 ′′ to 3 ′′ . By comparing predicted lensing probabilities with observations, we can examine the mass distributions of dark matter halos, in particular, their inner density slopes (Huterer & Ma 2004;Ricotti 2003).
Based on the Cold Dark Matter (CDM) model, which has become the standard theory of cosmic structure formation, the lensing probabilities strongly depend on the density profiles of CDM halos. The lensing model is usually described by a singular isothermal sphere (SIS), the Navarro-Frenk-White (NFW) model (Navarro et al. 1996(Navarro et al. , 1997, or generalized NFW (GNFW) density profiles of dark halos (Zhao 1996). Li & Ostriker (2002) employed a semianalytical approach to analyze the gravitational lensing of remote quasars by foreground dark halos and checked the plausibility of various lensing models. They found that no model can completely explain the current observations: the SIS models predict too many lenses with large splitting angles, while the NFW models predict too few small splitting angles. They therefore further developed a two-population halo model for lensing: small mass halos with a steep inner density slope and large mass halos with a shallow inner density slope, concluding that a combination of SIS and NFW halo models can reproduce the current observations reasonably well. Unlike previous work that directly models the density profiles of dark matter halos semi-analytically, Zhang (2004) generalized the density profiles of dark matter halos from high-resolution N-body simulations by means of generalized Navarro-Frenk-White (GNFW) models of three populations with slopes, α, of about -1.5, -1.3 and -1.1 for galaxies, groups and clusters, respectively. He presented the calculations of lensing probabilities using these GNFW profiles for three populations in various spatially flat cosmological models with a cosmological constant Λ. He showed that the compound model of density profiles does not match well with the lensing probabilities derived from the combined data of JVAS/CLASS. Recently, Metcalf (2004) compared predictions on small scale structure for the ΛCDM model by numerical simulations with observed flux ratios, and found that the disagreements between monochromatic flux ratios and simple lens models can be explained without any substructure in the dark matter halos of primary lenses. However, spectroscopic lensing observations of Q2237+0305 require more small mass dark halos than that expected in the ΛCDM model.
In particular, a power spectrum of primordial fluctuation, P p (k), should be assumed in advance in the calculation of lensing probabilities. Inflationary models predict a approximately scale-invariant power spectra for primordial density (scalar metric) fluctuation, P p (k) ∝ k n with index n = 1 (Guth & Pi 1982;Bardeen et al. 1983 and Lyman α forest) favor a ΛCDM cosmological model with a running index of the primordial power spectrum (RSI-ΛCDM), while the WMAP data alone still suggest a best-fit standard power-law ΛCDM model with the spectral index of n ≈ 1 (PL-ΛCDM) Peiris et al. 2003). However, there still exist the intriguing discrepancies between theoretical predictions and observations on both the largest and smallest scales. While the emergence of a running spectral index may improve problems on small scales, there remain a possible discrepancy on the largest angular scales. It is particularly noted that the running spectral index model suppress significantly the power amplitude of fluctuations on small scales Yoshida et al. 2003). This imply a reduction of the amount of substructure within galactic halos (Zentner & Bullock 2002). Yoshida et al. (2003) studied early structure formation in a RSI-ΛCDM universe using high-resolution cosmological Nbody/hydrodynamic simulations. They showed that the reduced small-scale power in the RSI-ΛCDM model causes a considerable delay in the formation epoch of low-mass minihalos (∼ 10 6 M ⊙ ) compared with the PL-ΛCDM model, although early structure still forms hierarchically in the RSI-ΛCDM model. Thus the running index probably affect the strong lensing process of distant sources by intervening dark matter halos because the lensing probabilities strongly depend on the abundance of dark halos formed in the evolution of the universe. In this letter, we will present the first calculation of lensing probabilities in a RSI-ΛCDM model to explore the effect of running spectral index of primordial fluctuation on strong lensing probabilities. We adopt a spatially flat ΛCDM cosmological model through this letter.
The reminder of this paper is organized as follows. We describe strong lensing probabilities and mass function of dark halos in Section 2 and 3 respectively. The cross-sections for producing multiple images are described in Section 4. Our results is shown in Section 5, while a conclusion and discussion are given in Section 6.
Strong Lensing Probability
The lensing probability with image separation larger than ∆θ is given by Schneider et al. (1992)
P (> ∆θ) = p(z s )dz s zs 0 dD p (z) dz dz ∞ 0n (M, z)σ(M, z)dM.(1)
In this expression, D p (z) = c/H 0 z 0 dz/(1+z)E(z) is the proper distance from the observer to the lens at redshift z where the expansion rate of the universe E(z) = Ω m (1 + z) 3 + Ω Λ in a spatially flat cosmological model, and p(z s ) is the redshift distribution of distant sources. The physical number densityn(M, z) of virialized dark halos of masses between M and M + dM is expressed asn(M, z) = n(M, z)(1 + z) 3 and σ(M, z) is the cross-section defined in the lens plane for forming multiple images.
Mass Function of Dark Halos
In the standard hierarchical theory of structure formation, the comoving number density of virialized dark halos per unit mass M at redshift z can be given by the Press and Schechter (PS) formula (Press & Schechter 1974): n(M, z) = dN/dM = ρ 0 f (M, z)/M where ρ 0 is the mean mass density of the universe today and, instead of PS formula in this letter, the mass function f (M, z) takes the form of an empirical fit from high-resolution simulation (Jenkins et al. 2001)
f (M, z) = 0.301 M d ln ∆ −1 (M, z) d ln M exp(−| ln ∆ −1 (M, z) + 0.64| 3.88 ).(2)
Here, ∆(M, z) = ∆(M)D(z) and D(z) = e(Ω(z))/e(Ω m )(1 + z) is the linear growth function of density perturbation (Carroll et al. 1992), in which e(x) = 2.5x/(1/70 + 209x/140 − x 2 /140 + x 4/7 ) and Ω(z) = Ω m (1 + z) 3 /E 2 (z). The present variance of the fluctuations within a sphere containing a mass M can be expressed as ∆ 2 (M) = 1 2π 2 ∞ 0 P (k)W 2 (kr M )k 2 dk, where W (kr M ) = 3[sin(kr M )/(kr M ) 3 − cos(kr M )/(kr M ) 2 ] is the Top-hat window function in Fourier space and r M = (3M/4πρ 0 ) 1/3 . The power spectrum of CDM density fluctuations is P (k) = P p (k)T 2 (k) where the matter transfer function T (k) is given by Eisenstein & Hu (1999), and P p (k) is the primordial power spectrum of density fluctuation. The scaleinvariant primordial power spectrum in the PL-ΛCDM model is given by P p (k) = Ak ns with index n s =1 and that in the RSI-ΛCDM model is assumed to be P p (k) = P (k 0 )(k/k 0 ) ns(k) , where the index n s (k) is a function of length scale
n s (k) = n s (k 0 ) + 1 2 dn s (k) d ln k ln k k 0 .(3)
The pivot scale k 0 =0.05 h Mpc −1 , n s (k 0 )=0.93, and dn s /d ln k=-0.03 are the best-fit values to the combination data of the recent CMB experiments and two other large-scale structure observations . For both PL-ΛCDM and RSI-ΛCDM models, the amplitude of primordial power spectrum, A and P (k 0 ), are normalized to σ 8 = ∆(r M = 8h −1 Mpc), which is the rms mass fluctuations when present universe is smoothed using a window function on a scale of 8h −1 Mpc.
The Cross-sections for Producing Multiple Images
The cross-section for producing multiple images relies on the density profile of dark matter halos. Based on previous work (Li & Ostriker 2002;Zhang 2004), we model dark halos as an SIS for M < M c1 and NFW profile for M > M c1 , respectively, where M c1 ∼ 10 13 h −1 M ⊙ corresponds to the cooling mass scale (Porciani & Madau 2000;Kochanek & White 2001).
The Cross-section for SISs
The mass density for a lens of SIS is ρ(r) = σ 2 v /2πGr 2 where σ v is the velocity dispersion (Schneider et al. 1992). Thus the surface mass density of the SIS satisfies Σ(ξ) = σ 2 v /2Gξ where ξ ≡ | ξ|, ξ is the position vector in the lens plane. By defining the length scales in the lens plane and the source plane as ξ
0 = 4π(σ v /c) 2 d A L d A LS /d A S , η 0 = ξ 0 d A S /d A L
respectively, we can simplify the lensing equation and obtain the image separation of SIS lensing
∆θ = 1.27 ′ ( d A LS d A S )M 2/3 15 E(z) 2/3 ,(4)
where M 15 = M/(10 15 h −1 M ⊙ ) is dimensionless halo mass. Here d A S and d A L are the angular diameter distances from the observer to the source and to the lens object respectively, while d A LS is the same quantity but from the lens to the source object. Therefore, the cross-section defined in the lens plane for forming two images by an SIS lens with splitting angle ∆θ > ∆θ 0 can be expressed as σ(M, z) = πξ 2 0 ϑ(∆θ − ∆θ 0 ) where ϑ is a step function.
The Cross-section for NFW Models
The GNFW density profile can be expressed in the form ρ(r) = ρ s r 3 s /r α (r + r s ) 3−α (Zhao 1996) where 0 < α < 3. The GNFW density profile reduces to the case of NFW if α = 1. The mass of a dark halo within r 200 can be defined as M = 4π r 200 0 ρr 2 dr = 4πρ s r 3 s f (c 1 ), and r 200 is the radius of a sphere around a dark halo within which the average mass density is 200 times the critical mass density of the universe. The function f (c 1 ) = c 1 0 x 2 dx/x α (1 + x) 3−α and c 1 = r 200 /r s is the concentration parameter and takes the form (Oguri et al. 2001(Oguri et al. , 2002Oguri 2004)
c 1 (M 15 , z) = c norm 2 − α 1 + z [10M 15 ] −0.13 ,(5)
where z is the redshift of halo, c norm = 8 (Bullock et al. 2001). So ρ s and r s can be related to mass M 15 and redshift z by
ρ s = ρ c 0 E 2 (z) 200 3 c 3 1 f (c 1 (M 15 , z)) , r s = 1.626 c 1 M 1/3 15 E 2/3 (z) h −1 Mpc,(6)
where ρ c 0 is the critical mass density of the universe today. The lensing equation for the GNFW profile is given by y = x − µ s g(x)/x; ξ = xr s and η = yr s d A S /d A L are the position vectors in the lens plane and the source plane respectively, g(
x) ≡ x 0 udu ∞ 0 (u 2 + z 2 ) −α/2 (u 2 + z 2 ) 1/2 + 1 −3+ and µ s ≡ 4ρ s r s Σ cr = 0.002( ρ s ρ c 0 )( r s 1h −1 Mpc )( d A R c/H 0 ),(7)
where µ s is a parameter on which the efficiency of producing multiple images is strongly
dependent. Here Σ cr = c 2 4πG d A S d A L d A LS
is the critical surface mass density, and
d A R = d A L d A LS /d A S .
The lensing equation curves are symmetrical with respect to the origin. Multiple images can be formed when |y| ≤ y cr , where y cr is the maximum value of y when x < 0 or the minimum value for x > 0. Generally speaking, there exist three images for |y| < y cr . We will just consider the outermost two images stretched by the splitting angle ∆θ when more than two images are formed. Therefore, we can write the cross-section as σ (M, z) ≈ πy 2 cr r 2 s ϑ (∆θ − ∆θ 0 ) with ∆θ > ∆θ 0 in the lens plane for multiple images produced by a GNFW lens at z. Here splitting angle ∆θ is given by ∆θ = r s ∆x/d A L ≈ 2x 0 r s /d A L and x 0 is the positive root of the lensing equation y(x) = 0.
Numerical Results
In this letter, we assume spatially flat ΛCDM models characterized by the matter density parameter Ω m , vacuum energy density parameter Ω Λ . For both PL-ΛCDM and RSI-ΛCDM models, we take cosmological parameters to be the new result from the WMAP: Hubble constant h = 0.71, Ω m = 0.27, σ 8 = 0.84 Spergel et al. 2003) and M c1 = 1.5 × 10 13 h −1 M ⊙ . As mentioned above, the lensing probability depend strongly on the abundance of viralized dark halos. The mass function of dark halos directly involve the calculation of primordial power of density fluctuation. According to Eq.(1), it is clear that the effect of running spectral index on mass function of halos cause the difference of lensing probabilities between the two models.
Although the redshift distribution of quasars in the JVAS/CLASS survey is still poorly known, the prediction of Dunlop & Peacock (1990) model and the CLASS lensing sub-sample redshift measurements suggest that the redshift distribution for CLASS unlensed sources can be modelled by a Gaussian distribution with mean redshift < z s >=1.27 (Marlow et al. 2000) and dispersion σ z =0.95 (Chae et al. 2002;Browne et al. 2003;Myers et al. 2003). Thus in this letter we adopt the Gaussian redshift distribution of quasars with the mean redshift < z s >=1.27 and dispersion σ z =0.95. In the definition of cross-section for forming two images, we just consider the criterion ∆θ, and neglect another one q r that is the flux density (brightness) ratio of multiple images (Schneider et al. 1992). In order to investigate the effect of central black holes or bulges on lensing probability, Chen (2003a,b) introduced q r into the calculation for lensing cross-section. Due to the existence of central black holes or galactic bulges, y cr becomes extremely large when |x| approaches zero. Thus y cr can be determined by the consideration of q r together with ∆θ. However for GNFW halo models in the absence of central black holes or galactic bulges, the lensing equation curves are so smooth that we do not need to define cross-section by q r . As for SISs, it is necessary to introduce q r because both the predicted lensing probabilities and the determination of the cosmic equation of state ω are quite sensitive to q r (Chen 2004a,b). Our objective in this letter is to examine the effect of running spectral index on the predicted lensing probabilities, so for explicit we neglect this selection criterion q r for forming multiple images. In such cases, we calculate lensing probabilities with image separations greater than ∆θ according to the compound model of SIS and NFW halo profile in both PL-ΛCDM and RSI-ΛCDM models. Our numerical results are shown in Fig.1 together with the observational one from JVAS/CLASS. As we expect, there is slight difference between the two models on large image separations, while this difference enlarges with the decrease of the splitting angle. More specifically, the RSI-ΛCDM model can reduce the predicted lensing probabilities on small splitting angle of less than about 4 ′′ compared with that of PL-ΛCDM model.
Conclusions and Discussion
Motivated by the new result on the index of primordial power spectrum from a combination of WMAP data with other finer scale CMB experiments and other large-scale structure observations, we present the first study on the predicted lensing probability in a ΛCDM model with a running spectral index. In a popular cosmological model from the new fit results mentioned above: h = 0.71, Ω m = 0.27, σ 8 = 0.84, we calculate lensing probabilities with image separations greater than ∆θ according to the compound model of SIS and NFW halo profile in both PL-ΛCDM and RSI-ΛCDM models. From the analysis above, we can see that the running spectral index mainly affect the predicted lensing probability on small image separation. It is well known that structures in the universe forms hierarchically in standard CDM models. Yoshida et al. (2003) found that although this hierarchical formation mechanism do not work well in RSI-ΛCDM model compared with that in PL-ΛCDM model and it also is not clear that the PS theory can be used in RSI-ΛCDM model, the mass function measured by high-resolution cosmological N-body/hydrodynamic simulations overall match the PS mass function for both RSI-ΛCDM and PL-ΛCDM model.
In addition, because the running spectral index model predicts a significant lower power of density fluctuation on small scales than the standard PL-ΛCDM model Yoshida et al. 2003), it should also attract considerable attention in studies on weak lensing by large-scale structure (Ishak et al. 2003), especially on skewness Zhang et al. 2003) which characterizes the non-Gaussian property of κ field in the nonlinear regime.
Fig. 1 .
1-The lensing probabilities for image separations greater than ∆θ. The solid-line histogram represents observed lensing probabilities from JVAS/CLASS. The solid line is the lensing probabilities for running spectral index ΛCDM model, while the dashed line is that for power law ΛCDM model.
). The combination of the first-year Wilkinson Microwave Anisotropy Probe (WMAP) data with other finer scale cosmic background (CMB) experiments (Cosmic Background Imager [CBI], Arcminute Cosmology Bolometer Array Receiver [ACBAR]) and two observations of large-scale structure (the Anglo-Australian Telescope Two-Degree Field Galaxy Redshift Survey [2dFGRS]
We are very grateful to the anonymous referee for constructive suggestion. T.J.Zhang would like to thank Ue-Li Pen, Peng-Jie Zhang, Xiang-Ping Wu, Bo Qin and CITA for their hospitality during his visits to the Canadian Institute for Theoretical Astrophysics(CITA), University of Toronto and the cosmology groups of the National Astronomical Observatories of P.R.China. This work was supported by the National Science Foundation of China (grants No.10473002 and 10273003).
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| []
|
[
"Quantum interaction between two gravitationally polarizable objects in presence of boundaries",
"Quantum interaction between two gravitationally polarizable objects in presence of boundaries"
]
| [
"Hongwei Yu \nDepartment of Physics\nSynergetic Innovation Center for Quantum Effects and Applications\nHunan Normal University\n410081ChangshaHunanChina\n",
"Zhao Yang \nDepartment of Physics\nSynergetic Innovation Center for Quantum Effects and Applications\nHunan Normal University\n410081ChangshaHunanChina\n",
"Puxun Wu \nDepartment of Physics\nSynergetic Innovation Center for Quantum Effects and Applications\nHunan Normal University\n410081ChangshaHunanChina\n\nCenter for High Energy Physics\nPeking University\n100080BeijingChina\n"
]
| [
"Department of Physics\nSynergetic Innovation Center for Quantum Effects and Applications\nHunan Normal University\n410081ChangshaHunanChina",
"Department of Physics\nSynergetic Innovation Center for Quantum Effects and Applications\nHunan Normal University\n410081ChangshaHunanChina",
"Department of Physics\nSynergetic Innovation Center for Quantum Effects and Applications\nHunan Normal University\n410081ChangshaHunanChina",
"Center for High Energy Physics\nPeking University\n100080BeijingChina"
]
| []
| We investigate, in the framework of the linearized quantum gravity and the leading-order perturbation theory, the quantum correction to the classical Newtonian interaction between a pair of gravitationally polarizable objects in the presence of both Neumann and Dirichlet boundaries. We obtain general results for the interaction potential and find that the presence of a boundary always strengthens in the leading-order the interaction as compared with the case in absence of boundaries. But different boundaries yield a different degree of strengthening. In the limit when one partner of the pair is placed very close to the Neumann boundary, the interaction potential is larger when the pair is parallel with the boundary than when it is perpendicular to, which is just opposite to the case when the boundary is Dirichlet where the latter is larger than the former. In addition, we find that the pair-boundary separation dependence of the higherorder correction term is determined by the orientation of the pair with respect to boundary, with the parallel case giving a quadratic behavior and the perpendicular case a linear one. | 10.1103/physrevd.97.026008 | [
"https://arxiv.org/pdf/1801.03669v1.pdf"
]
| 118,888,825 | 1801.03669 | 4f652a67246381bf84392b790054bef9f3dc35c5 |
Quantum interaction between two gravitationally polarizable objects in presence of boundaries
11 Jan 2018
Hongwei Yu
Department of Physics
Synergetic Innovation Center for Quantum Effects and Applications
Hunan Normal University
410081ChangshaHunanChina
Zhao Yang
Department of Physics
Synergetic Innovation Center for Quantum Effects and Applications
Hunan Normal University
410081ChangshaHunanChina
Puxun Wu
Department of Physics
Synergetic Innovation Center for Quantum Effects and Applications
Hunan Normal University
410081ChangshaHunanChina
Center for High Energy Physics
Peking University
100080BeijingChina
Quantum interaction between two gravitationally polarizable objects in presence of boundaries
11 Jan 2018PACS numbers: 0460Bc, 03
We investigate, in the framework of the linearized quantum gravity and the leading-order perturbation theory, the quantum correction to the classical Newtonian interaction between a pair of gravitationally polarizable objects in the presence of both Neumann and Dirichlet boundaries. We obtain general results for the interaction potential and find that the presence of a boundary always strengthens in the leading-order the interaction as compared with the case in absence of boundaries. But different boundaries yield a different degree of strengthening. In the limit when one partner of the pair is placed very close to the Neumann boundary, the interaction potential is larger when the pair is parallel with the boundary than when it is perpendicular to, which is just opposite to the case when the boundary is Dirichlet where the latter is larger than the former. In addition, we find that the pair-boundary separation dependence of the higherorder correction term is determined by the orientation of the pair with respect to boundary, with the parallel case giving a quadratic behavior and the perpendicular case a linear one.
I. INTRODUCTION
The classical Newtonian theory of gravity tells us that the interaction potential of two massive objects behaves as r −1 with r being the separation between them. This interaction is expected to be modified if gravity is quantized. However, a complete study of quantum corrections to the classical Newtonian interactions requires a full theory of quantum gravity which is elusive at the present. Even though, quantum gravity effects at the low energies can however be analyzed by treating the general relativity as an effective field theory or in the framework of linearized quantum gravity. For example, by summing one-loop Feynman diagrams with off-shell gravitons, it has been found that the monopole-monopole interaction provides a quantum correction, which behaves as r −3 , to the Newtonian force [1].
A direct consequence of quantization of gravity is the appearance of quantum vacuum fluctuations of gravitational fields, i.e., fluctuations of spacetime itself. These fluctuations are expected to induce instantaneous quadrupole moments in gravitationally polarizable objects. As a result, the induced quadrupole-quadrupole interaction produces a quantum correction to the classical Newtonian interaction, which has been studied in different contexts [2][3][4]. The quantum potential between gravitational quadrupoles is found to behave as r −11 and r −10 in the far and near regimes respectively. Recently, the quadruplequadruple interaction was extended to include the contribution of fluctuations of thermal gravitons at finite temperature [6]. In the high-temperature limit, the potential behaves like T /r −10 , thus the thermal fluctuations of gravitons produce a dominant contribution, while in the low-temperature limit, the zero-point fluctuations dominate the interaction and the thermal fluctuations only generate a small correction.
It is well known that field modes will be changed when boundaries are present [7][8][9], which leads to modifications of vacuum fluctuations. Changes in vacuum fluctuations can produce observable effects. The Casimir-Polder potential [10] between two neutral atoms near a perfectly conducting plate is an example of such effects that arise from the changes of vacuum modes of electromagnetic fields [11][12][13]. In the case of gravitation, one also finds that interesting effects appear when boundaries are present, for example, lightcone fluctuations are modified [14][15][16][17], which leads to flight time fluctuations of a probe light signal from its source to a detector [18].
In this paper, we shall examine the impact of plane boundaries on the induced quadrupole-quadrupole interaction between a pair of gravitationally polarizable objects in vacuum. Our approach is based upon the leading-order perturbation theory in the framework of linearized quantum gravity [14], which has been used to investigate quantum gravitational corrections in [4,6]. Throughout this paper, the Latin indices run from 0 to 3, while the Greek letter is from 1 to 3. The Einstein convention is assumed for repeated indices and = c = k B = 1 is set. Here, c is the light speed, is the reduced Planck constant and k B is the Boltzmann constant.
II. BASIC EQUATIONS
The system, which is shown in Fig. (1), consists of two gravitationally polarizable objects (A and B) in a bath of fluctuating quantum vacuum gravitational fields with a plane boundary at z = 0. For simplicity, we assume A and B to be described by twolevel harmonic oscillators with their Hamiltonians being
H A(B) = E 0 A(B) |0 A(B) 0 A(B) | + E 1 A(B) |1 A(B) 1 A(B) |.
For this system, the total Hamiltonian can be written as
H = H F + H A + H B + H AF + H BF ,(1)
where H F is the Hamiltonian of gravitational fields and
H A(B)F = − 1 2 Q ij A(B) E ij(2)
represents the interactions between the objects and gravitational fields. Here Q ij is the object's quadrupole moment induced by the gravitational vacuum fluctuations and the gravito-electric tensor E ij is defined as E ij = R 0i0j by analogy of linearized Einstein field equation with the Maxwell equations [19], where R µναβ is the Riemann tensor defined in terms of the metric tensor. A fluctuating metric tensor can be expanded in a flat background spacetime as g µν = η µν + h µν with h µν being the linearized perturbations which can be quantized as [14] h
ij (x, t) = k,λ [a λ (ω, x)f λ ij,k + H.c.],(3)
where H.c. denotes the Hermitian conjugate,
k = {k 1 , k 2 , k 3 }, x = {x, y, z}, a λ (ω, x)
is the gravitational field operator, which defines the vacuum a λ (ω, x)|{0} = 0, ω = k 2 1 + k 2 2 + k 2 3 , λ labels the polarization states, and f λ ij,k (x, t) is the field mode. Substituting the metric tensor into the Riemann tensor gives
E ij = 1 2ḧ ij ,(4)
where a dot denotes a derivative with respect to time t.
Using the leading-order perturbation theory, we find that the interaction potential between two objects, which is just the shift of the ground-state energy, arises from fourthorder perturbations [10,12,20] and can be expressed as
U AB (x A , x B ) = − I,II,III ′ 0|Ĥ AF +Ĥ BF |I| I|Ĥ AF +Ĥ BF |II (E I − E 0 )(E II − E 0 ) × II|Ĥ AF +Ĥ BF |III III|Ĥ AF +Ĥ BF |0 (E III − E 0 ) ,(5)
where |0 = |0 A |0 B |{0} is the ground state of the whole system, which is omitted in the summation as indicated by a prime, and the summation includes position and frequency integrals. |I , |II and |III are the intermediate states. In Ref. [4] it has been shown that there are ten possible combinations of intermediate states, which are listed in Table. (I). Summing up all of them, we obtain that the interaction potential for isotropically polarizable objects can be expressed as
U AB (x A , x B ) = − 1 4(ω A + ω B ) ∞ 0 dω ∞ 0 dω ′α AαB (ω A + ω B + ω) (ω A + ω)(ω B + ω) 1 ω + ω ′ − 1 ω − ω ′ ×G ijkl (ω, x A , x B )G ijkl (ω ′ , x A , x B ) ,(6)
where
ω A(B) = (ω 1 A(B) − ω 0 A(B) ) with ω 1 A(B) = E 1 A(B) and ω 0 A(B) = E 0 A(B) represents the transition frequency of the object,α A(B) ≡Q ij A(B)Q * ij A(B) = |Q ij A(B) | 2 withQ ij A(B) = 0 A(B) |Q ij A(B) |1 A(B) andQ * ij A(B) = 1 A(B) |Q ij A(B) |0 A(B) , and G ijkl (ω, x A , x B ) is the two-point correlation function of gravito-electric fields G ijkl (ω, x A , x B ) = 0|E ij (ω, x A )E kl (ω, x B )|0 .(7)
III.
NEUMANN BOUNDARY CONDITION
Now we consider what happens to the potential when a Neumann boundary is present.
For metric perturbations which satisfy the Neumann boundary condition ∂ z f λ ij,k | z=0 = 0, the field mode f λ ij,k can be expressed as
f λ ij,k (x, t) = 8πG 2ω(2π) 3 e ij (k, λ)e i(k·x−ωt) + e ij (k − , λ)e i(k − ·x−ωt) ,(8)
in the transverse tracefree (TT) gauge with e ij (k, λ) being polarization tensors. Here
k − = {k 1 , k 2 , −k 3 } ,
and G is the Newton's gravitational constant.
From Eqs. (3), (4), (7) and (8), one finds that the two-point correlation function of E ij has the form
G ijkl (r,r, ∆t) = 1 4 0|ḧ ij (x, t)ḧ kl (x ′ , t ′ )|0 (9) = G 8π 2 d 3 k ω 3 e iω∆t λ e ij (k, λ)e kl (k, λ)e ik·r + e ij (k, λ)e kl (k − , λ)e ik·r +e ij (k − , λ)e kl (k, λ)e ik − ·r + e ij (k − , λ)e kl (k − , λ)e ik − ·r .
Here r = |r|,r = |r|, and
r = {x − x ′ , y − y ′ , z − z ′ },r = {x − x ′ , y − y ′ , z + z ′ }.(10)
In the TT gauge, the summation of polarization tensors gives [14] λ
e ij (k, λ)e kl (k ′ , λ) = δ ik δ jl + δ il δ jk − δ ij δ kl +k ikjk ′ kk ′ l +k ikj δ kl +k ′ kk ′ l δ ij −k ik ′ l δ jk −k ik ′ k δ jl −k jk ′ l δ ik −k jk ′ k δ il ,(11)
wherek
i = k i ω .(12)
From this summation of polarization tensors, we can obtain two following relations
λ e ij (k, λ)e kl (k, λ)e ik·r = 1 ω 4 [(δ ik δ jl + δ il δ jk − δ ij δ kl )∇ 4 + (∂ i ∂ j δ kl + ∂ k ∂ l δ ij (13) −∂ i ∂ l δ jk − ∂ i ∂ k δ jl − ∂ j ∂ l δ ik − ∂ j ∂ k δ il )∇ 2 + ∂ i ∂ j ∂ k ∂ l ]e ik·r ≡ 1 ω 4ĝ r ijkl e ik·r ,
and λ e ij (k, λ)e kl (k − , λ)e ik·r = 1
ω 4 σ km σ ln [(δ im δ jn + δ in δ jm − δ ij δ mn )∇ 4 + (∂ i ∂ j δ mn + ∂ m ∂ n δ ij −∂ i ∂ n δ jm − ∂ i ∂ m δ jn − ∂ j ∂ n δ im − ∂ j ∂ m δ in )∇ 2 + ∂ i ∂ j ∂ m ∂ n ]e ik·r ≡ 1 ω 4 σ km σ lnĝr ijmn e ik·r ,(14)
whereĝ r ijkl is a differential operator whose definition straightforwardly follows from Eq. (13)
, σ = {{1, 0, 0}, {0, 1, 0}, {0, 0, −1}}, ∇ 2 = ∂ i ∂ i and ∂ i = ∂ x i . Substituting
Eqs. (13,14) into Eq. (9) and performing the Fourier transform, one has G ijkl (r,r, ω) = G 4π 2 dΩ ω ĝ r ijkl e iωr cos θ + σ km σ lnĝr ijmn e iωr cos θ = G π ĝ r ijkl sin(ωr) r + σ km σ lnĝr ijmn sin(ωr) r .
where Ω is the solid angle, and the relation dΩe ik·r = π 0 sin θdθ 2π 0 dφe iωr cos θ = 4π sin(ωr) ωr (16) has been used. Substituting Eq. (15) into Eq. (6) gives
U AB (r,r) = − G 2 4π 2 (ω A + ω B ) ∞ 0 dω ∞ 0 dω ′α AαB (ω A + ω B + ω) (ω A + ω)(ω B + ω) 1 ω + ω ′ − 1 ω − ω ′ × ĝ r ijkl sin(ωr) r + σ km σ lnĝr ijmn sin(ωr) r × ĝr ijkl sin(ω ′r ) r + σ km ′ σ ln ′ĝr ijm ′ n ′ sin(ω ′r ) r |r →r,r→r .(17)
Defining y(r, r ′ ) to be
y(r, r ′ ) = 1 (ω A + ω B ) ∞ 0 dω ∞ 0 dω ′α AαB (ω A + ω B + ω) (ω A + ω)(ω B + ω) 1 ω + ω ′ + 1 −ω + ω ′ × sin(ωr) r sin(ω ′ r ′ ) r ′ = 1 (ω A + ω B ) ∞ 0 dω sin(ωr) r ∞ −∞ dω ′α AαB (ω A + ω B + ω) (ω A + ω)(ω B + ω) × 1 ω + ω ′ + 1 −ω + ω ′ e iω ′ r ′ 2ir ′ = π (ω A + ω B ) ∞ 0 dωα AαB (ω A + ω B + ω) (ω A + ω)(ω B + ω) sin(ωr) cos(ωr ′ ) rr ′ ,(18)
and following an analogy with the electric polarizability of atoms [21] to define the object's ground-state polarizability as
α A(B) (ω) = lim ǫ→0 +α A(B) ω A(B) ω 2 A(B) − ω 2 − iǫω ,(19)
which satisfies Q ij (ω) = α(ω)E ij (ω, x), one can obtain that
y(r, r ′ ) = π 2 α A (0)α B (0) 1 rr ′ (r + r ′ ) ,(20)
when r ′ → r, and when r = r ′
y(r, r ′ ) = π 2 α A (0)α B (0) 1 rr ′ (r + r ′ ) + 1 rr ′ (r − r ′ ) ,(21)
where the approximate static polarizability has been assumed. Then, Eq.
+ σ km σ lnĝr ijmnĝ r ijkl 1 rr(r + r) +ĝr ijklĝr ijkl 1 rr(r +r) |r →r,r→r .
Here σ km σ lm = δ kl has been used.
After lengthy calculations, one can arrive at the interaction potential
U AB (r,r) = − G 2 4π α A (0)α B (0) 3987 r 11 + 3987
r 11 + 144 r 5r5 (r +r) 9 A + Br 4 cos 4θ (23) +4Cr 2 cos 2θ + 12Br 2r2 cos 2θ cos 2θ + 4Cr 2 cos 2θ + Br 4 cos 4θ , where A = 9(r 8 + 9r 7r + 37r 6r2 + 93r 5r3 + 198r 4r4 + 93r 3r5 + 37r 2r6 + 9rr 7 +r 8 ),
B = 3r 4 + 27r 3r + 83r 2r2 + 27rr 3 + 3r 4 , C = −3r 6 − 27r 5r − 100r 4r2 − 180r 3r3 + 60r 2r4 + 27rr 5 + 3r 6 , C = −3r 6 − 27rr 5 − 100r 2r4 − 180r 3r3 + 60r 4r2 + 27r 5r + 3r 6 .(24)
Here θ andθ are the angles of r andr with respect to the normal direction of the plane boundary, respectively. The potential includes three terms: the usual r −11 interaction potential between two objects in the absence of the plane boundary [2,4], ther −11 term which is the interaction between the object A and the image of object B reflected by the plane boundary, and the remaining term depending on both r andr.
A. Two special cases
Now we analyze the interaction potential in some special circumstances. The first special case is that two objects are placed in parallel with the plane boundary (z −z ′ = 0), which means that θ = π 2 ,θ = cos −1 2z r andr = √ r 2 + 4z 2 . When the two-object system is close to the boundary, i.e. when z ≪ r (r ∼r), we find that
U AB (r) = − G 2 4π α A (0)α B (0) 10242 r 11 − 119790 z 2 r 13 .(25)
It is easy to see that the boundary increases the potential about 2.6 times in the leadingorder since the coefficient in the case of flat spacetime without boundary is 3987 although the boundary do not change the behavior of r-dependence. The boundary also gives a negative higher-order correction term, which is dependent on z 2 . Now we consider that two objects are placed perpendicular to the boundary. Then, one has θ =θ = 0 andr = r + 2z. In the limit of z ≪ r, the potential becomes
U AB (r) = − G 2 4π α A (0)α B (0) 9252 r 11 − 101772 z r 12 .(26)
which is, in the leading-order, about 2.3 times that in the absence of the plane boundary, and is less than that in the parallel case. In addition, we find that the higher-order z−dependent correction term is different from that in the parallel case which relies on z 2 .
IV. DIRICHLET BOUNDARY CONDITION
For the Dirichlet boundary condition, the field mode satisfies f λ ij,k | z=0 = 0 and thus can be written as
f λ ij,k (x, t) = 8πG 2ω(2π) 3 1 i e ij (k, λ)e i(k·x−ωt) − e ij (k − , λ)e i(k − ·x−ωt)(27)
in the TT gauge. From the above equation, one can show that the two-point correlation function defined in (7) becomes G ijkl (r,r, ω) = − G 4π 2 dΩ ω ĝ r ijkl e iωr cos θ − σ km σ lnĝr ijmn e iωr cos θ = − G π ĝ r ijkl sin(ωr) r − σ km σ lnĝr ijmn sin(ωr) r
and then the interaction potential reads
U AB (r,r) = − G 2 8π α A (0)α B (0) ĝ r ijklĝr ijkl 1 rr(r +r) − σ km σ lnĝ r ijklĝr ijmn 1 rr(r +r)(29)
− σ km σ lnĝr ijmnĝ r ijkl 1 rr(r + r) +ĝr ijklĝr ijkl 1 rr(r +r) |r →r,r→r .
Following the same procedure as in the preceding section, we get that in the case of the Dirichlet boundary the interaction potential is
U AB (r,r) = − G 2 4π α A (0)α B(
A. Two special cases
For the special case of two objects placed in parallel with the plane boundary, we take the limit of z ≪ r and obtain
U AB (r) = − G 2 4π α A (0)α B (0) 5706 r 11 − 55638 z 2 r 13 .(31)
Thus, a Dirichlet boundary also reinforces the interaction, but it increases only about 1.4
time compared with the case without boundary, which is less than that in the case of a Neumann boundary. Another noteworthy difference is that the higher-order correction term is also less than that in the Neumann boundary case.
If objects A and B are placed in perpendicular to the plane boundary, we obtain
U AB (r) = − G 2 4π α A (0)α B (0) 6696 r 11 − 73656 z r 12(32)
in the limit of z ≪ r, which is about 1.7 times that in the absence of the plane boundary and is less than that from the Neumann boundary. Comparing Eqs (31) and (32) reveals that the leading term in the potential is larger when the pair of the objects is perpendicularly placed than when it is in parallel with the boundary, which is different from the Neumann boundary case where the former is less than the latter. Similar to the Neumann boundary case, the z-dependence of the higher-order correction term in the present case is also different from that of the parallel case.
V. CONCLUSION
In this paper, we have investigated the quantum correction to the classical Newtonian force between a pair of polarizable objects in the presence of plane boundaries in the framework of the linearized quantum gravity and the leading-order perturbation theory.
Two kinds of boundary conditions, i.e., Neumann and Dirichlet, are imposed. The general results are given in Eqs. (23) and (30). In both cases, the potentials consist of three terms,
i.e., the usual r −11 -dependent interaction potential between two objects in the absence of the plane boundary where r is the separation of the two objects, ther −11 term which is the interaction between the object A and the image of object B reflected by the plane boundary wherer is the distance between the object A and the image of object B, and the term depending on both r andr. Different boundary conditions in general lead to different interaction potentials, with the Neumann boundary yielding a larger interaction than the Dirichlet boundary.
When one partner of the pair is placed very close to the boundary (z ≪ r), where z is the distance between the boundary and the closer partner, we find, for both special cases, i.e., the pair is in parallel with or perpendicular to the plane boundary, that the boundary strengthens the interaction potential as compared with the case in the absence of a boundary. In the Neumann boundary case, the potential in the parallel case is larger than that of the perpendicular case, which is just opposite to the Dirichlet boundary case where the latter is larger than the former. In addition, we find that the sign of the higher-order correction term is negative and the pair-boundary separation dependence of the correction is determined by the orientation of the object pair, with the parallel case and the perpendicular case give a quadratic and a linear correction, respectively.
Finally, let us briefly comment on the issue of how to realize the boundary conditions considered in this paper in some specific physical setups. It is well known that ordinary materials can hardly reflect nor absorb gravitational waves [22], and thus the reflection coefficient for gravitational waves will be extremely small. However, recently, there have been interesting speculations that quantum matter such as superconducting films might behave like highly reflective mirrors that realize the Dirichlet boundary condition for gravitational waves, since the incident gravitational waves may be reflected effectively due to the so-called Heisenberg-Coulomb effect [23]. As for the Neumann boundary condition, we do not know of any specific physical setup that can realize it. So, at present, it only remains as a theoretical curiosity.
FIG. 1 :
1The system consists of objects A and B in a flat spacetime with a plane boundary at z = 0.
U
AB (r,r) = − G 2 8π α A (0)α B (0) ĝ r ijklĝr ijkl 1 rr(r +r) + σ km σ lnĝ r ijklĝr ijmn 1 rr(r +r)
) |0 A , 1 B |1. A , 0 B |11A , 0 B |1 (2) , 1 (3) |0 A , 1 B |1 (4)
A , 1 B |1 (1) |1 A , 1 B |{0} |0 A , 1 B |1. A , 1 B |1 (1) |1 A , 1 B |{0} |0 A , 1 B |1 (2)
TABLE I: Ten intermediate states contributing to the two-objects potential. TABLE I: Ten intermediate states contributing to the two-objects potential.
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| []
|
[
"Pricing index options by static hedging under finite liquidity",
"Pricing index options by static hedging under finite liquidity"
]
| [
"John Armstrong \nDepartment of Mathematics\nKing's College London\nWC2R 2LSStrand, LondonUnited Kingdom\n",
"Teemu Pennanen \nDepartment of Mathematics\nKing's College London\nWC2R 2LSStrand, LondonUnited Kingdom\n",
"Udomsak Rakwongwan \nDepartment of Mathematics\nKing's College London\nWC2R 2LSStrand, LondonUnited Kingdom\n"
]
| [
"Department of Mathematics\nKing's College London\nWC2R 2LSStrand, LondonUnited Kingdom",
"Department of Mathematics\nKing's College London\nWC2R 2LSStrand, LondonUnited Kingdom",
"Department of Mathematics\nKing's College London\nWC2R 2LSStrand, LondonUnited Kingdom"
]
| []
| We develop a model for indifference pricing in derivatives markets where price quotes have bid-ask spreads and finite quantities. The model quantifies the dependence of the prices and hedging portfolios on an investors beliefs, risk preferences and financial position as well as on the price quotes. Computational techniques of convex optimisation allow for fast computation of the hedging portfolios and prices as well as sensitivities with respect to various model parameters. We illustrate the techniques by pricing and hedging of exotic derivatives on S&P index using call and put options, forward contracts and cash as the hedging instruments. The optimized static hedges provide good approximations of the options payouts and the spreads between indifference selling and buying prices are quite narrow as compared with the spread between super-and subhedging prices. | 10.1142/s0219024918500449 | [
"https://arxiv.org/pdf/1803.02486v1.pdf"
]
| 158,078,132 | 1803.02486 | 0196d6c9d89750eb776b66d27fa301a5334e91df |
Pricing index options by static hedging under finite liquidity
March 8, 2018
John Armstrong
Department of Mathematics
King's College London
WC2R 2LSStrand, LondonUnited Kingdom
Teemu Pennanen
Department of Mathematics
King's College London
WC2R 2LSStrand, LondonUnited Kingdom
Udomsak Rakwongwan
Department of Mathematics
King's College London
WC2R 2LSStrand, LondonUnited Kingdom
Pricing index options by static hedging under finite liquidity
March 8, 2018
We develop a model for indifference pricing in derivatives markets where price quotes have bid-ask spreads and finite quantities. The model quantifies the dependence of the prices and hedging portfolios on an investors beliefs, risk preferences and financial position as well as on the price quotes. Computational techniques of convex optimisation allow for fast computation of the hedging portfolios and prices as well as sensitivities with respect to various model parameters. We illustrate the techniques by pricing and hedging of exotic derivatives on S&P index using call and put options, forward contracts and cash as the hedging instruments. The optimized static hedges provide good approximations of the options payouts and the spreads between indifference selling and buying prices are quite narrow as compared with the spread between super-and subhedging prices.
Introduction
In incomplete markets, the prices of financial products offered by an agent depend on subjective factors such as views on the future development of the underlying risk factors, risk preferences, the financial position as well as the trading expertise of the agent. A agent's prices also depend on the prices at which the agent can trade other financial products since that affects the costs of (partial) hedging when selling a product.
The indifference pricing principle provides a consistent way to incorporate the above factors into a pricing model. A classical reference on indifference pricing of contingent claims under transaction costs is [6]. In the insurance sector, where market completeness would be quite an unrealistic assumption, indifference pricing seems to have longer history; see e.g. [2]. A more recent account with further references can be found in [3].
Indifference pricing builds on an optimal investment model that describes the relevant sector of financial markets as well as the agent's financial position, views and risk preferences. Realistic models are often difficult to solve much like the investment problem they describe. This paper develops a computational framework for indifference pricing of European style options on the S&P500 index. Instead of the usual dynamic trading of the index and a cashaccount, we take index options as the hedging instruments. For the ease of implementation, we consider only buy-and-hold strategies in the options but we take actual market quotes as the trading costs. For the nearest maturities, there are some 200 strikes with fairly liquid quotes. This results in a convex stochastic optimization problem with one-dimensional uncertainty but over 400 decision variables. The model is solved numerically using discretization and an interior point solver for convex optimization. The indifference prices for a given payout are found within seconds so it is easy to study the effect of an agent's views, risk preferences and financial position on the indifference prices.
Much like the Breeden-Litzenberger formula, indifference pricing provides automatic calibration to quoted call option prices. While the Breeden-Litzenberger formula provides only a heuristic approximation in real markets with only a finite number of strikes and finite liquidity, the indifference approach finds the best static hedge given the quotes, the agents views and preferences. Moreover, the indifference approach gives explicit control of the hedging error in incomplete markets. Unlike the Breeden-Litzenberger formula would suggest, we find that in the presence of bid-ask spreads, the optimal hedges are often quite compressed portfolios of options taking positions only in few of the strikes. This is a significant benefit when implementing the hedges in practice. While the Breeden-Litzenberger formula applies only to options whose payouts are differences of convex functions of the underlying, the indifference pricing applies just as well to discontinuous payoffs such as digital options.
The market
We study contingent exchange traded claims with common maturity T and payouts that only depend on the value of the S&P500 index at T . This includes put and call options, forward contracts and cash. In general, lending and borrowing rates for cash are different, so the payout on cash depends nonlinearly on the position taken. Similarly, the forward rates available in the market depend on whether one takes a long or short position. For the options, on the other hand, the payout per unit held is independent of the position. The payoffs for holding x ∈ R units of an asset are given in Table 1.
Asset
Payoff with x units held Cash Table 1: The payoffs of holding x units of the assets. Here r a and r b and the borrowing and lending rates, respectively, X T is the value of the underlying at maturity, K a and K b are forward prices for long and short positions, respectively and K is the strike price of an option
min{e r a T x, e r b T x} Forward min {(X T − K a )x, (X T − K b )x} Call max {(X T − K), 0}x Put max {(K − X T ), 0}x
While the option payoffs are linear in the position, the cost of entering a position depends nonlinearly on the units x. For a long position x > 0, one pays the ask-price while for short position, one gets the bid-price. The cost of buying x units of cash is simply x while for the forward, the cost is zero.
For each contract, the market quotes come with finite quantities. For the nearest maturity, one can find quotes for some 400 options on S&P500.
The portfolio optimisation model
For given initial wealth and quotes on cash, forward and the options, our aim is to find a portfolio with optimal net payoff at maturity. In general, the payoff will depend on the value of the underlying at maturity so the optimality will depend on our risk preferences concerning the uncertain payoffs. The optimality of a portfolio also depends on our financial position which may involve uncertain cash-flows at time T .
We will denote our initial wealth by w ∈ R and assume that our financial position obligates us to pay c units of cash at time T . The collection of all traded assets (cash, forward, options) is denoted by J. The cost of buying x j units of asset j ∈ J is given by
S j 0 (x j ) := s j a x j if x j ≥ 0, s j b x j if x j ≤ 0,
where s j b ≤ s j a are the bid and ask prices of j. If j is cash, we simply have s j b = s j a = 1 while for the forward contract s j b = s j a = 0. The finite quantities for the best quotes mean that there are upper and lower bounds q j a and q j b , respectively, on the position x j one can take in asset j at the best available quotes. For example, the quotes for the forward contract in Table 2 mean that q j a = 377 while q j b = −258. We will denote the payout of holding x j units of asset j ∈ J by P j (x j ). The functions P j are given in Table 1. We model the value X T of the underlying at maturity as a random variable so that, in the case of forwards and the options, P j (x j ) will be random as well. We will assume that our financial before the trade obligates us to deliver a random amount c of cash at maturity.
Modelling our risk preferences with expected utility, the portfolio optimization problem can be written as
minimize Ev(c − j∈J P j (x j )) over x ∈ D subject to j∈J S j 0 (x j ) ≤ w,(P)
where
D := j∈J [q j b , q j a ]
is the set of feasible portfolios, E denotes the expectation and v(c) := −u(−c) with u being the utility function. In the terminology of [5], v : R → R is a loss function. The argument of v is the unhedged part of the claim c. Besides the available quantities, one could also include various margin requirements in the constraints. It is clear that problem (P) is highly subjective. Its optimum value and solutions depend on our • financial position described by the initial cash w and liability c,
• views on the underlying X T described by the probabilistic model,
• our risk preferences described by the loss function v.
The dependence will be studied numerically in the following sections. In pricing of contingent claims, the subjective factors will be reflected in the prices at which we are willing to trade the claims. The subjectivity is the driving force behind trading in practice but it is neglected e.g. by the traditional risk neutral pricing models.
Another important feature of (P) is that it is a convex optimization problem as soon as the loss function v is convex. The convexity simply means that we are risk averse. Convexity is crucial in numerical solution of (P) as well as in the mathematical analysis of the indifference prices based on the optimum value of (P).
Numerical portfolio optimization
The first challenge in the numerical solution of problem (P) is that the objective is given in terms of an integral which, in general, does not allow for closed form expressions that could be treated by numerical optimization routines. However, in applications where the liability c only depends on the the value of the underlying at maturity, the integral is one-dimensional which can be treated fairly easily with integration quadratures. This will be the case in the applications below where we study pricing and hedging of claims contingent on the underlying price at maturity. We will approximate the expectation by Gauss-Legendre quadrature which results in an objective given as a finite sum of convex functions of the portfolio vector x.
We will reformulate the budget constraint as two linear inequality constraints by writing the position in each asset as the sum of the long and short position. That is,
x j = x j + − x j − ,
where both x j + and x j − are constrained to be positive. This results in an inequality constrained convex optimization problem with the objective and constraints represented by smooth functions. The problem has 884 variables and 1769 constraints.
The resulting problem is solved with the interior-point solver of MOSEK [1] which is suitable for large-scale convex optimisation problems. To set up an instance of the optimization problem in MATLAB takes on average 11.20 seconds and its solution with MOSEK, 4.30 seconds on a PC with Intel(R) Core(TM) i5-4690 CPU @ 3.50GHz processor and 8.00 GB memory.
Quotes, views and preferences
We used quotes for S&P500 index options with maturity 17 June 2016. The quotes were obtained from Bloomberg on 8 April 2016 at 2:55:00PM when the value of S&P500 index was 2056.32. The available quantities at the best quotes are given in terms of lot sizes which are 50 for forwards and 100 for options. The lending and borrowing rates are 0.0043 and 0.03, respectively, which correspond to the 1-month LIBOR rate and the borrowing rate of Yorkshire bank that offered the most generous rate at the time.
As a base case, we modelled the logarithm of the S&P index at maturity with the Student t-distribution with the scale parameter σ and degrees of freedom ν estimated from 25 years of historical daily data. The mean µ was set to zero. The effect of varying the parameters will be studied later on. µ σ ν 0.0000 0.0554 4.8355 Table 3: The parameters for the Student t-distribution used to model the index value at maturity.
As for the objective, we used the loss function v(c) = e λc/w , where w is the initial wealth and λ > 0 is the risk aversion parameter. In other words, the risk preferences are described by exponential utility. It should be noted that, in general, the net position at maturity can take both positive as well as negative values which prevents the use of utility functions with constant relative risk aversion. The initial wealth w used in the examples was w = 100, 000USD. Figure 1 illustrates the optimized portfolios obtained with two different risk aversions, λ = 2 (blue line) and λ = 6 (red line). The bottom panels represent the optimal portfolios with the bars corresponding to the optimal positions in the assets. The top left plots the corresponding payoffs as functions of the index at maturity and the top right plots the kernel density estimates (computed using 10,000,000 simulated values of the index at maturity) of the payoff distributions. As expected, higher risk aversion results in a payoff distribution with a thinner left tail. Increasing the risk aversion also results in reduced quantities in the optimal portfolio compared with the portfolio of a less risk averse agent. An interesting feature of the optimal portfolios is that they are sparse in that our of more than 400 quoted options, the optimal portfolio has nonzero positions in less than 10 options. This is explained by the spreads between the quotes bid-and ask-prices. To illustrate this further, we repeated the optimization with risk aversion λ = 2 by optimizing two variants of the problem. In the first one, we increased the bid-ask spread by adding a 10% transaction cost on all trades and in the second, we set both the bid-and ask-prices equal to mid-prices. The results are illustrated in Figure 2. The addition of the transaction cost made the optimal portfolio only slightly sparser while removal of the bid-ask spread had a dramatic effect by giving a portfolio that takes large positions in almost all the quoted options. For many options, it was optimal to take maximal positions allowed by the available bid/ask quantities. Figure 2: The payoffs and optimal portfolios when an additional 10% transaction cost is added to all trades (left) and when the bid-ask spread is ignored by setting both bid-and ask-prices equal to the mid-price (right)
The results
To study the effect of views on the optimal portfolio, we reoptimized the portfolio after changing the parameters of the underlying t-distribution. The risk aversion was kept at λ = 2. Figure 3 plots the payouts of the optimal portfolios in three cases. The first one is the base case already presented in Figure 1. The second if obtained by increasing the scale parameter σ to 0.40 and the third one by increasing the degrees of freedom ν to 20. As expected, increasing σ results in a portfolio that gives higher payouts further in the tails (a straddle) while ν = 20 gives essentially a Gaussian distribution with thinner tails so the optimal portfolio has higher payouts near the median at the expense of lower payoffs in the tails. Table 4. The logarithm of the expected exponential utility is known as the entropic risk measure; see e.g. [5]. We see that the highest objective value is obtained with in the base case where the model parameters are estimated from historical data. An explanation of this could be that the option prices used in the model correspond to the market participants' views of the future behaviour of the underlying. If we use a model that is "inconsistent" with these prices, the option prices appear to offer profitable trading opportunities.
To explore this phenomenon more systematically, we repeated the optimization in the Gaussian case with ν = ∞ and the mean µ and volatility σ ranging over intervals. Figure 4 plots the corresponding logarithmic objective value, i.e. the entropic risk measure as a function of µ and σ. The risk seems to be concave as a function of (µ,σ) with the maximum around (µ, σ) = (−0.05, 0.08). The maximum value is −2.289.
Indifference pricing
We will denote the optimum value of (P) by
ϕ(w, c) := inf{Ev(c − j∈J P j (x j )) | x ∈ D, j∈J S j 0 (x j ) ≤ w}.
For an agent with financial position (w,c), the indifference price for selling a claim c is given by
π s (w,c; c) := inf{w | ϕ(w + w,c + c) ≤ ϕ(w,c)}.
This is the minimum price at which the agent could sell the claim c without worsening her financial position as measured by the optimum value of (P). Analogously, the indifference price for buying c is given by
π b (w,c; c) := sup{w | ϕ(w − w,c − c) ≤ ϕ(w,c)}.
We have π b (w,c; c) ≤ π s (w,c; c)
as soon as π s (w,c; 0) = 0. Indeed, it is easily checked that the function c → π s (w,c; c) is convex so
π s (w,c; 0) ≤ 1 2 π s (w,c; c) + 1 2 π s (w,c; −c)
while π s (w,c; −c) = −π b (w,c; c), by definition. We will compare the indifference prices with the super-and subhedging costs defined for a claim c by
π sup (c) := inf{ j∈J S j 0 (x j ) | x ∈ D, j∈J P j (x j ) − c ≥ 0 P -a.s.}, π inf (c) := sup{− j∈J S j 0 (x j ) | x ∈ D, j∈J P j (x j ) + c ≥ 0 P -a.s.}.
The superhedging cost is the least cost of a superhedging portfolio while the subhedging cost is the greatest revenue one could get by entering position that superhedges the negative of c. Whereas the indifference prices of a claim depend on our financial position, views and risk preferences described by (w, c), P and v, respectively, the super-and subhedging costs are independent of such subjective factors. In complete markets, the sub-and superhedging costs are equal for all claims c but, in general, the super-and subhedging costs are too wide apart to be considered as competitive quotes for a claim.
Recall that if c : R + → R is the difference of convex functions, then its right-derivative is of bounded variation and we have
c(X T ) = c(0) + c (0)X T + ∞ 0 (X T − K) + dc (K).
This might suggest that the payout c could be replicated by a buy-and-hold portfolio of c(0) units of a zero-coupon bond, c (0) units of the underlying and a continuum of call options weighted according to the Borel-measure associated with the BV function c . Even if one could buy and sell options with arbitrary strikes, it is not quite realistic to trade a continuum of them. Nevertheless, assuming that quotes for all strikes exist, the replication cost of c would become
c(0)P T + c (0)X 0 + ∞ 0 C(K) a dc + (K) − ∞ 0 C(K) b dc − (K),
where c + and c − denote the positive and negative variations, respectively, of c and C(K) b and C(K) a denote the bid-and ask-prices of a call with strike K.
The above formulas could be used to design approximate replication strategies given the finite number of quotes in real markets. We will find out that the hedges optimized for indifference pricing look quite different from what the above replication approach would suggest. Instead of aiming for approximate replication, indifference pricing optimizes the portfolios to the given quotes, risk preferences and the given probabilistic description of the underlying.
Numerical computation of indifference prices
The definitions of the indifference prices involve the optimum value function ϕ of problem (P) which can rarely be evaluated exactly. The definitions still make sense, however, if we replace the optimum value by the best value we are able to find numerically. Besides the financial position, future views and risk preferences of an agent, the indifference prices then also depend on the agents' expertise in portfolio optimization. In computations below, we will replace ϕ by the approximate value we find with the numerical techniques described in Section 4.1. The evaluation of the indifference prices then come down to a one-dimensional search over w. This can be done numerically by a line-search algorithm.
The computation of the super-and subhedging costs come down to solving linear programming problems where the constraints require the terminal position of the agent to be nonnegative in every scenario; see [7]. In the context of put and call options, the constraint can be written in terms of finitely many linear inequality constraints since we know that the net position will be linear between consecutive strike prices.
Pricing exotic options
We illustrate indifference pricing using the optimization model of Section 3 in the pricing of three "exotic" options namely, a digital option with payoff
c(X T ) = 10, 000 if X T ≥ K, 0 if X T < K
a "quadratic forward" with c(X T ) = |X T − K| 2 and a "log-forward" with c(X T ) = 100, 000 ln(K/X T ), all with strike K = 2050. Log-forwards have been used in the hedging of variance swaps; see e.g. [4]. To compare with a simper option, we also price a European call option with the same strike.
To make the last case nontrivial, we remove the call from the set of hedging instruments.
We compute the indifference selling prices assuming thatw = 100, 000 andc = 0, that is, assuming the agent has initial position consisting only of 100,000 units of cash. The indifference prices together with the super-and subhedging costs are given in Table 5. Superhedging is imposed on the interval [100,5000]. Clearly, superhedging the quadratic and log-forwards with the given hedging instruments against all positive values of X T is impossible. The numbers reported in Table 5 Table 5: Indifference prices, together with super-and subhedging costs. Figures 5-8 illustrate the corresponding hedging strategies. Each figure gives the optimal portfolio before and after selling the option together with the payout of the"hedging portfolio" as a function of the underlying at maturity. The hedging portfolio is defined as the difference x −x, wherex and x are the optimal portfolios before and after the sale of the option.
Sensitivities
This section studies the sensitivities of the indifference prices with respect to some of the model parameters. Figure 9 plots indifference prices of a call option with strike 2000 as functions of the "volatility" σ (Since we model the underlying with the t-distributions, the variance of the log-price is σ 2 ν/(ν − 2)). Again, we have removed the call being priced from the set of hedging instruments when computing the prices. Instead of being monotone, the indifference prices achieve their minimums when σ is close to its historical estimate of 0.0554. The implied volatility computed with the classical Black-Scholes model from the mid-quote of the call is 0.1478.
Price (dollars)
indifference prices for buying and selling best bid and ask prices super and sub hedging costs Figure 9: Indifference prices as functions of volatility. The dotted lines give the best bid and ask quotes while the dashed lines give the super-and subhedging costs. Figure 10 plots the indifference prices as functions of the risk aversion. As the risk aversion increases, the gap between the indifference prices widens. The indifference price for selling a call option is more sensitive to the risk aversion. This seems quite natural as shorting a call results in unbounded downside risk unless the call is superhedged. Figure 11 illustrates the dependence of the indifference prices on an agent's initial position. While in earlier cases, the agent's initial position was assumed to consist only of cash, in this case, we consider an agent with both cash and call options of the same type as the one being priced. Figure 11 plots the indifference prices as functions of the number of call options the agent holds before the trade. As one might expect, an agent who already has exposure to the option would assign a higher price to the option. A seller would increase her exposure to the option payout while for a buyer, the option would be a natural hedge and thus worth paying a higher price for. To illustrate the nonlinearity of the indifference prices as functions of the claim, we computed the prices for different multiples M of the call. Figure 12 plots the indifference prices per option as functions of the multiplier M . The figure plots the indifference prices also in a market model where the best quotes are assumed to come with unlimited quantities. As the multiplier M increases, the quantity constraints become binding thus worsening the prices. indifference prices for selling without quantity constraints indifference prices for selling with quantity constraints Figure 12: Indifference prices of a call option per unit as a function of the quantity traded. Buying price on the left and selling price on the right. The solid line gives the prices when quantity constraints are ignored
Further developments
The developed indifference pricing framework should be taken merely as an illustration of the computational techniques that are available for portfolio optimization. The presented model could be extended in various ways in practice. For example, it would be straightforward to include margin requirements as portfolio constraints in the model, as long as the requirements are given as explicit convex constraints on the portfolio. One could also study options with different maturities by including the relevant maturities in the underlying probabilistic model. Such a multiperiod model, could also incorporate dynamic trading strategies of the underlying and cash.
Figure 1 :
1The optimal portfolios obtained with risk aversions λ = 2 and λ = 6, respectively (bottom), the payoffs of the optimal portfolios as functions of the index at maturity (top-left) and the kernel-density estimates of the payoff distributions of the optimal portfolios (top-right)
Figure 3 :Figure 4 :
34Distributions of the underlying (bottom) and optimal payoffs (top) in the base case (solid line), ν = 20 (dotted) and σ = 0.40 (dashed). All other model parameters were unchanged. The entropic risk of the optimal portfolios as a function of the mean µ and volatility σ when ν = ∞ The logarithms of the objective values obtained with the three models of the underlying in Figure 3 are given in
Figure 5 :
5of the hedging portfolio the payoff of one contract of a call option with strike 2050 Optimal portfolios before (bottom left) and after (bottom right) the sale of a call option. The top panel gives the payoff of the hedging portfolio (solid line) together with the payoff of the claim being priced (dotted line).
Figure 6 :
6Optimal portfolios before (bottom left) and after (bottom right) the sale of a digital option. The top panel gives the payoff of the hedging portfolio (solid line) together with the payoff of the claim being priced (dotted line).
Figure 7 :
7of the hedging portfolio the payoff of one contract of a square forward with strike 2050 Optimal portfolios before (bottom left) and after (bottom right) the sale of a quadratic forward. The top panel gives the payoff of the hedging portfolio (solid line) together with the payoff of the claim being priced (dotted line).
Figure 8 :
8Optimal portfolios before (bottom left) and after (bottom right) the sale of a log-forward. The top panel gives the payoff of the hedging portfolio (solid line) together with the payoff of the claim being priced (dotted line).
Figure 10 :
10for buying and selling best bid and ask prices super and sub hedging costs Indifference prices of a call option with strike 2000 as functions of risk aversions.
Figure 11 :
11Indifference prices of a call option with strike 2000 as functions of initial position in the same call
Table 2
2gives an example of quotes available on the 8 April 2016 at 14:55:00 for contracts expiring on 17 June 2016.Ticker
Type
Bid quantity Bid price Ask price Ask quantity
ESM6 Index
Forward
258
2048.75
2049
377
SPX US 6/17/2016 C2095 Index
Call
623
26.90
28.20
506
SPX US 6/17/2016 P2095 Index
Put
27
72.60
74.70
22
Table 2 :
2Market quotes on 8 April 2016 at 14:55:00 for the forward, a call and a put option maturing 17 June 2016. For the forward, the bid and ask price quotes are the forward prices for entering a short or a long position, respectively. The data was extracted from Bloomberg.
are the costs of hedging over the interval [100,5000].Claim
subhedging buying price selling price superhedging
call
51.2333
51.7338
51.7399
53.0483
digital call
5280.00
6082.35
6160.65
6885.71
quadratic forward
20383.68
20979.84
22044.92
24542.01
log-forward
322.28
358.49
404.67
499.69
price per one option (dollars)0
20
40
60
80
100
120
140
160
180
200
quantity (contracts)
85.49
85.5
85.51
85.52
85.53
85.54
85.55
85.56
85.57
85.58
85.59
price per one option (dollars)
indifference prices for buying
without quantity constraints
indifference prices for buying
with quantity constraints
0
20
40
60
80
100
120
140
160
180
200
quantity (contracts)
86.3
86.4
86.5
86.6
86.7
86.8
86.9
87
87.1
The MOSEK optimization toolbox for MATLAB manual. Version 7.1 (Revision 28). Mosek Aps, MOSEK ApS. The MOSEK optimization toolbox for MATLAB manual. Version 7.1 (Revision 28)., 2015.
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Indifference pricing: theory and applications. Princeton series in financial engineering. R Carmona, Princeton University PressPrinceton, NJR. Carmona, editor. Indifference pricing: theory and applications. Prince- ton series in financial engineering. Princeton University Press, Princeton, NJ, 2009.
Towards a theory of volatility trading. Volatility: New Estimation Techniques for Pricing Derivatives. P Carr, D Madan, RISK Publications. P. Carr and D. Madan. Towards a theory of volatility trading. Volatility: New Estimation Techniques for Pricing Derivatives. RISK Publications, London, 1998.
Stochastic finance. H Föllmer, A Schied, Walter de Gruyter & CoBerlinextended edition. An introduction in discrete timeH. Föllmer and A. Schied. Stochastic finance. Walter de Gruyter & Co., Berlin, extended edition, 2011. An introduction in discrete time.
Optimal replication of contingent claims under transaction costs. S D Hodges, A Neuberger, Reviev of Futures Markets. 8S. D. Hodges and A. Neuberger. Optimal replication of contingent claims under transaction costs. Reviev of Futures Markets, 8:222-239, 1989.
Calibrated option bounds. A J King, M Koivu, T Pennanen, Int. J. Theor. Appl. Finance. 82A. J. King, M. Koivu, and T. Pennanen. Calibrated option bounds. Int. J. Theor. Appl. Finance, 8(2):141-159, 2005.
| []
|
[
"THE CALDERÓN PROBLEM FOR QUASILINEAR ELLIPTIC EQUATIONS",
"THE CALDERÓN PROBLEM FOR QUASILINEAR ELLIPTIC EQUATIONS"
]
| [
"Claudio Muñoz ",
"Gunther Uhlmann "
]
| []
| []
| In this paper we show uniqueness of the conductivity for the quasilinear Calderón's inverse problem. The nonlinear conductivity depends, in a nonlinear fashion, of the potential itself and its gradient. Under some structural assumptions on the direct problem, a real-valued conductivity allowing a small analytic continuation to the complex plane induce a unique Dirichletto-Neumann (DN) map. The method of proof considers some complex-valued, linear test functions based on a point of the boundary of the domain, and a linearization of the DN map placed at these particular set of solutions. | null | [
"https://arxiv.org/pdf/1806.09586v1.pdf"
]
| 119,656,169 | 1806.09586 | 7ec5a725ae05daa6ec68ee5ead720276291b23d0 |
THE CALDERÓN PROBLEM FOR QUASILINEAR ELLIPTIC EQUATIONS
25 Jun 2018
Claudio Muñoz
Gunther Uhlmann
THE CALDERÓN PROBLEM FOR QUASILINEAR ELLIPTIC EQUATIONS
25 Jun 2018
In this paper we show uniqueness of the conductivity for the quasilinear Calderón's inverse problem. The nonlinear conductivity depends, in a nonlinear fashion, of the potential itself and its gradient. Under some structural assumptions on the direct problem, a real-valued conductivity allowing a small analytic continuation to the complex plane induce a unique Dirichletto-Neumann (DN) map. The method of proof considers some complex-valued, linear test functions based on a point of the boundary of the domain, and a linearization of the DN map placed at these particular set of solutions.
Introduction
Setting of the problem.
Let Ω ⊂ R n , n ≥ 2, be a smooth C 2,α bounded domain, for some 0 < α < 1. Acting on Ω, we will consider a nonlinear, uniformly (in Ω) positive function a : Ω × R × R n → (0, ∞), a = a(x, s, p) ≥ a 0 (s, p) > 0, a 0 given.
(1.1)
The purpose of this paper is to describe the Calderón's inverse problem for a quasilinear conductivity a(·), that is to say, the study of the quasilinear scalar equation div x a(x, u(x), ∇u(x)) ∇u(x) = 0, x ∈ Ω. (1.2) Here u = u(x) is assumed to be a function defined from Ω into R. In order to determine a possibly unique u, we will impose a boundary condition
u ∂Ω = f,
for some fixed f in an space of smooth functions, to be specified below.
The standard and well-known Calderón's problem, namely the determination of the conductivity a = a(x) for the problem div x [a(x) ∇u(x)] = 0, x ∈ Ω,
u ∂Ω = f,(1.3)
under the knowledge of the Dirichlet-to-Neumann map (DN) H 1/2 (∂Ω) ∋ f −→ a ∇u · ν ∂Ω ∈ H −1/2 (∂Ω), ν unit outer normal to Ω, has attracted the attention of many researchers during the past thirty years. Outstanding results in this area are the works by Calderón [3], Sylvester the second author [12,13], Nachman [10], Astala and Päivärinta [1], among many others. The survey [17] is a suitable source for a historical account on the developments of the Calderón's problem.
However, in nonlinear media applications (see [15] for a detailed survey), the conductivity a(x) is usually a nonlinear function, not only depending on the point x but also on the function u(x), and more importantly, on its gradient ∇u(x). It is for this reason that problem (1.2) is a natural step towards the understanding of several inverse problems coming from different applied scientific problems.
More precisely, by quasilinear inverse problem associated to (1.2), we mean the following question: under which conditions on the conductivity a, the boundary values f , and a related Dirichlet-to-Neumann map associated to f , a and u, we can recover the coefficient a = a(x, s, p), where (x, s, p) ∈ Ω × R × R n .
Note that we must recover a scalar function depending on 2n + 1 variables, where n ≥ 2 is the dimension of space. This inverse problem for (1.2) is in some sense hard to tackle down because of the gradient term ∇u inside the conductivity a, which makes the problem effectively quasilinear, and standard methods do not apply except for very particular situations where the coefficient a has particular properties. For example, in the case where the coefficient a does not depend on the gradient ∇u, namely a = a(x, s) only, Sun [14] showed that the knowledge of the DN map C 2,α (∂Ω) ∋ f −→ Γ a [f ] := a(x, u)∇u · ν ∂Ω ∈ C 1,α (∂Ω), (1.4) where ν is the outer normal in ∂Ω, and u = u f is solution of the equation div(a(x, u)∇u) = 0 in Ω, u ∂Ω = f, determines the strictly positive coefficient a(x, s). The fundamental step in their proof is to linearize the DN map (1.4), following the original idea of Isakov [8]. The objective is then to show that equality of quasilinear DN maps leads to a corresponding equality at the level of linearized DN maps, where a much better developed theory is available. Afterwards, Sun and the second author [16] extended this result by considering anisotropic conductivities: the DN map ν · (A(x, u)∇u) ∂Ω determines the matrix-valued conductivity A(x, s), in the case where u solves the equation div(A(x, u)∇u) = 0 in Ω, u ∂Ω = f, and A is a symmetric, positive definite matrix. Note that uniqueness is obtained up to a change of coordinates that leaves invariant the boundary: if Φ : Ω → Ω is a C 1 diffeomorphism that satisfies Φ = Id on the boundary ∂Ω, then
A Φ (x, u) := |DΦ| −1 DΦ T A(Φ −1 x, u(Φ −1 x)) (1.5)
is another conductivity that has the same DN map. Here DΦ is the Jacobian matrix of Φ, and |DΦ| its Jacobian determinant. Later, Hervas and Sun [6] considered the problem for the quasilinear problem div(A(x, u, ∇u)) = 0 in Ω, u ∂Ω = f, and where A satisfy one of the following two conditions: either A(x, u, ∇u) = A 0 (x)∇u (linear case on ∇u, no dependence on u), or A(x, u, ∇u) = A 0 (∇u) (no dependence on x nor u at all). In both cases, uniqueness is obtained up to a diffeomorphism that changes coordinates, similar as in (1.5). Then, the natural question is the following: can on improve Sun-Uhlmann and Hervas-Sun results by allowing a complete quasilinear conductivity as in (1.2)?
A simple but somehow naive approach to this question should be to extend Hervas-Sun's result [6] by allowing the conductivity to depend on x, u and ∇u. However, one can easily deduce that the problem is in some formal sense undetermined, because one has to recover a scalar function depending on n + 1 + n = 2n + 1 variables, and the corresponding DN map provides much less information. As far as we understand, this problem is completely open. A second issue comes from the fact that even the solvability theory for the direct problem is not completely well-understood in the classical sense, and additional conditions are usually needed: either (i) one has solvability for u with small gradient and a few mild assumptions on the conductivity a, or (ii) the conductivity is taken having plenty of constraints (with almost no grow in the variable ∇u); however one can recover now solutions with large gradients. In the following, we will precisely specify which of these constraints are needed in our work.
1.2.
Assumptions. Let us come back to equation (1.2). The purpose of this paper is to give a first insight on the resolubility of the Calderón's inverse problem for the most possible general quasilinear problem. However, unlike equation (1.3), the solvability (i.e. existence, uniqueness) of the direct problem (1.2) is not guaranteed in general. Indeed, before stating our main results, we will need to assume some standard structural assumptions 1 on the conductivity a(·, ·, ·) that will ensure the existence and uniqueness of a solution for the quasilinear direct problem. These are standard sufficient conditions, stated e.g. in Gilbarg and Trudinger's monograph [4], but for the sake of completeness we give full details on their meaning in Section 2. Some of these conditions are necessary, meaning that the lack of a particular assumption leads to nonexistence or non uniqueness of the quasilinear solution. The reader may also consult [6] for similar conditions, in the case of a conductivity only depending on x and ∇u.
Structural assumptions. Recall that we have assumed that Ω ⊆ R n is an open, bounded domain, of class C 2,α , for some 0 < α < 1 fixed, and also that n ≥ 2. Additionally, let us assume the following:
(S1) (Smoothness and nonnegativity) a ∈ C 1,α (Ω × R × R n ), and a(x, s, p) > 0 for all (x, s, p) ∈ Ω × R × R n .
(S2) (Ellipticity) Let a ij be the symmetric n × n matrix
a ij (x, s, p) := 1 2 ((∂ pi a)(x, s, p) p j + (∂ pj a)(x, s, p) p i ).
Assume that a ij is elliptic in Ω, (1.6) which means that, for all (x, s, p) ∈ Ω × R × R n ,
0 < λ(x, s, p)|ξ| 2 ≤ a ij (x, s, p)ξ i ξ j ≤ Λ(x, s, p)|ξ| 2 < +∞,
(see Definition 2.1 and (2.2) for more details and a general definition).
(S3) (Growth conditions) Additionally, we will assume the following growth conditions: for any (x, s, p)
∈ Ω × R × R n , λ(x, s, p) ≥ λ 0 (|s|) > 0, |p||∇ p a(x, s, p)| + |a(x, s, p)| ≤ µ 0 (|s|), (1 + |p|)|∂ s a(x, s, p)| + |∇ x a(x, s, p)| ≤ µ 0 (|s|)|p|, p · A(x, s, p) ≥ |p| β − |a 1 s| β − a β 2 ,(1.7)
for functions λ 0 (resp. µ 0 ) positive and non-increasing (resp. non-decreasing) in |s|, and constants β > 1, a 1 , a 2 > 0. It is stated in [4] (see more details in Section 2, and in particular, the descriptions in Theorem 2.17) that, under assumptions (S1)-(S3), the quasilinear problem for u :
Ω → R real-valued, div(a(x, u, ∇u)∇u) = 0 in Ω, u ∂Ω = f ∂Ω , f ∈ C 2,α (Ω), (1.8)
is uniquely solvable for u in C 2,α (Ω), with standard a-priori estimates. 2 Moreover, the DN map
C 2,α (∂Ω) ∋ f −→ Γ a [f ] := a(x, u, ∇u)∇u · ν ∂Ω ∈ C 1,α (∂Ω),(1.9)
where u = u f is the solution to the equation (1.8), is well-defined and continuous (Corollary 2.19).
Remark 1.1. Conditions (S1)-(S3), although sufficient for solvability of the quasilinear problem (1.2), are in some sense also necessary, because some a-priori estimates on the boundary (needed for the existence part) may fail if one of these conditions is lifted, see Remark 2.16 for more details.
Remark 1.2. The conditions imposed in (S3) may seem too strong compared with the standard theory for linear scalar elliptic problems, but they are required with the purpose of having solutions with large gradients. Less restrictive assumptions on the growth of a(x, s, p) as a function of p are certainly possible, but an a-priori restriction on the size of the gradient ∇u of the solution will be probably needed.
Inverse problem assumptions. In addition to the previous estimates, we will need three additional, non-structural assumptions. We call them non-structural assumptions because these are sufficient conditions for showing that the DN map in (1.9) is one-to-one. Although needed for the method of proof, we also believe that some of them are in a certain sense also necessary conditions, but we have no rigorous proof of this claim. In what follows, we shall assume the following hypotheses (H1) The conductivity a is homogeneous in space: it only depends on s and p: a = a(s, p), and a(s, p) > 0 for all (s, p) ∈ R × R n .
(H2) There are constants r 0 , R 0 > 0, such that a(s, p) has an analytic continuation as a function of (s, p) ∈ R × B r0 (0), where 3
R := R + i[−R 0 , R 0 ] ⊆ C (1.10) 2
The assumption f ∈ C 2,α (Ω) is standard for the theories developed in [4], and it avoids the problem of finding a smooth extension of a boundary value condition f defined only on ∂Ω. For the sake of simplicity, we will assume this simplification as well as in [4]. 3 Br(p 0 ) := {p ∈ C n : |p − p 0 | < r}. Note also that R is unbounded.
is a (not necessarily small) band around the real line, and with a(s, p) ∈ C being real-valued for s and p real-valued.
(H3) Under assumption (H2), the real part of the complex-valued function a(s, p) is positive, in the sense that there exist 0 < λ < Λ < +∞ (depending continuously on s, p) such that 4
0 <λ ≤ λ(s, p) ≤ Re a(s, p) |a(s, p)| + |∇ p a(s, p)| ≤ Λ, for all (s, p) ∈ R × B r0 (0). (1.11)
Some preliminary remarks about these conditions are absolutely necessary.
Remark 1.3. Hypothesis (H1) can be understood as a reduction on the number of variables to be found: we are looking for a conductivity depending on n + 1 variables, but still improving in some sense each of the results in [14,16,6], which recover ≤ n + 1 variables (and in the case of a nontrivial gradient, the problem simplifies to an almost linear situation). However, we also believe that hypothesis (H1) can be relaxed to allow "near to a constant" inhomogeneous conductivities, as in standard "direct" elliptic theory. Hypothesis (H2) is needed for the method of proof, and probably can be lifted after one can construct some equivalent realvalued test functions to the ones we will mention in this paper. Finally, hypothesis (H3) is needed to preserve the ellipticity of a suitable complex-valued quasilinear problem, and it is certainly an essential condition for us.
Examples. Some examples of conductivities of this type are, for p small if necessary, the conductivity generalizing the minimal surface equation:
a 1 (s, p) := f (s) √ 1 + p · p ,
among many others (in this last example one must rewrite the equation without fractional terms). See also [4, pp. 260-263] for more details on the class of quasilinear equations appearing from different applied problems. 13) fully and uniquely determines the quasilinear coefficient a(s, p) in problem (1.12). More precisely, if Γ a1 ≡ Γ a2 , then a 1 ≡ a 2 in R × B r0 (0).
C 2,α (∂Ω) ∋ f −→ Γ a [f ] = a(u, ∇u)∂ ν u ∂Ω ∈ C 1,α (∂Ω),(1.
Remark 1.5. This result can be seen as the first example of uniqueness for the quasilinear Calderón's problem where the conductivity depends on both u and ∇u in a nontrivial fashion.
Remark 1.6. In principle, hypothesis (H1) may seem too restrictive, but as mentioned before, recovering a conductivity depending on x, u and ∇u could be considered as a problem with too many degrees of freedom, and uniqueness may not hold for the case of large gradients. On the other hand, we also believe that the analyticity condition (H2) can be relaxed to allow less restrictive conductivities.
1.4.
Ideas of the proof. The proof of Theorem 1.4 relies on the introduction of a new class of solutions for (1.12) which have nontrivial gradient. Recall the hypothesis (H2), that ensures that a(s, p) is analytic in a particular tubular neighborhood of the real case. Assume that 0 ∈ ∂Ω, otherwise we translate the domain (or change the following argument by a suitable space translation). Under this framework, we fix s ∈ C and p ∈ C n such that p·p = 0, and introduce the following set of functions
u s,p (x) := s + x · p ∈ C, x ∈ Ω.
(1.14)
Note that x · p is the standard inner product between the (real-valued) vector x and the complex-valued vector p. A first important property of these functions is the following: each u s,p solves (1.12) in the classical sense, provided p · p = 0: div(a(u s,p , ∇u s,p )∇u s,p ) = 0 in Ω.
This last identity shows precisely a nontrivial bifurcation, along a complex analytic manifold (p · p = 0) of the standard constant solutions (in this case, u s,0 ), which where mostly considered by Z. Sun and coauthors [16,6].
Solving in general the direct quasilinear problem (1.12) for a(·) complex-valued is a hard problem. Very few results are available in the literature, and they mostly consider the linear case only. Some recent breakthroughs on the regularity problem are the works by Hofmann et al. [7], and subsequent papers. See also the work by Barton [2] for more details on this approach. In our case, we will only consider solutions that are close enough to a particular exact solution (u s,p ), which in some sense simplifies the solvability theory.
In Theorem 4.4 we will show the existence and uniqueness of complex-valued solutions in the neighborhood of each u s,p , provided p is chosen small. In proving this result we will invoke the Implicit Function Theorem. A suitable complex-valued Dirichlet-to-Neumann map arises from this construction.
The second ingredient of the proof is the linearization technique above mentioned, applied this time to the complex-valued case. We will show in Corollary 5.4 that if two complex-valued DN maps coincide, then their respective linearization are well-defined and must coincide, at least for small gradients.
A third ingredient of the proof is the uniqueness a particular set of complexvalued conductivities in a linearized Calderón problem. Unlike the standard realvalued Calderón problem, the proof of the former result relies on simple and elementary evaluation techniques, and no CGO solutions are needed (although each u s,p may be recast as a very particular CGO solution), see the proof of Theorem 6.6 for the corresponding details. We also emphasize that our techniques do not apply for the standard real-valued Calderón problem, because of the absence of a particular zeroth order term that makes things work in our case.
The last part of the argument consists in comparing the real and complex-valued DN maps. By hypothesis, we only have information about the real-valued one, and some information must be transferred from the real to the complex one. In order to show this fact we will prove that the complex-valued DN map is the unique continuation of the real-valued one, around each u s,p , to the complex n-dimensional space p ∈ C n , p small. We will use here the fact that the manifold p · p = 0 is analytic outside the origin. As a consequence, complex-valued DN maps are equal for each boundary value data. Consequently, linearized DN maps coincide, and therefore, from the previous results (Theorem 6.6), both conductivities are the same everywhere.
Organization of this paper. In Section 2, and in order to make this paper self-contained, we review the standard solvability theory of quasilinear problems. Then, in Section 3 we show the existence and uniqueness of a complex-valued, linear elliptic problem. In Section 4 we extend this result to the nonlinear case, and show the existence of well-defined DN maps. Section 5 is devoted to the linearization of the DN map, and the transference of uniqueness from nonlinear to the linear regime. Section 6 deals with the uniqueness for the Calderón problem associated to a particular linear, complex-valued equation resulting from the linearization of the quasilinear problem. Finally, in Section 7 we show the main result, Theorem 1.4.
Notations. Through this paper, we will assume the following conventions:
• Given p ∈ C n , p T denotes its transpose vector.
• C α (Ω; C), C α (∂Ω; C) denote Hölder spaces of complex-valued functions of exponent α. • B r (x 0 ) will denote the open ball centered at x 0 ∈ C n (or R n ), of radius r > 0. • ν(x) ∈ S n−1 denotes the outer unit normal to a point x ∈ ∂Ω.
• Given an m × n matrix A, we will denote its norm as
A 2 := i,j |a ij | 2 .
Acknowledgments. C. M. would like to thank the Mathematics Department of the U. Washington for its kind hospitality during the elaboration of this work. He also would like to thank the Laboratoire de Mathématiques d'Orsay for his kind hospitality during past years, and where part of this work was completed. Finally, C.M. was partially funded by ERC Blowdisol (France), Fondecyt no. 1150202 Chile, Fondo Basal CMM (U. Chile), and Millennium Nucleus Center for Analysis of PDE NC130017. G. U. was partly supported by NSF and a Si-Yuan Professorship at IAS, HKUST.
2. Review on the real-valued, quasilinear direct problem 2.1. Preliminaries. In this section we recall some well-known results on quasilinear scalar equations. For the sake of completeness, we state (without proofs) all interesting results, even if they are not essentially needed. Some good references for these results are the monographs by Ladyzenskaja and Ural'ceva [9], and also Gilbarg and Trudinger [4]. Most of the results below are stated for operators in divergence form, but they have general counterparts, see the aforementioned monographs for more details and general statements.
Recall that we assumed Ω being a smooth (C 2,α for instance, 0 < α < 1), bounded domain in dimension n ≥ 2. The regularity of the boundary is essentially needed in one specific statement, Theorem 2.12. Let Q = Q u be an operator in divergence form Q u u = Qu := div A(x, u, ∇u),
u = u(x) ∈ C 2 (Ω). (2.1)
Here the vector field A = A(x, s, p), defined in Ω × R × R n with values in R n , is assumed to be at least differentiable, although in practice we will need C 1 regularity, see (S1) in page 3. We start with the following definition, which is standard.
Definition 2.1 (Ellipticity, see [4], eqn. (10.5), p. 259). We say that Q as in
(2.1) is elliptic in Ω if there are constants 0 < λ < Λ < +∞, depending on (x, s, p) ∈ Ω × R × R n , such that for all ξ ∈ R n , 0 < λ(x, s, p)|ξ| 2 ≤ a ij (x, s, p)ξ i ξ j ≤ Λ(x, s, p)|ξ| 2 , (2.2)
and where
a ij (x, s, p) := 1 2 (∂ xj A i + ∂ xi A j )(x, s, p). (2.3)
Similarly, we will say that Q is elliptic in Ω if (2.2) holds for all (x, s, p) ∈ Ω × R × R n , and uniformly elliptic if λ and Λ do not depend on x ∈ Ω.
Examples. Some examples of vector fields A that we will see through this paper are the following:
A 1 (x, s, p) := a(x, s, p)p, a(·) scalar valued.
and for (s, p) ∈ R × C n fixed, and if a = a(s, p) is differentiable in its variables (s, p),
A 2 (x, z, q) := a(s + p · x, p)q + p {(∇ p a)(s + p · x, p) · q + (∂ s a)(s + p · x, p)z}.
This second vector field A 2 can be recast as a sort of linearization of the nonlinear field A 1 , around a particular solution.
Remark 2.2 (Complex-valued case). Let us assume now that A = A(x, s, p) is complex-valued, differentiable in x ∈ Ω and analytic in a region of points M ∋ (s, p) ∈ C × C n . We will say that Q is elliptic in Ω × M if there are constants 0 < λ < Λ < +∞, depending on (x, s, p) ∈ R n × M, such that for all ξ ∈ C n ,
0 < λ(x, s, p)|ξ| 2 ≤ Re a ij (x, s, p)ξ i ξ j ≤ Λ(x, s, p)|ξ| 2 ,(2.4)
and where, as usual,
a ij (x, s, p) = 1 2 (∂ xj A i + ∂ xi A j )(x, s, p). (2.5)
As in the real-valued case, we will say that A is uniformly elliptic in Ω × M if both λ and Λ are functions not depending on x ∈ Ω.
2.2. Uniqueness. We first mention a uniqueness result. The following result is a slight modification of Theorem 10.7, p. 268 in [4], adapted to our needs.
Theorem 2.3 (Uniqueness). Assume that Qu = Qv = 0 for u, v ∈ C 2 (Ω), with Q in (2.1) elliptic in Ω (see inequalities (2.
2)), and A = A(x, s, p) continuously differentiable with respect to the s and p variables.
If u = v on ∂Ω, then u ≡ v in Ω.
The next result explains the maximum principle for operators in the form of divergence, adapted to our setting. Note that no ellipticity assumption is needed, although a different coercivity assumption is imposed. Theorem 2.4 (See Theorem 10.9, p. 272 in [4]). Assume that u ∈ C 0 (Ω) ∩ C 1 (Ω) satisfies Qu = 0 in the weak sense in Ω, 5 and suppose that for some β > 1, and a 1 , a 2 > 0,
p · A(x, s, p) ≥ |p| β − |a 1 s| β − a β 2 , for all (x, s, p) ∈ Ω × R × R n . (2.6)
Then one has the estimate
sup Ω |u| ≤ C(a 2 + a 1 sup ∂Ω |u|) + sup ∂Ω |u|, C = C(n, β, a 1 , |Ω|) > 0.
Before continuing, some remarks are essentially needed.
Remark 2.5. This result is useful because it gives a-priori C 0 estimates for any sufficiently smooth solution u of Qu = 0 in Ω, only in terms of its values on the boundary. In that sense, this is the first step for establishing a-priori estimates for solutions to the quasilinear problem Qu = 0.
Remark 2.6. Assume that A(x, s, p) = a(x, s, p)p, with a ≥ 1 uniformly in (x, s, p) ∈ Ω × R × R n . Then we have p · A(x, s, p) = a(x, s, p)|p| 2 ≥ |p| 2 ,
which implies that (2.6) is satisfied with β = 2 > 1, and a 1 = a 2 = 0. Therefore, in this particular case, one simply has the pure C 0 estimate sup Ω |u| ≤ sup ∂Ω |u|.
2.3.
A priori estimates. The next step is how to establish existence of solutions for the quasilinear problem Qu = 0, where Q is as in (2.1). Recall the definition of ellipticity in Ω, see Definition 2.2. For the next result, we will assume, for |p| → +∞, the following structural conditions on A:
ν(|s|)(1 + |p|) τ ≤ λ(x, s, p), (see (2.2)), ∇ p A(x, s, p) ≤ µ(|s|)(1 + |p|) τ (1 + |p|)|∂ s A(x, s, p)| + D x A(x, s, p) ≤ µ(|s|)|p| τ +2 ,(2.7)
for some τ > −1, and ν (resp. µ) positive and non-increasing (resp. non-decreasing) in |s|.
The next result, essentially Theorem 14.1 in [4] (p. 337), establishes boundary gradient estimates for solutions to Qu = 0 in Ω. Before announcing this, we need some standard notations and definitions. Assume that u ∈ C 2 (Ω), and that Q is elliptic as in Definition 2.1. Recall that (2.1) can be written, using Einstein's summation convention, as
Qu = div x A(x, u, ∇u) = ∂ xi A i (x, u, ∇u) = (∂ xi A i )(x, u, ∇u) + (∂ s A i )(x, u, ∇u) ∂ xi u + ∂ pj A i (x, u, ∇u) ∂ 2 xixj u. Since u ∈ C 2 (Ω), we have (∂ 2 xixj u) i,j symmetric, which implies that Qu = 1 2 (∂ pj A i (x, u, ∇u) + ∂ pi A j (x, u, ∇u)) ∂ 2 xixj u + (∂ xi A i )(x, u, ∇u) + (∂ s A i )(x, u, ∇u) ∂ xi u = a i,j (x, u, ∇u) ∂ 2 xixj u + b(x, u, ∇u), where b(x, s, p) := (∂ xi A i )(x, s, p) + (∂ s A i )(x, s, p) p i ,E(x, s, p) := a i,j (x, s, p)p i p j , so that, since Q is elliptic, 0 < λ(x, s, p)|p| 2 ≤ E(x, s, p) ≤ Λ(x, s, p)|p| 2 .
Having this in mind, we also have Remark 2.8. The uniform exterior sphere condition for Ω is needed in order to construct suitable barriers, which is a standard technique in elliptic theory. See [4,Chapter 14] for more details.
|p|Λ(x, s, p) ∼ |p| a i,j (x, s, p) ∼ |p| D p A(x, s, p) ≤ µ(|s|)(1 + |p|) τ +1 ,(2.µ(|s|)(1 + |p|) τ +1 ≤μ(|s|)λ(x, s, p)(1 + |p|) ≤μ(|s|)E(x, s, p), for p large. Additionally, if p is large, |b(x, s, p)| ≤ |(div x A)(x, s, p)| + (∂ s A)(x, s, p) |p| ≤ µ(|s|)|p| τ +2 ≤μ(|s|)E(x, s, p),(2.
The following result is Theorem 15.9 in [4]. Recall the definitions of M and N in (2.11). Theorem 2.9. Let u ∈ C 2 (Ω) ∩ C 0 (Ω) satisfy (2.10) in Ω bounded and assume (2.7) valid for τ > −1. Assume additionally that Ω satisfies the exterior sphere condition and that u = f on ∂Ω, with f ∈ C 2 (Ω). Then we have [4]). Assume u ∈ C 2 (Ω) is such that Qu = 0 in Ω, with Q elliptic in Ω, and A ∈ C 1 (Ω × R × R n ). Finally, assume that ∂Ω ∈ C 2 and u = f on ∂Ω, where f ∈ C 2 (Ω). Then
[∇u] α,Ω ≤ C n, K,
Λ K λ K , µ K λ K , Ω, f C 2 (Ω) , K := u C 1 (Ω) ,
and α = α(n, Λ K /λ K , Ω) > 0 (see (13.4) in [4]).
2.4.
Existence. Leray-Schauder fixed point argument. The following Leray-Schauder type result is the key tool to prove existence of solutions for Qu = 0. Note that the existence is proven in Hölder classes, however, this condition could be relaxed by allowing a less regular class of solutions (and a different notion of solution).
Theorem 2.11. Let Ω be a bounded domain in R n , with Q as in (2.10) elliptic in Ω, with coefficients
a ij ∈ C α (Ω × R × R n ), b ∈ C α (Ω × R × R n ), 0 < α < 1.
Let ∂Ω ∈ C 2,α and f ∈ C 2,α (Ω). If there exists a constant M, independent of u and σ ∈ [0, 1], such that for every C 2,α (Ω) solution of the Dirichlet problems The following result is strictly contained in Theorem 15.11 in [4] (p. 381).
Q σ u := a ij (x, u, ∇u)∂ 2 i,j u + σb(x, u, ∇u) = 0 in Ω, u ∂Ω = σf, σ ∈ [0, 1],
Theorem 2.12 (Existence).
Let Ω be a bounded domain in R n and suppose that Q is elliptic in Ω, with A ∈ C 1,γ (Ω × R × R n ), 0 < γ < 1, satisfying (2.7) and (2.6) with the additional restriction β = τ + 2. Then if ∂Ω ∈ C 2,γ , and for any f ∈ C 2,γ (Ω) there exists a solution u = u f ∈ C 2,γ (Ω) of the problem Qu = 0 in Ω, u ∂Ω = f .
In the following remarks, we essentially explain how Theorem 2.12 is proved.
Remark 2.13. The assumption β = τ + 2 is required to reconcile condition (2.6) with the first estimate in (2.7).
Remark 2.14. For the proof of Theorem 2.12, several intermediate steps are needed, parts of a main strategy invoking the Leray-Schauder fixed point theorem in Hölder spaces. In order to apply this result, one needs to show some Ladyzenskaja-Ural'ceva a-priori interior and boundary estimates in C 1,γ , for solutions to the problem Qu = 0 in Ω, which are established through Chapters 10 , 13, 14 and 15 in [4]. See the comments after Theorem 15.11 in [4] for full details.
Remark 2.15. The assumption ∂Ω ∈ C 2,γ is needed to ensure the so-called exterior sphere condition for every point in ∂Ω. [4]) that show that the absence of some of these assumptions leads to nonexistence results. Usually, the estimate that fails is the control of the derivative of u on the boundary.
2.5.
Applications. We will apply Theorems 2.3 and 2.12 to show the existence of a unique solution for the direct problem associated to the quasilinear problem (1.2). A first result deals with the solvability problem for scalar quasilinear problems. Before we need some notations. Assume that A(x, s, p) := a(x, s, p)p, a ≥ 1.
(2.13)
Then a ij in (2.3) is given by
a ij (x, s, p) := 1 2 ((∂ pi a)(x, s, p) p j + (∂ pj a)(x, s, p) p i ).
We will assume that a ij is elliptic in Ω, (2.14) as in (2.2), with involved parametric constants λ(x, s, p) and Λ(x, s, p) respectively (see (2.2)). Additionally, we will assume that
λ(x, s, p) ≥ ν(|s|) > 0, |p||∇ p a(x, s, p)| + |a(x, s, p)| ≤ µ(|s|), (1 + |p|)|∂ s a(x, s, p)| + |∇ x a(x, s, p)| ≤ µ(|s|)|p|, (2.15)
for ν (resp. µ) positive and non-increasing (resp. non-decreasing) in |s|. These conditions essentially say that a must be bounded with derivatives of the right order.
Theorem 2.17. Consider the quasilinear problem posed in a bounded domain Ω ⊆ R n of class C 2,α , 0 < α < 1, and n ≥ 2, for u : Ω → R real-valued: div(a(x, u, ∇u)∇u) = 0 in Ω, u ∂Ω = f.
(2.16)
Assume that a ∈ C 1,α (Ω × R × R n ), and a ≥ 1, and that (2.14) and (2.15) are satisfied. Then the direct problem (2.16) is uniquely solvable for u in C 2,α , i.e., for any f ∈ C 2,α (∂Ω), there exists a unique solution u = u f ∈ C 2,α (Ω).
Proof. The existence part is essentially Theorem 2.12 with A given by (2.13), and α = 2, τ = 0. Note also that conditions (2.7) are satisfied by assuming the conditions in (2.15). For the uniqueness part, it is enough to invoke Theorem 2.3.
Remark 2.18. Let us comment about the meaning of assumptions (2.15). They state, among other things, that the conductivity must be bounded uniformly in x ∈ Ω and p ∈ R n . In other words, it is not allowed to have e.g.
a(x, s, p) ∼ |p| 2 .
This requirement can be understood as a smallness condition for the gradients of solutions to Qu = 0. We will see later that this condition appears in a different form in our main results.
One of the main consequences of the previous result is the following existence result for the DN map.
C 2,α (∂Ω) ∋ f −→ Γ a [f ] := a(x, u, ∇u)∇u · ν ∂Ω ∈ C 1,α (∂Ω),(2.
17)
where u = u f is the solution of (2.16), is well-defined and bounded.
It seems reasonable now to deal with the inverse problem associated to the quasilinear problem (2.16). However, we will see later in this paper that, if we want to recover the conductivity a = a(x, s, p) for p = 0, it is better to consider complexvalued solutions for (2.16). However, the solvability theory for this type of solutions is, as far as we know, far from being completely understood. For this reason, we will have to make a digression from the standard theory and prove some particular existence theorems for complex-valued solutions of (2.16). The fact that there are explicit solutions in some particular cases will be essential for the uniqueness proof.
Before treating in detail the full quasilinear problem, it is somehow better to understand a simplified, complex-valued coefficients, linear problem.
Solvability for a complex-valued linear problem
3.1. Preliminaries. Let g ∈ C 0,α (Ω) and h ∈ C 2,α (Ω), 0 < α < 1 denote two fixed "source" functions. The purpose of this Section is to develop a solvability theory for the linear direct problem for the unknown function v = v(x) div x a(u s,p , p)∇v + p (∇ p a)(u s,p , p) · ∇v + (∂ s a)(u s,p , p)v = g in Ω, v ∂Ω = h. Note that u s,p = s + x · p, with s ∈ R and p ∈ C n is such that p · p = 0 (see (1.14)). From hypothesis (H2) in p. 4, when considering the expression a(u s,p , p) we are using the fact that a admits an analytic continuation to the region R × B r0 (0), for p ∈ B r0 (0) and r 0 small if needed. In that sense, this problem has complex-valued coefficients, but it still preserves its divergence form. Finally, note that the terms (∇ p a)(u s,p , p), (∂ s a)(u s,p , p), denote the functions ∇ p a(s, p) and ∂ s a(s, p) evaluated at the point (u s,p , p), respectively (and if no confusion arises, we will drop the parentheses).
Problem (3.1) is the key element to understand in this paper. It will be essential to show solvability for the quasilinear case studied in Section 4, and additionally, it will play an important role in the associated, quasilinear inverse problem (cf. Sections 5 and 6).
It is not difficult to identify the main symbol of the problem above. It turns out that we can write (3.1) as div x ( A(x, s, p)∇v + b(x, s, p)v) = g, in Ω, v ∂Ω = h, and where (I n is the n × n identity matrix)
A(x, s, p) := a(u s,p , p)I n + p ∇ p a(u s,p , p) T ∈ C n×n , b(x, s, p) := ∂ s a(u s,p , p) p ∈ C n .
(3.2)
Both coefficients have complex-valued components. The n × n matrix A ij is not symmetric nor diagonal, and the term p ∇ p a(u s,p , p) T may be a very large perturbation (in terms of its absolute value) of the main part a(u s,p , p)I n , in such a form that it is very probable that A is no longer elliptic, so that the nature of the Dirichlet boundary value problem may be completely different to a standard one.
The following result states that for p small enough, the matrix A from (3.2) is uniformly elliptic. Proof. Since from (1.11) we have Re a(s, p) > λ(s, p) > 0, we only have to show that for p small this inequality is preserved. We have
Re A ij (x, s, p)ξ i ξ j = Re a(u s,p , p)|ξ| 2 + p i ∂ pj a(u s,p , p)ξ i ξ j = Re a(u s,p , p)|ξ| 2 + Re(p i ∂ pj a(u s,p , p)ξ i ξ j .
Recall that we have |p| < r 0 . Now, since Ω is bounded, u s,p = s + x · p lies inside a narrow horizontal band of the complex plane, of the form Consequently, for r 0 small,
Re A ij (x, s, p)ξ i ξ j ≥ 9 10λ |ξ| 2 . (3.3)
On the other hand, note that in the region R × B r0 (0),
A(x, s, p) ≤ Λ + C|p| ≤ 2Λ,(3.
3.2.
Existence for a linear complex-valued problem. Now we will apply Lemma 3.1 to show existence for the problem (3.1).
Theorem 3.3. Let g ∈ C 0,α (Ω) and h ∈ C 2,α (Ω), 0 < α < 1, be two fixed data. Under the assumptions of Lemma 3.1, problem (3.1) has a unique, complex-valued solution v = v g,h in the class C 2,α (Ω). Moreover, one has the estimate
v C 2,α ≤ C( h C 2,α (∂Ω) + g C 0,α (Ω) ), C = C(λ, Λ, r 0 ). (3.6)
Proof. The proof of this result is based in the standard procedure to show solvability of linear elliptic PDEs.
We will see later, in Chapters 5 and 6, that the linear problem (3.1) appears naturally in the study of the quasilinear inverse problem (1.12). Theorem 3.3 will be applied in the next section in order to get the desired solvability for problem (1.12).
Solution for the quasilinear complex case
4.1. A model example. Now we make a small digression from the main subject of this paper. In this subsection we will consider the Calderón direct problem in
Ω ⊆ R n bounded div x (a(x)∇u) = 0 in Ω, u ∂Ω = f, (4.1)
where a ∈ C 1,α (Ω) is uniformly positive, and f ∈ C 2,α (Ω), with ∂Ω ∈ C 2,α , for some 0 < α < 1. Clearly (4.1) has a unique real-valued solution u = u f ∈ C 2,α . Moreover, there exists a solution in standard Sobolev spaces even if f is assumed less regular than Hölder.
The problem now is to get some insight about the same problem when now a(·) is assumed to be complex-valued, namely a : Ω −→ C. Since solvability for (4.1) in the real and complex-valued case is related to the Riesz Theorem (or Lax-Milgram Theorem), a sufficient condition to find a solution in a Sobolev space is the ellipticity condition (see Remark 2.2)
0 < λ ≤ Re a(x) ≤ Λ < +∞, x ∈ Ω.
The case where a(·) depends now on x, u(x) and ∇u(x) will have require some new (but standard, in view of Section 2) restrictions, because λ may now depend on u and ∇u. It turns out that, for some particular reasons, it is good to have a good control of the dependence on u of the lower bound λ(u, ∇u). This control will be important to obtain a desired ellipticity for our problem.
4.2.
Existence close to a given solution. We will consider s ∈ R and p ∈ C n fixed, with p · p = 0. Recall the linear affine function u s,p (x) defined in (1.14). The following is clearly satisfied: Assume that a = a(s, p) is a complex-valued, homogeneous conductivity, analytic for (s, p) ∈ C × C n . Then u s,p solves (1.12) with f = u s,p , that is, div x (a(u s,p (x), ∇u s,p (x))∇u s,p (x)) = 0 in Ω. Remark 4.3. Note additionally that, for any x 0 ∈ R n , the modified test functioñ u s,p := s + p · (x − x 0 ) is also solution to the first equation in (1.12). This sort of "degeneracy" in the choice of x 0 is completely absorbed by simply assuming that
Claim 4.1.x 0 = 0 ∈ ∂Ω.
The function u s,p is an example of a complex-valued solution to the quasilinear problem (4.2), revealing the existence of a general family of solutions beyond the ones mentioned by the existence theorems in Section 2. We would like to find new solutions to (1.12) around this explicit solutions. For this reason, we set u = u s,p + v, v unknown, and we will write (1.12) in terms of the new variable v. The following result is the main objective of this Section, and it can be seen as an extension of Claim 4.1. Before, recall the definition of R in (1.10).
Theorem 4.4.
Let h ∈ C 2,α (Ω) be a fixed, complex-valued function. Assume that the conductivity a = a(s, p) is analytic in R × B r0 (0), and consider the quasilinear
problem in Ω ⊆ R n , n ≥ 2, for v = v(x) complex-valued: div x (a(u s,p + v, p + ∇v)(p + ∇v) − a(u s,p , p)p) = 0 in Ω v ∂Ω = h. (4.3)
Finally, assume that (2.4) and (2.15) are satisfied. Then for any small h C 2,α (∂Ω) and small p ∈ C n such that p · p = 0, the direct problem (4.3) is uniquely solvable for v = v h in C 2,α (Ω).
Unlike Section 2, we will prove Theorem 4.4 following other steps: by the application of the Implicit Function Theorem to show existence and uniqueness. Although the methods of proof are somehow similar to the ones in Section 2, it will be clear from the beginning that the addition of a complex-valued conductivity will lead to several problems, and the smallness condition on the gradients will be essential for this approach.
Proof of Theorem 4.4. We will make use of the Implicit Function Theorem below. Assume t ∈ R and h ∈ C 2,α (Ω) given. Let us write
v(x) = h(x) + w(x), x ∈ Ω, w unknown.
Then (4.3) writes in terms of w, div x a(u s,p + h + w, p + ∇h + ∇w)(p + ∇h + ∇w) − a(u s,p , p)p = 0 in Ω,
w ∂Ω = 0.
In what follows, we denote by C 2,α 0 (Ω) the Banach space of complex-valued functions in C 2,α (Ω) which are zero at the boundary (but not necessarily its derivatives). Let us define the map F = F s,p,h such that
F : C 2,α 0 (Ω) × C 2,α 0 (Ω) → C 0,α (Ω),(4.4)
and F [h, w] := div x a(u s,p + h + w, p + ∇h + ∇w)(p + ∇h + ∇w) − a(u s,p , p)p . (4.5)
Clearly F is well-defined, and F [0, 0] ≡ 0. The fact that F is of class C 1 is a direct computation.
Now, forw ∈ C 2,α 0 (Ω) fixed, we compute D w F [0, 0](w). Since F is continuously differentiable, we have
D w F [0, 0](w) = d dσ F [0, σw] σ=0 = d dσ div x (a(u s,p + σw, p + σ∇w)(p + σ∇w) − a(u s,p , p)p) σ=0
= div a(u s,p , p)∇w + p (∇ p a(u s,p , p) · ∇w) + (∂ s a)(u s,p , p)w .
Thanks to Theorem 3.3, if |p| < r the homogeneous Dirichlet problem div x a(u s,p , p)∇w + p (∇ p a(u s,p , p) · ∇w) + (∂ s a)(u s,p , p)w = g ∈ C α (Ω),
w ∂Ω = 0,
has a unique solutionw ∈ C 2,α 0 (Ω), with uniform bounds. Therefore, applying the Implicit Function Theorem, for each h ∈ C 2,α (Ω) small in norm it is possible to find a unique small solution w = w[h] ∈ C 2,α (Ω) of F [h, w[h]] ≡ 0, which solves our problem.
4.3.
Another proof for uniqueness. Although the following result can be stated for more general conductivities, as in Theorem 2.3 (but under additional assumptions), we will only consider the following simple statement for uniqueness. Theorem 4.5 (Uniqueness). Let r 0 , R 0 > 0 be given by hypothesis (H2) (see (1.10)). Assume that v 1 , v 2 ∈ C 2 (Ω), complex-valued, solve (4.3) with same boundary condition, and satisfying
|p| + v 1 C 2 (Ω) + v 2 C 2 (Ω) < r 0 ,(4.
6)
Then v 1 ≡ v 2 in Ω.
Remark 4.6. The smallness condition on v 1 and v 2 in (4.6) is a sufficient condition for having a positive functional for the difference of two solutions. From the existence part, we see once again that smallness is in some necessary.
Remark 4.7. The proof of Theorem 4.5 is not based on any type of pointwise comparison principle (compare with section 2 and the monograph [4, Chapter 10]), since the former simply does not work for general, complex-valued equations. In that sense, an energy method is suitably more reasonable for taking into account the oscillatory behavior of complex-valued solutions.
Proof of Theorem 4.5. We follow the ideas of the proof of Theorem 10.7 in [4], with some important differences because several inequalities do not hold for complexvalued functions. 6 In that sense, the smallness character of the involved perturbation v will be a key ingredient.
Defining w := u 1 − u 2 = v 1 − v 2 , and v t := tv 1 + (1 − t)v 2 , for t ∈ [0, 1]. Finally, for s ∈ R and p ∈ C n fixed, denote A(x, z, q) := a(u s,p + z, p + q)(p + q) − a(u s,p , p)p, z ∈ C, q ∈ C n .
Clearly A is analytic in the (z, q) variables. Then we have
0 = div(A(x, u s,p + v 1 , p + ∇v 1 ) − A(x, u s,p + v 2 , p + ∇v 2 )) = ∂ i (a ij (x)∂ j w + b i (x)w) (Einstein's summation convention), (4.7)
where, for t ∈ [0, 1],
a ij (x, s, p) := 1 2 1 0 (∂ pj A i + ∂ pi A j )(x, v t , ∇v t )dt b i (x, s, p) := 1 0 (∂ s A i )(x, v t , ∇v t )dt.
A further simplification reveals that 7
a ij (x, s, p) = δ ij 1 0 a(u s,p + v t , p + ∇v t )dt + 1 2 1 0 (p j ∂ pi a + p i ∂ pj a)(u s,p + v t , p + ∇v t )dt, and b i (x, s, p) = p i 1 0 (∂ s a)(u s,p + v t , p + ∇v t )dt (4.8)
Using (H3) (see p. 5) and the smallness hypotheses on the perturbation (4.6), we note that a ij is still elliptic, in the sense of Remark 2.4:
0 < λ|ξ| 2 ≤ Re a ij (x, s, p)ξ i ξ j ≤ Λ|ξ| 2 , and |a ij | ≤ Λ, |b i | ≤ C.
Therefore, using (4.7) with test function ϕ = w,
0 = Re (a ij (x)∂ j w + b i (x)w)∂ i w = Re a ij (x)∂ j w∂ i w + Re b i (x)w∂ i w.
The first integral above is nonnegative (ellipticity), and the second one is small, because of the smallness assumptions: from (4.8) and the fact that |p| < r 0 ,
sup x∈Ω | Re b(x)| ≤ Cr 0 .
Therefore, using (2.4) and the Poincaré's inequality (since Ω is bounded), we have for r small
0 ≥ λ |∇w| 2 − Cr 2 0 |w| 2 |∇w| 2 .
Therefore, w ≡ 0.
As in Section 2, Corollary 2.19, one of the main consequences of Theorem 4.4 is the existence of a suitable complex-valued DN map around the solution u s,p . 7 δ ij represents the standard Kronecker's identity matrix. Corollary 4.8. Under the assumptions (and conclusions) of Theorem 4.4, the following holds. For any s ∈ R, h C 2,α (∂Ω) small, and p ∈ C n also small, such that p · p = 0 is satisfied, the Dirichlet-to-Neumann map
C 2,α (∂Ω) −→ C 1,α (∂Ω; C) h −→ Γ a,s,p [h] (4.9)
where Γ a,s,p [h] := a(u s,p + v, p + ∇v)(p + ∇v) − a(u s,p , p)p · ν ∂Ω (4.10)
and v = v h is the solution of (4.3), is well-defined and bounded.
Now that we have a well-defined DN map for the quasilinear problem (4.3), it is time for asking the differentiability properties of Γ a,s,p .
Linearization of the DN map
Differentiability of the DN map.
In what follows, we prove that the DN map Γ a,s,p defined in (4.9)-(4.10) has a well-defined Gateaux-derivative at the origin.
Given h ∈ C 2,α (Ω) fixed, s ∈ R, p ∈ C n and t ∈ R small enough (such that Theorem 4.4 holds for the boundary data th), and consider the DN operator Γ a,s,p (th). The following result shows that this nonlinear operator has a well-defined Gâteauxderivative around u s,p .
Lemma 5.1. Given s, p ∈ C × C n and t ∈ R fixed, and h ∈ C 2,α (Ω). As a functional from t ∈ R into C 1,α (∂Ω), we have that t → Γ a,s,p [th] is Gâteaux differentiable at t = 0. Moreover, one has (cf. 3.2) for any h ∈ C 2,α (Ω) and A, b defined in (3.2). = a(u s,p + th, p + t∇h)(p + t∇h) − a(u s,p , p)p · ν.
lim t→0 1 t ( Γ a,s,p [th] − Γ a,s,p [0]) − A(·, s, p)∇h + b(·, s, p)h · ν
The term above can be expanded as follows:
a(u s,p + th, p + t∇h)(p + t∇h) − a(u s,p , p)p = = a(u s,p , p) + t∂ s a(u s,p , p)h + t∇ p a(u s,p , p) · ∇h (p + t∇h) − a(u s,p , p)p + t 2 D 2 s,p a(s,p)[h, ∇h] 2 (p + t∇h) = t a(u s,p , p)∇h + p∂ s a(u s,p , p)h + p∇ p a(u s,p , p) · ∇h + t 2 ∂ s a(u s,p , p)h + ∇ p a(u s,p , p) · ∇h ∇h
+ t 2 D 2 s,p a(s,p)[h, ∇h] 2 (p + t∇h).
Since h ∈ C 2,α (Ω), and a is analytic in the considered region, we have then
lim C 1,α ,t→0 1 t [ Γ a,s,p [th] − Γ a,s,p [0]] = a(u s,p , p)∇h · ν + {∇ p a(u s,p , p) · ∇h + ∂ s a(u s,p p)h}p · ν.
Therefore, t → Γ a,s,p [th] is differentiable at t = 0 and its derivative is given by (5.1).
The linearized Calderón's problem.
In what follows, we consider the following linear direct problem: given h ∈ C 2,α (Ω), find v ∈ C 2,α (Ω) such that div x a(u s,p , p)∇v + p ∇ p a(u s,p , p) · ∇v + ∂ s a(u s,p , p)v = 0 in Ω, (5.2) under the Dirichlet boundary condition v ∂Ω = h.
(5.3)
Recall that in this problem, the coeffcients a(u s,p , p), p ∇ p a(u s,p , p), and p ∂ s a(u s,p , p), are functions of (x, s, p), but independent of v. Thanks to Theorem 3.3, this problem has a unique solution v = v h ∈ C 2,α (Ω), provided p is chosen small enough, a condition that will be assumed from now on. Proof. Direct from (5.1) and (5.4).
A second result translates the information from Γ a,s,p into Γ ℓ,a,s,p .
Corollary 5.4. Assume that Γ a1,s,p ≡ Γ a2,s,p for given coefficients a 1 , a 2 satisfying (H1)-(H3) in p. 4. Fix (s, p) ∈ R × C n , with p · p = 0 and p small. Finally let Γ ℓ,a1,s,p and Γ ℓ,a2,s,p be the corresponding DN maps obtained form the previous result for a 1 and a 2 , respectively. Then, for each fixed (s, p) one has Γ ℓ,a1,s,p ≡ Γ ℓ,a2,s,p .
Proof. We follow the proof in [16]. Fix h ∈ C 2,α (Ω), and pick t ∈ R small. We know by hypothesis that, for any suitable (s, p), 6. Uniqueness of a linear Calderón's problem 6.1. Setting. Let us assume, as already required in this paper, that s ∈ R is fixed and p ∈ C n is a small, complex-valued vector satisfying the compatibility condition p · p = 0. The purpose of this Section is the study of the Calderón's inverse problem for the complex-valued, matrix-valued, linear equation (5.2)-(5.3)-(5.4). We also assume h ∈ C 2,α (Ω) in (5.3).
Γ
Consider the associated DN map Γ ℓ,a,s,p [h], h as above, see (5.4). Now the question is the following: can the knowledge of Γ ℓ,a,s,p for a sufficiently large class of boundary data h determine the linearized, complex-valued conductivity a = a(z, q)?
Let us start by recalling the following well-known result, valid for the scalar, real-valued conductivity case, but whose extension to the complex-valued case is immediate from the proposed proof. Theorem 6.1 (Sylvester and Uhlmann, [13]). Assume that n ≥ 3 and a = a(x) is an unknown C 2 (Ω), real-valued conductivity. Consider the associated Calderón's inverse problem for a(x) :
div(a(x)∇v) = 0 in Ω, v = h on ∂Ω,(6.1)
and let us assume that two conductivities a 1 , a 2 above produce the same DN map:
H 1/2 (∂Ω) ∋ h − − → a j (x)∇u · ν ∂Ω ∈ H −1/2 (∂Ω).
Then a 1 ≡ a 2 .
Remark 6.2. The proof of this result is mainly based on the use of complex geometric optics solutions (CGO). Later, we will see that our uniqueness proof will not use CGO solutions, since a very helpful, zero-order linear term will appear in the linearization of the DN map. Such a term is not present in Theorem 6.1, which makes our proof not suitable for (6.1).
The purpose of this section is to extend Theorem 6.1 to the complex-valued case given in (5.2)-(5.3). Since this new problem is no longer a scalar one, we will need some different techniques. Before proving this result, we need an auxiliary lemma. Lemma 6.3. Assume that 0 ∈ ∂Ω, ν(0) being the outer unit normal to 0 ∈ ∂Ω, and let A(r 0 ) and C(r 0 ) be the sets of the form A(r 0 ) := p ∈ C n : |p| < r 0 , p · p = 0 , (6.2) and C(r 0 ) := p ∈ C n : |p| < r 0 , p · p = 0 and p · ν(0) = 0 , (6.3) where ν(0) is the outer unit normal to Ω at the point x = 0. Then,
(1) For any n ≥ 2, A(r 0 ) is not open, but A(r 0 )\{0} is an analytic manifold.
(2) For any n ≥ 2, the vector 0 ∈ C n satisfies 0 ∈ C(r 0 ).
(3) For any n ≥ 2, C(r 0 ) ⊂ B r0 (0)\{0} in C n .
(4) For any n ≥ 2, C(r 0 ) is a complex analytic manifold.
(5) If p ∈ C(r 0 ) and λ ∈ C\{0} is such that |λ| < 1, then λp ∈ C(r 0 ). (6) If n = 2, C(r 0 ) is of the form
p = p 0 + ip ⊥ 0 , p 0 ∈ R 2 \{0}, p ⊥ 0 · p 0 = 0, |p 0 | < r 0 √ 2 . (6.4) (7)
If n ≥ 3, the plane passing by the origin and determined by the directions p r , p i ∈ R n , with p = p r + ip i ∈ C(r 0 ), is not the plane orthogonal to ν(0).
Proof. The first assertion is a consequence of the fact that on A(r 0 )\{0}, ∇(p · p) = p = 0. The second and third assertions are direct. The proof of (4) is similar to the proof of (1), and the fact that p · ν(0) = 0 is an open set. Let us show (5). Assume p ∈ C n small. Writing p = p r + ip i , with p r , p i ∈ R n , we have p · p = 0 if and only if |p r | 2 = |p i | 2 , p r · p i = 0. (6.5)
Additionally, the condition p · ν(0) = 0 reads p r · ν(0) = 0 or p i · ν(0) = 0. (6.6)
In two dimensions, condition (6.6) is always satisfied if p = 0, therefore, the set of complex-valued vectors p ∈ C n for which (6.5) and (6.6) are satisfied is of the form (6.4).
In dimensions n ≥ 3, the set of points p r , p i ∈ R n for which p r · ν(0) = 0 and p i · ν(0) = 0 lie on a plane in R n passing through zero. Therefore, any pair of orthogonal, equal-size vectors p r , p i ∈ R n \{0} for which one of them is not in the plane orthogonal to ν(0), form a satisfactory p = p r + ip i . This shows (7).
Recall the region R defined in (1.10). Theorem 6.4. Assume that n ≥ 2, s ∈ R and p ∈ C n , p · p = 0 and |p| < r 0 small enough such that Theorem 3.3 is valid. Consider two conductivities a 1 (s, p) and a 2 (s, p), defined in R × B r0 (0), and satisfying hypotheses (H1)-(H3) in p. 4. Consider the Calderón's inverse problem associated to the linear equation (5.2)-(5.3)-(5.4), and let us assume that two conductivities a 1 , a 2 produce the same linearized DN map:
Γ ℓ,a1,s,p [h] = Γ ℓ,a2,s,p [h] ∈ C 1,α (∂Ω),
for all h ∈ C 2,α (Ω).
Then a 1 ≡ a 2 in the region R × C(r 0 ).
Proof of Theorem 6.4. The proof is simple and does not require a deep understanding or improvement of Theorem 6.1. Indeed, assume that given h ∈ C 2,α (Ω), we have knowledge of the DN map Γ ℓ,a,s,p [h] ∈ C 1,α (∂Ω). (6.7)
note that, since p · p = 0 (p ∈ A(r 0 )), every constant v = c ∈ C is solution to (5.2)-(5.3) with h ≡ c small. Since the solution to this problem is unique for |p| small (Theorem 3.3), we have that necessarily (see (5.4)) Γ ℓ,a,s,p [c](x) = c ∂ s a(u s,p , p)(p · ν)(x).
If c = 0 and (p · ν)(x) = 0 for a fixed x ∈ ∂Ω, we will have (∂ s a)(s + x · p, p) = Γ ℓ,a,s,p [c](x) c(p · ν)(x) .
Since 0 ∈ ∂Ω (otherwise we translate the domain or define u s,p = s + p · (x − x 0 ), with x 0 ∈ ∂Ω fixed), we have (∂ s a)(s, p) = Γ ℓ,a,s,p [c](0) c(p · ν)(0) , for all p ∈ C(r 0 ) (see (6.3)), and any s ∈ R. From now we fix a constant c = c 0 ∈ C\{0}, small if necessary. Note that the denominator is independent of s ∈ R.
Certainly we have lot of information about a(s, p), because we have proved Moreover, a(s, p) is completely known from the DN map, being the same for a 1 and a 2 .
Hence a 1 (s, p) = a 2 (s, p) for any s ∈ R and p ∈ C(r 0 ) small, except for a function depending on p only. In order to show that the additional function a(0, p) is identically the same for a 1 and a 2 , we consider now the function v p (x) := p · x, p ∈ C(r 0 ).
Note that v p is clearly a solution to (5.2)-(5.3) with h = p · x. Indeed, div x a(u s,p , p)∇v p + p ∇ p a(u s,p , p) · ∇v p + ∂ s a(u s,p , p)v p = = div x a(u s,p , p)p + p ∇ p a(u s,p , p) · p + ∂ s a(u s,p , p)(p · x) = (∂ s a)(u s,p , p)(p · p) + ∂ xj (∂ pi a(u s,p , p))p i p j + (p · p)(∂ s a)(u s,p , p) + (p · p)(∂ 2 s a)(u s,p , p)(p · x) = ∂ 2 s,pi a(u s,p , p))p i (p · p) = 0. Since the solution to this problem is unique for |p| small (Theorem 3.3), we have that necessarily (see (5.4)) Γ ℓ,a,s,p [p · x](x) = a(u s,p , p)p + p ∇ p a(u s,p , p) · p + ∂ s a(u s,p , p)(p · x) · ν ∂Ω . Since a(s, p) is almost completely explicit, except for a function of p, we have, for x ∈ ∂Ω, Γ ℓ,a,s,p [p · x](x) = a(u s,p , p)(p · ν) + a(0, p)(p · ν) + (p · ν) ∇ p a(u s,p , p) · p + ∇ p a(0, p) · p + ∂ s a(u s,p , p)(p · x) .
Evaluating at x = 0, we have Γ ℓ,a,s,p [p · x](0) = (p · ν(0)) a(s, p) + ∇ p a(s, p) · p + a(0, p) + ∇ p a(0, p) · p , or a(0, p)+ ∇ p a(0, p)·p = 1 (p · ν(0)) Γ ℓ,a,s,p [p·x](0)− (p·ν(0)) a(s, p)+ ∇ p a(s, p)·p .
The right side above is known, and we only need to find a(0, p). Now, for any η > 0 we have d dη (η a(0, ηp)) = a(0, ηp) + ∇ p a(0, ηp) · ηp = 1 ηp · ν(0) Γ ℓ,a,s,ηp [ηp · x](0)
− (ηp · ν(0)) a(s, ηp) + ∇ p a(s, ηp) · ηp , so that a(0, p) = lim θ→0 1 θ Γ ℓ,a,s,ηp [ηp · x](0) − (ηp · ν(0)) a(s, ηp) + ∇ p a(s, ηp) · ηp ηp · ν(0) dη.
Note that the first term in the integral above must converge near η = 0 since a is by hypothesis analytic near the origin. Hence we have a 1 ≡ a 2 in a set of the form (s, p) ∈ R × C(r 0 ), which is a complex (noncompact) analytic manifold. Since both a 1 and a 2 are analytic as functions of s ∈ R only, we conclude that both coincide in the region R × C(r 0 ). The proof is complete.
We need to extend the equality between a 1 and a 2 from the set R × C(r 0 ) to a larger set. This is a sort of unique continuation property. Theorem 6.6. One has a 1 ≡ a 2 in R × B r0 (0).
Proof.
In what follows, we fix s ∈ R and p ∈ C(r 0 ). Note that from Theorem 6.4, a 1 (s, p) = a 2 (s, p). Since {0} × C(r 0 ) ⊆ R × C(r 0 ) is an analytic manifold of codimension two in C n+1 , n + 1 ≥ 3, by Riemann's second extension Theorem [5, Theorem 2, p. 30], we get the desired result. 7. Uniqueness for the nonlinear problem 7.1. Preliminaries. In this Section we finally prove Theorem 1.4. The main idea of the proof is to find the correct link between the DN maps Γ a and Γ a,s,p already defined in (2.17) and (4.9)-(4.10). We start with a simple result.
Proof. We must show that for all h ∈ C 2,α (Ω) with sufficiently small norm,
1. 3 .
3Main results. Now we state our main result. Let u = u f be a solution of the equation div(a(u, ∇u)∇u) = 0 in Ω, u ∂Ω = f. (1.12) Theorem 1.4. Under the hypotheses (S1)-(S3) and (H1)-(H3), the knowledge of the DN map
see also Definition 2.1. Gilbarg and Trudinger define (see eqn. (10.3)) the principal part of Q as E:
x, s, p) + |b(x, s, p)| ≤μ(|s|)E(x, s, p). These are the "structure" conditions imposed in[4, eqn. (14.9)], and are clearly satisfied thanks to (2.7). Consequently, we haveTheorem 2.7 (Boundary estimates). Let u ∈ C 2 (Ω) ∩ C 1 (Ω) satisfy Qu = a i,j (x, u, ∇u) ∂ 2 xixj u + b(x, u, ∇u) = 0 (2.10) in Ω and u ∂Ω = f ∈ C 2 (Ω). Suppose that Ω satisfies the uniform exterior sphere condition, with uniform radius δ > 0. Then, under assumptions (2.7), one has sup ∂Ω |∇u| ≤ C(n, M, µ(M ), N, δ), M := sup Ω |u|, N := f C 2 (Ω) . (2.11)
C(n, τ, ν(M ), µ(M ), ∂Ω, N, P ), P :recall some Hölder estimates for the gradient of u. Let us remind the Hölder seminorm: [u] α,Ω := sup x,y ∈Ω, x =y |u(x) − u(y)| |x − y| α .
Dirichlet problem Qu = 0, u ∂Ω = f has a solution in C 2,α (Ω).
Corollary 2 . 19 .
219Consider the quasilinear problem (2.16). Under the assumptions and conclusions of Theorem 2.17, the Dirichlet-to-Neumann map
Lemma 3 . 1 .
31Assume that (s, p) ∈ R × C n , and x ∈ Ω. Assuming r 0 > 0 in (1.10) smaller if necessary, the following is satisfied. For all |p| < r 0 , the complex valued matrix A in (3.2) is elliptic in the sense of Remark 2.4, and the vector field b in (3.2) is also uniformly bounded.
R
× [−Cr 0 , Cr 0 ], C = C(Ω) > 0. Therefore, if r 0 is chosen small enough, R × [−Cr 0 , Cr 0 ] ⊆ R (see (1.10)). Consequently, by hypothesis (H2) (see p. 4), a and its derivatives are well-defined and bounded in the set R × B r (0). Re(p i ∂ pj a(u s,p , p)ξ i ξ j ≤ r 0 |ξ| 2 × sup (s,p)∈R×Br 0 (0) |∇ p a(s,p)| ≤ Cr 0 |ξ| 2 .Additionally, using (1.11) and the continuity of λ (taking r 0 smaller if necessary),Re a(u s,p , p) ≥ inf (s,p)∈R×Br 0 (0)
b(x, s, p)| ≤ C|p| ≤ Cr 0 .(3.5) Remark 3.2. Lemma 3.1 and hypothesis (H3) (see (1.11)) can be weakened by asking forλ depending on s, under suitable assumptions on a(·, ·) and it first derivatives, in such a form that (3.3) is satisfied with a positive lower boundλ depending on s also.
( 4 . 2 )
42Remark 4.2. For a conductivity a(s, p) defined only in a portion of the complex space C × C n (see e.g. (1.10)), we need additional restrictions on (s, p), depending on x ∈ Ω. However, even in this case it is possible to show that Claim 4.1 do hold for a subset of possible (s, p).
Remark 5.2. Identity (5.1) can be recast as a directional derivative: D Γ a,s,p [0](h) = A(·, s, p)∇h + b(·, s, p)h · ν.
Claim 6. 5 .
5For any s ∈ R and p ∈ C(r 0 ),
Remark 2.16. Conditions (2.7) are only sufficient for obtaining existence for the problem Qu = 0, however, there are examples (see Chapter 14, Section 14.4 in
a1,s,p [th] = Γ a2,s,p [th].In particular, thanks to Corollary 5.3, both Gateaux-derivatives coincide:
d
dt
Γ a1,s,p [th]
t=0
=
d
dt
Γ a2,s,p [th]
t=0
,
namely
Γ ℓ,a1,s,p [h] = Γ ℓ,a2,s,p [h],
as desired.
Γ a1,s,p [h] = Γ a2,s,p [h]. Since Γ a1 [h] = Γ a2 [h], we have Γ a1 [u s,0 + h] = Γ a2 [u s,0 + h]. From Lemma 7.1 we have for h small, and all s ∈ R, Γ a1,s,0 [h] = Γ a2,s,0 [h]. Hence, we conclude thanks to Proposition 7.4. Proof of Theorem 1.4. From Proposition 7.5, we have Γ a1 [u s,p + h] = Γ a2 [u s,p + h], and from Definition 7.2, this means that Γ a1,s,p [h] = Γ a2,s,p [h]. Hence, using Corollary 4.8, Lemma 5.1, Corollary 5.3 and Corollary 5.4, we have Γ ℓ,a1,s,p [h] = Γ ℓ,a2,s,p [h]. The final conclusion comes from Theorem 6.6. CNRS and Departamento de Ingeniería Matemática DIM, Universidad de Chile, Chile E-mail address: [email protected], [email protected] Department of Mathematics, University of Washington, Box 354350 Seattle, Washington 98195, USA 8 E-mail address: [email protected]
This terminology comes from Gilbarg-Trudinger's monograph[4].
Note that we are not asking for a being entire, but only bounded on a bounded domain.
This means the equation Qu = 0, with Q and u as in (2.1), is tested against a C ∞ 0 (Ω) function.
In particular, no comparison principle seems to hold in the complex-value case.
Also affiliated to Jockey Club Institute for Advanced Study, HKUST, Acknowledgement Clear Water Bay, Kowloon, Hong Kong, China.
Let Γ a the real-valued DN map from Corollary 2.19, and Γ a,s,p the complex-valued DN. Lemma, Then, for any s ∈ R, and for any small, real-valued h ∈ C 2,α (Ω), Γ a. u s,0 + h] = Γ a,s,0 [hLemma 7.1. Let Γ a the real-valued DN map from Corollary 2.19, and Γ a,s,p the complex-valued DN introduced in (4.9)-(4.10). Then, for any s ∈ R, and for any small, real-valued h ∈ C 2,α (Ω), Γ a [u s,0 + h] = Γ a,s,0 [h].
Since u s,0 = s + h is real-valued, and since p · p = 0 if p = 0, Theorems 2. Proof, 17 and 4.4 apply, with Γ a,s,0 [h] real-valued. From the uniqueness of the solutions in those theorems, we conclude (7.1)Proof. Since u s,0 = s + h is real-valued, and since p · p = 0 if p = 0, Theorems 2.17 and 4.4 apply, with Γ a,s,0 [h] real-valued. From the uniqueness of the solutions in those theorems, we conclude (7.1).
The next definition says that it is possible to extend Γ a in a particular, complexvalued case. Recall the definition of A(r 0 ) in (6.2)The next definition says that it is possible to extend Γ a in a particular, complex- valued case. Recall the definition of A(r 0 ) in (6.2).
Fix h ∈ C 2,α (Ω) with sufficiently small norm. Then, for any s ∈ R and p ∈ A(r 0 ) we define the function (s, p) − − → Γ a. Definition 7.2 (Extension of Γ a. u s,p + h] ∈ C 1,α (∂Ω; C) as follows: Γ a [u s,p + h] := Γ a,s,p [h] (cf. (4.10Definition 7.2 (Extension of Γ a ). Fix h ∈ C 2,α (Ω) with sufficiently small norm. Then, for any s ∈ R and p ∈ A(r 0 ) we define the function (s, p) − − → Γ a [u s,p + h] ∈ C 1,α (∂Ω; C) as follows: Γ a [u s,p + h] := Γ a,s,p [h] (cf. (4.10)).
) be small enough. Fix x ∈ ∂Ω. For each (s, p) ∈ R × A(r 0 ), the complex-valued function given by (s, p) − − → Γ a,s. h ∈ C 2,α (Ωp [h](x) is the unique analytic continuation of Γ a,s,0 [h], s ∈ R, to the complex-valued subset R × A(r 0 )\{0} in C n+1h ∈ C 2,α (Ω) be small enough. Fix x ∈ ∂Ω. For each (s, p) ∈ R × A(r 0 ), the complex-valued function given by (s, p) − − → Γ a,s,p [h](x) is the unique analytic continuation of Γ a,s,0 [h], s ∈ R, to the complex-valued subset R × A(r 0 )\{0} in C n+1 .
First note that, for each x ∈ ∂Ω fixed, the several complex-valued function R × A(r 0 ) ∋ (s, p) − − → Γ a,s,p [h](x) ∈ C is complex-valued analytic. Fix h sufficiently small such that Γ a,s,p [h] is well-defined for s ∈ R and p ∈ A(r 0 ). This is just a consequence of the analytic character of the DN map [3] with respect to the conductivity (see (4.10. and composition arguments.. Recall that a several complex-valued function is analytic if on each coordinate it is itself a complex-valued analytic function.Proof. Fix h sufficiently small such that Γ a,s,p [h] is well-defined for s ∈ R and p ∈ A(r 0 ). First note that, for each x ∈ ∂Ω fixed, the several complex-valued function R × A(r 0 ) ∋ (s, p) − − → Γ a,s,p [h](x) ∈ C is complex-valued analytic. This is just a consequence of the analytic character of the DN map [3] with respect to the conductivity (see (4.10)), and composition arguments. (Recall that a several complex-valued function is analytic if on each coordinate it is itself a complex-valued analytic function.)
Consequently, given another analytic continuation of Γ a,s,0 [h] to the set R × A(r 0 )\{0}, and since this last set is an analytic manifold. LemmaConsequently, given another analytic continuation of Γ a,s,0 [h] to the set R × A(r 0 )\{0}, and since this last set is an analytic manifold (Lemma 6
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| []
|
[
"Extension of MIH to Support FPMIPv6 for Optimized Heterogeneous Handover",
"Extension of MIH to Support FPMIPv6 for Optimized Heterogeneous Handover"
]
| [
"Jianfeng Guan [email protected] \nState Key Laboratory of Networking and Switching Technology\nBeijing University of Posts and Telecommunications\n100876BeijingChina\n",
"Vishal Sharma [email protected] \nDepartment of Information Security Engineering\nSoonchunhyang University\nAsan-siRepublic of Korea\n",
"Ilsun You [email protected] \nDepartment of Information Security Engineering\nSoonchunhyang University\nAsan-siRepublic of Korea\n",
"Mohammad Atiquzzaman \nSchool of Computer Science\nUniversity of Oklahoma\nNormanOK\n"
]
| [
"State Key Laboratory of Networking and Switching Technology\nBeijing University of Posts and Telecommunications\n100876BeijingChina",
"Department of Information Security Engineering\nSoonchunhyang University\nAsan-siRepublic of Korea",
"Department of Information Security Engineering\nSoonchunhyang University\nAsan-siRepublic of Korea",
"School of Computer Science\nUniversity of Oklahoma\nNormanOK"
]
| []
| Fast handover for Proxy Mobile IPv6 (FPMIPv6) can reduce handover delay and packet loss compared with Proxy Mobile IPv6 (PMIPv6). However, FPMIPv6 still cannot handle heterogeneous handovers due to the lack of unified Layer 2 triggering mechanism along with the booming of emerging wireless technologies. Media Independent Handover (MIH) can provide heterogeneous handover support, and a lot of integration solutions have been proposed for it. However, most of them focus on the integration of MIH and PMIPv6, and require the additional mechanisms, which are out of the scope the MIH and difficult to standardize the operations. Therefore, in this paper, we propose an integration solution of FPMIPv6 and MIH by extending the existing MIH standards, and adopt the city section mobility model to analyze its performance under different scenarios. The analytical results show that the proposed solution is capable of reducing the handover delay and the signaling cost compared with the standard as well as the fast handover solutions. | 10.1016/j.future.2019.03.031 | [
"https://arxiv.org/pdf/1705.09835v1.pdf"
]
| 27,475,387 | 1705.09835 | 58ffc041e10180f05682d19234286f5d82620075 |
Extension of MIH to Support FPMIPv6 for Optimized Heterogeneous Handover
Jianfeng Guan [email protected]
State Key Laboratory of Networking and Switching Technology
Beijing University of Posts and Telecommunications
100876BeijingChina
Vishal Sharma [email protected]
Department of Information Security Engineering
Soonchunhyang University
Asan-siRepublic of Korea
Ilsun You [email protected]
Department of Information Security Engineering
Soonchunhyang University
Asan-siRepublic of Korea
Mohammad Atiquzzaman
School of Computer Science
University of Oklahoma
NormanOK
Extension of MIH to Support FPMIPv6 for Optimized Heterogeneous Handover
*Corresponding authorMIHFPMIPv6IntegrationHeterogeneous handoverfast handover
Fast handover for Proxy Mobile IPv6 (FPMIPv6) can reduce handover delay and packet loss compared with Proxy Mobile IPv6 (PMIPv6). However, FPMIPv6 still cannot handle heterogeneous handovers due to the lack of unified Layer 2 triggering mechanism along with the booming of emerging wireless technologies. Media Independent Handover (MIH) can provide heterogeneous handover support, and a lot of integration solutions have been proposed for it. However, most of them focus on the integration of MIH and PMIPv6, and require the additional mechanisms, which are out of the scope the MIH and difficult to standardize the operations. Therefore, in this paper, we propose an integration solution of FPMIPv6 and MIH by extending the existing MIH standards, and adopt the city section mobility model to analyze its performance under different scenarios. The analytical results show that the proposed solution is capable of reducing the handover delay and the signaling cost compared with the standard as well as the fast handover solutions.
Introduction
The massive development of wireless technologies such as Wi-FI, WiMAX, LTE, 5G promotes the booming of various mobile devices with multiple wireless network interfaces. The coexistence and mutual complementation of multiple wireless networks derive the demand for higher bandwidth with enhanced QoE and QoS. This has spurred the related researches of mobility management to support seamless mobility services for media rich applications.
According to Cisco report [1], the global mobile data traffic has grown 74% in 2015, and the expected mobile data traffic will be 30.6 EB/Month by 2020, in which mobile video content will occupy about 23 EB/Month. Mobile video has higher QoE requirements than data services, which makes the users tend to adopt high-speed wireless network with guaranteed QoS. Considering that more and more mobile devices have equipped multiple network interfaces, it is important for mobile users to select the most appropriate interface to improve the bandwidth and reduce the cost. However, due to the limited coverage, heterogeneous handover will happen when users leave from one wireless network to another. Keeping the continuous ongoing session during the handover is an urgent problem.
Various solutions have been proposed [3] to provide mobility for the emerging mobile devices, in which IP mobility management solutions have got more attentions and many schemes have been proposed [4]. These solutions can be classified into the host-based mobility and the network-based mobility in terms of handover initiator; single-host mobility and sub-network mobility according to the mobility types; and the central mobility management solutions and distributed mobility management in terms of the distribution of mobility-related entities. Among these solutions, Mobile IPv6 (MIPv6) [5] is most cited scheme published by IETF, in which Home Agent (HA) manages the registered Mobile Nodes (MNs) and maintains the bi-directional tunnel for each MN that leaves the home network. Due to the triangle routing problem, Hierarchical Mobile IPv6 (HMIPv6) [6] was proposed to divide topology into different domains to reduce the signaling cost for the MNs with micro-mobility. In HMIPv6, Mobility Anchor Point (MAP) is introduced as the "HA" of the given domain to reduce MNs' updating cost. Another important improvement is the Mobile IPv6 Fast Handovers (FMIPv6) [7], which reduce the handover delay and packet loss with the aid of link layer triggering and pre-registration mechanisms. More specially, FMIPv6 sets up a tunnel between previous attachment point and new attachment point to forward buffered packets during handover to reduce packet loss. However, these solutions require the involvement of MNs which may result in excessive energy consumption for energy-limited MNs. Therefore, Proxy MIPv6 (PMIPv6) [13] is proposed by introducing Mobile Access Gateway (MAG) to perform the mobility-related operations on behalf of MNs, and it can reduce handover delay and signaling cost [14] [15] compared with the previous solutions. PMIPv6 can be further improved by combining the fast handover mechanism which is also known as the Fast handover for PMIPv6 (FPMIPv6) [16]. However, the basic FPMIPv6 does not provide the support for heterogeneous handovers.
For MN with dual radios, Liehbsch et al. [8] introduce a new transient binding mobility option in PBU/PBA, which sets up a temporary Binding Cache Entry (BCE) to forward packets to PMAG and NMAG simultaneously, which can avoid superfluous packet forwarding delay or even packet loss. Transient binding can be initiated by NMAG or LMA, and some studies [9][10] have adopted it to improve handover performance. Due to the adoption of multiple interfaces, it is easy to support heterogeneous handover.
Besides, IEEE has published Media Independent Handover (MIH) framework (IEEE-802.21) to support heterogeneous handover. MIH can optimize the network selection, realize the seamless roam, and lower the power consumption. The previous heterogeneous handover schemes [17] generally combine the PMIPv6 and MIH, while in this paper, we propose an integration model of FPMIPv6 and MIH to further improve handover performance. The main contributions are summarized as follows.
(1) An integration model of FMIPv6 and MIH is proposed.
(2) MIH TLV is extended to contain the related information used for pre-registration without introducing any additional signaling messages.
The rest of this paper is organized as follows. Section 2 introduces FPMIPv6, MIH, and investigates the related work. Section 3 describes the proposed integration model of FPMIPv6 and MIH. Section 4 evaluates its performance and compares it with the related solutions. Section 5 summarizes this work and analyzes the potential challenges and further research directions.
Related works
Overview of FPMIPv6
FPMIPv6, as an enhancement of PMIPv6, adopts pre-registration and short time tunnel to reduce the handover delay and the packets loss. Figure 1 shows atypical reference network model of FPMIPv6, which consists of LMA, PMAG, NMAG, P-AN and N-AN.LMA is the topological anchor point that performs home agent function for each registered MNs and manages MN's binding and reachability. MAG is a function on an access router to manage mobility-related signaling for each attached MN, and performs mobility management on behalf of MNs by recording their movements. In a PMIPv6 domain, there are multiple MAGs, and AN is a network composed of link-layer access devices such as AP or BS which connects to MAG.
FPMIPv6 adopts the similar idea of FMIPv6 which sets up a bi-directional tunnel between PMAG and NMAG to tunnel the packets during the handover procedure.
On basis of FMIPv6 [7], FPMIPv6 extends Handover Initiate (HI) and Handover Acknowledge (HAck) messages to carry MN's Network Access Identifier (NAI), Home Network Prefix (HNP) and IPv4 Home Address from PMAG. However, for Router Solicitation, PIPv6 does not apply "Proxy Advertisement" (RtSolPr), the "Proxy Router Advertisement" (PrRtAdv), "Fast Binding Update" (FBU), "Fast Binding Acknowledgment" (FBack), and the "Unsolicited Neighbor Advertisement" (UNA)and for that the MN is not directly involved in FPMIPv6, and these messages, therefore, will be ignored by MAGs.
2.1.1FPMIPv6Operation Modes
FPMIPv6 has two operation modes based on the setup timing of bi-directional tunnel between PMAG and NMAG: the predictive mode in which the tunnel is established prior to attachment of the MN to the NMAG, and the reactive mode in which tunnel is established after the MN attaches to the NMAG. In both models, all the MAGs have to buffer the packets during the detachment procedure. Figure 2 shows the operation flow of P-FMIPv6 in a predictive mode which is also called the PMAG-initiated handover. The detailed operation flow is explained below: (1) MN detects an imminent handover and reports its ID and New AP ID (Target AP) by access technology specific methods.
(2) P-AN sends a handover indication including MN ID and New AP ID to PMAG based on access technology specific methods.
(3) PMAG sends an HI message to NMAG with 'P' flag including MN ID, HNP(s) and LMA address, and MN link-layer ID such as MN LL-ID and MN LLA-IID to initiate the pre-registration procedure.
(4) NMAG replies with a HAck message to PMAG with 'P' flag and code value that indicates the pre-handover result.
(5) NMAG sends the HI message with 'U' or 'F' flag to optionally request the PMAG to buffer or forward packet at a later and appropriate time, respectively. (6) PMAG will set up a bi-directional tunnel between PMAG and NMAG with Previous CoA and New CoA, and forwards the packets destined for MN if it receives an HI message with 'F'. These packets will be buffered at NMAG before MN is attached.
(7) MN performs the handover from P-AN to N-AN based on the specific access technology operations.
(8) MN sets up a physical-layer connect with the N-AN and NMAG, and configures IPv6 address.
(9) NMAG forwards the packets to MN via N-AN. (10) MN sends uplink packets to NMAG and NMAG forwards them to PMAG, while NMAG sends downlink packets to MN directly.
(11) NMAG updates the binding cache by sending PBU to LMA. (12) If the binding is a success, LMA updates the tunnel between LMA and serving MAG via PBA, and all the packets will be forwarded via NMAG. Figure 3 shows the reactive mode of P-FMIPv6 which is also called the NMAG-initiated handover.
(1) At first, the MN handovers from P-AN to N-AN.
(2) MN establishes a new connection with N-AN, and transfers the MN ID and old AP ID to NMAG to identify the PMAG (substituted for UNA and FBU).
(3) NMAG sends an HI message with 'P' flag including MN ID to PMAG. If NMAG wants to set up a tunnel between PMAG and NMAG, it should also set 'F' flag in HI.
(4) PMAG replies with a HAck message to NMAG with 'P' flag including HNP(s), IPv4-MN-HoA, MN LLID (optional), the address of LMA and the other information requested by NMAG.
(5) If the HI message sent by NMAG is with 'F' flag, a bi-directional tunnel between PMAG and NMAG will be set up, and used to forward the packets destined for MN.
(6) MN sends uplink packets to NAMG via N-AN, and forwards to PMAG, and finally deliveries to LMA.
(7) NMAG updates the related binding cache by sending PBU to LMA. (8) LMA updates the tunnel between LMA and serving MAG by reply a PBA, and all the packets will be forwarded via NMAG.
FPMIPv6Problems
FPMIPv6 is a network layer mobility support solution which provides a fast handover interaction framework and defines the related signaling messages format to reduce the handover delay and packets loss. However, to implement and deploy FPMIPv6 in large scale, it should consider the link layer operations and specific access technology. Therefore, it still has the following problems.
(1) Lack of definition of handover triggers events. For example, FPMIPv6 just gives a report message to notify the imminent handover in PMAG-initiated mode, which does not provide the operation in detail.
(2) Lack of candidate network discovery and selection mechanism which may result in the handover failure.
(3) Lack of handover execution procedure and link-layer specific operations in detail.
(4) Lack of detailed explicit heterogeneous handover mechanism.
Due to these problems, FPMIPv6 should incorporate other mechanisms such as MIH to support heterogeneous handover.
Overview of MIH
IEEE 802.21 (MIH) [11] aims to realize heterogeneous handover without service disruption, which can be a complementation of FPMIPv6. MIH contains handover initiation and preparation to search new link and set up a new link. The adaptation of MIH can facilitate service and maximize handover efficiency by combining with upper layer mobility management solutions.
MIH maintains a global network map which records available networks such as 802.11/16/22, 3G/4G, and their link layer information including neighbor maps, and higher layer services. More specifically, MIH locates between the layer 2 and the layer 3, and it abstracts the information exchange procedure of different link layer protocols. It also defines a unified interface Service Access Points (SAP) to provide services for upper layer by hiding the heterogeneity of the different link layer protocols in terms of topological and location information. Figure 4 shows the typical MIH framework which consists of MIH users such as mobility management protocols, a mobile node such as a smartphone, and networks such as 3GPP/3GPP2. All information exchanges are through SAPs, which have three types: a)
The MIH_SAP-which is the interface between MIHF and its users. It provides sharing of MIHF-generated events.It also communicates the link-layer events. b) The MIH_LINK_SAP-which is an abstract media dependent Figure 5 PMIPv6 assisted MIH using fast handover procedure [12] [19] [21] interface between the MIHF and lower layer protocols stacks and it provides the media-specific SAPs for different link-layer technologies. c) The MIH_NET_SAPwhich is an abstract media dependent interface of the MIHF responsible for transport services over the data plane. It handles the information and messages exchanges with remote MIHFs.
Related Work of Integration Models
IEEE 802.21 provides two integration schemes of MIH and PMIPv6, namely network-initiated handover and mobile-initiated handover, whose handover procedures generally consist of information query, resource availability check, resource preparation, new L2 connection establishment, link up indication, IP connectivity restored, higher layer handover execution (PMIPv6 handover operations) and resource release. In the resource preparation phase, both of them introduce the pre-registration mechanism to reduce the handover delay and buffer the packets in target MAG. However, they have to acquire the MN's profile from AAA server or LMA, which may introduce unexpected delay. Besides, both of them are in faced with the packet disorder caused by buffer and larger handover delay problems due to lack of efficient L2 trigger mechanism. Therefore, several enhanced schemes were proposed, which are generally based on L2optimized mechanism, pre-authentication and pre-registration to realize fast handover.
Considering L2 scanning delay as one main component of handover delay, Kim et al. [19] proposed a low latency proactive handover scheme for PMIPv6 with MIH, which can reduce the overall L2 scanning time. In fact, this reduction of L2 scanning time is mainly specific to Wi-Fi based on MIHF which provides the channel configuration information of each AP, so that MN can only scan the configured channels in each AP, not for all channels [20]. However, this scheme requires the buffering packets from both the LMA and the nMAG, which may result in the out-of-order problem. Therefore, they improved it by transient binding [21] and modified pre-registration procedure. After that, to provide the QoS guaranteed real-time services, Kim et al [9], optimized predictive PFMIPv6 handover scheme which supports the pre-registration via MIH_Net_HO_Candidate_Query request and response messages, and supports the pre-authentication with EAP. At the same time, to reduce the handover delay, it adopts the prediction-based smart channel scanning [22]. Figure 5 shows its handover procedure [12] [19] [21] (In the following part, we note this scheme as fast handover solution).
The detailed operation flow is presented below:
(1) Once an MN detects the link going down (it detects the signal strength becoming weak), it sends MIH_Link_Going_down message to its serving network to search the available neighbor networks. The serving network queries the related information from MIIS server through the MIH_Get_Information request message and MIH_Get_Information reply message.
(2) Resource checking and target network selection stage:
Serving MAG and MN negotiates the candidate MAGs through MIH_Net_HO_Candidate_Query request and response messages, and serving MAG checks the resources availability of all the candidate MAGs by MIN_N2N_HO_Candidate_Query request and response messages. By this way, the target MAG is selected.
(3) Handover execution and new network entry stage:
The serving MAG sends MIH_N2N_HO_Commit request message to request the target network to allocate resources, and then serving MAG and target MAG use the HI/HAck messages to transfer the context information including MN ID, MN IID, LMAA, and perform the pre-registration. Once the target network received the HI message, it will set up the transient BCE via PBU/PBA with a transient binding option to simultaneous reception from both networks during handover. After that, serving MAG informs MN about the target network information and indicates the completion of pre-authentication.
(4) PMIPv6 binding and resource release: MN sends the MIH_Link_Up message to the target network to capture the buffered packets. The target MAG and LMA complete the binding update via PBU and PBA. After that, the serving network will tear down the bi-directional tunnel to LMA via De-registration PBU/PBA, and finally, the target MAG sends the MIH_N2N_HO_Complete to serving MAG to release the resource.
This fast handover scheme can improve the performance of heterogeneous handover, however, it still has the following problems: 1) It is a mobile-initiated method for that it introduces five signals between MN and serving MAG, which may increase the power consumption on MN and affects its execution effectiveness. Also, it does not comply with the design principle of PMIPv6 that reduces the involvements of MN in mobility management.
2) It does not distinguish MN's network interfaces which make it difficult to describe cooperation mechanism among different network interfaces, and unclear to support heterogeneous handover. 3) It is mainly used for 802.11, which can be further extended to a universal network by IE_POA_CHANNEL_RANGE in MIIS.
The other solution such as PMIPv6 assisted MIH using MIHF at network side only scheme [17] is dependent on the network to exchange the signaling messages. In such case, it is difficult to support heterogeneous handover as the S-PoS does not know the wireless interfaces information of mobile nodes and it, therefore, cannot select the heterogeneous candidate networks. To make it clear, we summarized related existing schemes as shown in Table 1. (2) large handover delay due to the lack of the fast handover support.
(1) Lack of the trigger mechanism;
(2) S-PoS selects the target network which is difficult to support trigger;
(3) lack of fast handover;
(4) larget packet during the handover (1) Network side selects target network which is difficult to support heterogeneous handover;
(2) Lots of information is handled by S-PoS which can reduce the function requirement of MN
(1) The function of ND has been involved in MIH;
(2) unclear how to select the target network;
(3) lack of the resource check.
(1) Support thepre-resgristatio n and pre-authentication;
(2) Bi-casting can reduce packets loss; (2) Only focus on single interface, lack of the analysis of multiple interfaces. In this paper, we extend these operations and integrate FPMIPv6 with MIH. Considering that the link layer support is beneficial to the predictive fast handover mode, we mainly focus on the integration scheme of predictive mode and MIH.
The Proposed Solution
Compared with PMIPv6, FPMIPv6 optimizes the handover procedure by L2 trigger and pre-registration. In the proposed integration model, it performs the MIH discovery procedure to find an MIHF's capabilities of MIH services through the standard protocol or media-specific mechanisms (i.e., IEEE 802.11 Beacon frames, IEEE 802.16 downlink channel descriptor (DCD), IEEE 802.11management frames, or IEEE 802.16 management messages).
The main idea is to extend the MIH messages to carry the required information for triggering the pre-registration procedure without the additional MIH user signaling messages, such as HI/HACK. More specifically, we extend MIH_N2N_HO_Commit request message to carry MN ID, MN LLA-IID, LMAA and HNP to trigger the binding between nMAG, and extend MIH_N2N_HO_Commit response message to indicate the handover status. Besides, the transient binding is introduced which will be initiated by LMA to support transient BCE for both pMAG and nMAG to reduce the packet loss.
MIH message extensions 3.1.1 Extension of MIH_N2N_HO_Commit request
According to [18] Annex L, the TLV code has been assigned to 100, so the parameters of extended MIH_N2N_HO_Commit request message are shown in Table 2. In this message, we extended three parameters as shown in following.
(1) MN LLA-IID We define a new TLV value as 101 to represent MN's link-local address interface identifier, which is used by MAG to associate PMIPv6 tunnel with the access link that MN attached in case of the point-to-point link.
(2) LMA Address This extended TLV with value 102 carries the address of LMA that can be IPv6 or IPv4.
(3) Home Network Prefix It is used to identify the list of MN's IPv6 home network prefix (es) assigned by LMA to the MN's target link. This is a list of HNPs which consists of 1 octet to identify the length of HNPs and HNPs.
The extended message format of MIH_N2N_HO_Commit request is shown in Figure 6. By introducing these fields, the MN's profile information can be delivered to the potential target networks to perform the pre-register and set up a tunnel.
3.1.2Extension of MIH_N2N_HO_Commit response
This message is extended to include the handover statuses that are defined by IANA. The current statuses defined by MIH only consist of success (value=0), unspecified failure (value=1), rejected (value=2), authorization failure (value=3) and network error (value=4), while others are undefined (value = 5~255). These statuses is simpler compared with the IANA, therefore in the proposed scheme, we extended the existing status code by adopting the status codes defined by IANA which are shown in Table 3. Figure 7 shows the format of an extended MIH_N2N_HO_Commit response message. By using these extensions, the HI/Hack can be replaced by MIH_N2N_HO_Commit messages. Besides, by introducing the transient binding initiated by LMA, the packet of downlink packets can be transmitted to pMAG and nMAG to reduce the packet loss. Besides, there is no tunnel, so the traffic overhead will be reduced. Furthermore, our solutions can also be improved by using the optimized L2 mechanism when applied in the specific wireless network types.
3.2The proposed scheme operation flow
The proposed integration model is based on the following assumptions:
(1) All the network entities support MIH function;
(2) MIIS server is already set up and records the neighboring network information in advance;
(3) All the MIH entities have been configured and registered;
(4) MN has multiple network interfaces, and each network interface can work at the same time, but only one interface is used to transmit the packets. This assumption considers that an MN is energy-limited and the use of multiple network interfaces may reduce its available time although it can increase the throughput. Figure 8 shows the proposed integration model, in which IF-S notifies the current serving interface, and the IF-C notifies the candidate network interface.
Assuming that each interface can detect their link status independently, and MN is equipped with two kinds of interfaces called serving interface (IF-S) and candidate interface (IF-C). These network interfaces can be homogeneous or heterogeneous. In the following section, the serving network is, in fact, the pMAG of PMIPv6, while the target network is the nMAG.
(1) In bootstrap stage, each PoA/PoS acquires its neighboring network information through MIH Information Server via MIH_Get_Information request and response messages to facilitate seamless handover. These neighboring networks contain not only the homogeneous access networks but also the heterogeneous access networks. The serving network will update its neighboring network information once an MN is attached or periodically refreshed.
(2) The IF-S of MN detects the link status and sends an MIH_Link_Going_down indication message to its serving network once it finds the current link quality is going down.
(3) The serving network sends the MIH_Net_HO_Candidate_Query request message to the MN for acquiring the candidate networks, and MN replies MIH_Net_HO_Candidate_Query response message that carries the MN's preferred link and PoS list in sequence.
(4) After receiving the response message, the serving network requires the resource preparation to the networks in the list (candidate networks) by MIH_N2N_HO_Query_Resource request and response messages. (5) After that, the serving network will select the target network based on the available resource of candidate networks and sends extended MIH_N2N_HO_Commit request/response messages to the candidate network to prepare the resource. By this operation, the MN's related profile information will be exchanged between serving network and the target network. At the same time, the target network will send PBU message with a transient flag to LMA to set up the transient BCE. During the handover, the downlink packets will be buffered in the target network via LMA and serving network to reduce the packet loss.
(6) Once the resource is successfully prepared, the serving network sends the MIH_Net_HO_Commit request message to command MN for handover to the target network.
(7) The MIHF will send an MIH_Link_Up indication message to IF-C to set up the L2 connection and replies an MIH_Net_HO_Commit response message to the serving network.
(8) After that, MN sends an Unsolicited Neighbor Advertisement (UNA) message to the target network. Once target MAG receives this message, the link layer and IP layer will be established. At the same time, this message will trigger target MAG to forward buffered packets to MN.
(9) Once the IP layer is established, target MAG and LMA will perform the PBU/PBA procedure to update the BCE and bi-directional tunnel between target network and LMA. After updating the BCE, the target network will become the serving network and forwards the native packets to MN.
(10) Once MN gets the native packets from the new network, it will send the MIH_Link_Down to the old serving network to tear up the original binding and release it. (11) After that, the serving network sends the MIH_N2N_HO_Complete request/response to release the resource.
Performance Evaluations
In our performance analysis, we compare our solution with the standard handover scheme and the fast handover solution in terms of handover delay and signaling cost.
To simplify the analysis, we note the distance between X and Y as Hx-y which equals to Hy-x. The delay between X and Y is denoted as Dx-y. Besides, the signaling and delay between PoA and PoS are neglected. The related parameter setting is shown in Table 4, and some values of them are based on [23] [24].
The signaling cost is computed by hop*message_size. The typical MIH message size for the Events and Commands Services ranges between 50 to 100 bytes, and therefore, the transport of MIH message should solve the message fragmentation of UDP and the message concatenation problems. In the analysis, the message size is subjected to the numbers of neighboring networks, candidate networks, HNPs and so on. To simplify the notations, we adopt the abbreviations specified in Table 5. Besides, to make the analytical results more realistic, the wireless link failure probability (Pf)is considered.
Handover delay analysis
The handover delay is the time interval between the moments when an MN loses connectivity with its serving MAG until the moment it receives the first packets from the target MAG. For the pre-registration mechanisms, the first packet can be the buffered packet.
To evaluate handover delay, we adopt the method in [24]. Assuming that τis interframe time, ρf is the frame error rate over the wireless link, Lp and Lf are the packet size and frame size, respectively, and Dwl is the wireless link delay mainly depending on the L2 technology being used.
The one-way packet transport delay over the wireless link dwl(Lp) can be expressed as
( ) ( 1) wl p frame d L d k (1)
Where p f k L L , and dframe is the one-way frame transport delay of wireless link, which can be expressed as:
, 1 1 (1 ) (2 2( 1) ) n i frame wl f i j wl i j d D p i D j (2)
Where pi,j is given as:
2 2 (( ) 2 1) , (1 ) ((2 ) ) i i j i j f f f f p (3)
Assuming that BWwiredand Dwired are the bandwidth and the latency of wired links, respectively, the one-way packet transportation delay over a wired link through h hops can be expressed as (4) Figure 9 shows the handover delay of standard handover procedure (mobile-initiated and network-initiated method), fast handover and the proposed solution. From Figure 9, it can be noticed that the fast handover and the proposed method have the similar handover delay.
Based on Figure 9, the handover delay of mobile-initiated method and network-initiated method of standard handover scheme can be expressed as
2 _ _ ( , )( ( ) ( ) , ) ( )
Similarly, the handover delay of fast handover solution is shown in (6). Since the packets are buffered in target MAG, it should include the additional tunneling header which is 40 bytes.
2 ( ) ( 40) FH L wl R l S w D t T d d L M (6)
The proposed solution can be expressed as
2 ( 40) ( ) OH L wl UN l A w D t T d d L M (7)
Figure 10shows the handover delay versus different frame error rates. It is obviously that the increasing of frame error rate increases the handover due to retransmission. However, the handover delay of the proposed solution is less than others. Figure 11 shows the handover delay versus the wireless layer delay. The increasing wireless layer delay results in the larger handover delay. Similar to Figure 10, the handover delay of the proposed method is less than the existing solutions.
Signaling cost analysis
In this paper, we adopt the city mobility model [25][27] to analyze the performance of the proposed scheme. In this model, mobile node moves in an epoch-based pattern, in which it starts at a defined point on the street, and then it randomly selects a destination. Once it reaches the destination, it will pause for a random time.
The city is supposed to be a rectangle of a*b. Let dx and dy represent the distance between adjacent horizontal roads and vertical roads, respectively. Both of them can be used to reflect the road density.
According to [25] [27], the expected epoch length can be expressed as
( 1)( 1) ( 1)( 1) ( ) 3 3 y v v x h h h v d N N d N N E L N N (8) Where h x N a d and v y N b d
The expected number of subnet crossing can be expressed as
( ) ( ) ( ) t x y E N E N E N (9)( ) ln ( ) t E L V V E T V V (12)
Assuming that mobile node in the destination follows the uniform distribution between [0, Tmax], then the expected pause time E(Tp) is calculated by 0.5*Tmax.
Therefore, the number of handover per unit time can be calculated as follows.
( Figure 12 The number of handover per unit time versus cell radius and distance between adjacent road Figure 12 depicts the number of handovers per unit time versus cell radius and distance between the adjacent roads. The city section is a rectangular area of 36000m*24000m, and the speed is set to [1,50] m/s. The number of handovers increases as the distance between adjacent roads as well as the cell radius reduces. Differenttothepreviousanalysiswhichassumesthatthesi zeofallsignalingmessagesissame, we distinguish the sizes of different messages as shown in Table 5.
) ( ) ( ) 2 ( ) t c t p E N E N E T E T (13)
According to [26], the wireless link transmission fail probability is Pf, therefore the signaling cost of the given solution can be expressed as
_ _ ( ) *( / (1-) * ) c f f FH WL FH W S E N P P S S (14)
Where SFH_WL means the signaling cost in the wireless link and SFH_W is the signaling cost in the wired link.
(1) Standard handover solution
For standard handover solution, the handover delay can be expressed as follows.
The messages between MN and serving MAG/target MAG (HMN_MAG) includes: M3, M4, M5, M6, M15, M16, MRS and MRA.
The messages between serving MAG and MIIS (HMAG_MIIS) includes: M3 and M4.
The messages between serving MAG and candidate MAG (HMAG_MAG) includes: M7and M8. In fact, there may be multiple candidate MAGs in the resource check phase. While in the handover delay analysis, we ignore the sending interval of M7 and M8.
The messages between serving MAG and target MAG (HMAG_MAG) includes: M9, M10, M13, M14, MPBU and MPBA.
The message between serving MAG and LMA (HMAG_LMA) includes: AAA Query (M17), AAA Reply (M18), MPBU and MPBA. AAA Query and AAA Reply messages are used for pre-registration procedure to query about MN's profile.
The message between target MAG and LMA (HMAG_LMA) includes: MPBU and MPBA.
Therefore, the signaling cost of standard handover can be expressed as
( ( ) ) ( ) 2 ( ) (17)
Therefore, we can get the signaling cost based on Eq. (14) - (17). Figure 13 depicts the signaling cost versus the distance among MAGs (HMAG_MAG). The wireless link failure probability is set to 0.5, the cell radius is set to 100m, the speed is set to [1,50] m/s, and the distance between adjacent roads is set to 10m. We can find that increasing HMAG_MAG results in higher signaling cost. When the HMAG_MAG is less than 8, the standard handover solution has the highest signaling cost than the standard and fat handover solutions. When HMAG_MAG is more than 8, the fast handover solution has the highest signaling cost as it needs additional signaling delivery between the serving MAG and the target MAG to perform the pre-registration. In each case, the proposed solution has the lowest signaling cost as it carries the related information in regular MIH messages. Figure 13 Signaling cost versus the distance among MAGs Figure 14 illustrates the signaling cost versus wireless link failure probability. The cell radius is set to 100m, the speed is set to [1,50] m/s, and the distance between adjacent roads is set to 10m. The increasing wireless link failure probability results in higher mobility signaling cost, and the proposed solution has lower signaling cost than the standard handover and fast handover solutions. Figure 16 shows the signaling cost versus cell radius. The wireless link failure probability is set to 0.5, the cell radius is set to 100m, the speed is set to [1,50] m/s, and the distance between adjacent roads is set to 10m. The increase of cell radius reduces the probability of handover as well as the signaling cost. The signaling cost of fast handover is slightly higher than the standard handover, while the proposed solution is lower than others. Figure 14 Signaling cost versus wireless link failure probability Figure 15 shows the signaling cost versus moving speed of MNs. The cell radius is set to 100m and the distance between adjacent roads is set to 10m. The maximum speed is set to 50m/s, and the minimum speed is set to [1,36]m/s. When MN is moving at high speed, it may cross more cell roads, and incurs more handovers. Accordingly, the signaling cost is increased. Figure 15 Signaling cost versus moving speed Similar, as that shown in Figure 16, the signaling cost of the proposed solution is lower than others, and that of fast handover is slightly higher than standard handover. Figure 17 illustrates the signaling cost versus the distance between adjacent roads. The wireless link failure probability is set to 0.5, the cell radius is set to 100m, the speed is set to [1,50] m/s. To simplify the analysis, we use d to represent dx and dy, which can be used as a metric to evaluate the road network density of the city. The increasing of d results in higher signaling cost. This is because the probability of crossing cell is increased when the road network density is sparse as an MN can only move along the road. The signaling cost of the proposed solution is lower than other two, while for fast handover solution, signaling cost is highest. Figure 16 Signaling cost versus cell radius Figure 17 Signaling cost versus the distance between adjacent roads
Signaling cost
Conclusions
This paper proposes an integration model of MIH and FPMIPv6 with the fast handover support, which can reduce the handover delay without introducing additional signaling cost compared with the standard handover solution and the fast handover solution. This is obtained by extending related MIH messages to include the related information for pre-registration, and adopting the transient binding mechanism.
Although the proposed solution can improve the performance of heterogeneous handover, it still has some limitations. It is a Centralized Mobility Management (CMM) which is dependent on the central mobility entity such as HA, Local Mobility Anchor (LMA). However, with the rapid increase of MNs, these central entities may become the single point of failure and result in serious service degradation. To solve this problem, Distributed Mobility Management (DMM) is already available in the literature with the objective to distribute the traffic in more flat architecture with optimal routing. DMM distributes the traffic on different mobility anchors to solve the signal point failure problem, and it can be used in host-based and network-based mobility management. In host-based DMM, it maintains multiple CoAs for each session and sets up the tunnels between MN and its each session's MAs. While in network-based DMM, it maintains multiple tunnels but introducing a CMD to maintain for the sessions for each MN.
This paper uses city section mobility model to evaluate its performance, which is more realistic for its deployment scenarios. The numerical results show that the proposed solution is better than the standard handover solution and the fast handover solution. In the future, we plan to combine the DMM with MIH to analyze their integration model and performance.
Figure 1
1The reference network model of FPMIPv6
Figure 2
2The predictive fast handover flow
Figure 3
3The reactive fast handover flow
Figure 4
4The MIH frameworkmodel[11]
Figure 6 Figure 7
67The message format of extended MIH_N2N_HO_Commit request The format of extended MIH_N2N_HO_Commit response message
Figure 8
8The proposed FPMIPv6 and MIH integration model operation flow
Figure 9
9The
Figure 10
10Handover delay versus the frame error rate
Figure 11
11Handover delay versus wireless layer delay
Table 1
1The summary of existing PMIPv6 and MIH integration modelSchemes
Metric
The PMIPv6
assisted MIH
using standard
approach
PMIPv6 assisted
MIH using
wireless modified
MIHF signals
PMIPv6 assisted
MIH using MIHF at
network side only
PMIPv6
assisted MIH
with neighbor
discovery
PMIPv6 assisted
MIH with
Handover
Coordinator (HC)
PMIPv6 assisted
MIH using fast
handover scheme
Triggermechanis
m
MN's MIHF
No
L2 trigger
MN trigger
MN trigger
MN trigger
Initiation
method
MN
Network
Network
undefined
HC
Network
Candidate
Network
provider
MN
S-PoS
S-PoS
undefined
S-PoS
MN
Resource check
point
S-PoS
S-PoS
S-PoS
No
Undefined, maybe
is HC
S-PoS
target network
selector
MN
S-PoS
S-PoS
undefined
S-PoS
S-PoS
Packet buffer
point
LMA
LMA
LMA
S-PoS
Undefined
C-PoS (DL)
MN (UL)
features
(1) large power
comsumption for
that MN is
involved in too
many signaling
exchanges;
Table 2
2The parameters of extended MIH_N2N_HO_Commit request message It is MN link-local address interface identifier which is used to identify the attached interface of a MN, and it can be MAC address, cell ID or other link addressParameter
Type&Value
Data type
Description
MN
LLA-IID
MN LLAID
(Extended)
Value=101
LINK ADDR or
Interface identifier
LMAA
LMA Address
(Extended)
Value=102
TRANSPORT_ADDR
Identify the address of LMA which can be IPv4 or IPv6.
HNP
Home Network Prefix
(Extended)
Value=103
LIST(HNP)where
HNP=SEQUENCE(
UNSIGNED_INT(1)
OCTET_STRING(16))
Identify the list of MN's IPv6 home network prefix(es)
assigned to the Target MN link.
UNSIGENED_INT: IP_PREFIX_LEN
Table 3
3Comparison of status code defined in MIH and IANA[7][17] Status code
MIH definition
IANA definition
0
success
Handover accept or success
1
Unspecified Failure
Handover Accepted, NCoA not valid [RFC5568]
2
Rejected
Handover Accepted, NCoA assigned [RFC5568]
3
Authorization failure
Handover Accepted, use PCoA
[RFC5568]
4
Network error
Message sent unsolicited [RFC5568]
5
Unassigned
Context Transfer Accepted or Successful [RFC5949]
6
Unassigned
All available Context Transferred [RFC5949]
7-127
Unassigned
Unassigned
128
Unassigned
Handover Not Accepted, reason unspecified [RFC5568]
129
Unassigned
Administratively prohibited [RFC5568]
130
Unassigned
Insufficient resources [RFC5568]
131
Unassigned
Requested Context Not Available [RFC5949]
132
Unassigned
Forwarding Not Available [RFC5949]
133-255
Unassigned
Unassigned
Table 4
4The parameters listSymbol
Value(s) Description
a
36000m City section length
b
24000m City section width
R
100m
Cell radius
V
1~50
m/s
Average speed of MN
Α
1
Unit transmission cost over wired link
Β
1.5
Unit transmission cost over wireless link
N
6
Average number of neighboring
networks
m
6
Average number of preferred PoAs
Pf
0.5
Wireless link failure probability
τ
20ms
Interframe time
ρf
0.1
Frame error rate (FER) over wireless link
d
10m
Distance between adjacent roads
Tmax
70s
Maximum pause time in a location
TL2
45.35ms The link layer handover delay
HMN-MAG
1 hop
Average distance between MN and MAG
HMAG-LMA
10 hops
Average distance between MAG and
LMA
HMAG-MIIS 10 hops
Average distance between MAG and
MIIS
HMAG-MAG 10 hops
Average distance between MAG and
MAG
Let Vminand Vmax represent the minimum speed and maximum speed of mobile node, respectively. The expected time for an epoch isWhere
1
1
1
2
(
1)
( )
(6
4
3)
6
x
h
h
m m
K
E N
N
mK K
N
(10)
Where
1
2
x
K
r d
2
2
2
2
(
1)
( )
(6
4
3)
6
y
v
v
m m
K
E N
N
mK K
N
(11)
Where
2
2 y
K
r d
max
min
max
min
Table 5
5The sizes of messages used in the analysis The messages sizes only consider the protocol header, while ignore the outer header such as IPv6 header.Message name
Type
Size
Abbreviation
MIH_Link_Going_down
Event service
78
M1
MIH_Link_Up
Event service
95
M2
MIH_Get_Information request
Information service
1500
M3
MIH_Get_Information response
Information service
1500
M4
MIH_Net_HO_Candidate_Query resquest
Command service
63+11*n+8*m*n
M5
MIH_Net_HO_Candidate_Query response
Command service
77+101*m
M6
MIH_N2N_HO_Query Resource request
Command service
150+11*m
M7
MIH_N2N_HO_Query Resource response
Command service
165
M8
MIH_N2N_HO_Commit request
Command service
213
M9
MIH_N2N_HO_Commit request (Extended)
Command service
264
M9e
MIH_N2N_HO_Commit response
Command service
92
M10
MIH_N2N_HO_Commit response (Extended)
Command service
92
M10e
MIH_Net_HO_Commit request
Command service
122
M11
MIH_Net_HO_Commit response
Command service
103
M12
MIH_N2N_HO_Complete request
Command service
109
M13
MIH_N2N_HO_Complete response
Command service
112
M14
MIH_MN_HO_Commit request
Command service
75
M15
Similarly, we can get that of fast handover solution as follows.The signaling cost of the proposed solution is shown as follows.
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"Adversarial training for multi-context joint entity and relation extraction",
"Adversarial training for multi-context joint entity and relation extraction"
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"Giannis Bekoulis \nIDLab Department of Information Technology\nGhent University -imec\n\n",
"Johannes Deleu \nIDLab Department of Information Technology\nGhent University -imec\n\n",
"Thomas Demeester \nIDLab Department of Information Technology\nGhent University -imec\n\n",
"Chris Develder \nIDLab Department of Information Technology\nGhent University -imec\n\n"
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"IDLab Department of Information Technology\nGhent University -imec\n",
"IDLab Department of Information Technology\nGhent University -imec\n",
"IDLab Department of Information Technology\nGhent University -imec\n",
"IDLab Department of Information Technology\nGhent University -imec\n"
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| Adversarial training (AT) is a regularization method that can be used to improve the robustness of neural network methods by adding small perturbations in the training data. We show how to use AT for the tasks of entity recognition and relation extraction. In particular, we demonstrate that applying AT to a general purpose baseline model for jointly extracting entities and relations, allows improving the state-of-the-art effectiveness on several datasets in different contexts (i.e., news, biomedical, and real estate data) and for different languages (English and Dutch). | 10.18653/v1/d18-1307 | [
"https://www.aclweb.org/anthology/D18-1307.pdf"
]
| 52,056,726 | 1808.06876 | 2391d8bcef01fc55d928c44ce3b35462cb34a85d |
Adversarial training for multi-context joint entity and relation extraction
Giannis Bekoulis
IDLab Department of Information Technology
Ghent University -imec
Johannes Deleu
IDLab Department of Information Technology
Ghent University -imec
Thomas Demeester
IDLab Department of Information Technology
Ghent University -imec
Chris Develder
IDLab Department of Information Technology
Ghent University -imec
Adversarial training for multi-context joint entity and relation extraction
Adversarial training (AT) is a regularization method that can be used to improve the robustness of neural network methods by adding small perturbations in the training data. We show how to use AT for the tasks of entity recognition and relation extraction. In particular, we demonstrate that applying AT to a general purpose baseline model for jointly extracting entities and relations, allows improving the state-of-the-art effectiveness on several datasets in different contexts (i.e., news, biomedical, and real estate data) and for different languages (English and Dutch).
Introduction
Many neural network methods have recently been exploited in various natural language processing (NLP) tasks, such as parsing , POS tagging (Lample et al., 2016), relation extraction (dos Santos et al., 2015), translation (Bahdanau et al., 2015), and joint tasks (Miwa and Bansal, 2016). However, Szegedy et al. (2014) observed that intentional small scale perturbations (i.e., adversarial examples) to the input of such models may lead to incorrect decisions (with high confidence). Goodfellow et al. (2015) proposed adversarial training (AT) (for image recognition) as a regularization method which uses a mixture of clean and adversarial examples to enhance the robustness of the model. Although AT has recently been applied in NLP tasks (e.g., text classification (Miyato et al., 2017)), this paper -to the best of our knowledge -is the first attempt investigating regularization effects of AT in a joint setting for two related tasks.
We start from a baseline joint model that performs the tasks of named entity recognition (NER) and relation extraction at once. Previously proposed models (summarized in Section 2) exhibit several issues that the neural network-based baseline approach (detailed in Section 3.1) overcomes: (i) our model uses automatically extracted features without the need of external parsers nor manually extracted features (see Gupta et al. (2016); Miwa and Bansal (2016); Li et al. (2017)), (ii) all entities and the corresponding relations within the sentence are extracted at once, instead of examining one pair of entities at a time (see Adel and Schütze (2017)), and (iii) we model relation extraction in a multi-label setting, allowing multiple relations per entity (see Katiyar and Cardie (2017); Bekoulis et al. (2018a)). The core contribution of the paper is the use of AT as an extension in the training procedure for the joint extraction task (Section 3.2).
To evaluate the proposed AT method, we perform a large scale experimental study in this joint task (see Section 4), using datasets from different contexts (i.e., news, biomedical, real estate) and languages (i.e., English, Dutch). We use a strong baseline that outperforms all previous models that rely on automatically extracted features, achieving state-of-the-art performance (Section 5). Compared to the baseline model, applying AT during training leads to a consistent additional increase in joint extraction effectiveness.
Related work
Joint entity and relation extraction: Joint models (Li and Ji, 2014;Miwa and Sasaki, 2014) that are based on manually extracted features have been proposed for performing both the named entity recognition (NER) and relation extraction subtasks at once. These methods rely on the availability of NLP tools (e.g., POS taggers) or manually designed features leading to additional complexity. Neural network methods have been exploited to overcome this feature design issue and usually involve RNNs and CNNs (Miwa and Bansal, 2016;Zheng et al., 2017). Specifically, Miwa and Bansal (2016) as well as Li et al. (2017) apply bidirectional tree-structured RNNs for different contexts (i.e., news, biomedical) to capture syntactic information (using external dependency parsers). Gupta et al. (2016) propose the use of various manually extracted features along with RNNs. Adel and Schütze (2017) solve the simpler problem of entity classification (EC, assuming entity boundaries are given), instead of NER, and they replicate the context around the entities, feeding entity pairs to the relation extraction layer. Katiyar and Cardie (2017) investigate RNNs with attention without taking into account that relation labels are not mutually exclusive. Finally, Bekoulis et al. (2018a) use LSTMs in a joint model for extracting just one relation at a time, but increase the complexity of the NER part. Our baseline model enables simultaneous extraction of multiple relations from the same input. Then, we further extend this strong baseline using adversarial training.
Adversarial training (AT) (Goodfellow et al., 2015) has been proposed to make classifiers more robust to input perturbations in the context of image recognition. In the context of NLP, several variants have been proposed for different tasks such as text classification (Miyato et al., 2017), relation extraction (Wu et al., 2017) and POS tagging (Yasunaga et al., 2018). AT is considered as a regularization method. Unlike other regularization methods (i.e., dropout (Srivastava et al., 2014), word dropout (Iyyer et al., 2015)) that introduce random noise, AT generates perturbations that are variations of examples easily misclassified by the model.
Model
Joint learning as head selection
The baseline model, described in detail in Bekoulis et al. (2018b), is illustrated in Fig. 1. It aims to detect (i) the type and the boundaries of the entities and (ii) the relations between them. The input is a sequence of tokens (i.e., sentence) w = w 1 , ..., w n . We use character level embeddings to implicitly capture morphological features (e.g., prefixes and suffixes), representing each character by a vector (embedding). The character embeddings are fed to a bidirectional LSTM (BiLSTM) to obtain the character-based representation of the word. We also use pre-trained word embeddings. Word and character embeddings are concatenated to form the final token representation, which is then fed to a BiLSTM layer to extract sequential information.
For the NER task, we adopt the BIO (Beginning, Inside, Outside) encoding scheme. In Fig. 1, the B-PER tag is assigned to the beginning token of a 'person' (PER) entity. For the prediction of the entity tags, we use: (i) a softmax approach for the entity classification (EC) task (assuming entity boundaries given) or (ii) a CRF approach where we identify both the type and the boundaries for each entity. During decoding, in the softmax setting, we greedily detect the entity types of the tokens (i.e., independent prediction). Although independent distribution of types is reasonable for EC tasks, this is not the case when there are strong correlations between neighboring tags. For instance, the BIO encoding scheme imposes several constraints in the NER task (e.g., the B-PER and I-LOC tags cannot be sequential). Motivated by this intuition, we use a linear-chain CRF for the NER task (Lample et al., 2016). For decoding, in the CRF setting, we use the Viterbi algorithm. During training, for both EC (softmax) and NER tasks (CRF), we minimize the cross-entropy loss L NER . The entity tags are later fed into the relation extraction layer as label embeddings (see Fig. 1), assuming that knowledge of the entity types is beneficial in predicting the relations between the involved entities.
We model the relation extraction task as a multi-label head selection problem (Bekoulis et al., 2018b;. In our model, each word w i can be involved in multiple relations with other words. For instance, in the example illustrated in Fig. 1, "Smith" could be involved not only in a Lives in relation with the token "California" (head) but also in other relations simultaneously (e.g., Works for, Born In with some corresponding tokens). The goal of the task is to predict for each word w i , a vector of headsŷ i and the vector of corresponding relationsr i . We compute the score s(w j , w i , r k ) of word w j to be the head of w i given a relation label r k using a single layer neural network. The corresponding probability is defined as: P(w j , r k | w i ; θ) = σ(s(w j , w i , r k )), where σ(.) is the sigmoid function. During training, we minimize the cross-entropy loss L rel as:
n i=0 m j=0 − log P(y i,j , r i,j | w i ; θ)(1)
where m is the number of associated heads (and thus relations) per word w i . During decoding, the most probable heads and relations are selected using threshold-based prediction. The final objective for the joint task is computed as L JOINT (w; θ) = L NER + L rel where θ is a set of parameters. In the case of multi-token entities, only the last token of the entity can serve as head of another token, to eliminate redundant relations. If an entity is not involved in any relation, we predict the auxiliary "N" relation label and the token itself as head.
Adversarial training (AT)
We exploit the idea of AT (Goodfellow et al., 2015) as a regularization method to make our model robust to input perturbations. Specifically, we generate examples which are variations of the original ones by adding some noise at the level of the concatenated word representation (Miyato et al., 2017). This is similar to the concept introduced by Goodfellow et al. (2015) to improve the robustness of image recognition classifiers. We generate an adversarial example by adding the worst-case perturbation η adv to the original embedding w that maximizes the loss function:
η adv = argmax η ≤ L JOINT (w + η;θ)(2)
whereθ is a copy of the current model parameters. Since Eq.
(2) is intractable in neural networks, we use the approximation proposed in Goodfellow et al. (2015) defined as: η adv = g/ g , with g = ∇ w L JOINT (w;θ), where is a small bounded norm treated as a hyperparameter. Similar to Yasunaga et al. (2018), we set to be α √ D (where D is the dimension of the embeddings). We train on the mixture of original and adversarial examples, so the final loss is computed as: L JOINT (w;θ) + L JOINT (w + η adv ;θ).
Experimental setup
We evaluate our models on four datasets, using the code as available from our github codebase. 1 Specifically, we follow the 5-fold crossvalidation defined by Miwa and Bansal (2016) for the ACE04 (Doddington et al., 2004) dataset. For the CoNLL04 (Roth and Yih, 2004) EC task (assuming boundaries are given), we use the same splits as in Gupta et al. (2016); Adel and Schütze (2017). We also evaluate our models on the NER task similar to Miwa and Sasaki (2014) in the same dataset using 10-fold cross validation. For the Dutch Real Estate Classifieds, DREC (Bekoulis et al., 2017) dataset, we use train-test splits as in Bekoulis et al. (2018a). For the Adverse Drug Events, ADE (Gurulingappa et al., 2012), we perform 10-fold cross-validation similar to Li et al. (2017). To obtain comparable results that are not affected by the input embeddings, we use the embeddings of the previous works. We employ early stopping in all of the experiments. We use the Adam optimizer (Kingma and Ba, 2015) and we fix the hyperparameters (i.e., α, dropout values, best epoch, learning rate) on the validation sets. The scaling parameter α is selected from {5e−2, 1e−2, 1e−3, 1e−4}. Larger values of α (i.e., larger perturbations) lead to consistent performance decrease in our early experiments. This can be explained from the fact that adding more noise can change the content of the sentence as also reported by Wu et al. (2017).
We use three types of evaluation, namely: (i) S(trict): we score an entity as correct if both the entity boundaries and the entity type are correct (ACE04, ADE, CoNLL04, DREC), (ii) B(oundaries): we score an entity as correct if only the entity boundaries are correct while the entity type is not taken into account (DREC) and (iii) R(elaxed): a multi-token entity is considered correct if at least one correct type is assigned to the tokens comprising the entity, assuming that the Table 1: Comparison of our method with the stateof-the-art in terms of F 1 score. The proposed models are: (i) baseline, (ii) baseline EC (predicts only entity classes) and (iii) baseline (EC) + AT (regularized by AT). The and symbols indicate whether the models rely on external NLP tools. We include different evaluation types (S, R and B).
boundaries are known (CoNLL04), to compare to previous works. In all cases, a relation is considered as correct when both the relation type and the argument entities are correct. Table 1 shows our experimental results. The name of the dataset is presented in the first column while the models are listed in the second column. The proposed models are the following: (i) baseline: the baseline model shown in Fig. 1 with the CRF layer and the sigmoid loss, (ii) baseline EC: the proposed model with the softmax layer for EC, (iii) baseline (EC) + AT: the baseline regularized using AT. The final three columns present the F 1 results for the two subtasks and their average performance. Bold values indicate the best results among models that use only automatically extracted features. For ACE04, the baseline outperforms Katiyar and Cardie (2017) by ∼2% in both tasks. This improvement can be explained by the use of: (i) multi-label head selection, (ii) CRF-layer and (iii) character level embeddings. Compared to Miwa and Bansal (2016), who rely on NLP tools, the baseline performs within a reasonable margin (less than 1%) on the joint task. On the other hand, Li et al. (2017) use the same model for the ADE biomedical dataset, where we report a 2.5% overall improvement. This indicates that NLP tools are not always accurate for various contexts. For the CoNLL04 dataset, we use two evaluation settings. We use the relaxed evaluation similar to Gupta et al. (2016); Adel and Schütze (2017) on the EC task. The baseline model outperforms the state-of-the-art models that do not rely on manually extracted features (>4% improvement for both tasks), since we directly model the whole sentence, instead of just considering pairs of entities. Moreover, compared to the model of Gupta et al. (2016) that relies on complex features, the baseline model performs within a margin of 1% in terms of overall F 1 score. We also report NER results on the same dataset and improve overall F 1 score with ∼1% compared to Miwa and Sasaki (2014), indicating that our automatically extracted features are more informative than the hand-crafted ones. These automatically extracted features exhibit their performance improvement mainly due to the shared LSTM layer that learns to automatically generate feature representations of entities and their corresponding relations within a single model. For the DREC dataset, we use two evaluation methods. In the boundaries evaluation, the baseline has an improvement of ∼3% on both tasks compared to Bekoulis et al. (2018a), whose quadratic scoring layer complicates NER. Table 1 and Fig. 2 show the effectiveness of the adversarial training on top of the baseline model. In all of the experiments, AT improves the predictive performance of the baseline model in the joint setting. Moreover, as seen in Fig. 2, the performance of the models using AT is closer to maximum even from the early training epochs. Specifically, for ACE04, there is an improvement in both tasks as well as in the overall F 1 performance (0.4%). For CoNLL04, we note an improvement in the overall F 1 of 0.4% for the EC and 0.8% for the NER tasks, respectively. For the DREC dataset, in both settings, there is an overall improvement of ∼1%. Figure 2 shows that from the first epochs, the model obtains its maximum performance on the DREC validation set. Finally, for ADE, our AT model beats the baseline F 1 by 0.7%.
Results
Our results demonstrate that AT outperforms the neural baseline model consistently, considering our experiments across multiple and more diverse datasets than typical related works. The im- provement of AT over our baseline (depending on the dataset) ranges from ∼0.4% to ∼0.9% in terms of overall F 1 score. This seemingly small performance increase is mainly due to the limited performance benefit for the NER component, which is in accordance with the recent advances in NER using neural networks that report similarly small gains (e.g., the performance improvement in Ma and Hovy (2016) and Lample et al. (2016) on the CoNLL-2003 test set is 0.01% and 0.17% F 1 percentage points, while in the work of Yasunaga et al. (2018), a 0.07% F 1 improvement on CoNLL-2000 using AT for NER is reported). However, the relation extraction performance increases by ∼1% F 1 scoring points, except for the ACE04 dataset. Further, as seen in Fig. 2, the improvement for CoNLL04 is particularly small on the evaluation set. This may indicate a correlation between the dataset size and the benefit of adversarial training in the context of joint models, but this needs further investigation in future work.
Figure 1 :
1Our model for joint entity and relation extraction with adversarial training (AT) comprises (i) a word and character embedding layer, (ii) a BiLSTM layer, (iii) a CRF layer and (iv) a relation extraction layer. In AT, we compute the worst-case perturbations η of the input embeddings.
Figure 2 :
2F 1 performance of the baseline and the AT models on the validation sets from 10-30 epochs onwards depending on the dataset. The smoothed lines (obtained by LOWESS smoothing) model the trends and the 95% confidence intervals.
https://github.com/bekou/multihead_ joint_entity_relation_extraction
ConclusionWe proposed to use adversarial training (AT) for the joint task of entity recognition and relation extraction. The contribution of this study is twofold: (i) investigation of the consistent effectiveness of AT as a regularization method over a multi-context baseline joint model, with (ii) a large scale experimental evaluation. Experiments show that AT improves the results for each task separately, as well as the overall performance of the baseline joint model, while reaching high performance already during the first epochs of the training procedure.
AcknowledgmentsWe would like to thank the anonymous reviewers for the time and effort they spent in reviewing our work, and for their valuable feedback.
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"Supersymmetric Vacua in N = 2 Supergravity",
"Supersymmetric Vacua in N = 2 Supergravity"
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"Jan Louis [email protected] \nII. Institut für Theoretische Physik\nUniversität Hamburg\nLuruper Chaussee 149D-22761HamburgGermany\n\nZentrum für Mathematische Physik\nUniversität Hamburg\nBundesstrasse 55D-20146HamburgGermany\n",
"Paul Smyth [email protected] \nInstitut de Théorie des Phénomènes Physiques\nCH-1015LausanneEPFLSwitzerland\n",
"Hagen Triendl [email protected] \nInstitut de Physique Théorique\nCEA Saclay Orme des Merisiers\nF-91191Gif-sur-YvetteFrance\n",
"\nIntroduction\n\n"
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"II. Institut für Theoretische Physik\nUniversität Hamburg\nLuruper Chaussee 149D-22761HamburgGermany",
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"Institut de Théorie des Phénomènes Physiques\nCH-1015LausanneEPFLSwitzerland",
"Institut de Physique Théorique\nCEA Saclay Orme des Merisiers\nF-91191Gif-sur-YvetteFrance",
"Introduction\n"
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| We use the embedding tensor formalism to analyse maximally symmetric backgrounds of N = 2 gauged supergravities which have the full N = 2 supersymmetry. We state the condition for N = 2 vacua and discuss some of their general properties. We show that if the gauged isometries leave the SU(2) R-symmetry invariant, then the N = 2 vacuum must be Minkowski. This implies that there are no AdS backgrounds with eight unbroken supercharges in the effective N = 2 supergravity of six-dimensional SU(3) × SU(3) structure compactifications of type II string theory and M-theory. Combined with previous results on N = 1 vacua, we show that there exist N = 2 supergravities with a given set of gauged Abelian isometries that have both N = 2 and N = 1 vacua. We also argue that an analogue of our analysis holds in five and six spacetime dimensions.April 20121 Related work on tensor fields in N = 2 supergravity has been performed in[4]. | 10.1007/jhep08(2012)039 | [
"https://arxiv.org/pdf/1204.3893v3.pdf"
]
| 119,105,454 | 1204.3893 | 31c0bcd801b6cc14089f44af0cd78c650470c8f0 |
Supersymmetric Vacua in N = 2 Supergravity
19 Nov 2012 April 2012
Jan Louis [email protected]
II. Institut für Theoretische Physik
Universität Hamburg
Luruper Chaussee 149D-22761HamburgGermany
Zentrum für Mathematische Physik
Universität Hamburg
Bundesstrasse 55D-20146HamburgGermany
Paul Smyth [email protected]
Institut de Théorie des Phénomènes Physiques
CH-1015LausanneEPFLSwitzerland
Hagen Triendl [email protected]
Institut de Physique Théorique
CEA Saclay Orme des Merisiers
F-91191Gif-sur-YvetteFrance
Introduction
Supersymmetric Vacua in N = 2 Supergravity
19 Nov 2012 April 2012
We use the embedding tensor formalism to analyse maximally symmetric backgrounds of N = 2 gauged supergravities which have the full N = 2 supersymmetry. We state the condition for N = 2 vacua and discuss some of their general properties. We show that if the gauged isometries leave the SU(2) R-symmetry invariant, then the N = 2 vacuum must be Minkowski. This implies that there are no AdS backgrounds with eight unbroken supercharges in the effective N = 2 supergravity of six-dimensional SU(3) × SU(3) structure compactifications of type II string theory and M-theory. Combined with previous results on N = 1 vacua, we show that there exist N = 2 supergravities with a given set of gauged Abelian isometries that have both N = 2 and N = 1 vacua. We also argue that an analogue of our analysis holds in five and six spacetime dimensions.April 20121 Related work on tensor fields in N = 2 supergravity has been performed in[4].
Introduction
The analysis of Minkowski and Anti-de Sitter (AdS) supersymmetric vacua in gauged extended supergravity has received much attention in recent years. In this paper we consider such maximally-symmetric backgrounds of N = 2 supergravities in four spacetime dimensions (d = 4) and their "cousins" in d = 5, 6 which also have eight supercharges.
The general conditions for N = 2 vacua in electrically gauged N = 2 supergravities, together with a few illustrative examples, were given recently in [1]. By using the embedding tensor formalism, introduced in [2] and applied to N = 2 gauged supergravity in [3], 1 we extend the analysis of [1] by allowing for the possibility of electrically and magnetically charged fields in the spectrum. We derive the conditions for N = 2 vacua with Abelian and non-Abelian factors in the gauge group and show that solutions generically exist. However, it is not guaranteed that these solutions lie inside the physical domain of the Kähler cone and thus are physically acceptable. For the special case of hypermultiplets that are gauged with respect to isometries which do not induce an SU(2) R-symmetry rotation, we show that AdS vacua with eight unbroken supercharges are not possible. It is straightforward to extend our analysis to spacetimes with d = 5, 6.
We shall specifically study the class of isometries that are present in quaternionic-Kähler manifolds which are in the image of the c-map and appear at the tree level of type II compactifications in string theory [5,6]. These manifolds can be viewed as a graded Heisenberg algebra fibred over a special-Kähler base. We show that no N = 2 AdS vacua can occur for gauged isometries in the fibre, which in turn implies that there are no AdS vacua in the low-energy effective N = 2 action of six-dimensional SU(3)×SU(3)-structure compactifications of type II string theory and M-theory that preserve eight supercharges. This means in particular that SU(3) × SU(3)-structure backgrounds with four-and fivedimensional N = 2 AdS vacua as found in [7][8][9] do not have any description in terms of N = 2 gauged supergravity. The conditions for N = 2 Minkowski vacua are linear in the fibre coordinates and holomorphic in the coordinates on the special-Kähler manifold suggesting that generically a solution exists. However, the Kähler cone condition is not automatically satisfied for these solutions. N = 2 supergravities with N = 1 vacua were first discovered in Refs. [10][11][12] and later systematically analysed in [13][14][15]. It is of interest to determine under what conditions these supergravities can also admit N = 2 vacua in their field space. We again find that the conditions are linear in the fibre coordinates and holomorphic in the special-Kähler coordinates, leaving the Kähler cone condition as the non-trivial requirement to find a physically acceptable solution. We give two examples of special-Kähler manifolds with cubic prepotential, one of which contains either an N = 1 or an N = 2 vacuum inside the Kähler cone but never both at the same time. The second example can accommodate both N = 1 and N = 2 vacua inside the Kähler cone, as long as the charges are chosen appropriately.
This paper is organised as follows. In Section 2 we briefly introduce N = 2 gauged supergravity in order to set the stage for the analysis. In Section 3 we record the conditions for vacua with the full N = 2 supersymmetry and determine some of their properties. In Section 4 we extend the analysis to supergravities with eight supercharges in d = 5, 6. In Section 5 we consider the special case of gauged isometries in the fibre of quaternionic-Kähler manifold which are in the image of the c-map. Finally, in Section 6 we address the question of simultaneously having N = 2 and N = 1 vacua in the same gauged supergravity.
Gauged supergravity with eight supercharges
Let us start with a brief summary of gauged N = 2 supergravity in d = 4. 2 Its spectrum consists of a gravitational multiplet, n v vector multiplets and n h hypermultiplets. 3 The gravitational multiplet contains the spacetime metric g µν , two gravitini Ψ µA , A = 1, 2 and the graviphoton A 0 µ . Each vector multiplet contains a vector A i µ , two gaugini λ iA and a complex scalar t i , where i = 1, . . . , n v labels the vector multiplets. 4 Finally, a hypermultiplet consists of two hyperini ζ α and four scalars q u , where α = 1, . . . , 2n h and u = 1, . . . , 4n h . The scalar field space is parametrised by (t i , q u ) and splits into the product
M = M v × M h .(1)
The first component M v is a special-Kähler manifold of complex dimension n v spanned by the scalars t i in the vector multiplets. This implies that the metric obeys
g i = ∂ i ∂K v , with K v = − ln i X Λ Ω ΛΣ X Σ ,(2)
where X Λ = (X I , F I ), I = 0, . . . , n v is a 2(n v + 1)-dimensional symplectic vector that depends holomorphically on the t i . F I = ∂F /∂X I is the derivative of a holomorphic prepotential F which is homogeneous of degree 2 and Ω ΛΣ is the standard symplectic metric. The physical range of the coordinates t i is restricted to the Kähler cone defined by i X Λ Ω ΛΣ X Σ > 0 .
The second component of the field space M h , spanned by the scalars q u in the hypermultiplets, is quaternionic-Kähler and of real dimension 4n h . These manifolds admit a triplet of almost complex structures I x , x = 1, 2, 3 satisfying I x I y = −δ xy 1 + ǫ xyz I z , with the metric being Hermitian with respect to all three I x . The associated two-forms K x are the field strength of the SU(2) connection ω x , i.e.
K x = dω x + 1 2 ǫ xyz ω y ∧ ω z .(4)
In gauged supergravities the multiplets can be charged under a set of electric and magnetic gauge fields. The corresponding covariant derivatives of the scalars read
D µ q u = ∂ µ q u − A Λ µ Θ λ Λ k u λ , D µ t i = ∂ µ t i − A Λ µΘλ Λ k î λ ,(5)
where A Λ µ = (A I µ , B µ I ) is a symplectic vector of electric and magnetic gauge fields and k u λ (k î λ ), λ = 1, . . . , n Kh , (λ = 1, . . . , n Kv , ) are Killing vectors on M h (M v ) respectively.
Finally, the charges or group theoretical representations of the scalars are specified by the embedding tensors Θ λ Λ ,Θλ Λ . Note that the t i transform in the adjoint representation of the gauge group and thus for any non-Abelian factor the gauged k î λ have to be nontrivial. Moreover, if the gauged isometries are non-Abelian, the embedding tensor has to transform covariantly, which is ensured by the quadratic constraint
fλ σρΘσ ΛΘρ Σ +Θσ Λ (kλ) Γ ΣΘλ Γ = 0 .(6)
Here (kλ) Γ Σ is the symplectic transformation induced by the Killing vector k î λ via
k î λ ∂ i X Λ = (kλ) Λ Σ X Σ ,(7)
and fλ σρ are the structure constants [kσ, kρ] = fλ σρ kλ .
Note that both (kλ) Γ Σ and fλ σρ are independent of the coordinates t i . The gauging of isometries requires additional terms in the supersymmetry variations. Since we are looking for maximally symmetric backgrounds it is sufficient to focus on the scalar parts of the fermionic supersymmetry variations given by
δ ǫ Ψ µA = D µ ǫ * A − S AB γ µ ǫ B + . . . , δ ǫ λ iA = W iAB ǫ B + . . . , δ ǫ ζ α = N A α ǫ A + . . . ,(9)
where ǫ A are the supersymmetry parameters and
S AB = 1 2 e K v /2 X Λ Θ λ Λ P x λ (σ x ) AB , W iAB = ie K v /2 g i (∇X Λ )Θ λ Λ P x λ (σ x ) AB + e K v /2 ǫ ABX ΛΘλ Λ k î λ , N A α = 2e K v /2X Λ Θ λ Λ U A αu k u λ .(10)
Here U Aα are the vielbein one-forms on M h , the (σ x ) A B are the Pauli matrices, and
∇ i X Λ := ∂ i X Λ + (∂ i K v )X Λ . Finally, P x λ are the Killing prepotentials defined by − 2k u λ K x uv = ∇ v P x λ ,(11)
where ∇ v is the SU(2)-covariant derivative and the two-forms K x are defined in (4). The matrices given in (10) also determine the scalar potential V in the Lagrangian
V = −6S ABS AB + 1 2 g i W iAB W AB + N A α N α A .(12)
To conclude, a gauged supergravity is specified by the spectrum of vector-and hypermultiplets, their respective field spaces and the embedding tensor which determines the charged directions in field space.
Vacua with N = 2 supersymmetry
We shall now give the conditions for vacua which have the full N = 2 supersymmetry. This requires that all fermionic supersymmetry variations (9) vanish, which, for a maximally symmetric spacetime, translates into the conditions
S AB ǫ B = 1 2 µǫ * A , W iAB = 0 , N αA = 0 ,(13)
where Λ = −3|µ| 2 is the cosmological constant of the N = 2 vacuum. These conditions have been discussed before for electric gaugings in [1].
Let us start by analysing the second condition in (13). Since (σ x ) AB and ǫ AB are linearly independent, this condition together with the definition (10) implies [1]
(∇ i X Λ ) Θ λ Λ P x λ = 0 ,(14)X ΛΘλ Λ k î λ = 0 .(15)
Equation (15) only depends on the t i and has a trivial solution k î λ = 0 with the property that any non-Abelian factor of the gauge group is unbroken in the vacuum. If, on the other hand, the background has k î λ = 0, the gauge group is spontaneously broken and (15) can only be fulfilled by tuning some of the t i 's appropriately. Contracting (15) with ∂ i X Σ and using (7) yieldsX
ΛΘλ Λ (kλ) Σ Γ X Γ = 0 ,(16)
which, upon further multiplication withΘρ Σ and use of (6), results in
iX Λ (Θλ Λ fρ λσΘσ Γ )X Γ = 0 .(17)
This gives a number of real quadratic equations for X Λ , which fix n r = rk(T (t,t)) real degrees of freedom at some point t i , where we defined the n Kv × (4n v + 4)-matrix
Tρ Λ (t,t) = −Θλ Λ fρ λσΘσ Γ Im(X Γ (t)),Θλ Λ fρ λσΘσ Γ Re(X Γ (t)) ,Λ = 1, . . . , 4n v + 4 . (18)
As a consequence n r gauge bosons become massive by "eating" n r real scalar degrees of freedom leaving n r massive short BPS vector multiplets. 5
Before we analyse (14) let us turn to the third condition in (13). Since the vielbein on the quaternionic-Kähler manifold is invertible we infer from (10) that N αA = 0 implies
X Λ Θ λ Λ k u λ = 0 ,(19)
which is similar to (15) but now couples the vector-and hypermultiplet sector. Furthermore, in contrast to (15), equations (19) are holomorphic conditions on the t i . As before there is the trivial solution k u λ = 0 but (19) can also be satisfied by tuning further vector scalars t i appropriately. More precisely, from the Killing vectors k λ that are non-zero at the vacuum locus n c = rk(Θ λ Λ k u λ ) holomorphic conditions arise for the vector multiplet scalars t i which in turn imply that there are n c further massive gauge boson. 6 As we shall see shortly, these massive gauge bosons reside in long non-BPS vector multiplets. Note that the combined conditions following from (15) and (19) have to be compatible and solvable by tuning at most n v complex scalars. Now let us turn to (14) which can be nicely combined with the first equation in (13). Noting that the matrix (X I , ∇ i X I ) is invertible in special geometry we can rewrite the two conditions together as [13]
(Θ λ I − F IJ Θ Jλ )P x λ = −e −K v /2 (∂ I K v )μ a x ,(20)
where a x is an arbitrary real vector on S 2 andμ is related to µ by a phase. From the definition of the Kähler potential (2) we have X I ∂ I K v = 1 and (∂ i X I )∂ I K v = 0, which means that the right-hand side in (20) gives only a contribution to the gauging of the graviphoton X I Im F IJ A J µ . 7 The non-vanishing prepotential of this gauging therefore determines the cosmological constant, while the prepotentials of all other gaugings should vanish in an N = 2 vacuum. We can easily solve (20) for the prepotentials. Since Im F IJ is required by special geometry to be invertible, (20) is equivalent to
Θ λ Λ P x λ = − 1 2 e K v /2 Ω ΛΣ Im(μX Σ ) a x ,(21)
where we used (2) and Θ λ Λ = (Θ λ I , −Θ Jλ ). In general (21) corresponds to 3n c real conditions for the hypermultiplet scalars which in turn become massive. As we observed above n c gauge bosons also become massive by each eating the forth real scalar field of a hypermultiplet. We thus see that the Higgs mechanism leads to a long massive vector multiplet which contains altogether five massive scalars -three from hypermultiplets and two from vector multiplets. For N = 2 Minkoswki vacua those scalar fields which do not participate in the Higgs mechanism are flat directions of the vacuum and thus define its N = 2 moduli space. For N = 2 AdS vacua both (19) and (21) generate further scalar masses so that the actual moduli space can be much smaller. Note that we need n h ≥ n c in order to have an N = 2 vacuum.
Let us now consider the special case of isometries k λ which do not induce an SU(2) R-symmetry rotation on the fermions, i.e. isometries of M h whose Lie derivative on the SU(2) connection vanishes L k ω x = 0 .
For such isometries the Killing prepotentials are given in terms of the SU(2) connection by [17]
P x = ω x (k) .(23)
Inserted into S AB the hyperino condition (19) implies
S AB ∼ X Λ Θ λ Λ P x λ (σ x ) AB = ω x (X Λ Θ λ Λ k λ )(σ x ) AB = 0 .(24)
From Eq. (13) we then infer that the cosmological constant must vanish and all N = 2 vacua in such theories are necessarily Minkowski. It can be easily checked that the isometries in the fibre of quaternionic-Kähler manifolds which are in the image of the c-map, and which we discuss in more detail in Section 5, have this property [13,17]. Note, however, that there are also examples where (22) is not fulfilled [1,19].
Before we proceed, let us address the issue of the SU(2)-covariance of our result. Both (22) and (23) do not transform covariantly under local SU(2) rotations and therefore one might worry that they only hold for a particular choice of coordinates. 8 Indeed, the Killing prepotentials can be written more generally as [18]
P x λ = ω x (k λ ) + W x λ ,(25)
where W x λ is the so-called compensator field that makes the right-hand side transform non-trivially as an SU(2) vector and that is defined via
L k λ K x = ǫ xyz K y W z λ .(26)
As a consequence of (22) the left-hand side of this equation vanishes and the compensator field vanishes in this particular SU(2) frame. However, in the N = 2 locus (19) implies that
X Λ Θ λ Λ P x λ N =2 = X Λ Θ λ Λ W x λ N =2 ,(27)
where each side transforms as an SU (2) vector. This means that
X Λ Θ λ Λ L k λ K x N =2 = 0 ,(28)
is an SU(2)-covariant condition (see also appendix A.3 of [13] for similar manipulations), which follows from the non-covariant equation (22). Furthermore, the condition (28) implies that S AB is vanishing and that the N = 2 vacuum must be Minkowski.
N = 2 supergravities in d = 5, 6
The analysis of the previous section can be repeated in five and six dimensions for supergravities with the same number (eight) of supercharges. The hypermultiplet sector is unchanged while the vector multiplets have only one real scalar in d = 5 or none at all in d = 6. As a result the matrices appearing in fermionic supersymmetry variations (9) change.
Five-dimensional N = 2 gauged supergravity has been discussed for example in [20][21][22] and references therein. Here we will restrict to the case with no tensor multiplets and comment on the more general case later. The N = 2 vacua again arise as solutions of (13), with the major difference relative to d = 4 being that there are no magnetically charged fields, as there are no magnetic gauge vectors. In addition, the scalar matrices previously defined in (10) now read
S AB =h I Θ λ I P x λ (σ x ) AB , W iAB = − √ 3 √ 2 g ij ∂ j h I Θ λ I P x λ (σ x ) AB , N A α = √ 6 4 h I Θ λ I U A αu k u λ ,(29)
and depend in the vector multiplets only on a set of real coordinates h I (instead of the complex coordinates X I ) that obey the cubic condition
d IJK h I h J h K = 1 .(30)
Analogously to the derivation of (19), the hyperino condition N A α = 0 leads to
h I Θ λ I k u λ = 0 .(31)
These are n r = rk(Θ λ I k u λ ) real equations on the h I which fix the scalars of n r vector multiplets. Furthermore, (h I , ∂ j h I ) is again an invertible matrix so that, similarly to (21), we can combine the gaugino and gravitino equation into
Θ λ I P x λ = d IJK h J h K µa x ,(32)
where µ is real and d IJK h J h K replaces ∂ I K in (21) by virtue of the cubic condition (30). This fixes 3n r hypermultiplet scalars, consistent with the Higgs mechanism and we end up with n r long massive vector multiplets. Note that, analogously to four dimensions, a supersymmetric AdS vacuum exists only if the Lie derivative on the SU(2) connection is non-zero for at least one of the gauged Killing vectors. The story gets more involved in the presence of tensor multiplets [20]. However, let us stress that the cosmological constant is only affected by gaugings in the hypermultiplets, and therefore our discussion concerning the existence of supersymmetric AdS vacua still applies.
We now turn to gauged supergravities with eight supercharges in d = 6 which are discussed, for example, in [23,24]. In this case there are no scalars in the vector multiplet sector. Moreover, due to chirality of the supergravity no scalar contributions arise in the hyperino or gravitino variation, in contrast to (9). From the gaugino variation one finds similarly to (19)
the condition Θ λ i P x λ = 0 ,(33)
which again are 3 rk(Θ) real conditions on the hypermultiplet scalars, as required by the Higgs mechanism. Furthermore, supersymmetric AdS is not a solution, as gaugings do not give a contribution to the cosmological constant.
Gauging the isometries of the c-map
A large class of known quaternionic-Kähler manifolds are those that lie in the image of the c-map [5,6]. These manifolds are fibrations of a graded Heisenberg algebra over a special-Kähler manifold and they are of interest as the fibre admits a large number of isometries. Furthermore, they appear in the low-energy effective action of type II and M-theory compactifications on six-dimensional SU(3) × SU(3) structure manifolds where fluxes, torsion and non-geometric fluxes precisely gauge these isometries (see e.g. [17,[25][26][27][28][29][30]). Therefore the vacua in these gauged supergravities coincide with the vacua for SU(3) × SU(3)-structure compactifications of type II and M-theory to four and five dimensions, respectively.
Let us denote the (n h − 1) complex coordinates of the special-Kähler base space by z a , the analogue of the holomorphic symplectic vector X Λ by ZΛ = (Z A , G A ) and the corresponding Kähler potential by K h . The c-map adds an additional (2n h + 2) real fibre coordinates (φ,φ, ξΛ) where ξΛ = (ξ A ,ξ A ) is a 2n h -dimensional symplectic vector. 9 The isometries of the fibre are generated by the Killing vectors 10
kφ = −2 ∂ ∂φ , kΛ = ∂ ∂ξΛ + ΩΛΣξΣ ∂ ∂φ ,(34)
which form a graded Heisenberg algebra with the only non-trivial commutator being
[kΛ, kΣ] = ΩΛΣkφ .(35)
Since these Killing vectors are everywhere linearly independent, eq. (19) simplifies to
X Λ ΘΛ Λ = 0 , X Λ Θφ Λ = 0 .(36)
This gives n c = rk(Θ) holomorphic conditions on M v , giving a mass to n c vector multiplets in the Higgs mechanism. Furthermore, Eq. 36 defines the N = 2 vector moduli space of the vacuum.
Let us continue with the constraints in the hypermultiplet sector. The isometries generated by (34) fulfil (22) and therefore the N = 2 vacuum is necessarily Minkowski. Inserting (23) into (21) we arrive at
ΘΛ Λ ω x (kΛ) = 0 , Θφ Λ ω x (kφ) = 0 .(37)
The explicit form of the SU(2) connection is given by [6,13]
ω 1 − i ω 2 = 2e K h /2+φ Z A (dξ A − F AB dξ B ) , ω 3 = 1 2 e 2φ (dφ +ξ A dξ A − ξ A dξ A ) − i e K h Z A (Im G AB )dZ B −Z A (Im G AB )dZ B .(38)
Inserted into (37) yields
ΘΛ Λ ΩΛΣZΣ = 0 ,(39)ΘΛ Λ ΩΛΣξΣ = Θ Λφ .(40)
The first equation is completely analogous to (36) and gives n c holomorphic conditions on the special-Kähler base of M h . The second equation leads to n c real conditions on the fibre of M h . The other n c fibre scalars are eaten by the gauge vectors so that altogether there are n c long massive vector multiplets leaving n v − n c vector and n h − n c hypermultiplets unfixed and massless.
Note that (36) and (39) are holomorphic equations of the special-Kähler coordinates and (40) gives real, linear equations for the fibre. Therefore they are generically solvable but it is not automatic that the solution lies inside the Kähler cones for both the X I and Z A (cf. (3)). We will see this feature more explicitly in the next section when we discuss some examples. 6 N = 2 and N = 1 vacua in the same gauged supergravity In Ref. [13] the issue of spontaneous N = 2 → N = 1 supersymmetry breaking was considered and the possible N = 1 vacua of N = 2 supergravities were classified. It is of interest to determine under which conditions a given gauged supergravity can have simultaneously N = 2 and N = 1 vacua in its field space. 11 Supersymmetry then implies that both vacua are completely stable [32,33]. In the following we derive these conditions and give two explicit examples. As we will see, they are separated in scalar field space and can lie in the same or in different chambers of the Kähler cone.
We will concentrate in the following on supergravities that are in the image of the c-map. For this class the N = 1 Minkowski solutions of [13] can be stated in terms of the embedding tensor as
ΘΛ Λ = Re C Λ DΛ , Θφ Λ = Re C ΛD ,(41)
where the solution is parametrised by two complex lightlike vectors C Λ and DΛ satisfyinḡ
C Λ Ω ΛΣ C Σ = 0 ,DΛΩΛΣDΣ = 0 ,(42)
and
C J F JI (t N =1 ) = C I , D B G BA (z N =1 ) = D A , DΛ ΩΛΣ ξΣ N =1 =D .(43)
D is a constant and the last equation fixes two of the scalars ξΣ. The first two equations in (43) generically fix all scalars t i and z a , but for special theories there can be a moduli space spanned by t N =1 and z N =1 , respectively [15]. The structure of the embedding tensor given in (41) Clearly, the embedding tensor in (41) has just rank two, so that generically there should also exist an N = 2 vacuum. Inserting the N = 1 solutions (41) into (36), (39) and (40) we find the N = 2 condition to be
X Λ N =2 Re C Λ DΛ = 0 , X Λ N =2 Re C ΛD = 0 , Re C Λ DΛ ΩΛΣ ZΣ N =2 = 0 , Re C Λ DΛ ΩΛΣ ξΣ N =2 = Re C ΛD ,(44)
where the subscript N = 2 indicates that we evaluate the quantity in the N = 2 vacuum. Using the fact that (43) holds at some point in scalar field space and that Im F IJ and Im G AB are invertible, it follows that neither of the complex vectors C Λ and DΛ are proportional to a real vector. Therefore, the most general solution of (44) is
X Λ N =2C Λ = X Λ N =2 C Λ = 0 , DΛ ΩΛΣ ZΣ N =2 =DΛ ΩΛΣ ZΣ N =2 = 0 , DΛ ΩΛΣ ξΣ N =2 =D .(45)
We see that the condition on the fibre coordinates ξΣ is the same for N = 1 and N = 2 vacua while the conditions on the scalars t i and z a are less restrictive for N = 2 vacua. Generically, two complex t i and two complex z a are fixed by (45). Therefore, gauged supergravities which admit an N = 1 vacuum could easily also have an N = 2 vacuum. However, it is not obvious that both vacua lie within the same Kähler cone where (3) holds.
Before we discuss examples where both types of vacua are realised, let us discuss their positions in field space. On the one hand one expects that different vacua should not intersect in field space. On the other hand one easily imagines a point in field space which could fulfil both the N = 2 and N = 1 conditions (45) and (43) simultaneously. However, the Kähler cone condition (3) ensures that N = 1 and N = 2 vacua are always separated in field space. To see this we combine (43) and (45) to arrive at
X I (Im F ) IJ C J = 0 ,(46)while (43) impliesC I (Im F ) IJ C J = 0 .(47)
Eq. (47) states that C I is lightlike while (46) means that C I and X I are orthogonal to each other. In the Kähler cone defined by (3), X I is timelike, contradicting one of these two statements. Therefore, both conditions cannot be fulfilled simultaneously as long as (3) holds. Hence, N = 1 and N = 2 vacua can only coincide outside the physical region of the Kähler cone. Of course, the same reasoning also holds for the special-Kähler base space in the hypermultiplet sector.
We shall now consider the STU model as a first example, where the scalar manifolds are given by
M v = Sl(2, R) SO(2) 3 , M h = SO(4, 4) SO(4) 2 .(48)
This means that both the special-Kähler manifold M v for the vector multiplets as well as the special-Kähler base underlying the quaternionic-Kähler manifold M h are described by the holomorphic prepotential
F = X S X T X U X 0 = ST U ,(49)
where we have defined the complex coordinates S = X S X 0 , T = X T X 0 , U = X U X 0 and chosen X 0 = 1. Since the equations (43) and (45) are identical for both special-Kähler manifolds, we will only focus on M v in the following. The discussion for M h is completely analogous. The Kähler potential can be computed from (49) and is given by
K = − ln(− i(S − S)(T − T )(Ū − U)) ,(50)
so that the Kähler cone condition (3) reads
Im S Im T Im U > 0 .(51)
This gives various domains where either all imaginary parts are positive or two imaginary parts are negative and the third one is positive. In [15] we already discussed the N = 1 vacuum of this model. In order to find a vacuum inside the Kähler cone, we choose
C S = C T C U C 0 , C T = C S C U C 0 , C U = C S C T C 0 , C 0 = − C S C T C U (C 0 ) 2 ,(52)
with C 0 = 0. Furthermore, condition (42) gives
Im C S C 0 Im C T C 0 Im C U C 0 = 0 .(53)
This means that one of the three imaginary parts, say Im C U C 0 , must vanish. Then the N = 1 solution is at [15]
S N =1 = C S C 0 , T N =1 = C T C 0 ,(54)
with U arbitrary. On the other hand, from (45) we infer that a possible N = 2 vacuum would be located at
S N =2 = C S C 0 , T N =2 =C T C 0 , or at S N =2 =C S C 0 , T N =2 = C T C 0 .(55)
Checking the Kähler cone condition (51) we see that the N = 1 and N = 2 solutions can never be both in the same chamber of the Kähler cone. Therefore, we find either an N = 1 or an N = 2 vacuum inside the Kähler cone, depending on the choice of C I .
Let us now give an example where N = 1 and N = 2 vacua do exist in the same theory and, moreover, in the same domain of the Kähler cone. We consider a supergravity with the field space
M v = Sl(2, R) SO(2) × SO(2, n + 2) SO(2) × SO(n + 2)
, M h = SO(4, n + 4) SO(4) × SO(n + 4)
.
M h is in the image of the c-map where the special Kähler base coincides with M v [5]. Thus the holomorphic prepotential for both spaces is given by F = X S (X T X U + X m X m ) X 0 = ST U + Sy m y m , m = 1, . . . , n ,
where again the first expression is in terms of X I and the second one in terms of holomorphic coordinates with X 0 = 1. As before, we will focus on M v in the following with the discussion for M h being completely analogous. The Kähler potential is given by
K = − ln i(S − S) − ln − (T −T )(U −Ū ) − (y m −ȳ m )(y m −ȳ m ) ,(58)
so that the Kähler cone condition (3) reads Im S(Im T Im U + Im y m Im y m ) > 0 .
In the following we will concentrate on the domain where Im S > 0 , Im T Im U + Im y m Im y m > 0 .
In [15] the condition (43) was discussed in detail for the example (57). The vector C Λ parametrising the embedding tensor was defined to be
C S = C T C U C 0 , C T = S C U , C U = S C T , C m = 2 S C m , C 0 = − S C T C U C 0 , C S = S C 0 ,(61)
with C 0 = 0. The N = 1 vacuum is located at
S = S , (T − C T C 0 )(U − C U C 0 ) + (y m − 2C m C 0 ) y m = 0 .(62)
If Im S > 0, condition (42) gives
Im C T C 0 Im C U C 0 = − C mC m 2|C 0 | 2 .(63)
If we take Im C T C 0 > 0, then one point of the N = 1 vacuum is given by
T = C T C 0 , y m = 0 ,(64)
and therefore an N = 1 vacuum exists. Now let us discuss the N = 2 vacuum. From (45) we obtain two equations that read
(S − S ) (T − C T C 0 )(U − C U C 0 ) + y m (y m − 2C m C 0 ) = 0 , (S − S ) (T − C T C 0 )(Ū − C U C 0 ) +ȳ m (ȳ m − 2C m C 0 ) = 0 .(65)
The first one is easily satisfied by S = S . The second one is then more difficult to solve since (60) demands Im S > 0. Here we only display one point of the N = 2 vacuum to prove that it exists inside the Kähler cone. This point is S = S , U = Re C U C 0 + 3 i Im
C U C 0 , T = Re C T C 0 + 3 i Im C T C 0 , y m = 2 i Im C m C 0 ,(66)
where we set Re C m C 0 = 0. By using (53), one can check that the point (66) solves (65) and therefore gives an N = 2 solution. Furthermore, (66) lies inside the Kähler cone defined by (60). Therefore, we have an N = 1 and an N = 2 vacuum in the same N = 2 gauged supergravity.
defines the gauged supergravity and the conditions (42) and (43) ensure that it has N = 1 vacua. Let us now consider under what conditions these supergravities can also have N = 2 vacua.
For a more comprehensive review see, for example, Ref.[16].3 We neglect the possibility of tensor multiplets, as they can be dualised into hypermultiplets (or vector multiplets, if they are massive).4 Strictly speaking, the definition of the graviphoton is X I Im F IJ A J µ , which can be read off from the gravitino variation and depends on the scalar fields in the vector multiplets.
Note that for k î λ = 0 we have Tρ Λ = 0 and therefore n r = 0 so that the gauge group remains unbroken.6 Note that electric gaugings give rise to linear equations, while magnetic gaugings are non-linear in the standard coordinates on M v .
This explicit expression for the graviphoton is found from its appearance in the gravitino variation.
We thank the referee and S. Vandoren for drawing our attention to this subtlety.
For more details see, for example,[5,6,13].
We neglect the Killing vector in the φ direction, as this isometry is broken in string compactifications by one-loop corrections[31].
We thank Z. Komargodski for a remark which inspired the following analysis.
AcknowledgementsWe would like to thank Zohar Komargodski, Thomas van Riet, Diederik Roest and Bert Vercnocke for useful conversations. This work was partly supported by the German Science Foundation (DFG) under the Collaborative Research Center (SFB) 676 "Particles, Strings and the Early Universe". The work of P.S. is supported by the Swiss National Science Foundation. The work of H.T. is supported by the DSM CEA/Saclay, the ANR grant 08-JCJC-0001-0 and the ERC Starting Independent Researcher Grant 240210 -String-QCD-BH.
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[
"Effect of Correlated Noise on Source Shape Parameters and Weak Lensing Measurements",
"Effect of Correlated Noise on Source Shape Parameters and Weak Lensing Measurements"
]
| [
"Alexandre Refregier ",
"Scott T Brown ",
"\nDepartment of Astrophysical Sciences\nDepartment of Physics, Pupin Hall\nPrinceton University\n08544PrincetonNJ\n",
"\nColumbia University\n10027New YorkNY\n"
]
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"Department of Astrophysical Sciences\nDepartment of Physics, Pupin Hall\nPrinceton University\n08544PrincetonNJ",
"Columbia University\n10027New YorkNY"
]
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| The measurement of shape parameters of sources in astronomical images is usually performed by assuming that the underlying noise is uncorrelated. Spatial noise correlation is however present in practice due to various observational effects and can affect source shape parameters. This effect is particularly important for measurements of weak gravitational lensing, for which the sought image distortions are typically of the order of only 1%. We compute the effect of correlated noise on two-dimensional gaussian fits in full generality. The noise properties are naturally quantified by the noise autocorrelation function (ACF), which is easily measured in practice. We compute the resulting bias on the mean, variance and covariance of the source parameters, and the induced correlation between the shapes of neighboring sources. We show that these biases are of second order in the inverse signal-to-noise ratio of the source, and could thus be overlooked if bright stars are used to monitor systematic distortions. Radio interferometric surveys are particularly prone to this effect because of the long-range pixel correlations produced by the Fourier inversion involved in their image construction. As a concrete application, we consider the search for weak lensing by large-scale structure with the FIRST radio survey. We measure the noise ACF for a FIRST coadded field, and compute the resulting ellipticity correlation function induced by the noise. In comparison with the weak-lensing signal expected in CDM models, the noise correlation effect is important on small angular scales, but is negligible for source separations greater than about 1 ′ . We also discuss how noise correlation can affect weak-lensing studies with optical surveys. 1 | null | [
"https://arxiv.org/pdf/astro-ph/9803279v1.pdf"
]
| 1,072,597 | astro-ph/9803279 | 4396d70ddc5caf52746530c81aaaf9ddd6b96cb0 |
Effect of Correlated Noise on Source Shape Parameters and Weak Lensing Measurements
24 Mar 1998
Alexandre Refregier
Scott T Brown
Department of Astrophysical Sciences
Department of Physics, Pupin Hall
Princeton University
08544PrincetonNJ
Columbia University
10027New YorkNY
Effect of Correlated Noise on Source Shape Parameters and Weak Lensing Measurements
24 Mar 1998arXiv:astro-ph/9803279v1 Submitted to ApJSubject headings: gravitational lensing -methods: data analysisstatistical - techniques: image processinginterferometric
The measurement of shape parameters of sources in astronomical images is usually performed by assuming that the underlying noise is uncorrelated. Spatial noise correlation is however present in practice due to various observational effects and can affect source shape parameters. This effect is particularly important for measurements of weak gravitational lensing, for which the sought image distortions are typically of the order of only 1%. We compute the effect of correlated noise on two-dimensional gaussian fits in full generality. The noise properties are naturally quantified by the noise autocorrelation function (ACF), which is easily measured in practice. We compute the resulting bias on the mean, variance and covariance of the source parameters, and the induced correlation between the shapes of neighboring sources. We show that these biases are of second order in the inverse signal-to-noise ratio of the source, and could thus be overlooked if bright stars are used to monitor systematic distortions. Radio interferometric surveys are particularly prone to this effect because of the long-range pixel correlations produced by the Fourier inversion involved in their image construction. As a concrete application, we consider the search for weak lensing by large-scale structure with the FIRST radio survey. We measure the noise ACF for a FIRST coadded field, and compute the resulting ellipticity correlation function induced by the noise. In comparison with the weak-lensing signal expected in CDM models, the noise correlation effect is important on small angular scales, but is negligible for source separations greater than about 1 ′ . We also discuss how noise correlation can affect weak-lensing studies with optical surveys. 1
Introduction
The measurement of source morphologies in two-dimensional images is a fundamental problem in astronomy. The measurements of shape parameters are usually performed while assuming that the underlying noise is uncorrelated. However, spatial correlation of the noise is always present to some degree, and can significantly affect the derived parameters. In experimental situations, noise correlation can be produced by various effects such as convolution of background light with a beam (or point-spread function), interferometric imaging techniques, CCD readouts, etc.
This effect is particularly important for measurements of weak gravitational lensing. Weak lensing provides a unique opportunity to measure the gravitational potential of massive structures along the line-of-sight (for reviews see Schneider et al. 1992;Narayan & Bartelmann 1996). This technique is now routinely used to map the potential of clusters of galaxies (see Fort & Mellier 1994;Kaiser et al. 1994, for reviews). Detections of the more elusive effect of lensing by large-scale structure have been reported by Villumsen (1995) and Schneider et al. (1997) in small optical fields. The search for a strong detection of the effect on larger angular scales is currently being attempted with present and upcoming wide-field CCDs in the optical band (e.g. Stebbins et al. 1995;Kaiser 1996;Bernardeau et al. 1997), and with the FIRST radio survey Refregier et al. 1998). The main challenge comes from the fact that weak lensing induces image distortions of only about 1%, and thus requires high-precision measurements of source-shape parameters. Correlated noise can produce spurious image distortions and correlations in the shapes of neighboring sources, and therefore must be carefully accounted for in weak lensing studies.
The effect of correlated noise is also particularly important for radio surveys performed with interferometric arrays. In such surveys, images are produced by Fourier inversion of the visibilities from a set of antenna pair. As a result of the incomplete visibility coverage, part of the noise on the image plane contains long range correlations which can extend across an entire field. The effect of these correlations is particularly relevant in the context of recent large radio surveys such as FIRST White et al. 1996) and NVSS (Condon et al. 1997). While the present analysis was motivated by our attempt to measure weak lensing by large-scale structure with the FIRST radio survey, most of the following is general and can be applied to any wavelength and imaging technique. Condon (1997) computed the errors in gaussian fit parameters in the absence of noise correlation. He also gave a semi-quantitative treatment of the noise correlation which relies on simulations. In the present paper, we generalize his approach to include a general analytic treatment of the noise correlation. We show how the spatial correlation of the noise can be naturally characterized by the noise Auto-Correlation Function (ACF). We explicitly derive the effect of noise correlation on source parameters for a general two-dimensional fit and for a two-dimensional gaussian fit. We also consider correlations induced in the parameters of nearby sources. In particular, we focus on the ellipticity correlation functions, which are used in searches for weak lensing by large-scale structure. We apply our formalism to the case of the FIRST radio survey. After measuring the noise ACF for one FIRST coadded field, we compare the noise-induced ellipticity correlations to those expected for weak lensing by large-scale structure. We show that, while they are important on small angular scales, noise correlation effects are negligible for source separations greater than about 1 ′ . This paper is organized as follows. In §2, we define the noise ACF and show how it can be measured in practice. In §3, we consider a general two-dimensional least-square fit. We derive the bias produced by the noise correlation on the mean, variance and covariance of the fit parameters, and the correlation of the parameters for two neighboring sources. In §4, we apply these results to the case of two-dimensional gaussian fits and compute the error matrix. We explicitly derive the ellipticity correlation function induced by the noise correlation. In §5, we consider the concrete case of the FIRST radio survey. We measure the noise ACF for a FIRST field, compute the induced ellipticity correlation function, and compare the latter to that expected for weak lensing. In §6, we discuss the implications of this effect for optical surveys and related searches for weak lensing by large-scale structure. Finally, §7 summarizes our conclusions.
Characterization of the Noise
Noise Auto-Correlation Function
Let us consider a two-dimensional image with intensity I(x), where x = (x 1 , x 2 ) is the pixel position. The total intensity can generally be decomposed into
I(x) = S(x) + N (x),(1)
where S(x) is the intensity of detected sources and N(x) is that of the noise. We take the term "noise" to broadly refer to any intensity which is not associated with detected sources. Depending on the context, it can include not only instrumental noise, but also the background light produced by undetected sources.
We assume that, for each pixel, N (x) has mean and variance given by
N (x) ≡ 0 and (2) N (x) 2 ≡ σ 2 N ,(3)
respectively. The brackets denote an ensemble average which, in practice, is well approximated by a sky average. If necessary, the first equation can be enforced be subtracting a constant term from N (x). The noise is generally correlated from pixel to pixel. This is quantified by the noise ACF which we define as
η(x ab ) ≡ N (x a )N (x b ) ,(4)where x ab ≡ x b − x a .
In the case of uncorrelated noise, the noise ACF becomes
η(x) = σ 2 N , if x = 0 (uncorrelated) 0, otherwise(5)
In the continuous limit, i.e. in the limit of small pixel separation h, this can be written as
η(x) ≃ σ 2 N h 2 δ (2) (x),(uncorrelated)
where δ (2) is the two-dimensional Dirac-delta function.
If the noise is gaussian and homogeneous (i.e. statistically invariant under translations), its statistical properties are completely characterized by η(x). Generally, however, the noise is non-gaussian, and higher correlation functions are required for a full description. Nevertheless, since our analysis only involves the two-point correlation function η(x), it is also valid in the non-gaussian case. In addition, the noise is generally not homogeneous. For instance, the sensitivity and beam shape can vary accross the field-of-view and thus cause the noise properties to depend on position. In this analysis, we assume that this effect is small and that η(x) provides an adequate characterization of the noise, at least in an average sense.
Practical measurement of the noise auto-correlation function
The noise ACF, η(x) is easily measured in practice. To do so, discrete sources are (iteratively) removed above a given threshold. Then, η(x) is measured by computing the average of equation (4) over pairs of pixels separated by η(x). Figure 1 shows a portion of a field in the FIRST radio survey. The contrast was enhanced to make the noise more apparent. Stripe-like patterns in the noise are clearly visible and indicate that the noise is correlated. Figure 2 shows our measurement of the noise ACF for this field. Sources were iteratively removed by excising pixels with intensities above 4σ, and by also removing an area 5.4 ′′ (1 beam width) in radius surrounding each pixel. The measurement of η(x) was then performed by randomly choosing pairs of pixels with separations equal to x. The long range correlations reminiscent of the VLA antenna pattern are apparent. In §5, we discuss the case of the FIRST survey in detail.
It is easy to show that, in the absence of correlation, the standard deviation, σ[η(x)], of the measured ACF is
σ[η(x)] = σ 2 N / N pairs (x),(7)
Fig. 1.-Noise patterns for the coadded field 07210+29486E in the FIRST survey. A 9 ′ × 9 ′ region in the 46.5 ′ × 34.5 ′ of field is shown. The contrast for the noise was enhanced by clipping pixels with intensities higher than 2 mJy beam −1 . The residual rms intensity is σ N ≃ .15 mJy beam −1 , for pixels at least 5.4 ′′ away from pixels with intensities greater than 0.6 mJy beam −1 . Apart from bright sources, "stripes" in the noise are clearly apparent. -Normalized noise auto-correlation function measured for the coadded field 07210+29486E in the FIRST radio survey. For clarity, only the inner 2 ′ ×2 ′ arcmin region is shown. The correlation function was normalized to 1 at x = 0. For this field, η(0) ≡ σ 2 N ≃ 2.08 × 10 −8 mJy 2 beam −2 . For this measurement, the SNR for η(0) is 60.
where N pairs (x) is the number of pixel pairs with separation x. This provides a measure of the signal-to-noise ratio (SNR) of the ACF measurement, namely
SNR[η(x)] ≡ η(x)/σ[η(x)] = η(x) η(0) N pairs (x).(8)
In the case of figure 2, the number of pairs was chosen to be N pairs = 3600 for all x. As a result, the SNR for η(0) is 60. This high value of the SNR was achieved owing to the large number of pixels in the FIRST field.
Noise Correlation from Beam Convolution
As Condon (1997) remarked, a natural example of noise correlation arises when part of the noise is convolved with a beam or Point-Spread Function (PSF). The total noise is then a sum of a convolved component (e.g. background light which gets smoothed by the PSF) and an unconvolved component (e.g. Poisson noise from the detector). For simplicity, we ignore the latter component and suppose that the totality of the noise is convolved as
N (x) = d 2 x ′ B(x − x ′ )N (x ′ ),(9)
whereN (x) is the intrinsic noise which is assumed to be uncorrelated. The convolution beam B(x) is assumed to be normalized as d 2 xB(x) ≡ 1.
As is easy to show using equation (4) and equation (6) forN , the resulting noise ACF is then
η(x) = σ 2 N β(x),(10)
with
β(x ab ) ≡ h 2 d 2 xB(x − x a )B(x − x b ),(11)
where, again, x ab = x b − x a . The function β(x) can be thought of as the beam ACF. Figure 3 shows the dirty beam for a typical FIRST grid pointing. (The dirty beam is the image of a point source placed at the center of the field prior to CLEANing; see discussion in §5). The coadded field of figure 1 is a weighted sum of 13 similar grid pointings, and therefore does not have a well defined dirty beam. It is nevertheless instructive to compare the noise correlation function of figure 2 to this dirty beam. Qualitatively, the resemblance is striking and reveals that most of the noise is effectively convolved with the dirty beam.
General Fit with Correlated Noise
There exist several methods for measuring the shape parameters of sources. The first method consists in fitting a model (usually a gaussian) to the two-dimensional source profile. Another -Typical dirty beam for a grid pointing in the FIRST survey. This particular beam was derived using AIPS for the grid pointing 07000+50218 and was normalized to 1 at θ = 0. This pattern is typical of VLA snapshot images in which limited coverage of the u-v plane is reflected in relatively intense sidelobes along the projected arms of the VLA. possibility is to measure the multipole moments of the source. The latter method is usually implemented using various window functions to ensure proper convergence (e.g. Schneider & Seitz 1995;Kaiser et al. 1995). The first method is more common in radio interferometric imaging (e.g. AIPS software package; see http://www.cv.nrao.edu/aips/), while the second is usually used with optical images (e.g. FOCAS, SExtractor packages; see Jarvis & Tyson 1981;Bertin & Arnouts 1996, respectively) Here, we only consider the effect of correlated noise on model fitting. An extension of our results to the multipole moment method is, however, straightforward. In §3.1, we derive the parameter corrections from the noise correlation for the case of a general two-dimensional χ 2 -fit. We include corrections up to second order in the inverse SNR of the source. In §3.2, we then compute the resulting bias in the mean ( §3.2), variance, and covariance of the parameters ( §3.3). In §3.4, we compute the induced correlation in the parameters of pairs of neighboring sources. Finally, in §3.5, we consider general functions of the parameters. This analysis builds on the results of Condon (1997) to include an analytical treatment of the noise correlation.
Least Square Fit
Let us consider a sector of an image consisting of a single source superimposed on noise. We then consider a fit to the two-dimensional source profile with a function F (x; a), where a = (a 1 , a 2 , ...) is a vector of parameters. In §4, we will consider the case where F (x; a) is a two dimensional gaussian with a consisting of the gaussian normalization, peak location, and size parameters (see Eq.
[51]).
We assume that the fit is "good", i.e., that the source profile S(x) is well described by
S(x) ≃ F (x;â),(12)
whereâ is the parameter vector that would be measured in an ideal measurement without noise. The total image intensity (Eq.
[1]) thus becomes
I(x) = F (x;â) + N (x).(13)
We assume that the fit is performed using the method of least squares, without accounting for the noise correlation. This is usually the case for most fitting routines (e.g. AIPS). In other words, the fit is performed by computing the usual functional
χ 2 (a) = p [I(x p ) − F (x p ; a)] 2 σ 2 N ,(14)
where the sum runs over all the pixels in the image and x p are the position of the pixel centers. The best fit parameters a are found by minimizing χ 2 (a). This minimum occurs when ∂χ 2 ∂a l (a) = 0, that is, for the value of a which satisfies
p [F (x p ;â) − F (x p ; a) + N (x p )] ∂F ∂a l (x p ;â) = 0.(15)
In the noiseless limit (N → 0), an obvious, if expected, solution is a =â. Note that in the most general case, there could exist more solutions if the system of equation is degenerate. We will not consider this complication here and assume that this solution is unique.
In the presence of noise, the solution a will deviate fromâ. In most cases, the integrated source flux is much larger than the integrated flux of the noise. This is quantified by the source signal-to-noise ratio, SNR s which we take to be much larger than 1. We can thus treat the noise N (x p ) as a perturbation in equation (15). For this purpose, we rewrite N (x) as αN (x), where α is a dimensionless parameter of the order of α ∼ SNR −1 s . We then expand the parameter vector in powers of α as a =â + αa (1) + α 2 a (2) + O(α 3 ).
After inserting these expansions in equation (15), Taylor expanding, collecting terms in powers of α, and setting α = 1, we obtain
a i =â i + D il P l + C imn P m P n + E ilkm P m Q lk + O(SNR −3 s ),(16)
where
D ij = (H −1 ) ij ,(17)H ij = p ∂F ∂a i (x p ;â) ∂F ∂a j (x p ;â),(18)B ijk = p ∂F ∂a i (x p ;â) ∂ 2 F ∂a j ∂a k (x p ;â),(19)C imn = − B k,rj + 1 2 B r,kj D ir D km D jn ,(20)E ilkm = D il D km ,(21)P i = p ∂F ∂a i (x p ;â)N (x p ), and(22)Q ij = p ∂ 2 F ∂a i ∂a j (x p ;â)N (x p ).(23)
Unless otherwise specified, the summation convention is assumed throughout this paper. For reasons which will become clear below, the matrix D is called the error matrix (see Condon 1997).
In the oversampling regime, i.e. when the pixel spacing h is small compared to the source size, the sums above can be turned into integrals using the substitution
p ≃ 1 h 2 d 2 x p .(24)
Bias in Fit Parameters
We first focus on the bias induced on the source parameters by the noise correlation. By taking an ensemble average of equation (16), we find the mean of the parameter a i to be
a i =â i + D il P l + C imn P m P n + E ilkm P m Q lk + O(SNR −3 s ).(25)
But from equation (2), we see that P l = 0 so that the first order term vanishes. Using the definition of η(x) (Eq.
[4]) we thus find
a i =â i + C imn V mn (0) + E ilkm U mlk (0) + O(SNR −3 s ),(26)
where
V ij (0) ≡ p q ∂F ∂a i (x p ;â) ∂F ∂a j (x q ;â)η(x pq ), and(27)U ijk (0) ≡ p q ∂F ∂a i (x p ;â) ∂ 2 F ∂a j ∂a k (x q ;â)η(x pq ).(28)
The operand "(0)" was added for future convenience. In the matrices V and U, the noise ACF η(x) is averaged over the pixel pairs weighted by parameter derivatives of the source profile. The noise correlation therefore produces a bias in the fit parameters of the order of SNR −2 s .
Note however, that this bias does not entirely disappear in the event of uncorrelated noise. Inserting equation (5) in the previous equations, we find that, for uncorrelated noise
V ij (0) = σ 2 N H ij (uncorrelated),(29)U ijk (0) = σ 2 N B ijk (uncorrelated).(30)
Thus, the ensemble averaged parameter becomes
a i =â i − 1 2 σ 2 N B lkj D li D kj + O(SNR −3 s ) (uncorrelated).(31)
Therefore, even for uncorrelated noise, the χ 2 -fit parameters are biased to second order in SNR s . This is not surprising: while the χ 2 -fit is always unbiased if F (x; a) is linear in a (see e.g. Lupton 1993), no such guarantee exist if it is non-linear. It is easy to check that a i is indeed unbiased in the linear case. However, most two-dimensional source models, including the gaussian model, are non-linear in their parameters. The correlation in the noise then makes this bias generally more pronounced.
Parameter Variance and Covariance
We can also compute the covariance of the fit parameters
cov[a i , a j ] ≡ (a i − a i )(a j − a j ) .(32)
Inserting equation (16) in this definition, we obtain
cov[a i , a j ] = D il D jk V lk (0) + O(SNR −3 s ).(33)
In particular, the diagonal elements yield the variance (or squared error) of each of the fit parameters taken separately
σ 2 [a i ] = D il D ik V lk (0) + O(SNR −3 s ),(34)
where the index i is not to be summed over on the right hand side.
In the case of uncorrelated noise, the covariance reduces to
cov[a i , a j ] = σ 2 N D ij + O(SNR −3 s ) (uncorrelated),(35)
in agreement with Condon (1997), and thereby justifying the term "error matrix" applied to D.
The variance obviously becomes
σ 2 [a i ] = σ 2 N D ii + O(SNR −3 s ) (uncorrelated),(36)
where, again, the index i is not to be summed over.
Parameter Correlation Function
Let us now consider two distinct sources, S 1 and S 2 , in a single field. We assume that the sources are sufficiently distant from each other that the intensity of one source is negligible at the position of the other. If the noise in the field is uncorrelated, the parameter fit to each of the sources would thus be independent. But if the noise is spatially correlated, the parameters of the two sources will be correlated. This effect is particularly relevant for weak lensing studies in which one searches for pairwise correlations in the ellipticities of sources.
To quantify the effect, we define the parameter correlation function as
w aa ij (x 12 ) = (a 1 i − a 1 i )(a 2 j − a 2 j ) ,(37)
where a r is the fit-parameter vector for source S r , andx 12 is the separation between the (noiseless) source centroids.
Inserting equation (16) into this definition yields
w aa ij (x 12 ) = D 1 il D 2 jk V 12 lk (x 12 ) + O(SNR −3 s ),(38)
where,
D r ij ≡ p ∂F ∂a i (x p ;â r ) ∂F ∂a j (x p ;â r ),(39)
and
V 12 ij (x 12 ) = p q ∂F ∂a i (x p ;â 1 ) ∂F ∂a j (x q ;â 2 )η(x pq ).(40)
The fit-parameter correlation is therefore of order SNR −2 .
In the absence of noise correlation (Eq.
[5]), V 12 ij (x 12 ) becomes
V 12 ij (x 12 ) = σ 2 N p ∂F ∂a i (x p ;â 1 ) ∂F ∂a j (x q ;â 2 ).(41)
As is the case for a gaussian fitting function, the parametric partial derivatives, ∂F ∂a i , are often spatially localized. In this case, the above sum vanishes, if the sources are sufficiently distant from each other. Thus, for |x| much larger than the source sizes, and for well-behaved fitting functions
w aa ij (x) ≃ 0 (uncorrelated).(42)
As expected, the fit parameters of the two sources are then uncorrelated if the noise is itself spatially uncorrelated.
Functions of the Parameters
We can also compute the effect of correlated noise on functions of the parameters. Let us consider a set of such functions, t i (a). In §4.3, we will consider an application were the t i 's are the 2 components of the ellipticity of the source. In the presence of noise, the value of the functions are perturbed as (see Eq.
[16])
t i (a) = t i (â) + ∂t i ∂a j (â)D jk P k + O(SNR −2 s ).(43)
The covariance of the t i 's is thus
cov[t i , t j ] ≡ (t i − t i ) (t j − t j ) = K ik K jl V kl (0) + O(SNR −3 s ).(44)
where we have defined
K ij ≡ ∂t i ∂a k (â)D kj .(45)
In the absence of noise correlation (Eq.
[29]), this covariance matrix becomes
cov[t i , t j ] = σ 2 N ∂t i ∂a k (â) ∂t j ∂a l (â)D kl + O(SNR −3 s ),(uncorrelated)
in accordance with Condon (1997).
For a pair of two distinct sources, S 1 and S 2 , the correlation function for the t i 's is easily found to be
w tt ij (x 12 ) ≡ t i (a 1 ) − t i (a 1 ) t j (a 2 ) − t j (a 2 ) = K 1 ik K 2 jl V 12 kl (x 12 ) + O(SNR −3 s ).(47)
where
K r ij ≡ ∂t i ∂a k (â r )D r kj .(48)
In the absence of noise correlation, and for distinct sources with well-behaved fitting functions (Eq.
[41]), this correlation function is simply
w tt ij (x) ≃ 0 (uncorrelated)(49)
Gaussian Fit
In the previous section, we derived the bias, covariance, and correlation function of parameters for a general fit in the presence of correlated noise. In this section, we apply these results to the case which is of most practical interest, namely that of a two-dimensional gaussian fit.
Gaussian Parametrization
As the fitting function F (x; a), we consider the the following parametrization of a two-dimensional elliptical gaussian
G(x, a) ≡ Ae − 1 2 (x−x a ) T A(x−x a ) ,(50)
where A is the amplitude, x a is the centroid vector, A is a symmetric positive-definite 2 × 2 matrix which defines the shape and orientation of the gaussian, and the superscript T denotes the transpose operation. We choose the parameter vector to be
a ≡ (x a 1 , x a 2 , A, A 11 , A 12 , A 22 ).(51)
Note that this parametrization is different from that of Condon (1997), and was to chosen because it is easier to relate to the parameters used in weak-lensing measurements.
It is convenient to compute the first few multipole moments G (n) of G. With the above definition, we find
G (0) ≡ d 2 xG(x, a) = πA|A| −1 ,(52)G (1) i G (0) ≡ 1 G (0) d 2 xx i G(x, a) = x a i ,(53)G (2) ij G (0) ≡ 1 G (0) d 2 x(x i − x a i )(x j − x a j )G(x, a) = (A −1 ) ij ,(54)
where |A| is the determinant of A. Note that
J ≡ A −1 = |A| −1 A 22 −A 12 −A 12 A 11(55)
is exactly equal to the normalized quadrupole moments of G. It can be diagonalized as
J = R(−α) a 2 0 0 b 2 R(−α) T(56)
where a, b are the (1σ) major and minor axes, and α is the position angle measured counter-clockwise from the positive x-axis. The rotation matrix R is defined as
R(ϕ) ≡ cos ϕ sin ϕ − sin ϕ cos ϕ .(57)
Inverting these relations yields
(a 2 , b 2 ) = 1 2 J 11 + J 22 ± (J 11 − J 22 ) 2 + 4J 2 12 ,(58)tan 2α = 2J 12 J 11 − J 22 .(59)
Source and Noise Matrices
We are now in a position to compute the source and noise matrices in equations (17-23). For this purpose, we first compute the partial derivatives of G. We find,
∂G ∂a = G [(x 1 − x a 1 )A 11 + (x 2 − x a 2 )A 12 ], [(x 2 − x a 2 )A 22 + (x 1 − x a 1 )A 12 ], A −1 , − 1 2 (x 1 − x a 1 ) 2 , −(x 1 − x a 1 )(x 2 − x a 2 ), − 1 2 (x 2 − x a 2 ) 2(60)
In the continuous limit (Eq.
[24]), the components of the matrix H (Eq.
[18]) have a closed form, i.e.,
H ≃ πA 2 h 2 |A| × 1 2 A 11 1 2 A 12 0 0 0 0 1 2 A 12 1 2 A 22 0 0 0 0 0 0 A −2 − 1 4 |A| −1 A −1 A 22 1 2 |A| −1 A −1 A 12 − 1 4 |A| −1 A −1 A 11 0 0 − 1 4 |A| −1 A −1 A 22 3 16 |A| −2 A 2 22 − 3 8 |A| −2 A 22 A 12 1 16 |A| −2 A 2 0 0 1 2 |A| −1 A −1 A 12 − 3 8 |A| −2 A 22 A 12 1 4 |A| −2 A 2 − 3 8 |A| −2 A 11 A 12 0 0 − 1 4 |A| −1 A −1 A 11 1 16 |A| −2 A 2 − 3 8 |A| −2 A 11 A 12 3 16 |A| −2 A 2 11 ,(61)
where A 2 ≡ (A 11 A 22 + 2A 2 12 ). Note that the amplitude and shape parameters mix with each other, but not with the position parameters.
By inverting the above matrix, we find the error matrix D to be
D ≃ 2h 2 |A| πA 2 |A| −1 A 22 −|A| −1 A 12 0 0 0 0 −|A| −1 A 12 |A| −1 A 11 0 0 0 0 0 0 A 2 AA 11 AA 12 AA 22 0 0 AA 11 4A 2 11 4A 11 A 12 4A 2 12 0 0 AA 12 4A 11 A 12 2(A 11 A 22 + A12 2 ) 4A 12 A 22 0 0 AA 22 4A 2 12 4A 22 A 12 4A 2 22 .(62)
The B matrix of equation (19) can be computed in a similar way. The calculation for this 6 3 component matrix is however cumbersome. Since it does not enter in the parameter correlation function on which we will now focus, we will not consider it further.
The noise matrices P (Eq. [22]) and Q (Eq. [23]) can not be computed without knowledge of the noise ACF, η(x). In practice, η(x) has a complicated x-dependence. As a result, P and Q must be computed numerically using the direct measurement of η(x), along with the expressions for the derivatives of G (Eq. [60]). In the next section, we study the effect of the noise correlation on the source ellipticity, a combination of the source parameters which is particularly relevant for weak lensing measurements. (By source ellipticity we mean the ellipticity of the observed source image and not the ellipticity in the source plane, as is common in the lensing nomenclature).
Source Ellipticities
The shear produced by weak gravitational lensing produces distortions in the shapes of background sources. The most direct way to detect this effect is to look for correlations in the ellipticities of sources. Measurements of the ellipticity involve integrals that are more spatially extended than that of the flux and position of a source. Consequently, we expect the ellipticity to be particularly sensitive to the spatial correlation of the noise. Here, we compute the magnitude of this effect on the ellipticities and on the ellipticity correlation function.
A number of ellipticity measures have been used in weak lensing studies (e.g. Schneider 1995;Bonnet & Mellier 1995). Here, we will use the ellipticity measure considered by Miralda-Escudé (1991), Kaiser & Squires (1993) and others, which has the advantage of being easily related to the gaussian parameters of equation (50). In this definition, the ellipticity is defined as a two component "vector" ǫ whose components are given by
ǫ 1 = J 11 − J 22 J 11 + J 22 , ǫ 2 = 2J 12 J 11 + J 22 .(63)
This can be rewritten in terms of the major axis, minor axis, and position angle of the source as ǫ = a 2 −b 2 a 2 +b 2 (cos 2α, sin 2α). The component ǫ 1 , sometimes written as ǫ + , describes stretches along the x and y-axes, while ǫ 2 , sometimes written as ǫ × , describes stretches at ±45 • with respect to these axes. In a coordinate system (x ′ 1 , x ′ 2 ) rotated by ϕ counter-clockwise from the positive x 1 -axis, the components of ǫ are
ǫ ′ i = R ij (2ϕ)ǫ j ,(64)
where R is the rotation matrix defined in equation (57). This shows that ǫ is not a true vector since it has a period of π in ϕ. (In fact, ǫ can be written as a symmetric traceless 2 × 2 tensor).
The ellipticity can be rewritten in terms of the gaussian fit parameters (see Eq.
[55]) as
ǫ 1 = A 22 − A 11 A 11 + A 22 , ǫ 2 = −2A 12 A 11 + A 22 .(65)
The partial derivative matrix of ǫ is thus
∂ǫ ∂a = 2(A 11 + A 22 ) −2 0 0 0 −A 22 0 A 11 0 0 0 A 12 −(A 11 + A 22 ) A 12 .(66)
Note that the position parameters, x a 1 and x a 2 , have decoupled from the ellipticity matrices and can thus be dropped. There are kept here for consistency. It is then straightforward to compute the K matrix (Eq. [48]). We find
K ≃ − 16h 2 |A| 3 2 πA 2 (A 11 + A 22 ) 2 0 0 0 A 11 0 −A 22 0 0 0 A 12 1 2 (A 11 + A 22 ) A 12 .(67)
With this result and a measurement of the noise ACF η(x), the covariance cov[ǫ i , ǫ j ] and correlation function w ǫǫ ij (x) of the ellipticity are readily calculated using equations (44) and (47).
We now focus on w ǫǫ ij (x) which is of interest for measurements of weak lensing by large-scale structure. To be explicit, this correlation function is defined as
w ǫǫ ij (x 12 ) ≡ ǫ 1 i ǫ 2 j(68)
In practice, it is usually sufficient to consider the correlation of two sources which are identical and circular. In this simplified case, the unperturbed parameters for source S 1 and S 2 arê a 1 = (A,x 1 1 ,x 1 2 , a −2 , 0, a −2 ), andâ 2 = (A,x 2 1 ,x 2 2 , a −2 , 0, a −2 ), where a is the (1σ) radius of the sources, A is their amplitude, andx r i gives their (unperturbed) centroid position. In this case, the ellipticity correlation function reduces to
w ǫǫ ij (x 12 ) ≃ 4h 4 π 2 a 8 A 2 1 h 4 d 2 x 1 d 2 x 2 X ij (x 1 −x 1 , x 2 −x 2 )e − (x 1 −x 1 ) 2 +(x 2 −x 2 ) 2 2a 2 η(x 12 ) + O(SNR −3 s ),(69)
where
X(ξ 1 , ξ 2 ) ≡ [(ξ 1 1 ) 2 − (ξ 1 2 ) 2 ][(ξ 2 1 ) 2 − (ξ 2 2 ) 2 ] 2ξ 2 1 ξ 2 2 [(ξ 1 1 ) 2 − (ξ 1 2 ) 2 ] 2ξ 1 1 ξ 1 2 [(ξ 2 1 ) 2 − (ξ 2 2 ) 2 ] 4ξ 1 1 ξ 1 2 ξ 2 1 ξ 2 2 .(70)
It is also convenient to consider the ellipticities measured with respect to axes parallel and perpendicular to x 12 , the vector connecting two sources ). Let us write this vector in polar coordinates as x 12 = (θ 12 , ϕ 12 ), where θ 12 is the norm of the vector and ϕ 12 the angle it substands with the x 1 -axis. From equation (64), the components of the ellipticities rotated into this coordinate system are ǫ r i ≡ R ij (2ϕ 12 )ǫ j . The correlation function of the rotated ellipticities is then w rr ij (θ 12 , ϕ 12 ) ≡ ǫ r,1 i ǫ r,2 j = R ik (2ϕ 12 )R jl (2ϕ 12 )w ǫǫ kl (θ 12 , ϕ 12 ).
(71)
Following the notation of Kaiser (1992), we identify w rr 11 (x) ≡ C 1 (x), and w rr 22 (x) ≡ C 2 (x). We can also define a third independent correlation function w rr 12 (x) ≡ w rr 21 (x) ≡ C 3 (x), which should vanish if the noise is invariant under parity conservation.
These correlation functions can then be averaged over ϕ 12 by defininḡ
w rr ij (θ) ≡ 1 π π 0 dϕw rr ij (θ, ϕ).(72)
In the above notation, we simply write the azimuthally averaged correlation functions as w rr 11 (θ) ≡ C 1 (θ),w rr 22 (θ) ≡ C 2 (θ) andw rr 12 (θ) ≡w rr 21 (θ) ≡ C 3 (θ). In the context of weak lensing studies, C 1 (θ) and C 2 (θ) can be directly related to the power spectrum of density perturbations along the line of sight (see e.g. Kaiser 1992). Figure 4 shows the rotated correlation functions for the FIRST coadded field of figures 1 and 2. A source size of a = 2.45 ′′ and signal-to-noise ratio of SNR s = 5 were chosen. Figures 5, 6, and 7 show the corresponding azimuthally averaged correlation functions. In the next section, we will describe these results in detail and discuss their implications for weak lensing measurements.
Application to the FIRST Radio Survey
In this section, we apply the general formalism developed above to the specific case of the FIRST radio survey White et al. 1996). The current version of the FIRST survey contains about 4 × 10 5 sources, and covers about 4.3 × 10 3 square degrees. The survey fields were observed at 1.4 GHz with the VLA in the B configuration. The 5σ detection limit of the survey is about 0.75 mJy. The pixel size for the survey maps is 1.8 ′′ , while the restoring beam is 5.4 ′′ (FWHM). The restoring beam is thus reasonably well sampled. This allows us to take the continuous limit (Eq. [24]) in our calculation. In addition, the relatively high angular resolution allows an accurate measurement of source morphologies. We are in the process of attempting to detect the effect of weak lensing by large-scale structure with this unique database Refregier et al. 1998).
Because of the observing time limitations for a survey of this magnitude, the observations had to be performed in the snapshot mode with integration times of 5 seconds. The interferometric Fig. 4.-Ellipticity correlation functions for the coadded field 07210+29486E. Each correlation function C 1 , C 2 and C 3 is plotted as a function of the separation vector x in panels a, b, and c, respectively. For each source pair member, source signal-to-noise ratio was set to SNR s = 5, while the 1σ source size was set to a = 2.45 ′′ . This corresponds to FWHM convolved and deconvolved diameters of 5.76 ′′ and 2 ′′ , respectively. The clean beam for the FIRST survey has a FWHM diameter of 5.4 ′′ . data (or "UV" data) for each snapshot pointing was then Fourier transformed to construct a "dirty" image. The resulting pointing images were CLEANed using algorithms adapted from the standard AIPS deconvolution package. The final survey fields were produced by a weighted sum of the grid pointing images. At a given point, approximately 12 pointings contribute to the coadded field. Sources were then detected and their shape parameters measured, using a two-dimensional gaussian fit.
Even though the grid pointing strategy was designed to optimize the signal-to-noise ratio of the coadded images, the final maps still suffer from the limited UV-coverage of the snapshot pointings. As a result, typical coadded maps contain small but noticeable "stripes" in the noise (see figure 1). In other words, the noise is correlated. As we discussed above, this affects the source parameters derived by the fitting routine. In particular, the effect of correlated noise is a potential systematic effect in a search for correlations in the source ellipticities produced by weak lensing.
Noise Auto-Correlation Function
As a specific case, we considered the coadded field 07210+29486E in the FIRST survey. This field covers a solid angle of 46.5 ′ × 34.5 ′ , with a pixel size of 1.8 ′′ . Figure 1 shows a portion of this field with the contrast enhanced to make the noise more apparent. Bright radio sources are apparent in the right hand corner. In addition, the stripe-like nature of the noise is visible. For this analysis, pixels with intensities above .6 mJy beam −1 (corresponding to about 4σ) were excised from the field. The excised regions were then padded by 5.4 ′′ to avoid contamination of the noise by the source side wings. The residual noise has a standard deviation of σ N ≃ .15 mJy beam −1 . (Intensities are quoted in mJy beam −1 , were the beam refers to the restoring beam with a FWHM diameter of 5.4 ′′ .) The median noise standard deviation for the FIRST survey is 0.14 mJy beam −1 , with 95% of the FIRST area having a noise standard deviation less than 0.17 mJy beam −1 (White et al. 1996). This makes the coadded field 07210+29486E somewhat noisy, yet representative of the FIRST survey. Figure 2 shows our measurement of the noise ACF, η(x), for this field. For this measurement, the SNR for η(0) is 60 (see Eq.
[8]). For convenience, η(x) was normalized to 1 at x = 0 in this figure. In addition to a prominent central peak, the ACF shows marked radial structures. These structures are reminiscent of the VLA antenna pattern and characterize the "stripes" which are visible in the noise.
Each grid pointing is characterized by a dirty beam. (It is the Fourier transform of the UV coverage and is equal to the dirty image produced by a point source at the phase tracking center). In general, the dirty beam shape varies from one grid pointing to another due to differing geometries, flagged antennas, etc. As a result, coadded fields, which are composed of several grid pointings, do not have a well defined dirty beam. It is nevertheless instructive to qualitatively compare our measurement of the noise ACF with a typical grid pointing dirty beam. Figure 3 shows such a dirty beam for the grid pointing 07000+50218. The qualitative resemblance with η(x) is striking. This shows that a large fraction of the noise is effectively convolved with the dirty beam.
Ellipticity Correlation Functions
Of practical interest for weak lensing measurements, is the effect of the correlated noise on the ellipticity correlation functions. Figure 4 shows the rotated correlation functions C 1 (x), C 2 (x) and C 3 (x) (Eq [71]) for the coadded field of figure 1. A source size of a = 2.45 ′′ was chosen. This corresponds to a FWHM diameter of 5.76 ′′ , that is to a deconvolved FWHM diameter of 2 ′′ , for a restoring beam of 5.4 ′′ . The source signal-to-noise ratio was set to SNR s = 5. These parameters correspond to both the detection and resolution limit of the FIRST survey and can thus be considered as "worse-case" parameters for a weak lensing search.
The symmetric patterns of η(x) (figure 2) are also clearly apparent on figure 4. However, the patterns are different for each of the correlation functions. The first correlation function, C 1 (x) exhibits a pronounced star-like pattern with radial elongation on each arm. On the other hand, C 2 (x) is mostly characterized by a negative correlation close to the center. Finally, C 3 (x) exhibits star-like patterns but of much smaller amplitude than that of C 1 (x) and C 3 (x). The last fact is expected for a non-parity violating noise pattern. Figure 5 shows the azimuthally averaged correlation functions, C 1 (θ), C 2 (θ) and C 3 (θ) as a function of θ. These are simply the radial profiles of the rotated correlation functions of figure 4. They were derived by Monte-Carlo averaging over 4 × 10 5 separation vectors x picked randomly within a radius of 50 ′ . The thick lines correspond to a source size of a = 2.45 ′′ , while the thin lines corresponds to a = 3.40 ′′ . In both cases, the source signal-to-noise ratio was set to SNR s = 5, as before. For θ ∼ < 0.4 ′ , C 1 (θ) is positive while C 2 (θ) is negative and drops off faster. Note that both C 1 (θ) and C 2 (θ) are significantly non-zero well beyond the source radius a. This is of course due to the long-range features in η(x) (figure 2). At all angles, C 3 (θ) remains smaller and oscillates around 0. As we remarked above, this is expected for non-parity violating noise.
A comparison between the thick and thin lines in figure 5 shows how the ellipticity correlation functions depend on the source size. For larger source sizes, C 1 (θ) and C 2 (θ) keep the same qualitative behavior but have a lower maximum amplitude and drop off more slowly. This is expected since larger source sizes effectively smooth out the sharp features in the noise ACF (figure 2; see Eq.
[69]). Figure 6 shows the long range behavior of the ellipticity correlation functions. For angles larger than θ = 10 ′ , a mosaic of 4 fields contiguous with 07210+29486E was used to compute the noise ACF η(x). The error bars shown for C 1 (θ) are the error in the mean derived from the Monte-Carlo sampling described above. Since the separation plane was undersampled, these error -Azimuthally-averaged ellipticity correlation functions for the same coadded field. Each correlation functions, C 1 , C 2 , and C 3 , are plotted as a function of the separation radius θ. A source signal-to-noise ratio of SNR s = 5 was assumed. The correlation functions scale as C i ∝ SNR −2 s . The correlation functions are plotted for a source size of a = 2.45 ′′ (thick lines) and of a = 3.40 ′′ (thin lines).
bars provide a good estimate of the uncertainty in the correlation functions. As is apparent on this figure, all three correlation functions are consistent with 0 for θ ∼ > 1 ′ .
Consequences for Weak Lensing Searches
As was demonstrated above, correlated noise affects the ellipticity correlation functions in the FIRST survey. We now compare the amplitude of this systematic effect with the signal expected for weak lensing by large-scale structure. Figure 7 shows the noise-induced correlation functions, C 1 (θ) and C 2 (θ) on a log-log scale. The absolute value of the correlation functions are plotted, with positive and negative values indicated by squares and stars, respectively. For comparison, the (absolute value of the) correlation functions expected for weak lensing in CDM models are also displayed. The four models correspond to the COBE-normalized CDM models in table 2 and figure 4 in Kamionkowski et al. (1998). Models 1-3 correspond to a standard, tilted, and lambda CDM model, respectively. Model 4 corresponds to a standard CDM model with a smaller Hubble constant than that of Model 1. In all models, only linear evolution of density perturbations were considered.
As we noted above, noise correlation produces significant biases in C 1 (θ) and C 2 (θ) for θ ∼ < 1 ′ . In this angular range, the noise-induced correlation functions exceed those expected for weak lensing for all models. (The inclusion of nonlinear evolution of density perturbations increases the estimates of the weak lensing signal for θ ∼ < 10 ′ by only a factor of a few and therefore does not significantly alter this conclusion; see Jain & Seljak 1997.) For angles θ ∼ > 1 ′ , we showed above that the noise-induced correlation functions are consistent with zero, within the Monte-Carlo uncertainties. This can be seen by the alternation of squares and stars in figure 7. The noise-induced lines in this figure should thus be interpreted as 1σ upper limits in this angular range. (A larger number of Monte-Carlo simulations could of course reduce this upper limit even further but is computationally cumbersome.) We notice that, for 1 ′ ∼ < θ ∼ < 50 ′ , these upper limits are almost one order of magnitude smaller than model 3, the most pessimistic CDM model. As we showed above, the noise-induced correlation functions depend only moderately on the source size (see figure 5). Most of the sources used in our weak lensing searches have sizes close to the resolution threshold (a ≃ 2.45 ′′ , or, equivalently, FWHM deconvolved diameter of 2 ′′ ), and hardly ever exceed a ∼ 5 ′′ . As a result, the noise-induced correlation functions in such a survey will be close to that shown on figure 7.
The Monte-Carlo realizations also allow us to compute the rms standard deviation σ[C i (θ)] of the ellipticity correlation functions. For 10 ′ < θ < 50 ′ , we find σ[C 1 (θ)] ≃ σ[C 2 (θ)] ≃ 1.9 × 10 −3 . This is almost two orders of magnitude below the standard deviation produced by the intrinsic ellipticities of radio sources (σ[C 1 (θ)] ≃ σ[C 2 (θ)] ≈ 0.4 2 = 0.16; see Refregier et al. 1998). Thus, Kamionkowski et al. (1998). The top and bottom curves for each model correspond to C 1 and C 2 , respectively. The broken lines correspond to the ellipticity correlation functions induced by the noise correlation, for sources with a = 2.45 ′′ and SNR s = 5. In all cases, the absolute value of the correlation function is shown. To preserve a sense of the noisy behavior of the noise-induced correlation functions, positive and negative values are indicated by squares and stars, respectively. statistical fluctuations in the noise-induced correlation functions are negligible for θ ∼ > 1 ′ .
The estimation of the noise-induced correlation functions for θ ∼ > 50 ′ requires a large mosaic of coadded fields and is therefore computationally cumbersome. However, we expect the correlation functions to continue to be consistent with 0 at large angles. In addition, a search for weak lensing will comprise an average over the whole survey, as opposed to an average over a few coadded fields in the present study. The dirty beam, and therefore η(x), varies from one coadded field to the next. As a result, we expect any long-range features in the noise ACF to average out, leading to even smaller ellipticity correlations.
We conclude that, in the FIRST survey, noise-induced ellipticity correlations dominate over the expected weak lensing signal for θ ∼ < 0.5 ′ , but are negligible for θ ∼ > 1 ′ .
Correlated Noise in Optical Images
While the effect is likely to be less severe than for intereferometric radio surveys, noise correlation may also affect source shapes in optical images. Most weak lensing studies are performed in the optical band and are thus potentially sensitive to this effect. Noise correlation in optical images can be produced by several effects. First, noise correlation can arise from the convolution of background light with the PSF. Because field distortions, diffraction spikes, etc, can produce anisotropy of the PSF, this can induce ellipticity correlations. In addition, CCD readouts and charge-transfer efficiency can potentially produce linear features in the noise ACF. Finally, observing and image processing techniques such as drift-scanning, shifting-and-adding, and "drizzling" (Fruchter & Hook 1997) can also produce features in the ACF.
Measurements of the noise ACF in CCD images have recently been performed by Van Waerbeke et al. (1997), in the context of a novel technique to measure weak lensing shear with the noise ACF. In one of their fields, they find, at the center of the noise ACF, evidence for a cross pattern which they tentatively attribute to charge transfer efficiency and/or the shift-and-add procedure. Another application of their technique by Schneider et al. (1997) also indicates the presence of similar instrumental effects at the center of the noise ACF for the three CCD fields they considered. In one of the fields, the central region of the noise ACF is clearly elongated along one of the CCD axes. In another context, Vogeley (1997) measured the noise ACF of the Hubble Deep Field (HDF) in a search for fluctuations in the Extragalactic Background Light. Preliminary tests indicate that the extended wings of the PSF of the Hubble Space Telescope (HST) produce as much as 10-40% of this noise ACF signal. These wings are probably due to internally scattered light. Other effects such as flat-field errors could also contribute to the ACF but are likely to be less important.
The effect of noise correlation should not be overlooked in weak lensing searches with optical images. We have shown that the effect of correlated noise is of second order in the inverse source SNR. One must therefore be cautious in correcting for systematic shape distortions using bright stars. While this technique, which is standard in weak lensing studies in the optical, ensures proper correction of the effect of the convolution of the PSF with the sources, it would miss the effect of noise correlation. Indeed, for stars with high SNR, the effect of the correlated noise is negligible, while it could be significant for fainter galaxies used in weak lensing searches.
It would thus be instructive to measure the noise ACF in optical images with high SNR and to compute the ensuing ellipticity correlations. This is one of the potential systematic effect in future high precision searches of weak lensing by large-scale structure with optical surveys (see Kaiser 1996;Bernardeau et al. 1997). In particular, the drift-scanning technique involved in the processing of the Sloan Digital Sky Survey could produce noise correlations which would need to be corrected for in future searches of weak lensing with this database (Stebbins et al. 1995). The effect of noise correlation could also be relevant for the recent detection of galaxy-galaxy lensing in the HDF (Hudson et al. 1997), since this detection involves measurements of ellipticities at small pair separations (a few arcsec) with the under-sampled PSF of the HST.
Conclusions
We have studied the effect of noise correlation on the shape parameters in two-dimensional images. The noise correlation is conveniently described by the noise ACF which can easily be measured in practice. We derived the magnitude of the effect for a general two-dimensional least-square fit. The noise correlation can produce a bias, and affect the variance and the covariance of source parameters. In addition, it can produce systematic correlation in the parameters of pairs of sources. We find that the effect is of second order in the inverse SNR of the sources.
We applied these general results to the case of most practical interest, namely a twodimensional gaussian fit. We computed the relevant matrices explicitly. In addition, we explicitly derived the systematic bias produced by this effect on the ellipticity correlation function of source pairs. This is particularly relevant for weak lensing studies.
As a concrete example, we studied the effect of correlated noise on the shape of sources in the FIRST radio survey. We measured the noise ACF and found long range features which extend beyond the central maximum. We computed the resulting systematic effect on the ellipticity correlation function. We find that, in the FIRST survey, noise-induced ellipticity correlations dominate over the expected weak lensing signal for θ ∼ < 0.5 ′ , but are negligible for θ ∼ > 1 ′ .
We discussed the consequences of noise correlation for optical surveys. In optical images, noise correlation can arise from various effects such as the convolution of background light with the PSF, CCD read outs, shift-and-add preprocessing, drift-scanning, etc. Because the effect is quadratic in the source SNR, the effect could be overlooked if systematic distortions are monitored solely with bright stars. The effect of noise correlation could thus be important for searches of galaxy-galaxy lensing and of weak lensing by large-scale structure in optical surveys.
Fig. 2 .
2Fig. 2.-Normalized noise auto-correlation function measured for the coadded field 07210+29486E in the FIRST radio survey. For clarity, only the inner 2 ′ ×2 ′ arcmin region is shown. The correlation function was normalized to 1 at x = 0. For this field, η(0) ≡ σ 2 N ≃ 2.08 × 10 −8 mJy 2 beam −2 . For this measurement, the SNR for η(0) is 60.
Fig. 3 .
3Fig. 3.-Typical dirty beam for a grid pointing in the FIRST survey. This particular beam was derived using AIPS for the grid pointing 07000+50218 and was normalized to 1 at θ = 0. This pattern is typical of VLA snapshot images in which limited coverage of the u-v plane is reflected in relatively intense sidelobes along the projected arms of the VLA.
Fig. 4b .
4b-[See caption above] -21 -Fig. 4c.-[See caption above]
Fig. 5 .
5Fig. 5.-Azimuthally-averaged ellipticity correlation functions for the same coadded field. Each correlation functions, C 1 , C 2 , and C 3 , are plotted as a function of the separation radius θ. A source signal-to-noise ratio of SNR s = 5 was assumed. The correlation functions scale as C i ∝ SNR −2 s . The correlation functions are plotted for a source size of a = 2.45 ′′ (thick lines) and of a = 3.40 ′′ (thin lines).
Fig. 6 .
6-Same as the previous figure but this time focusing on the large-angle behavior of the correlation functions. The correlation functions are only plotted for a source size of a = 2.45 ′′ . The error bars are the 1σ errors in the mean derived from the Monte-Carlo integration. For clarity, they were only shown for C 1 (θ). The correlation functions are all consistent with 0 for θ ∼ > 1 ′ . Fig. 7.-Comparison with the weak lensing correlation functions expected in CDM models. The smooth curves show the ellipticity correlation functions for the four COBE-normalized CDM models of
We first thank our collaborators in the FIRST weak-lensing project for numerous discussions and exchanges. They include David Helfand, Marc Kamionkowski, Catherine Cress, Arif Babul, Richard White, and Robert Becker. We are also indebted to Robert Lupton, Michael Vogeley and Jim Gunn for useful discussions. AR was supported by the NASA MAP/MIDEX program. STB was supported at Columbia University with funds from an NSF Research Experience for Undergraduates supplement to AST-94-19906 and NASA LTSA grant NAG 5-6035. This work was also supported by the NASA ATP grant NAG5-7154.
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| []
|
[
"Cross-Lingual Cross-Platform Rumor Verification Pivoting on Multimedia Content",
"Cross-Lingual Cross-Platform Rumor Verification Pivoting on Multimedia Content"
]
| [
"Weiming Wen [email protected] \nUniversity of California\nDavis\n",
"Songwen Su \nUniversity of California\nDavis\n",
"Zhou Yu \nUniversity of California\nDavis\n"
]
| [
"University of California\nDavis",
"University of California\nDavis",
"University of California\nDavis"
]
| [
"Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing"
]
| With the increasing popularity of smart devices, rumors with multimedia content become more and more common on social networks. The multimedia information usually makes rumors look more convincing. Therefore, finding an automatic approach to verify rumors with multimedia content is a pressing task. Previous rumor verification research only utilizes multimedia as input features. We propose not to use the multimedia content but to find external information in other news platforms pivoting on it. We introduce a new features set, cross-lingual cross-platform features that leverage the semantic similarity between the rumors and the external information. When implemented, machine learning methods utilizing such features achieved the state-of-theart rumor verification results. | 10.18653/v1/d18-1385 | [
"https://www.aclweb.org/anthology/D18-1385.pdf"
]
| 52,009,862 | 1808.04911 | 3886fc8278d4623d5de4345d4b59f0af76fb3473 |
Cross-Lingual Cross-Platform Rumor Verification Pivoting on Multimedia Content
Association for Computational LinguisticsCopyright Association for Computational LinguisticsOctober 31 -November 4. 2018. 2018
Weiming Wen [email protected]
University of California
Davis
Songwen Su
University of California
Davis
Zhou Yu
University of California
Davis
Cross-Lingual Cross-Platform Rumor Verification Pivoting on Multimedia Content
Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing
the 2018 Conference on Empirical Methods in Natural Language ProcessingBrussels, BelgiumAssociation for Computational LinguisticsOctober 31 -November 4. 2018. 20183487
With the increasing popularity of smart devices, rumors with multimedia content become more and more common on social networks. The multimedia information usually makes rumors look more convincing. Therefore, finding an automatic approach to verify rumors with multimedia content is a pressing task. Previous rumor verification research only utilizes multimedia as input features. We propose not to use the multimedia content but to find external information in other news platforms pivoting on it. We introduce a new features set, cross-lingual cross-platform features that leverage the semantic similarity between the rumors and the external information. When implemented, machine learning methods utilizing such features achieved the state-of-theart rumor verification results.
Introduction
Social network's unmoderated nature leads to the spread and emergence of information with questionable sources. With the increasing popularity of the social media, we are exposed to a plethora of rumors. Here we borrow the rumor definition from DiFonzo and Bordia (2007) as unverified information. Unmoderated rumors have not only caused financial losses to trading companies but also panic for the public (Matthews, 2013). Especially if rumors contain multimedia content, the public generally accepts the multimedia information as a "proof of occurrence" of the event (Sencar and Memon, 2009). Readers usually don't have time to look through similar events across different platforms to make an informed judgment. Therefore, even if a credible platform, such as CNN, has debunked a rumor, it can still go viral on other social media platforms.
Intuitively, people believe fake rumors would contain fabricated multimedia content. Boididou et al. (2015b) used forensics features for detecting multimedia fabrication to verify rumors. However, these features did not lead to noticeable improvement. We suspect that this is because un-tampered multimedia content can still convey false information when paired with fake news from a separate event. For example, Figure 1 shows one fake post on MH 370 that used a real video about US Airways Flight 1549. Inspired by the fact that readers tend to search related information covered by different media outlets to garner an objective view, we propose to verify rumors pivoting on multimedia content to tackle such problems. Compared to keywords, searching information pivoting on the visual content is more effective and accurate.
In order to access information from different platforms easily, we created a new rumor verification dataset by expanding a Twitter rumor dataset to include webpages from different social media platforms using search engines. Previous rumor verification datasets are mainly monolingual, such as English (Derczynski et al., 2017) or Chinese (Wu et al., 2015). However, textual information in the native language where the rumor happened can be more helpful when it comes to ver-ifying worldwide rumors. Therefore, we not only indexed English webpages by searching Google with images but also included Chinese webpages via Baidu.
We next introduced our cross-lingual crossplatform features which capture the similarity and agreement among rumors with posts from different social media. We built an automatic verification model using the proposed features and achieved the state-of-the-art performance on the MediaEval 2015's Verifying Multimedia Use (VMU 2015) dataset (Boididou et al., 2015a) utilizing information from Google.
Collecting and annotating rumors in foreign languages is difficult and time-consuming, especially for languages with low rumor verification labeling. Finding out an automatic way to verify those rumors in an unsupervised way is also meaningful. Since our cross-lingual cross-platform features are adaptable to rumors in different languages, we demonstrated that these features could transfer learned knowledge by training on one language and testing on another. Such cross-lingual adaptation ability is especially useful for predicting rumors that have low annotation resource with available annotated rumors in languages such as English.
We published our code and dataset on GitHub 1 .
Related work
Previous research has utilized multimedia information for rumor verification in various ways. Zampoglou et al. (2015); Jin et al. (2015); Boididou et al. (2015b) verified rumors by leveraging forensic features which are extracted to ensure the digital images are not tempered (Sencar and Memon, 2009). However, none of these studies found such information useful for rumor verification on a Twitter-based multimedia dataset. Jin et al. (2017) incorporated image features using a pre-trained deep convolutional neural network (Krizhevsky et al., 2012) on the extended Twitter-based multimedia dataset. Although the image features improve their results, their framework cannot outperform methods other than multimodal fusing networks. One possible reason is that the multimedia content in fake rumors is borrowed from another real event and usually their content corresponds to the text of the rumors. In this case, the image itself is real but not real in the context of the fake news. We thus propose to leverage the multimedia information by finding the agreement and disagreement among posts that are from different social media platforms but share similar visual contents. The agreement between rumors and their comments is used heavily in automatic verification. Mendoza et al. (2010) declared that fake rumors tended to have more people question their validity. Later, Qazvinian et al. (2011) first annotated comments on tweets as supporting, denying or querying, and then used such stance information in the classification to leverage the "wisdom of crowds". Recently, the best performing system (Enayet and El-Beltagy, 2017) in RumourEval shared task at SemEval 2017 (Derczynski et al., 2017) also used such information. However, the crowd is not always wise. For example, Starbird et al. (2014) suspected the correctness of public opinions in rumors, pointing out some certain fake news received more support than questions. In our work, instead of using the "wisdom of crowds", we used the knowledge from different news platforms to assist rumor verification.
Computational journalism (Cohen et al., 2011) exploits external knowledge widely. Diakopoulos et al. (2012) first leveraged information from reliable sources in the context of journalism. They developed a tool for journalists to search for and assess sources in social media around breaking news events. Ciampaglia et al. (2015) utilized factual knowledge bases, such as Wikipedia, to assess the truth of simple statements. Shao et al. (2016) designed a system for tracking rumors on different platforms, which is probably the closest work to ours. However, they did not utilize cross-platform information for rumor verification. Our proposed method is able to leverage information on any platform to verify rumors as long as it has both textual and multimedia information.
CCMR dataset
We created a cross-lingual cross-platform multimedia rumor verification dataset (CCMR) to study how to leverage information from different media platforms and different languages to verify rumor automatically. CCMR consists of three sub-datasets: CCMR Twitter, CCMR Google, and CCMR Baidu.
CCMR Twitter is borrowed from VMU 2015 dataset (Boididou et al., 2015a). There are 17 events containing fake and real posts with images or videos shared on Twitter. We created CCMR Google and CCMR Baidu by searching Google and Baidu indexed webpages that share similar multimodal content with CCMR Twitter. The upper part of Figure 2 shows the collection process. We searched Google with every image (URL for video) in CCMR Twitter to get English webpages. Then we indexed those webpages to form CCMR Google. Similarly, we searched Baidu to get Chinese webpages and created CCMR Baidu. Two human annotators manually annotated both datasets. The annotation is for better analysis and dataset quality control. None of it is utilized during our feature extraction process. Annotators were asked to label collected webpages based on their title and multimedia content. If they are not enough to tell fake news from real news, the webpage is labeled as "others". The Cohen's kappa coefficient for two annotators is 0.8891 in CCMR Google and 0.7907 in CCMR Baidu. CCMR has 15,629 tweets indexed by CCMR Twitter (Twitter), 4,625 webpages indexed by CCMR Google (Google) and 2,506 webages indexed by CCMR Baidu (Baidu) related to 17 events. The webpages from Google and Baidu are in English and Chinese respectively. The statistics of the CCMR dataset with respect to each event is listed in Table 1.
Observation in Annotation
We observe that 15.7% of webpages are fake in CCMR Google while 20.3% in CCMR Baidu. We speculate that this is because all events in CCMR dataset took place outside China. Chinese web-pages searched via Baidu are thus more likely to mistake the information. In the manual annotation process, we found that many images are actually borrowed from other events, which confirms our assumption. Another interesting observation is that webpages indexed by Baidu tend to have more exaggerating or misleading titles to attract click rates. We labeled such webpages as fake if they also convey false information through their multimedia content. We also found that news in different languages has different distributions concerning the subtopics of the event. For example, in the Boston marathon bombing, Baidu indexed Chinese reports generally put more emphasis on a Chinese student who is one of the victims, while Google indexed English reports cover a wider range of subtopics of the event, such as the possible bomber. This phenomenon is understandable as social media from a specific country usually focus more on information related to their readers. Figure 2 describes the overview of our framework. After collecting CCMR dataset in Section 3, we first performed Twitter rumor verification leveraging Google in Section 6 as shown in the bottom left of the figure. We extracted cross-lingual crossplatform features for tweets in CCMR Twitter leveraging webpages from CCMR Google (TFG). Section 5 discusses the automatic construction of this feature set. We then use the features to verify rumors automatically.
Framework Overview
We then performed Twitter rumor verification leveraging Baidu in Section 7 to test if our method can verify rumors by borrowing information from different languages and platforms. This experiment is meant to demonstrate that our method is language and platform agnostic. Such an advantage also enables our method to use one language information to predict in another language (see the experiment in Section 8). We extracted crosslingual cross-platform features for tweets leveraging webpages from CCMR Baidu instead (TFB) and used it to verify tweets in Section 6.
In Section 8, we performed Baidu rumor verification via transfer learning to test the crosslingual adaptation ability of the cross-lingual cross-platform features. We treated Chinese webpages in CCMR Baidu as rumors and empirically verified them via transfer learning. We extracted cross-lingual cross-platform features for Baidu webpages leveraging Google (BFG) in Section 6. Since BFG and TFG are both cross-lingual crossplatform features leveraging Google, we adopted the classifier pre-trained with TFG on CCMR Twitter to verify webpages in CCMR Baidu using BFG, under the assumption that tweets and webpages follow a similar distribution.
Although we labeled webpages in CCMR Google and CCMR Baidu, we did not leverage the annotation here because annotation is timeconsuming and using annotation information is not generalizable to other datasets.
Cross-lingual Cross-platform Features
We propose a set of cross-lingual cross-platform features to leverage information across different social media platforms. We first embed both the rumor and the titles of the retrieved webpages into 300-dimension vectors with a pre-trained multilingual sentence embedding. It is trained using English-Chinese parallel news and micro-blogs in UM-Corpus (Tian et al., 2014). We encode English-Chinese parallel sentences with the same word dictionary, as they share some tokens such as URLs and punctuation. We then use a two-layer bidirectional gated recurrent unit (GRU) (Cho et al., 2014) to generate hidden states. We obtain the embedding by averaging the hidden states of the GRU. A pairwise ranking loss is used to force the cosine distance between embeddings of paired sentences to be small and unpaired sentences to be large. We train our multilingual sentence embedding on 453,000 pairs of English-Chinese parallel sentences and evaluated it on another 2000 sentence pairs. Our published code includes the implementation details of the multilingual sentence embedding.
After obtaining the embeddings of the rumor and the titles of the retrieved webpages, we further calculate the distance and agreement features between these embeddings to create a set of crosslingual cross-platform features. In total, there are 10 features, two for distance features and eight for agreement features.
Distance Features
We compute the cosine distances between the embeddings of the target rumors and the titles of the retrieved webpages. The distance indicates if the rumor has similar meaning with the retrieved webpages that have similar multimedia content. We calculate the mean and variance of the distance.
The mean of the distance indicates the average similarity between a rumor and its corresponding webpages from other platforms. A high value in mean suggests that the rumor is very different from the retrieved information from other platforms. We suspect that rumors with this property have a higher probability of being fake. Because the rumor might have borrowed the image from another event that was covered in the retrieved information. Meanwhile, the variance indicates how much these retrieved webpages are different from each other. A high variance indicates that the multimedia information is used by different events or is described in different statements. So the event or the statement the rumor covers could be fake.
Agreement Features
We first pre-train an agreement classifier on a stance detection dataset provided by the Fake News Challenge 2 . This dataset provides pairs of English sentences with their agreement annotations in "agree", "disagree", "discuss" and "unre-2 http://www.fakenewschallenge.org/ lated". Figure 3 shows four example body texts of a headline corresponding to each type of annotation in the dataset. During the training process, we embed the sentences in the Fake News Challenge dataset using our pre-trained multilingual sentence embedding. We then concatenate the embeddings of the sentence pair as the input. We use a multilayer perceptron to pre-train our agreement classifier. We randomly select a balanced development set containing 250 pairs for each label (1000 in total) and train our agreement classifier on the rest of 74,385 pairs. The agreement classifier achieves 0.652 in the macro-averaged F1-score on the development set. Our published code also includes the details. We calculate the agreement features using the mean and variance of the prediction probability between the rumor and all the retrieved webpages. There are in total four agreement labels. Therefore, we have eight agreement features in total. Agreement features capture information about if the rumor's statement agrees with the corresponding information in other platforms. Besides being able to gain similar benefits as distance features, our agreement features also capture the case that the information stance is portrayed differently by different resource rumors. Conflicting information will also be an indicator of fake news.
Rumor Verification Leveraging Cross-platform Information
We extracted cross-lingual cross-platform features for tweets in CCMR Twitter leveraging Google (TFG) and evaluated the effectiveness of TFG on rumor verification tasks.
We proposed a simple multi-layer perceptron classifier (MLP) to leverage the extracted features. MLP has two fully-connected hidden layers of 20 neurons with ReLU as the activation function. Each layer is followed by a dropout of 0.5. We evaluated TFG in two settings: task and event.
1) The task setting used event 1-11 for training and event 12-17 for testing according to (Boididou et al., 2015a). 2) The event setting evaluated the model performance on a leave-one-event-out cross-validation fashion. F1-score is used for evaluation metric. Since both collecting source rumors from Google and doing feature extraction for a given tweet can be done automatically, it is fair to compare the performance of our model with baselines described below.
Baselines
We adopted three best performing models in the VMU 2015 task as our baselines:
UoS-ITI (UoS) (Middleton, 2015) uses a natural language processing pipeline to verify tweets. It is a rule-based regular expression pattern matching method. It ranks evidence from Twitter according to the most trusted and credible sources. (Jin et al., 2015) is an approach including two levels of classification. It treats each image or video in the dataset as a topic and uses the credibility of these topics as a new feature for the tweets. They used the tweet-based and user-based features (Base), such as the number of hashtags in the tweet or the number of friends of the user who posted the tweet. (Boididou et al., 2015b) uses an agreement-based retraining scheme. It takes advantage of its own predictions to combine two classifiers built from tweet-based and userbased features (Base). Besides features provided by the task, it included some additional features such as the number of nouns in tweets and trust scores of URLs in tweets obtained from thirdparty APIs.
MCG-ICT (MCG)
CERTH-UNITN (CER)
Results
We describe the task setting results in Table 2, and detailed per-event results in Table 3. Although TFG does not achieve the highest F1-score in the task setting, it is mainly due to the split of the dataset. More than half of the tweets in the test set do not have images. Thus we can only leverage cross-platform information by searching videos' URLs, which results in less accurate crosslingual cross-platform features. In the event setting, which has a more fair comparison, TFG outperformed other methods with a big margin (p<0.001). It is surprising to see that only 10 features extracted from external resources indexed by search engines leveraged by a simple classifier can bring such a big performance boost.
To further explore the quality of the crosslingual cross-platform features, we calculated the Pearson correlation coefficient (PCC) between each feature with respect to the tweet's label (fake or real). We evaluated both TFG and features used by baseline models. Table 4 lists the top six features with the highest absolute PCC values. A positive value indicates this feature positively correlates with fake news. We can see four out of the top six features are cross-lingual cross-platform features. The variance of the unrelated probability (unrelated variance) has the highest score, which further validates our design intuition that tweets might convey false information when they have different agreement with all other webpages that shared similar multimedia content. The second feature, "distance var" is also highly correlated with fake news. This result supports our hypothesis that if there is a large information dissimilarity across different platforms, there is a high probability of fake information involved. The only feature from baselines (56 features in total) in the top six features is whether a tweet contains an exclamation mark or not (containsExclaimationMark).
Feature
PCC unrelated variance 0.306 distance variance 0.286 agree variance 0.280 discuss mean -0.231 unrelated mean 0.210 containsExclamationMark 0.192 Table 4: Top six features correlated with fake news.
Analysis
We found that Google webpages usually cover a complete set of different information. There are usually both fake news and real news that debunk these fake ones. As a result, there is a big information variance among all those webpages' titles, which is captured by the cross-lingual crossplatform features. Therefore, TFG performs much better than baselines in a number of events, such as Event 03 (Sochi Olympics). The statistics of the CCMR dataset, described in Table 1, also supports our observation that the labels of posts in CCMR Google are distributed more evenly compared to other media sources.
However, the F1-score of TFG is very low in Event 15 (Garissa Attack). Gunmen stormed the Garissa University College in this event. We analyzed the Google webpages' titles that share the same image in this event. Although some titles are related to the event, more of them are talking about completely unrelated information such as "Daily Graphic News Sun 20th Oct, 2013 -GhHeadlines Total News ...". This webpage's title only shows its published date and the name of the website. Such noise hurt the performance of the cross-lingual cross-platform features. Since we did not perform any manual labeling or filtering, sometimes the crawled webpages can be misleading. To analyze the prevalence of such noise, we randomly picked 100 Google webpages and 100 Baidu webpages from CCMR and counted the number of noisy posts. The ratio of noise is 22% in Google and 18% in Baidu. However, even with such noise, our proposed methods can still outperform current state-of-the-art methods.
Rumor Verification Leveraging
Cross-lingual Information
We tested if our cross-lingual cross-platform features are able to leverage external information from another language for rumor verification. We simply replaced the Google webpages with Baidu webpages to extract features for tweets (TFB), because we have a pre-trained multilingual sentence embedding that can project Chinese and English to a shared embedding space. We used the same classifier, MLP to evaluate the performance of TFB with both baselines and TFG.
Results
Experiment results using the task setting are shown in Table 2 and the detailed per-event results are listed in Table 3. Similar to the problem in TFG, we can not obtain any Chinese webpages related to events such as Syrian boy, Varoufakis and zdf, which cover most tweets in the test set. Those missing features make TFB perform poorly in the task setting. However, TFB performs better than two of the baselines in the event setting. If we exclude events without Baidu webpages (event 10, 16 and 17), the average F1-score of UoS, MCG and CER are 0.130, 0.732 and 0.660, which are all lower than TFB's. The performance of TFB proves that our method can be generalized across languages or platforms.
To further test the robustness of our crosslingual cross-platform features, we also examined if it would still work when leveraging external information that contains different languages. We extracted the cross-lingual cross-platform features for tweets leveraging Google and Baidu webpages together (Combo) and accessed the performance of Combo using MLP similarly. The performance of Combo is also listed in Table 2. Since Combo would contain noise introduced from combining webpages indexed by different search engines, it is not surprising that Combo performs slightly worse than TFG extracted from Google webpages which already cover a wide range of information solely. However, Combo performs much better than TFB which only leverages Baidu webpages. It proves that our cross-lingual cross-platform features are robust enough to utilize combined external information from different languages and platforms.
Analysis
We checked the actual titles of webpages from CCMR in certain events to analyze the reason for TFB's worse performance exhaustively. We found that those Baidu webpages' titles often talk about subtopics different from the target rumor's on Twitter, while Google webpages are more related. For example, the F1-score of TFB is much lower than TFG's in event 02 (Boston Marathon bombing). This performance corresponds to our observation in Section 3 that Baidu webpages mainly focus on a subtopic related to the Chinese student instead of other things discussed on Twitter.
Low-resource Rumor Verification via Transfer Learning
We extracted cross-lingual cross-platform features of webpages in CCMR Baidu leveraging information from Google (BFG). Then we applied Transfer, MLP in Section 6 trained on the whole CCMR Twitter using TFG, to verify those webpages. Because this pre-trained model is for binary classification, only webpages labeled as real or fake in CCMR Baidu are involved. Since webpages in CCMR Baidu do not share the same features with tweets, such as the number of likes and retweets, we adopted a random selection model as our baseline. It would predict a rumor as real or fake with the same probability. We compared the performance of the Transfer model with this baseline on each event. F1-score is also used for evaluation metric. Table 5 lists the detailed results of our transfer learning experiment. We achieved much better performance compared to the baseline with statistical significance (p<0.001), which indicates that our cross-lingual cross-platform feature set can be generalized to rumors in different languages. It enables the trained classifier to leverage the information learned from one language to another.
Results
ID
Analysis
In event 11 (Pig fish), Transfer achieves much higher performance than the random baseline. Generally, Baidu webpages' titles are semantically different from tweets. However, in this particular event, the textual information of those titles and tweets are semantically close. As a result, models learned from English rumors can easily work on Chinese rumors, which is helpful for our transfer learning. Figure 4 shows three Twitter-Baidu rumor pairs with similar meaning in this event.
Transfer obtains pretty low F1-scores in event 07 (Passport hoax). The annotation conflict caused its weak performance. This event is about a Child drew all over his dads passport and made his dad stuck in South Korea. During the manual annotation process, we found out that it is a real event confirmed by official accounts according to one news article from Chinese social media 3 , while CCMR Twitter labeled such tweets as fake. Since Transfer is pre-trained using Twitter dataset, it is not surprising that Transfer achieves 0 in F1-score on this event. The annotation conflict also brings out that rumor verification will benefit from utilizing cross-lingual and cross-platform information.
Conclusion
We created a new multimedia rumor verification dataset by extending a multimedia Twitter dataset with external webpages from Google and Baidu that share similar image content. We designed a set of cross-lingual cross-platform features that leverage the similarity and agreement between information across different platforms and languages to verify rumors. The proposed features are compact and generalizable across languages. We also designed a neural network based model that utilizes the cross-lingual cross-platform features and achieved state-of-the-art results in automatic rumor verification.
Figure 1 :
1A video of US Airways Flight 1549 was borrowed by news on Malaysia Airlines Flight 370.
Figure 2 :
2The information flow of our proposed pipeline. TFG represents the cross-lingual cross-platform features for tweets leveraging Google information, while TFB is similar but leverages Baidu information instead. BFG means cross-lingual cross-platform features for Baidu leveraging Google information.
Figure 3 :
3An example headline and its body texts of the Fake News Challenge dataset.
Figure 4 :
4Example parallel rumors in the Pig fish event.
Table 1 :
1CCMR dataset statistics.
Table 2 :
2The task and event settings performance.ID
UoS
MCG CER TFG TFB
01
0.658 0.594 0.718 0.715 0.704
02
0.007 0.494 0.745 0.557 0.448
03
0.057 0.882 0.595 0.956 0.822
04
0.538 0.826 0.717 0.856 0.678
05
0.000 0.988 0.947 0.969 0.956
06
0.555 0.949 0.916 0.912 1.000
07
0.000 1.000 0.475 0.989 0.989
08
0.000 0.870 1.000 0.960 1.000
09
0.000 0.772 0.996 1.000 0.996
10
0.000 0.615 0.821 0.875 -
11
0.000 0.963 0.000 0.963 0.667
12
0.000 0.655 0.754 0.677 0.656
13
0.000 0.954 0.795 0.998 0.850
14
0.000 0.330 0.419 0.409 0.430
15
0.000 0.130 0.156 0.145 0.154
16
1.000 0.990 0.999 0.996 -
17
1.000 0.827 1.000 0.992 -
Avg 0.224 0.756 0.693 0.822 0.739
Table 3 :
3F1-scores for each event.
Table 5 :
5Rumor verification performance on the CCMR Baidu, where -indicates there is no webpage in that event.
https://github.com/WeimingWen/CCRV
http://new.qq.com/cmsn/20140605/20140605002796
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"Supporting Information The Role of Intracellular Interactions in the Collective Polarization of Tissues and its Interplay with Cellular Geometry",
"Supporting Information The Role of Intracellular Interactions in the Collective Polarization of Tissues and its Interplay with Cellular Geometry"
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"Shahriar Shadkhoo \nKavli Institute for Theoretical Physics\nUniversity of California\nSanta BarbaraCaliforniaUSA\n\nPhysics Department\nUniversity of California\nSanta BarbaraCaliforniaUSA\n",
"Madhav Mani \nDepartment of Engineering Sciences and Applied Mathematics\nNorthwestern University\nEvanstonIllinoisUSA\n\nNSF-Simons Center for Quantitative Biology\nNorthwestern University\nEvanstonIllinoisUSA\n\nDepartment of Molecular Biosciences\nNorthwestern University\nEvanstonIllinoisUSA\n"
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"Kavli Institute for Theoretical Physics\nUniversity of California\nSanta BarbaraCaliforniaUSA",
"Physics Department\nUniversity of California\nSanta BarbaraCaliforniaUSA",
"Department of Engineering Sciences and Applied Mathematics\nNorthwestern University\nEvanstonIllinoisUSA",
"NSF-Simons Center for Quantitative Biology\nNorthwestern University\nEvanstonIllinoisUSA",
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| Supporting Information (SI) intends to (a) introduce rigorous mathematical formalism and definitions of the quantities introduced in the Main Text, (b) discuss secondary results, and (c) provide further evidence that support the findings. For pedagogical purposes, after introducing the formalism, we start by analyzing the one-dimensional version of the model. Studying 1D systems is insightful and helps us develop intuition about the conditions required for the polarization to become the stable fixed point of the differential reaction-diffusion equations. The results we obtain for the 1D systems, although not directly applicable to the 2D case, help us explore the mean-field solutions in 2D systems. In particular we introduce trivial and non-trivial solutions in 2D, and compare the solutions in systems with the SLCI and NLCI. We fully discuss how segregation is accomplished in systems with nonlocal interactions, and illustrate the inability of local interactions in separating the membrane proteins. The irregular patterns of polarity (swirls and crosses) and their possible origins are explored. A detailed investigation of the impact of local upregulating cytoplasmic interactions (LA-NLI) on the stability of polarization is presented, which is accompanied by a table illustrating the rose plots of dipoles' angular distributions for various combinations of length scales and geometric disorder; seeFig.(E). This is followed by a discussion on the gradient cues, its mathematical definition and further comments on the results.Elongated tissues are investigated in Sec.(3), where the nematic tensor of elongation is introduced, the eigenvalues and eigenvectors are calculated, and the expected value of angle of polarity is plotted against the magnitude of elongation inFig. (Fc). The last section (4) includes a brief discussion, and a table similar to that in the Main Text which illustrates the results of numerical simulations for all mutations of interest; see Fig. (G). | 10.1371/journal.pcbi.1007454 | null | 80,628,390 | 1803.09914 | a95cbb1aec9926da62684b253d60b5220f249cfa |
Supporting Information The Role of Intracellular Interactions in the Collective Polarization of Tissues and its Interplay with Cellular Geometry
Shahriar Shadkhoo
Kavli Institute for Theoretical Physics
University of California
Santa BarbaraCaliforniaUSA
Physics Department
University of California
Santa BarbaraCaliforniaUSA
Madhav Mani
Department of Engineering Sciences and Applied Mathematics
Northwestern University
EvanstonIllinoisUSA
NSF-Simons Center for Quantitative Biology
Northwestern University
EvanstonIllinoisUSA
Department of Molecular Biosciences
Northwestern University
EvanstonIllinoisUSA
Supporting Information The Role of Intracellular Interactions in the Collective Polarization of Tissues and its Interplay with Cellular Geometry
1 Formalism of The Generalized Reaction-Diffusion Model Like in the Main Text, we denote the polar complexes by F-G, namely F ≡ Fz:Fmi and G ≡ Fmi:Vang. Labeling two adjacent cells by i and j, the average concentrations of complexes F i -G j is denoted by u ij ; consistently the concentration of F i -G j is denoted by v ij = u ji . The junctions shared by cells i and j, are denoted by ij hereafter. The formation of complexes F i -G j is upregulated by the already-bound complexes of the same polarity, and downregulated by the opposite complexes, i.e. G i -F j . The key assumption in this
Supporting Information (SI) intends to (a) introduce rigorous mathematical formalism and definitions of the quantities introduced in the Main Text, (b) discuss secondary results, and (c) provide further evidence that support the findings. For pedagogical purposes, after introducing the formalism, we start by analyzing the one-dimensional version of the model. Studying 1D systems is insightful and helps us develop intuition about the conditions required for the polarization to become the stable fixed point of the differential reaction-diffusion equations. The results we obtain for the 1D systems, although not directly applicable to the 2D case, help us explore the mean-field solutions in 2D systems. In particular we introduce trivial and non-trivial solutions in 2D, and compare the solutions in systems with the SLCI and NLCI. We fully discuss how segregation is accomplished in systems with nonlocal interactions, and illustrate the inability of local interactions in separating the membrane proteins. The irregular patterns of polarity (swirls and crosses) and their possible origins are explored. A detailed investigation of the impact of local upregulating cytoplasmic interactions (LA-NLI) on the stability of polarization is presented, which is accompanied by a table illustrating the rose plots of dipoles' angular distributions for various combinations of length scales and geometric disorder; seeFig.(E). This is followed by a discussion on the gradient cues, its mathematical definition and further comments on the results.Elongated tissues are investigated in Sec.(3), where the nematic tensor of elongation is introduced, the eigenvalues and eigenvectors are calculated, and the expected value of angle of polarity is plotted against the magnitude of elongation inFig. (Fc). The last section (4) includes a brief discussion, and a table similar to that in the Main Text which illustrates the results of numerical simulations for all mutations of interest; see Fig. (G).
Formalism of The Generalized Reaction-Diffusion Model
Like in the Main Text, we denote the polar complexes by F-G, namely F ≡ Fz:Fmi and G ≡ Fmi:Vang. Labeling two adjacent cells by i and j, the average concentrations of complexes F i -G j is denoted by u ij ; consistently the concentration of F i -G j is denoted by v ij = u ji . The junctions shared by cells i and j, are denoted by ij hereafter. The formation of complexes F i -G j is upregulated by the already-bound complexes of the same polarity, and downregulated by the opposite complexes, i.e. G i -F j . The key assumption in this model is that such intracellular interactions, mediated by diffusing cytoplasmic proteins (Dsh, Dgo, and Pk), are not strictly local, but instead diffuse to mediate the nonlocal interaction between the complexes localized on different junctions. The magnitude of interaction, which is proportional to the local concentration of the diffusing messenger protein, decreases exponentially with the distance between the two complexes. The concentration u ij of F i -G j complexes at point r on the perimeter of cell i is governed by the following equation:
du ij (r) dt = κf ubd i g ubd j 1 + α {k}i i∩k dr K d (r − r )u ik (r ) − γu ij (r) 1 + β {k}i i∩k dr K d (r − r )v ik (r ) + η ij (r, t) + M ij (r) e −t/τm .(1)
In the above equation, {k} i stands for the labels of all neighbors k of a given cell i, and i∩j dr implies integration along the edge shared by cells i and j. The factors K u (r − r ) and K d (r − r ) are the kernels of up-and downregulating interactions, and γ −1 is the dissociation timescale of bound complexes. As mentioned in the Main Text, the kernels K(r) are assumed to be of the exponential form, which is deduced from the diffusive dynamics of the cytoplasmic proteins. The concentration of cytoplasmic protein obeys a diffusion equation with diffusion constant D and degradation time τ :
∂c(r, t) ∂t = D∇ 2 c(r, t) − τ −1 c(r, t).
Assuming the degradation of C is much faster than polarization dynamics, γτ 1, it suffices to only consider the steadystate solutions of c(r, t) in Eq. (2):
D∇ 2 c(r) − τ −1 c(r) = 0 =⇒ c(r) = dr c 0 (r ) exp(−|r − r |/λ).(3)
Here, r corresponds to the specific point on the cell membrane, where the the concentration of C is the linear superposition of the concentration of proteins diffused from all around the cell; hence integration dr . Next, λ = √ Dτ is the diffusion length of protein C before degradation. The diffusing proteins enhance/suppress the formation of like/unlike complexes. The linear dependence of cooperative terms on the concentration u(r ) originates from the fact that in Eq. (3), the source concentration of modified proteins, c 0 (r ), is assumed to be proportional to the local concentration u(r ). This dependency is a function of the binding affinities of cytoplasmic proteins and the membrane-bound proteins. Altogether, the coefficients are lumped into the phenomenological constants α and β. We shall emphasize that the above derivations are merely to provide a qualitative relationship between the degradation rate, diffusion constants, and the cytoplasmic interaction length scales. As mentioned in the Main Text we are aware of the complicated dependencies of λ on the molecular details which are not captured by the above equations.
The functional forms of the interaction kernels read: K u (r) = N −1 u exp(−|r|/λ u ) and K d (r) = N −1 d exp(−|r|/λ d ), where λ u , λ d are the characteristic length scales of up-and downregulating interactions, respectively. The prefactors N u and N d are normalization factors, to be determined shortly. Before calculating the normalization factors, we make a detour to explain an approximation we used to reduce the computational cost of simulations.
Uniform junctional distribution of proteins. The full integro-differential equation (1) couples every pairs of points on the perimeter of a cell; solving it on a 2D array of cells is computationally expensive. A simplifying approximation is to assume uniform junctional distributions of proteins, and break the integration on the cells' perimeters down to summation over the junctions. Thus the integrations are replaced by matrix products; see Eqs. (4) and (5). For notational convenience, we introduce the following: cells are labeled by the Latin letters, i, j, . . . , and comprise edges labeled by Greek letters, µ, ν, . . . . For example: i ≡ {µ, ν, . . . }, and consistently edges, that are the cell-cell junctions, are the intersections of two sets (cells): e.g. µ ≡ i ∩ j. The junctional averaging is then defined as:
u µ = −1 µ µ dr u µ (r), where the integral is taken over the concentration of complex u µ (r) ≡ u ij (r) = [F i -G j ](r), along the junction µ. Thus, one can recast Eq. (1) to du µ dt = κf ubd i g ubd j 1 + α µ ν∈i K µν u u ν − γu µ 1 + β µ ν∈i K µν d v ν + η µ (t) + M µ exp(−t/τ m ).(4)
In the above equation, η µ and M µ are the noise and the global cue averaged over the length of edge µ. The cooperative interactions are reduced to matrix products of K u and the vector u. Matrices K u are symmetric with respect, and their elements K µν u are purely geometrical constants obtained from the following relation:
K µν u = µ,ν dr ν dr µ K u (r ν − r µ ).(5)
A similar relation holds for downregulating interactions K µν d . In Eq. (4), coupling constants equal α µ = α/ µ , β µ = β/ µ , and for the kernels we have K µν = K νµ = µ,ν drdr K(r − r ). The self interactions, i.e. the diagonal elements equal
K µµ = 2 2 µ x −2 µ e −xµ + x µ − 1 ,(6)
in which x µ = µ /λ. We dropped the subscripts u and d for simplicity. The effective stochastic noise on a junction µ reads:
η µ (t)η µ (t ) = η 2 0 µ δ(t − t )
. It is noteworthy that in reality the core proteins, for example Flamingo (Fmi) and Frizzled (Fz), are observed to be persistently localized at the puncta, the subdomains of plasma membrane; see e.g. [1].
With this background, we now calculate the normalization factors introduced above. In order to discern the net effect of interaction ranges from the effective coefficients α, β, we choose the normalization factors N u and N d , such that the self-interaction of an edge of length µ equals α, namely: α K µµ u = α, or K µµ u = 1. The same relation holds for β and K µµ d . This choice of normalization ensures that the observed behavior upon changing λ's is purely due to nonlocal edge-edge coupling, not the effective coefficients of local interactions α, β. Satisfying this condition for all edges simultaneously is not possible, except for ordered tissues. The normalization constant is thus calculated for an edge with the average length of all edges, 0 ≡ µ . Using the definition of kernels for an edge of length 0 from Eq. (5), we obtain:
N (λ) = drdr exp(−|r − r|/λ) = 2λ 2 e − 0λ −1 + 0 λ −1 − 1 .(7)
Here, both r and r lie on a single edge with length 0 .
Limit of strictly local cytoplasmic interactions (SLCI). In the limit of small λ/ µ → 0, we get, K µν = δ µν , where δ µν is the Kronecker delta. Thus in the SLCI limit, the equations read,
du µ dt = κf ubd i g ubd j (1 + αu µ ) − γu µ (1 + βv µ ) + η µ (t).(8)
Junctional and cellular polarity. Planar cell polarity can be defined at the junctional and the cellular levels. For individual junctions, polarity is defined as the difference between the concentrations u ij = [F i − G j ] and the opposite dimer, u ji ,
namely p ij = u ij − u ji = u ij − v ij .
Cell polarity, on the other hand, is referred to as the asymmetric (anisotropic) distribution of membrane proteins on the cell perimeter, thus requires integration on the cell-cell junctions.
Depending on the pathway of interest, polarity can be defined either as a vector (vectorial polarity), or a nematic (axial polarity), corresponding to heterodimer and homodimer clusters, respectively. The former is a vector identified by an angle ∈ [0, 2π), whereas the latter is a traceless tensor with the angle in the ∈ [0, π) range. In our case of study, the polarization involves two distinct complexes, Fz:Fmi and Fmi:Vang, hence vectorial polarity. Below, we introduce both definitions.
(i) Vectorial Polarization: The polarization vector (also called dipole) associated with cell i is defined as:
P i = 1 2 i dr r − R i |r − R i | u i (r ) − v i (r ) = p x ix + p y iŷ .(9)
Here R i is a reference point within cell i, with respect to which the polarization is defined. We take this to be the geometrical center of mass of each cell. The magnitude of the cell polarity |P i |, and the angle φ p i ∈ [0, 2π) measured from the x-axis, can be computed using the following relations:
P i = |P i | = (p x i ) 2 + (p y i ) 2 , and φ p i = tan −1 (p y i /p x i ).(10)
The polarization of the tissue with N c cells, and its global order are characterized by the following quantities: (1) magnitude of average polarization, (2) average magnitude of polarization, and (3) the ratio of (1) and (2):
(1) : P = |P| = 1 N c Nc i=1 P i ,(2): Q = 1 N c Nc i=1 P i ,(3): O = P /Q.(11)
(ii) Axial Polarization: The second definition is a measure for cell polarity when dealing with homodimers like Fmi-Fmi complexes, and determines the axis of polarization:
P i = P i,1 P i,2 P i,2 −P i,1 where, P i,1 = 2π 0 dφ u i (r) cos(2φ), P i,2 = 2π 0 dφ u i (r) sin(2φ).(12)
In the above equation, φ is the polar angle of point r on the perimeter of cell i and is measured with respect to a reference axis. The magnitude of polarization equals P i = (P 2 i,1 + P 2 i,2 ) 1/2 . Its orientation is determined by the angle φ p i that satisfies cos(2φ p i ) = P i,1 /P i and sin(2φ p i ) = P i,2 /P i . In Sec.
(3), we use the same formalism for the elongation tensor.
Correlation function and correlation length. Correlation function is a measure of alignment of polarization field. In order to investigate the temporal behavior of the spatial extension of alignment, we define the equal-time correlation functions. Consider an arbitrary cell at R i , and a vector r connecting it to another cell at R j = R i + r. With no further assumption we can define a correlation function, that is dependent on the distance r = |r| and the relative angle of r and P(R i ); we call it θ r,p . The latter appears due to the vectorial nature of polarization field; there is no a priori reason for dipoles to be correlated equally in all directions. For a tissue of N c cells, the correlation function at time t reads,
S(r, θ r,p ; t) = N −1 c i P(R i ; t) · P(R i + r ; t).(13)
The above quantity calculates at each timepoint t, the average (over all cells i) of the conditional probability that the dipole at point R j = R i + r acquires the value and direction of P(R j ), given the dipole at point R i equals P(R i ).
For θ r,p = 0, π, and θ r,p = ±π/2, we get longitudinal and transverse correlations, respectively. In spite of this angular dependence, averaging the above correlation function over θ r,p ∈ [0, 2π), returns a weighted average of correlation as a function of r = |r|, which is bounded by the longitudinal and transverse correlations from above and below, respectively. In spite of the distinction between longitudinal and transverse correlation lengths, the qualitative behavior of the correlation length could be captured by the angular average of the correlation function. Further discussion in this regard can be found below in SI. Sec. (2.3). Thus, we define radial correlation function:
S(r; t) = N −1 c i 2π 0 dθ r,p 2π P(R i ; t) · P(R i + r ; t).(14)
Correlation length can be obtained from the above equation:
ξ(t) = Rc 0 dr r S(r; t) Rc 0 dr S(r; t) .(15)
Here, R c = 40 (cell diameter) is the largest distance between two given cells in our simulations. A perfectly correlated polarization field, is identified by ξ = R c /2. The time-dependent correlation lengths for three important cases of study are shown in the Main Text Fig. (3c2).
Geometric disorder. In 1D systems, the edge lengths are i = 0 (1+ i ), where i ∈ [− 0 , + 0 ] with uniform distribution, and i j = 2 0 δ ij /6. In 2D the level of quenched disorder is controlled by randomizing the sites of a triangular lattice, based on which a polygonal lattice is generated using Voronoi tessellation. Suppose the nodes of a triangular lattice are perturbed:
R i = R 0 i (1 + ∆ i ), with {R 0
i } the loci of the nodes in the ordered triangular lattice. The magnitude of the disorder term ∆ i is uniformly distributed in range [−∆ 0 , +∆ 0 ], with local correlations: ∆ i · ∆ j = ∆ 2 0 δ ij /3, where δ ij is the Kronecker delta function. For ∆ 0 = 0, the corresponding Voronoi lattice would be an ordered hexagonal lattice, hence 0 = 0. By randomly displacing the nodes of the triangular lattice, we can distort the resultant Voronoi lattice. The edge-length disorder of the Voronoi lattice i , and density of defects n d are obtained by ensemble averaging over 10000 realizations of disordered triangular lattices of size 50×50. Defects appear for disorder ∆ d 0.25, in the triangular lattice.
Physical considerations. Given the quantitative approach of this study, we find it crucial to clarify the term "long-range", used frequently throughout the paper. The Mermin-Wagner theorem states that "true long-range" ordering is prohibited in 2D systems with continuous (e.g. rotational) symmetries, except at zero stochastic noise. The long-range order is referred to as the algebraic decay of correlation functions with distance. The magnitude of noise in our system drops as 1/ √ N mol. , with N mol. the number of molecules participating in binding/unbinding reactions. Thus in the limit N mol. → ∞, long-range order is achieved. For finite N mol. , a state of quasi-long range order can potentially exist.
Mean-Field and Numerical Solutions
Mean-field (MF) solutions are obtained by assuming that for all junctions, e.g. shared by adjacent cells i and j, the products f ubd i g ubd j and f ubd j g ubd i are uniform across the tissue. This assumption is backed up by the following numerical analysis. We compute the root-mean squared fluctuations of f ubd i g ubd j , for all the junctions with fully randomized labels, and compare the initial and final values. Figures (Bb1) and (Bb2) show that in 2D systems for both SLCI and NLCI mechanisms, the final distribution of f ubd i g ubd j is noticeably narrower than the initial distribution. Intuitively this approximation can be justified by the fact that the linearized reaction-diffusion equations governing u and v, obey diffusion-like equations in the continuum limit (see [3] where 1D arrays were studied). Therefore, given that the total concentrations f 0 and g 0 are uniform across the tissue, the free proteins too, spread diffusively into a rather uniform state. Here we first elaborate on the MF solutions in 1D, then discuss the 2D case.
Mean-Field Solutions in 1D
In one dimension, cells are juxtaposed in an array and are separated by junctions. The proteins localize on both sides of these junctions and form dimers. A schematic of one-dimensional arrays can be seen in Fig. (Aa). The reaction-diffusion (RD) equations governing u and v complexes on a junction shared by cells i and i + 1 read:
du i,i+1 dt = κf ubd i g ubd i+1 (1 + αu i,i+1 ) − γu i,i+1 (1 + βu i+1,i ).(16)
A similar equation exists for the opposite complex v i,i+1 = u i+1,i . The general RD equations introduced in Eq. (1), reduce, in 1D with strictly local interactions, to the above equation that was originally proposed by Mani, et.al. in Ref. [3], where the properties of the solutions to Eq. (16) in 1D systems are investigated elaborately. Here we briefly review the main results and show that in the steady state, the mean field (MF) solutions of ordered systems exhibit a bifurcation at a critical value of the control parameter g 0 /f 0 , i.e. the ratio of total amounts of proteins F and G. We start by deriving equations for
p i,i+1 = u i,i+1 − v i,i+1 and s i,i+1 = u i,i+1 + v i,i+1
, by adding and subtracting Eq. (16) and its counterpart for the opposite complex: In the above equations we have:
dp i,i+1 dt = κ(f ubd i g ubd i+1 − f ubd i+1 g ubd i ) + κα(f ubd i g ubd i+1 u i,i+1 − f ubd i+1 g ubd i u i+1,i ) − γ(u i,i+1 − u i+1,i ),(17a)ds i,i+1 dt = κ(f ubd i g ubd i+1 +f ubd i+1 g ubd i )+κα(f ubd i g ubd i+1 u i,i+1 +f ubd i+1 g ubd i u i+1,i )−γ(u i,i+1 +u i+1,i )−2γβ−u i,i+1 u i+1,i . (17b)✏ 0 G 0 /F 0 (b) activation inhibition · · · · · · i,i+1 i 1 i i + 1 i 2 F Gf ubd i = F 0 − (u i,i+1 i,i+1 + u i,i−1 i−1,i ) i,i+1 + i−1,i = f 0,i − u i,i+1 i,i+1 + u i,i−1 i−1,i i,i+1 + i−1,i ,(18)
in which F 0 is the total amount of protein in a cell, assumed to be uniform across the tissue, and f 0,i is the concentration of free F available to the two junctions of cell i. A similar relation can be written for g ubd i and g 0,i . In ordered systems and within the MF approximation, all of the above variables become independent of index i. A few lines of simple algebra shows that the critical value reads:
p 2 = s 2 − 4 αβ ⇒ s * = 2 √ αβ ⇒ g * 0 = γ/κα 1 − 1/αβ + 1 αβ .(19)
For the values of the parameters used in the simulations, i.e. α = β = 5, and γ = 1 , κ = 10, the value of g * 0 is found to be g * 0 = 0.225, which can also be seen in Fig. (2) of Main Text. Note that in the above equations, for αβ → 0, the bifurcation point diverges, implying that cooperative interactions are essential to the emergence of collective polarization. A closer look at Eqs. (17) makes it clear that the relevant parameters in the local bistability of the polarization are f ubd i g ubd i+1 , and f ubd i+1 g ubd i . When normalized by the total concentration f 0 = F 0 / 0 as the unit of concentration, one can rewrite the above parameters in terms of f 0,i g 0,i+1 and f 0,i+1 g 0,i . Therefore, for disordered systems where f 0,i and g 0,i are no longer uniform, the local bifurcation point is randomized and the singular transition turns into a smooth cross-over.
Numerical solutions, Fig. (Ac) and (Ad), suggest that in the limit of small stochastic noise and initial bias, the steady state is not guaranteed to be uniformly polarized. The initial imbalance of protein distributions is defined as p 0 = |u 0 −v 0 |, with u 0 , v 0 , the spatial averages of initial dimers' concentrations. The bias δp 0 /p 0 , is the normalized magnitude of spatial fluctuations of initial polarity; small and large biases correspond to δp 0 /p 0 1 and δp 0 /p 0 1. While in ordered systems, a moderate initial bias suffices to achieve a uniform polarization, the patterns of polarity in highly disordered systems exhibit robustness, and are largely determined by the local geometry (disorder) of the array. Therefore as we observe in 1D, the quenched disorder imposes undesirable solutions, impairing the faithful transduction of directional information through PCP signaling. As we will see below, the situation gets only more complicated in two dimensions.
Mean-Field Solutions in 2D
The systems in one and two dimensions show inherently different behavior. In 1D, the proteins have only two junctions to localize. This limited number of choices and the resultant predictability are absent in two dimensions. Due to the large number of possible steady states in 2D, the initial configuration, as well as stochastic and geometrical disorders, influence the final state. We show, in this section, that nonlocal cytoplasmic interactions (NLCI) destabilize the vast majority of unpolarized fixed points in favor of the polarized ones. Furthermore, we find an optimal range of the NLCI length scale that assists with establishment of long-range alignment. In the following, we present the results for a set of parameters which lies within the polarized regime: in the units of f 0 = γ = 0 = 1, we set α = β = 5, κ = 10, and g 0 = 1.
The numerical solutions are presented below, but first let us attempt to gain some insight using analytical MF solutions for ordered tissues. Again, MF solutions imply uniform distributions of f ubd i g ubd j for all junctions {ij}, and recall that it may be justified by noting the diffusive nature of p, s dynamics, which in turn implies the diffusive dynamics of the free membrane proteins. We checked this assumption numerically. The value of f ubd i g ubd j , normalized by the mean, is plotted for all edges at the initial and final states in Fig. (Bb1) and (Bb2). In steady state, the standard deviation of the distribution is nearly independent of initial condition as well as the model parameters, and remains below 0.05, for both SLCI and NLCI regimes. Under this assumption, the RD equations take the same form as in 1D, except the total amount of proteins Fz and Vang are shared by six junctions instead of two. For ordered tissues, by virtue of sixfold symmetry, all junctions equally absorb the proteins, namely p, s are the same on all edges. Thus three edges would carry inward, and the other three carry outward dipoles.
Trivial mean-field solutions. Trivial mean-field solutions, in addition to the MF assumption discussed above, enjoy translational invariance of polarization along each of the three axes separately. The trivial MF solutions acquire two distinct configurations illustrated in Figs. (Ba1) and (Ba2), respectively. While the former represents a state with nonzero net polarization, the latter has zero net polarization. The solutions of the second type, are however destabilized when NLCI is included in the model. The net polarity in the polarized case can be calculated as follows:
p c = p e (1 − 2 cos θ),
where p e is the magnitude of polarization of one edge calculated above, and θ is the angle between the two adjacent edges in ordered hexagons, which equals p c = 2 s 2 − 4/αβ, for θ = 2π/3.
We also consider a different situation in which edges have unequal α, β's. At this point, different values of α, and β can have various origins, that are beyond the scope of our discussion. However in the Sec. (3), we argue that unequal parameters can be a consequence of nonlocal interactions in elongated cells. Assume the three pairs of parallel edges acquire coefficients α 1,2,3 and β 1,2,3 . We consider two scenarios: (i) α 1 > α 2 = α 3 , and (ii) α 1 = α 2 > α 3 . From the results we found in the case of sixfold symmetric lattices, the onset of bifurcation is inversely proportion to s * ∼ 1/ √ α. Thus in the first scenario, axis 1 is the first one to show instability upon increasing g 0 above g * 0 . Therefore, we have f ubd g ubd = γ/(kα 1 ), where,
f ubd = f 0 − s 1 1 + s 2 2 + s 3 3 2( 1 + 2 + 3 ) ,(20)
and similarly for g ubd . For the unpolarized edges we have:
s 2 2,3 + 2 γβ 2,3 γ − κf ubd g ubd α 2,3 s 2,3 − 2κf ubd g ubd = 0.(21)
Using kf ubd g ubd = γ/α 1 , we get:
s 2,3 = 1 β 2,3 1 − α 2,3 α 1 + 1 β 2 2,3 1 − α 2,3 α 1 2 + 2γ α 1 .(22)
Defining f eff 0 and g eff 0 ,
f eff 0 = f 0 − s 2 2 + s 3 3 2( 1 + 2 + 3 )
, thus:
f ubd = f eff 0 − s 1 1 2( 1 + 2 + 3 ) ,(23)
and similar expressions for g eff 0 and g ubd , we get for p 1 , s 1 :
p 2 1 = s 2 1 − 4 α 1 β 1 , s 1 = f eff 0 + g eff 0 − f eff 0 − g eff 0 2 + 4γ κα 1 .(24)
From the above analysis, we learn that in systems with unequal α's and β's, the problem essentially reduces to a one dimensional systems, with effective values for the concentrations of free proteins Fz and Vang. In the second scenario where α 1 = α 2 > α 3 , the third axis remains unpolarized as the other two are effectively more absorbent, and share the total bound proteins; thus are equally polarized with fourfold symmetry. As mentioned above, one situation of interest in which the coefficients α, β acquire unequal values on different edges, is the elongated tissues. In such systems, the above two cases correspond to Figs Nontrivial mean-field solutions. The nontrivial solutions in 2D satisfy a weaker constraint; they do not possess the translational invariance of cell dipoles. The full analytic treatment of this problem is cumbersome. We briefly touch upon the solutions to provide some insight into the richness of the basin of attraction for a system in SLCI limit. The constraint of nontrivial solutions demands the junctional polarization p and the total amount of proteins s to be identical for all junctions. Thus in practice the only constraint is that three junctions are positively polarized and the other three carry negative polarizations; i.e. three outgoing and three incoming dipoles; see Fig. (Ba3). The numerous configurations satisfying this condition, outnumber the trivial solutions by a large margin. Note that the trivial solutions also satisfy the above constraint, but are very special configurations. Therefore, if the segregation is not carried out systematically across the tissue, the membrane proteins are extremely unlikely to redistribute in a polarized fashion; unless the initial distribution to begin with is nearly polarized. As mentioned previously, nonlocal cytoplasmic interactions destabilize the majority of nontrivial solutions, and drive the system towards a segregated state where the polarized trivial solution appears. for randomized edges labels (ij), in SLCI and NLCI systems, respectively. The variance of the distributions decreases significantly over time, supporting the MF approximation.
Numerical Solutions in 2D
While in relatively ordered tissues with NLCI, and in the absence of orientational cues, the polarity is determined purely by chance (stochastic noise and initial conditions), highly disordered systems show robustness against such random factors. Instead, the fixed points of polarization fields are determined collectively by the geometry of the lattice. In finite-size systems, the geometrical disorder provides a bias towards one orientation over others. This effect gets progressively more pronounced with increasing range of NLCI and/or level of the quenched disorder. External cues of sufficiently large magnitudes, however, reorient the polarity. A typical configuration of the steady states is illustrated in Main Text Fig. (3a2). We find a range of interaction length scale, 0.2 λ/ 0 0.7, for which the NLCI guarantees the long-range alignment of polarization. The directional correlation shows a peak at around λ/ 0 0.5. This range of λ also depends on the geometric disorder, which hinders the establishment of polarization, and increases the lower bound of the range. For example, with 0 = 0.6, the range changes to 0.25 λ/ 0 0.7. Before addressing the segregation, we note that for λ/ 0 0.7, all edges within a cell strongly couple to each other, rendering the system fragile against stochastic noise.
Segregation mechanisms in SLCI vs. NLCI. Through a comparison of the dynamics of SLCI and NLCI in Figs. (3b1)
and (3b2) of the Main Text, the role of cytoplasmic nonlocal interactions in cell-cell interactions becomes evident. While the average polarization P (t) remains negligible in SLCI limit, it saturates to the average magnitude Q(t) in systems with NLCI, which reflects the angular correlation of cell-cell polarities. More elaborately, during the first stage of dynamics, the nonlocal cytoplasmic interactions prepare each and every cell for later intercellular communications. The coarsening and propagation of polarization is then carried out by cell-cell interactions, which increase with the magnitude of cellular dipoles Q i (t). Here, an important question arises, regarding the interpretation of Q(t): Can Q(t) be thought of as a measure of cellular segregation of proteins? A naïve guess would be that since Q i (t) is oblivious to the direction of polarity, it only measures the magnitude of the dipoles per cell, which is a candidate for quantifying segregation. This, however, warrants a careful investigation, as one can see that Q(t) shows arguably similar behavior in both SLCI and NLCI cases. Does this rule out the lack of segregation in SLCI? One possibility is that while the average Q(t) increases rapidly like in the NLCI case, the segregation is not achieved consistently in all cells. This is in part due to initial condition. Consider a tissue with randomly distributed proteins on the membranes. The magnitude of a cell's dipole increases due to the localization of some of the free proteins on the membranes. Above the polarization instability, namely for large enough g 0 , edges with higher initial concentration of a certain protein absorb more free protein of the same kind due to the cooperative interactions. Therefore, the final polarity depends, among other factors, on the initial condition. This is a separate effect from nonlocal interactions, and is built in the nonlinearity of RD equations regardless of the length-scale of nonlocal interactions. In order to directly measure the segregation, we define partial polarities as follows:
P F i = i dr r − R i |r − R i | u i (r),(25)
and similarly P G i is obtained by substituting v i (r) for u i (r). Perfect segregation then corresponds to P F i = −P G i in the steady state, regardless of the initial distributions. The less the level of segregation, the less the deviation from the initial condition over the time evolution: P F i ∼ P F i (t = 0) and P G i ∼ P G i (t = 0). Therefore, by comparing the final to initial partial polarities, one can easily signify the differences between NLCI and SLCI mechanisms, in terms of the cytoplasmic segregation. In order to quantify the segregation level, and compare that in the two mechanisms of SLCI and NLCI, we introduce the following quantities:
1. the spatial average of the magnitude of: S i = P F i + P G i . Note the vector sum, thus for perfect segregation we get: S i = 0. (Overlines mean spatial average over all cells at a given time). 2. the standard deviation of Q i normalized by its mean,
S(t)
Q(t) = |S i (t)| Q(t) = |P F i (t) + P G i (t)| Q(t) .(26)δQ(t) Q(t) ≡ |Q i (t) − Q(t)| 2 1/2 Q(t) ,(27)
The former characterizes the asymmetry of protein distributions, and the latter measures the consistency in segregation among the cells. We plot the above quantities as functions of time in Figs. (Ca1) and (Cb1), respectively. In (a1), while the ratio approaches zero in steady state for the system with NLCI, it remains finite ( 0.5) in the SLCI case, clearly showing the lack of segregation. In (b1) we see that in SLCI, the normalized standard deviation drops slightly from 0.5 at t = 0 to 0.4 in steady state, whereas in the NLCI case, it drops to nearly 0. The latter implies that segregation is fully achieved in all cells, and the individual cell polarities are very much close to the average polarity. Therefore, as suspected, in SLCI only the average value of Q grows, whereas cells are not coherently polarized across the tissue. In order to show explicitly the cell-by-cell distributions of initial and final values of S i and Q i , and compare SLCI and NLCI cases, we plotted these quantities in Fig. (C). In (a2) and (a3), the initial (red) and final (blue) cell-by-cell distributions of S i /Q i are illustrated for SLCI and NLCI, respectively. Again, the nearly zero final values of the ratio for NLCI shows perfect segregation. The cellular polarity normalized by spatial average at the corresponding time-point, i.e. initial and final, are shown in (b2) and (b3). The width of the distribution shrinks dramatically in NLCI, whereas it remains comparable to its initial value in the SLCI case.
Longitudinal vs. transverse correlations. As discussed above, and according to Eqs. (14) and (15), correlation function and length are dependent on the relative angle of the reference dipole and the connecting position vector, and at least in our case is stronger in the longitudinal compared to transverse directions. Intuitively, the polarity of a given cell points towards the edges with higher concentrations of localized proteins. These edges are shared with neighbors that are located rather longitudinally with respect to the axis of polarity. The polarities of these neighbors too, are influenced by the shared edges. Therefore longitudinal correlations are stronger than the transverse correlations. This discrepancy leads to formation of transient vortex-like structures, but the steady-state behavior of the correlation length remains qualitatively the same. We ignore this effect and compute the effective correlation length by averaging over all angles.
Vortices and saddles. Several theoretical [4,5,6] and experimental studies [7,8,9] have observed swirls and saddles as different forms of the so-called "topological defects". Such defects appear either as steady or transient patterns. Steady defects can be an indication of mutations of various origins; geometric [9], or genetic [10,11,4,12,13,14,11]. In Table (1) of the Main Text. (b) A system with g 0 = 0.25 (i.e. close to polarization threshold), and "wild-type" interaction length scale λ/ 0 = 0.5. Both systems exhibit swirling and crossing patterns that appear as long-lived steady patterns. We picked one ordered and one disordered tissue. However, in both cases of small λ and small g 0 , defects appear irrespective of the geometric disorder.
Drosophila wing, where Fz and Vang localize distally and proximally, respectively, the coupling to global cues is believed to be dependent on the presence of Fat [10,9]. While small fat − clones ( 10 cell diameters), exhibit little deviation from wild-type polarization, swirling patterns appear in larger clones, where the polarity is aligned over clusters of (roughly) 10 cells; also implying that Fz feedback loops are left intact in fat − patches. Therefore, the propagation of polarization across neighboring cells is carried out through Fz feedback loop, and the global alignment is achieved through coupling to the cues. Our model predicts both transient and steady swirls, depending on the values of the model parameters. We observe two distinct types of steady defects: (a) As λ is increased from SLCI to NLCI regime, there exists a narrow range 0.05 λ 0.15, over which vortices and saddles appear as long-lived patterns; Fig. (Da). (b) Another situation where swirls appear is when g 0 g * 0 , i.e. under-expression of one of the membrane proteins, say Vang, that is interpreted as a global mutation within the context of our model; Fig. (Db). Such patterns are indeed observed in Vang mutants [7]. Local mutations of various kinds are fully discussed in Main Text and below in SI. Sec. (4).
An interesting observation regarding the second type, i.e. g 0 g * 0 , is that for each specific realization of the geometrically disordered tissues, the characteristics of the steady-state pattern of polarity (e.g. positions of swirls) are left almost intact, as g 0 increases from 0.2 to 0.3. The magnitudes of dipoles, however, increase. Consequently, the swirls and branches gradually merge and align for g 0 0.3, and long-range polarity emerges. This behavior, in disordered tissues, is independent of the initial condition and stochastic noise, implying that the patterns of polarization are fixed by the the cell geometry of the tissue in regimes where the correlation lengths are of the order of a few cell diameters. In general, and as briefly mentioned above, in NLCI the geometrical information is detected through nonlocal interactions. The coupling to geometry provides local biases for cell dipoles. As g 0 increases, the correlation length increases and the global direction is chosen collectively.
Equal vs. unequal length scales of cytoplasmic interactions. In the Main Text we discussed the effects of unequal nonlocal cytoplasmic interactions that appear in upregulating and downregulating kernels, i.e. λ u and λ d . Here we elaborate on that discussion and consider various cases to investigate the role of these length scales, and the behavior of the angular correlation of polarity. In each case the geometry of the tissue is held fixed. The geometric disorder is denoted by 0 .
(1) For 0 = 0.5, find the angular correlation for different values of λ u = λ d = 0.01 -0.8. (1) For equal length scales, we observe that the angular correlation increases with λ. The standard deviation of polarity is found to be smaller than 30 • , for 0.2 λ 0.7, and is maximized at around λ 0.4 -0.5. For λ 0.7 the stochastic noise destroys the long-range polarization drastically. The reason is that the stabilizing and destabilizing forces compete at all points on the perimeter of each cell, namely all points are strongly coupled to one another. Thus the segregation becomes progressively more unstable as λ is increased; see (2),(3) For geometric disorder 0 0.45, and for 0.2 λ d 0.7 there is no detectable difference between the angular correlation of systems with different values of λ u 0.8. Beyond this value, the correlation declines slightly, and is eventually destroyed for larger values. In order to further examine the importance λ u in disordered tissues, we repeat the same simulations, but for larger geometric disorder 0 = 0.6. Interestingly we realize that larger disorder impedes the Directional cues. We consider two types of cues, bulk and boundary signals, each of which may be persistent or transient. Bulk cues couple to the F complex across the entire tissue, whereas boundary cues couple only at the boundaries. For bulk cues in, say +x direction, we use the gradient cues of constant slope in each cell: M ij (r) = M 0 (x ij − X i ); where M 0 is the slope of the cue, x ij is the x-coordinate of the points on junction (ij), and X i is that of the centroid of cell i. We simulated the response of the polarization field and observed that NLCI significantly enhances the sensitivity of the polarization field to such global cues. Before proceeding, we shall mention that there exist two time scales in this analysis: the response time scale of polarization field τ res , and the persistence time scale of the cue τ 0 . The results in a nutshell, are as follows:
✏ 0 = 0.6 ✏ 0 = 0.45 ✏ 0 = 0.6 ✏ 0 = 0.6 (a) (b) (c) (d) u = d u = d = 0.01 u = d = 0.1 u = d = 0.3 u = d = 0.5 u = d =
(1) Reorientation of dipoles for fixed τ res and τ 0 requires weaker cues in systems with NLCI than those with SLCI. For persistent cues (γτ m 100), NLCI responds to signals as small as M 0 0.05 over γτ res 2, whereas SLCI requires at least M 0 0.5. The minimum M 0 increases for smaller τ m 's. For example, in NLCI with M 0 0.05 a nearly persistent signal is required, whereas for larger cues M 0 1, even a rapid transient signal γτ m 1 is sufficient to rotate the dipoles over the same time scale. (2) In accord with (1), and due to the small correlation length in SLCI systems, the detection of a cue in these systems happens over exceedingly larger time scales, compared to NLCI with the same magnitude. (3) In the case of boundary cues (i.e. a column of polarized cells), in nearly ordered ( 0 0.2) tissues with zero stochastic noise, SLCI suffice to detect the signal. Presence of geometric disorder and/or stochastic noise, however, necessitates NLCI for the dipoles to align with the cue. Given that the onset of PCP alignment precede the geometrical ordering of the tissue [2], NLCI seems to be the key to the detection of directional cues. Finally, an interesting observation is that, (4) NLCI systems appear to detect sufficiently large initial boundary signals. Initial boundary signal is implemented by polarizing a column of cells, with significantly larger asymmetry of protein distributions compared to the bulk cells. This implies that a temporary boundary signal would in principle be able to rotate the dipoles, should the cytoplasmic interactions be nonlocal.
Elongated Cell Geometry
Measure of elongation. Elongation is commonly parametrized using a traceless matrix E. This matrix, the magnitude and the angle of elongation with respect to x-axis can be calculated as follows:
E i = ε i,1 ε i,2 ε i,2 −ε i,1 , E i = ε 2 i,1 + ε 2 i,2 , φ e i = 1 2 cos −1 (ε i,1 /E i ).(28)
For a tissue consisting of N c cells, and with ε 1 = N −1 c Nc i=1 ε i,1 , we get for the average elongation: In terms of the elongation index E, we plot in Fig. (Fb), the angle between y-axis and the steady-state polarization vector |Φ p − Φ e | (in degrees), versus different values of E, for a system with the same initial condition and same initial lattice (i.e. before stretching), but elongated along y-axis. Here, Φ p is defined as the angle of the axis between polarization and the x-axis, and over the domain of Φ p ∈ [0, π). Since Φ e = π/2 for elongation in the y direction, we have |Φ p −Φ e | ∈ [0, π/2]. Of course for E = 0, the axis of elongation is not defined, yet we measure it with respect to y-axis. It is important to note that this figure is only an example where the polarization for E = 0, happens to make a small angle with yaxis (25 • ). It is clear that depending on the geometry of the lattice and the initial condition, the polarization can be perpendicular to y-axis, even for E = 0. Therefore among different simulations, we chose one with a relatively large effect of elongation, so that the onset of perpendicular polarity becomes easily distinguishable. Illustrations of type I, II, and III mutants. The layout of the table is the same as that in the Main Text, and the red arrows show the directions of distortion with respect to the wild-type polarity. This table facilitates a more detailed comparison of the autonomous effects that were absent in Fig. (6) of the Main Text. In particular note the differences between the polarities within the putative dsh − and pk − clones, that were induced by small (α, β) and small λ, respectively.
E = N −1 c Nc i=1 E i , Φ e = 1 2 cos −1 (ε 1 /E).(29)
Elongation parameter E, measures the deviation from equilateral hexagons where E = 0. In ordered tissues, there are two possible choices for elongation axis: parallel to a pair of parallel edges, Fig. (Fa1); and perpendicular to a pair of edges; Figs. (Fa2,Fa3). While (a1) and (a2) are polarized perpendicular to the axis of elongation, (a3) exhibits parallel polarization. The latter, however, is destabilized by NLCI inhibiting localization of unlike complexes on adjacent junctions. In disordered tissues, elongation is a mixture of (a1) and (a2), both of which give rise to perpendicular polarization.
In anisotropic cells, the effective values of cooperative interactions are larger for the long junctions. Here, without deriving explicit expressions for the effective parameters in elongated system, we only argue that the longer junctions acquire larger coefficients α, β. As in the case of one dimension, searching for mean-field (MF) solutions, we assume that in the steady state the concentrations of bound proteins are translationally invariant along the three main axes of the lattice. In a self-consistent approach, the effective α, β are dependent on the concentrations of dimers on other edges. Therefore, assuming the system has reached its steady state, we can write the cooperative interactions as functions of the concentrations of proteins on all the edges, weighted by the geometrical factors originating from nonlocal interactions. For small ranges of NLCI, λ 0.7, the effect of other edges are negligible and only the self-interaction of each edge is to be taken into account. Intuitively this is because the self-interaction is a monotonically increasing function of the edge length. Therefore longer edges with NLCI, have larger values of α, β. Upon increasing λ, the interactions between all pairs of edges increase. However the qualitative behavior of effective α, β's for different edges does not change. With this in mind, and using the symmetry arguments between junctions with equal lengths, one can see that the cartoons in Figs. (Fa1), and (Fa2, Fa3) correspond to, i.e. α 1 > α 2 , α 3 and α 1 = α 2 > α 3 , respectively. Using the analysis sketched in Sec. (2.2), it is easy to see that in such systems the junctions with larger α, β, are more unstable towards polarization transition (remember the critical value is inversely related to α, β). Following the above discussion on different coefficients of cooperative self-interactions α s , and the equations derived in Sec. (2.2), we calculate these coefficients as functions of L/ 0 , for different λ's. Fig. (Fb) shows the numerical results for 0.2 ≤ λ/ 0 ≤ 1. In order to discern the effect of the elongation from that of nonlocal interactions, we normalize all the curves by the coefficients calculated for L = 0 , such that α s (1) = 1 for all λ's. The coefficient of self-interaction is a monotonically increasing function of the length of junction; hence polarity emerges perpendicular to the longer junctions, as depicted in Figs. (Fa1) and (Fa2).
Mutants and The Corresponding Phenotypes
In the Main Text we presented the schematics of phenotypes of three types of mutants. The schematics in the Main Text Fig. (9) only show the non-autonomous effects of each clone. Here we show the actual polarization field obtained from the simulations, in which the autonomous effects within the clones are also illustrated.
Type-I Mutants: Lack of membrane proteins. Type I lacks one or both of the membrane proteins, namely f 0 , g 0 . The non-autonomy is evident in all cases. In the first row of Fig. (G), non-autonomy is extended to multiple layers of cells from the clone boundaries. In the second row where the background lacks Vang as well, the non-autonomy is mostly limited to a single layer of surrounding cells.
Type-II Mutants: Lack of cytoplasmic interactions. The role of cytoplasmic proteins is deduced by comparing the in silico and in vivo phenotypes. We concluded that Dsh is mostly responsible for the local cooperative interactions with some contributions to the nonlocal part. Pk on the other hand is mainly involved in nonlocal interactions, through cytoplasmic diffusion. It can be seen in bottom left panel of Fig. (G), that in agreement with experiments, e.g. [4], the phenotypes show almost no non-autonomy. Their autonomous effects, however, are distinct. Lacking Dsh removes the cell polarity within the clone, whereas absence of Pk has only minor effects on the polarity of the clone.
Type-III Mutants: Disordered geometry. We tested the predictions of our model in the case of geometric irregularities. In Ref. [9], Ma, et.al. suggested that clones with enhanced geometric disorder-induced by under-expression of PTENobstruct the propagation of polarization field. This effect appears in tissues where global cues are absent. Polarity was observed to be retained when the global cue was added. As can be seen in the two right panels of the last row in Fig. (G), the system with lack of global cue shows swirling pattern of polarization. The wild-type polarity, however, reappears upon adding the gradient cue.
Figure A :
APolarization in one-dimensional arrays of cells. (a) cartoon of a 1D array of cells. Complexes with similar and opposite polarities, activate and inhibit each other on the interfaces. The edge lengths are denoted by i,i+1 . (b) shows the average polarizations against G 0 /F 0 , for different values of length disorder 0 = 0 to 0.6. In ordered arrays, the critical value is g * 0 0.23. The plot is obtained by ensemble averaging over 1000 realizations of quenched disorder in arrays of 1000 cells. For G 0 /F 0 = 0.3, (c) and (d) show the heatmaps of the cell polarities versus time (vertical axis), in an ordered array with a small bias, and in a highly disordered array ( 0 = 0.6) with a large initial bias.
. (Fa1), (Fa2), and (Fa3), respectively; see Sec. (3).
Figure B :
BMean-field solutions in two dimensions. (a1) and (a2) show the trivial MF solutions with nonzero and zero net polarities, respectively. While (a1) is a stable solution, (a2) is destabilized by NLCI. (a3) is an illustration of a nontrivial MF solution where cellular polarities point in random directions. (b1) and (b2) show the initial (red) and final (blue) distributions of f ubd i g ubd j
Figure C :
CCytoplasmic segregation in NLCI and SLCI regimes. (a1) The average value of the vector sum of the partial polarities defined in Eq. (26), divided by the average magnitude Q, as a function of time for SLCI and NLCI systems. Evidently, the ratio drops to zero for NLCI, implying full segregation. (b1) The normalized standard deviation of cell polarities defined in Eq. (27). Zero standard deviation in the presence of nonlocal interactions implies that, unlike in the SLCI case, segregation is achieved in these systems.
Figure D :
DSwirls and crosses in tissues with under-expression of cytoplasmic and membrane proteins. Two examples of persistent defects. (a) shows a system with λ/ 0 = 0.1. Other parameters are fixed at the values mentioned in
( 2 ) & ( 3 )
23For 0 = 0.45 and 0.6, with fixed λ d = 0.5, compare the angular correlation as λ u increases from 0.01 to 0.8. (4) For 0 = 0.5, and λ u = 0.5, compare the angular correlation for λ d = 0.01 -0.8.The findings are as follows:
Fig. (Ea).
Figure E :
EComparison of LA-NLI and NLA-NLI in disordered tissues. Rose plots illustrate, for different values of geometric disorder, 0 , the angular distributions of polarization fields in systems with equal and unequal length scales, λu and λ d . (a) Equal λ's, with 0 = 0.6. (b),(c) For λ d / 0 = 0.5 and 0.01 λu/ 0 0.8, the angular distributions are shown for 0 = 0.45 and 0 = 0.6. (d) Angular distributions for λu/ 0 = 0.5 and 0.01 λ d / 0 0.8, with geometric disorder 0 = 0.6. long-range correlation of polarity for λ u 0.4. The correlated polarity is retained for 0.4 λ u 0.7 and is declined again for larger values; see Fig. (Eb) and (Ec). (4) Systems with nonlocal activation and local inhibition are incapable of achieving long-range polarization. We fixed 0 = 0.6 and λ u = 0.5 and vary the inhibition length scale between λ d = 0.01 -0.8. The correlation grows as λ d increases; see Fig. (Ed). An intuitive argument for why small λ d fails to work is as follows: for λ d 0.8, the unlike complexes on the opposite sides of the cell inhibit each other, e.g. Fz on the right side, inhibits Vang on the left side.Thus, the Vang proteins sitting on the left side of the same cell are destabilized by Fz from across the cell. However, as far as polarity is concerned, the Vang protein on the opposite side of the cell is indeed contributing positively to the same direction of polarity as Fz. Therefore, for λ d 0.8, the polarization, even if initially correlated over long distances, becomes highly unstable against the stochastic noise. The same argument holds true for mutual stabilizing feedback of the like complexes if λ u 0.8, which contributes negatively to the alignment of the polarization field.Comparing the charts inFig. (E), we conclude that: (1) In general geometrical disorder distorts the alignment of polarization on large length scales. (2) Equal length scales of activation and inhibition is more efficient in retaining the long-range correlation, especially in highly disordered tissues. (3) Unlike the case of equal λ u = λ d , fixing λ u at an intermediate value, e.g. 0.5, allows for the other length scale to exceed λ d 0.7, without losing the angular correlation. Though the correlation eventually decays for larger values, λ d 0.9.
Figure F :
FElongated tissues and mean-field solutions. (a1) Shows the elongated system with elongation axis passing thorough a vertex. Since the cooperative interactions increase with length, the long junctions get polarized before the shorter edges can absorb complexes. This case is twofold symmetric, like a 1D array of cells that is extended perpendicular to the elongation axis. (a2) and (a3) show the alternative elongation axis perpendicular to one of the edges. In these cases there are four long edges competing to absorb membrane proteins. The possible configurations are now fourfold, two of which are polarized perpendicular, and the other two are polarized parallel to the axis of elongation; they are shown in (a1) and (a2), respectively. The latter is destabilized by the nonlocal cytoplasmic interactions. (b) For different values of λ/ 0 , the magnitude of cooperative self-interactions αs is plotted as a function of L/ 0 . (c) The angle between the average polarization and the axis of elongation, as a function of the average elongation index E (initial condition and geometry are held fixed). At E 0.1, the polarization and elongation axis are almost orthogonal; |Φp − Φe| 87 • .
Figure G :
GPatterns of polarity for different classes of in silico mutants.
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[
"Some Remarks on the Total CR Q and Q -Curvatures",
"Some Remarks on the Total CR Q and Q -Curvatures"
]
| [
"Integrability Symmetry ",
"Geometry "
]
| []
| [
"Methods and Applications SIGMA"
]
| We prove that the total CR Q-curvature vanishes for any compact strictly pseudoconvex CR manifold. We also prove the formal self-adjointness of the P -operator and the CR invariance of the total Q -curvature for any pseudo-Einstein manifold without the assumption that it bounds a Stein manifold. | 10.3842/sigma.2018.010 | [
"https://arxiv.org/pdf/1711.01724v2.pdf"
]
| 53,654,920 | 1711.01724 | 1a82650ce94a5dbdce5506cb02425f2cfd48b56b |
Some Remarks on the Total CR Q and Q -Curvatures
2018
Integrability Symmetry
Geometry
Some Remarks on the Total CR Q and Q -Curvatures
Methods and Applications SIGMA
1410201810.3842/SIGMA.2018.010Received November 09, 2017, in final form February 12, 2018;CR manifoldsQ-curvatureP -operatorQ -curvature 2010 Mathematics Subject Classification: 32V0552T15
We prove that the total CR Q-curvature vanishes for any compact strictly pseudoconvex CR manifold. We also prove the formal self-adjointness of the P -operator and the CR invariance of the total Q -curvature for any pseudo-Einstein manifold without the assumption that it bounds a Stein manifold.
Introduction
The Q-curvature, which was introduced by T. Branson [3], is a fundamental curvature quantity on even dimensional conformal manifolds. It satisfies a simple conformal transformation formula and its integral is shown to be a global conformal invariant. The ambient metric construction of the Q-curvature [9] also works for a CR manifold M of dimension 2n + 1, and we can define the CR Q-curvature, which we denote by Q. The CR Q-curvature is a CR density of weight −n − 1 defined for a fixed contact form θ and is expressed in terms of the associated pseudo-hermitian structure. If we take another contact form θ = e Υ θ, Υ ∈ C ∞ (M ), it transforms as
Q = Q + P Υ,
where P is a CR invariant linear differential operator, called the (critical) CR GJMS operator. Since P is formally self-adjoint and kills constant functions, the integral
Q = M Q,
called the total CR Q-curvature, is invariant under rescaling of the contact form and gives a global CR invariant of M . However, it follows readily from the definition of the CR Qcurvature that Q vanishes identically for an important class of contact forms, namely the pseudo-Einstein contact forms. Since the boundary of a Stein manifold admits a pseudo-Einstein contact form [5], the CR invariant Q vanishes for such a CR manifold. Moreover, it has been shown that on a Sasakian manifold the CR Q-curvature is expressed as a divergence [1], and hence Q also vanishes in this case. Thus, it is reasonable to conjecture that the total CR Q-curvature vanishes for any CR manifold, and our first result is the confirmation of this conjecture: Theorem 1.1. Let M be a compact strictly pseudoconvex CR manifold. Then the total CR Q-curvature of M vanishes: Q = 0.
For three dimensional CR manifolds, Theorem 1.1 follows from the explicit formula of the CR Q-curvature; see [9]. In higher dimensions, we make use of the fact that a compact strictly pseudoconvex CR manifold M of dimension greater than three can be realized as the boundary arXiv:1711.01724v2 [math.DG] 14 Feb 2018 of a complex variety with at most isolated singularities [2,10,11]. By resolution of singularities, we can realize M as the boundary of a complex manifold X which may not be Stein. In this setting, the total CR Q-curvature is characterized as the logarithmic coefficient of the volume expansion of the asymptotically Kähler-Einstein metric on X [15]. By a simple argument using Stokes' theorem, we prove that there is no logarithmic term in the expansion.
Although the vanishing of Q is disappointing, there is an alternative Q-like object on a CR manifold which admits pseudo-Einstein contact forms. Generalizing the operator of Branson-Fontana-Morpurgo [4] on the CR sphere, Case-Yang [7] (in dimension three) and Hirachi [12] (in general dimensions) introduced the P -operator and the Q -curvature for pseudo-Einstein CR manifolds. Let us denote the set of pseudo-Einstein contact forms by PE and the space of CR pluriharmonic functions by P. Two pseudo-Einstein contact forms θ, θ ∈ PE are related by θ = e Υ θ for some Υ ∈ P. For a fixed θ ∈ PE, the P -operator is defined to be a linear differential operator on P which kills constant functions and satisfies the transformation formula
P f = P f + P (f Υ)
under the rescaling θ = e Υ θ. The Q -curvature is a CR density of weight −n − 1 defined for θ ∈ PE, and satisfies Q = Q + 2P Υ + P Υ 2 for the rescaling. Thus, if P is formally self-adjoint on P, the total Q -curvature
Q = M Q
gives a CR invariant of M . In dimension three and five, the formal self-adjointness of P follows from the explicit formulas [6,7]. In higher dimensions, Hirachi [12,Theorem 4.5] proved the formal self-adjointness under the assumption that M is the boundary of a Stein manifold X; in the proof he used Green's formula for the asymptotically Kähler-Einstein metric g on X, and the global Kählerness of g was needed to assure that a pluriharmonic function is harmonic with respect to g. In this paper, we slightly modify his proof and prove the self-adjointness of P for general pseudo-Einstein manifolds: Theorem 1.2. Let M be a compact strictly pseudoconvex CR manifold. Then the P -operator for a pseudo-Einstein contact form satisfies
M f 1 P f 2 − f 2 P f 1 = 0 for any f 1 , f 2 ∈ P.
Consequently, the CR invariance of Q holds for any CR manifold which admits a pseudo-Einstein contact form: Theorem 1.3. Let M be a compact strictly pseudoconvex CR manifold which admits a pseudo-Einstein contact form. Then the total Q -curvature is independent of the choice of θ ∈ PE.
We note that Q is a nontrivial CR invariant since it has a nontrivial variational formula; see [13]. We also give an alternative proof of Theorem 1.3 by using the characterization [12, Theorem 5.6] of Q as the logarithmic coefficient in the expansion of some integral over a complex manifold with boundary M .
2 Proof of Theorem 1.1
We briefly review the ambient metric construction of the CR Q-curvature; we refer the reader to [9,12,13] for detail.
Let X be an (n + 1)-dimensional complex manifold with strictly pseudoconvex CR boundary M , and let r ∈ C ∞ (X) be a boundary defining function which is positive in the interior X. The restriction of the canonical bundle K X to M is naturally isomorphic to the CR canonical bundle
K M := ∧ n+1 (T 0,1 M ) ⊥ ⊂ ∧ n+1 (CT * M ). We define the ambient space by X = K X \ {0}, and set N = K M \ {0} ∼ = X| M .
The density bundles over X and M are defined by
E(w) = K X ⊗ K X −w/(n+2) , E(w) = K M ⊗ K M −w/(n+2) ∼ = E(w)| M for each w ∈ R.
We call E(w) the CR density bundle of weight w. The space of sections of E(w) and E(w) are also denoted by the same symbols. We define a C * -action on X by δ λ u = λ n+2 u for λ ∈ C * and u ∈ X. Then a section of E(w) can be identified with a function on X which is homogeneous with respect to this action:
E(w) ∼ = f ∈ C ∞ X | δ * λ f = |λ| 2w f for λ ∈ C * .
Similarly, sections of E(w) are identified with homogeneous functions on N . Let ρ ∈ E(1) be a density on X and (z 1 , . . . , z n+1 ) local holomorphic coordinates. We set
ρ = |dz 1 ∧ · · · ∧ dz n+1 | 2/(n+2) ρ ∈ E(0) and define J [ρ] := (−1) n+1 det ρ ∂ z j ρ ∂ z i ρ ∂ z i ∂ z j ρ .
Since J [ρ] is invariant under changes of holomorphic coordinates, J defines a global differential operator, called the Monge-Ampère operator. Fefferman [8] showed that there exists ρ ∈ E(1) unique modulo O(r n+3 ) which satisfies J [ρ] = 1 + O(r n+2 ) and is a defining function of N . We fix such a ρ and define the ambient metric g by the Lorentz-Kähler metric on a neighborhood of N in X which has the Kähler form −i∂∂ρ.
Recall that there exists a canonical weighted contact form θ ∈ Γ(T * M ⊗ E(1)) on M , and the choice of a contact form θ is equivalent to the choice of a positive section τ ∈ E(1), called a CR scale; they are related by the equation θ = τ θ. For a CR scale τ ∈ E(1), we define the CR Q-curvature by
Q = ∆ n+1 log τ | N ∈ E(−n − 1),
where ∆ = − ∇ I ∇ I is the Kähler Laplacian of g and τ ∈ E(1) is an arbitrary extension of τ . It can be shown that Q is independent of the choice of an extension of τ , and the total CR Q-curvature Q is invariant by rescaling of τ .
The total CR Q-curvature has a characterization in terms of a complete metric on X. We note that the (1, 1)-form −i∂∂ log ρ descends to a Kähler form on X near the boundary. We extend this Kähler metric to a hermitian metric g on X. The Kähler Laplacian ∆ = −g ij ∇ i ∇ j of g is related to ∆ by the equation
ρ ∆f = ∆f, f ∈ E(0) (2.1)
near N in X \ N . In the right-hand side, we have regarded f as a function on X.
For any contact form θ on M , there exists a boundary defining function ρ such that With this formula, we prove the following characterization of Q.
ϑ| T M = θ, |∂ log ρ| g = 1 near M in X,(2.
Lemma 2.1 ([15, Proposition A.3]).
For an arbitrary defining function ρ, we have
lp ρ> vol g = (−1) n (n!) 2 (n + 1)! Q,
where lp denotes the coefficient of log in the asymptotic expansion in .
Proof . Since the coefficient of log in the volume expansion is independent of the choice of ρ [15, Proposition 4.1], we may assume that ρ satisfies (2.2) for a fixed contact θ on M . We take τ ∈ E(1) such that ρ = τ ρ. Then, θ is the contact form corresponding to the CR scale τ | N . By the same argument as in the proof of [12, Lemma 3.1], we can take F ∈ E(0), G ∈ E(−n − 1) which satisfy
∆ log τ + F + Gρ n+1 log ρ = O ρ ∞ , F = O(ρ), G| N = (−1) n n!(n + 1)! Q.
We set G := τ n+1 G ∈ E(0). By (2.1) and the equation ρ ∆ log ρ = n + 1, we have
∆ log ρ − F − Gρ n+1 log ρ = n + 1 + O(ρ ∞ ).
Then, by using (2.4), we compute as
(n + 1) lp ρ> vol g = lp ρ> ∆ log ρ − F − Gρ n+1 log ρ vol g = −(n!) −1 lp ρ= N log ρ − F − Gρ n+1 log ρ · (1 + O( )) −n ϑ ∧ (dϑ) n = n + 1 n! M G θ ∧ (dθ) n = (−1) n (n!) 3 Q.
Thus we complete the proof.
Proof of Theorem 1.1. Let ρ be an arbitrary defining function of M , and τ ∈ E(1) the density on X defined by ρ = τ ρ. Then α := −i∂∂ log τ is a closed (1, 1)-form on X. The volume form of g is given by vol g = ω n+1 /(n + 1)! with the fundamental 2-form ω = ig jk θ j ∧ θ k . Near the boundary M in X, we have
ω = −i∂∂ log ρ = −i∂∂ log ρ + α.
Since the logarithmic term in the volume expansion is determined by the behavior of vol g near the boundary, we compute as
(n + 1)! lp ρ> vol g = lp ρ> (−i∂∂ log ρ + α) n+1 = lp ρ> α n+1 + lp ρ> n+1 k=1 n + 1 k (−i∂∂ log ρ) k ∧ α n+1−k .
The first term in the last line is 0 since α is smooth up to the boundary. Using −i∂∂ log ρ = d(ϑ/ρ) and dα = 0, we also have
lp ρ> (−i∂∂ log ρ) k ∧ α n+1−k = lp −k ρ= ϑ ∧ (dϑ) k−1 ∧ α n+1−k = 0.
Thus, by Lemma 2.1 we obtain Q = 0.
Proof of Theorem 1.2
We will recall the definitions of the P -operator and the Q -curvature. A CR scale τ ∈ E(1) is called pseudo-Einstein if it has an extension τ ∈ E(1) such that ∂∂ log τ = 0 near N in X.
The corresponding contact form θ is called a pseudo-Einstein contact form and characterized in terms of associated pseudo-hermitian structure; see [12,13,14]. If τ is a pseudo-Einstein CR scale, another τ is pseudo-Einstein if and only if τ = e −Υ τ for a CR pluriharmonic function Υ ∈ P. For any f ∈ P, we take an extension f ∈ E(0) such that ∂∂ f = 0 near M in X and define
P f = − ∆ n+1 f log τ | N ∈ E(−n − 1).
We note that the germs of τ and f along N is unique, and P f is assured to be a density by ∆ f | N = 0. The Q -curvature is defined by
Q = ∆ n+1 (log τ ) 2 | N ∈ E(−n − 1).
Here, the homogeneity of Q follows from the fact ∆ log τ | N = 0.
To prove the formal self-adjointness of P , we use its characterization in terms of the metric g. We define a differential operator ∆ by ∆ f = −g ij ∂ i ∂ j f . Since g is Kähler near the boundary, ∆ agrees with ∆ near M in X. . Let τ ∈ E(1) be a pseudo-Einstein CR scale and τ ∈ E(1) its extension such that ∂∂ log τ = 0 near N in X. Let ρ = ρ/ τ be the corresponding defining function. Then, for any f ∈ C ∞ (X) which is pluriharmonic in a neighborhood of M in X, there exist F, G ∈ C ∞ (X) such that F = O(ρ) and
∆ f log ρ − F − Gρ n+1 log ρ = (n + 1)f + O ρ ∞ . Moreover, τ −n−1 G| M = (−1) n+1 (n+1)!n! P f holds.
In the statement of [12,Lemma 4.4], the Laplacian ∆ is used, but we may replace it by ∆ since they agree near the boundary in X.
Proof of Theorem 1.2. We extend f j to a function on X such that ∂∂f j = 0 in a neighborhood of M in X. Let τ be a pseudo-Einstein CR scale and ρ = ρ/ τ the corresponding defining function. Then we have ω = −i∂∂ log ρ near M in X. We take F j , G j as in Lemma 3.1 so that u j := f j log ρ − F j − G j ρ n+1 log ρ satisfies ∆ u j = (n + 1)f j + O(ρ ∞ ). We consider the coefficient of log in the expansion of the integral
I = Re ρ> i∂f 1 ∧ ∂u 2 ∧ ω n + i∂f 2 ∧ ∂u 1 ∧ ω n − f 1 f 2 ω n+1 ,
which is symmetric in the indices 1 and 2. Since dω = 0, ∂∂f 2 = 0 near M in X, we have
i∂f 1 ∧ ∂u 2 ∧ ω n = d if 1 ∂u 2 ∧ ω n − if 1 ∂∂u 2 ∧ ω n + inf 1 ∂u 2 ∧ dω ∧ ω n−1 = d if 1 ∂u 2 ∧ ω n + 1 n + 1 f 1 ∆ u 2 ω n+1 + (cpt supp), i∂f 2 ∧ ∂u 1 ∧ ω n = −d iu 1 ∂f 2 ∧ ω n + (cpt supp),
where (cpt supp) stands for a compactly supported form on X. Thus,
I = ρ> 1 n + 1 f 1 ∆ u 2 − (n + 1)f 2 ω n+1 + Re ρ= i(f 1 ∂u 2 − u 1 ∂f 2 ) ∧ ω n + ρ> (cpt supp).
The first and the third terms contain no log terms. Since ω = d(ϑ/ρ) near M in X, the second term is computed as
Re ρ= i(f 1 ∂u 2 − u 1 ∂f 2 ) ∧ ω n = −n Re ρ= if 1 ∂ f 2 log ρ − F 2 − G 2 ρ n+1 log ρ ∧ (dϑ) n − i f 1 log ρ − F 1 − G 1 ρ n+1 log ρ ∧ ∂f 2 ∧ (dϑ) n + O ∞ .
The logarithmic term in the right-hand side is
log ρ= (n + 1)f 1 G 2 ϑ ∧ (dϑ) n + 2 −n log Re ρ= if 1 ∂f 2 ∧ (dϑ) n + O( log ).
The coefficient of log in the first term is
(−1) n+1 (n!) 2 M f 1 P f 2 . (3.1)
The second term is equal to 2 −n log Re ρ> i∂f 1 ∧ ∂f 2 ∧ (dϑ) n + −n log ρ> (cpt supp).
The first term in this formula is symmetric in the indices 1 and 2 while the second term gives no log term. Therefore, (3.1) should also be symmetric in 1 and 2, which implies the formal self-adjointness of P .
Proof of Theorem 1.3
The formal self-adjointness of the P -operator implies the CR invariance of the total Q -curvature. When n ≥ 2, the CR invariance can also be proved by the following characterization of Q in terms of the hermitian metric g on X whose fundamental 2-form ω = ig jk θ j ∧ θ k agrees with −i∂∂ log ρ near M in X:
Theorem 4.1 ([12, Theorem 5.6]). Let τ ∈ E(1) be a pseudo-Einstein CR scale and τ ∈ E(1) its extension such that ∂∂ log τ = 0 near N in X. Let ρ = ρ/ τ be the corresponding defining function. Then we have lp r> i∂ log ρ ∧ ∂ log ρ ∧ ω n = (−1) n 2(n!) 2 Q (4.1)
for any defining function r.
In [12,Theorem 5.6], it is assumed that X is Stein and ω = −i∂∂ log ρ globally on X, but as the logarithmic term is determined by the boundary behavior, it is sufficient to assume ω = −i∂∂ log ρ near M in X as above.
Proof of Theorem 1.3. Let τ , ρ be as in Theorem 4.1 and let ρ be the defining function corresponding to another pseudo-Einstein CR scale τ . Then we can write as ρ = e Υ ρ with Υ ∈ C ∞ (X) such that ∂∂Υ = 0 near M in X.
Using the defining function ρ for r in the formula (4.1), we compute as which implies that the third term is also 0. Thus, Q is independent of the choice of a pseudo-Einstein CR scale τ .
H := Ker ∂ρ ⊂ T 1,0 X. Then, N := Re ξ is smooth up to the boundary and satisfies N ρ = 1, ϑ(N ) = 0. Moreover, ν := ρN is ( √ 2) −1 times the unit outward normal vector filed along the level sets of ρ. By Green's formula, for any function f on X we have ρ> ∆f vol g = ρ= νf ν vol g . (2.3) Since the Monge-Ampère equation implies that g satisfies vol g = −(n!) −1 (1 + O(ρ))ρ −n−2 dρ ∧ ϑ ∧ (dϑ) n , the formula (2.3) is rewritten as ρ> ∆f vol g = −(n!) −1 ρ= N f · (1 + O( )) −n ϑ ∧ (dϑ) n . (2.4)
lp ρ> i∂ log ρ ∧ ∂ log ρ ∧ ω n = lp ρ> i(∂ log ρ + ∂Υ) ∧ (∂ log ρ + ∂Υ) ∧ ω n = lp ρ> i∂ log ρ ∧ ∂ log ρ ∧ ω n + lp ρ> i∂Υ ∧ ∂Υ ∧ ω n + 2 Re lp ρ> i∂ log ρ ∧ ∂Υ ∧ ω n .The second term in the last line is lp ρ> i∂Υ ∧ ∂Υ ∧ ω n = lp ρ= iΥ∂Υ ∧ ω n + lp ρ> (cpt supp) = 0.Since ω = d(ϑ/ρ) near M in X, we have ρ> i∂ log ρ ∧ ∂Υ ∧ ω n = log ρ= i∂Υ ∧ ω n +
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Variation of total Q-prime curvature on CR manifolds. K Hirachi, T Marugame, Y Matsumoto, 10.1016/j.aim.2016.11.005arXiv:1510.03221Adv. Math. 306Hirachi K., Marugame T., Matsumoto Y., Variation of total Q-prime curvature on CR manifolds, Adv. Math. 306 (2017), 1333-1376, arXiv:1510.03221.
Pseudo-Einstein structures on CR manifolds. J M Lee, 10.2307/2374543Amer. J. Math. 110Lee J.M., Pseudo-Einstein structures on CR manifolds, Amer. J. Math. 110 (1988), 157-178.
Volume renormalization for complete Einstein-Kähler metrics. N Seshadri, 10.1016/j.difgeo.2007.02.004math.DG/0404455Differential Geom. Appl. 25Seshadri N., Volume renormalization for complete Einstein-Kähler metrics, Differential Geom. Appl. 25 (2007), 356-379, math.DG/0404455.
| []
|
[
"Wide-Field CCD Photometry of the Globular Cluster M30",
"Wide-Field CCD Photometry of the Globular Cluster M30"
]
| [
"Eric L Sandquist [email protected] ",
"Michael Bolte [email protected] ",
"G E Langer [email protected] ",
"James E Hesser [email protected] ",
"C Mendes De Oliveira ",
"\nDepartment of Physics\nUCO/Lick Observatory\nUniversity of California\n95064Santa CruzCA\n",
"\nDominion Astrophysical Observatory\nInstituto Astronômico e Geofísico (IAG)\nColorado College\nHerzberg Institute of Astrophysics\nNational Research Council of Canada\n5071 West Saanich Road, Victoria, Av. Miguel Stefano 4200 CEP: 04301-90480903, V8X 4M6São PauloColorado Springs, CO, BCCanada;, Brazil\n"
]
| [
"Department of Physics\nUCO/Lick Observatory\nUniversity of California\n95064Santa CruzCA",
"Dominion Astrophysical Observatory\nInstituto Astronômico e Geofísico (IAG)\nColorado College\nHerzberg Institute of Astrophysics\nNational Research Council of Canada\n5071 West Saanich Road, Victoria, Av. Miguel Stefano 4200 CEP: 04301-90480903, V8X 4M6São PauloColorado Springs, CO, BCCanada;, Brazil"
]
| []
| We present new V I photometry for the halo globular cluster M30 (NGC 7099 = C2137-174), and compute luminosity functions (LFs) in both bands for samples of about 15,000 hydrogen-burning stars from near the tip of the red giant branch (RGB) to over four magnitudes below the main-sequence (MS) turnoff. We confirm previously observed features of the LF that are at odds with canonical theoretical predictions: an excess of stars on subgiant branch (SGB) approximately 0.4 mag above the turnoff and an excess number of RGB stars relative to MS stars.Based on subdwarfs with Hipparcos-measured parallaxes, we compute apparent distance moduli of (m − M) V = 14.87 ± 0.12 and 14.65 ± 0.12 for reddenings of E(V − I) = 0.06 and 0.02 respectively. The implied luminosity for the horizontal branch (HB) at these distances is M HB V = 0.11 and 0.37 mag. The two helium indicators we have been able to measure (R and ∆) both indicate that M30's helium content is high relative to other clusters of similar metallicity. M30 has a larger value for the parameter ∆V HB T O than any of the other similarly metal-poor clusters for which this quantity can be reliably measured. This suggests that M30 has either a larger age or higher helium content than all of the other metal-poor clusters examined. The color-difference method for measuring relative ages indicates that M30 is coeval with the metal-poor clusters M68 and M92. | 10.1086/307268 | [
"https://export.arxiv.org/pdf/astro-ph/9810259v2.pdf"
]
| 55,434,898 | astro-ph/9810259 | 58b4ebe4e59edcb371c65267b2828bdd19dc4392 |
Wide-Field CCD Photometry of the Globular Cluster M30
arXiv:astro-ph/9810259v2 17 Oct 1998
Eric L Sandquist [email protected]
Michael Bolte [email protected]
G E Langer [email protected]
James E Hesser [email protected]
C Mendes De Oliveira
Department of Physics
UCO/Lick Observatory
University of California
95064Santa CruzCA
Dominion Astrophysical Observatory
Instituto Astronômico e Geofísico (IAG)
Colorado College
Herzberg Institute of Astrophysics
National Research Council of Canada
5071 West Saanich Road, Victoria, Av. Miguel Stefano 4200 CEP: 04301-90480903, V8X 4M6São PauloColorado Springs, CO, BCCanada;, Brazil
Wide-Field CCD Photometry of the Globular Cluster M30
arXiv:astro-ph/9810259v2 17 Oct 1998Received ; acceptedSubject headings: globular clusters: individual (M30) -stars: luminosity function -stars: abundances -stars: distances -stars: interiors
We present new V I photometry for the halo globular cluster M30 (NGC 7099 = C2137-174), and compute luminosity functions (LFs) in both bands for samples of about 15,000 hydrogen-burning stars from near the tip of the red giant branch (RGB) to over four magnitudes below the main-sequence (MS) turnoff. We confirm previously observed features of the LF that are at odds with canonical theoretical predictions: an excess of stars on subgiant branch (SGB) approximately 0.4 mag above the turnoff and an excess number of RGB stars relative to MS stars.Based on subdwarfs with Hipparcos-measured parallaxes, we compute apparent distance moduli of (m − M) V = 14.87 ± 0.12 and 14.65 ± 0.12 for reddenings of E(V − I) = 0.06 and 0.02 respectively. The implied luminosity for the horizontal branch (HB) at these distances is M HB V = 0.11 and 0.37 mag. The two helium indicators we have been able to measure (R and ∆) both indicate that M30's helium content is high relative to other clusters of similar metallicity. M30 has a larger value for the parameter ∆V HB T O than any of the other similarly metal-poor clusters for which this quantity can be reliably measured. This suggests that M30 has either a larger age or higher helium content than all of the other metal-poor clusters examined. The color-difference method for measuring relative ages indicates that M30 is coeval with the metal-poor clusters M68 and M92.
Introduction
This paper is the second in a series investigating the evolved-star populations in nearby globular clusters. With the large-field CCD imagers now available it is possible to measure nearly complete samples of giant stars in clusters, and at the same time measure stars faint enough that we can normalize the luminosity functions (LFs) to the unevolved main sequence. Because the LFs for evolved stars directly probe the timescales and fuel consumed in the different phases of stellar evolution, they provide a stringent test of the models for the evolution of low-mass stars. These models are the basis for our use of globular clusters to set a lower limit to the age of the Universe and are a fundamental tool in the interpretation of the integrated spectra and colors of elliptical galaxies.
The subject of this study is M30 (NGC 7099 = C2137-174), a relatively nearby cluster (∼ 7 kpc; Peterson 1993) at high galactic latitude (b = −46 • .8). M30 has a high central density (log (ρ 0 /(M ⊙ /pc 3 )) = 5.9), a moderate total mass (log(M/M ⊙ ) = 5.3; Pryor & Meylan 1993), and is at the metal-poor end of the cluster [Fe/H] distribution. It is one of approximately 10% of clusters that have cusps at the core of their surface brightness profiles, and it also has one of the largest radial color gradients of any cluster (Stetson 1991b).
Previous studies of the LF for stars in metal-poor clusters have uncovered unexpected features. In a LF formed from the combination of CCD-based observations of the clusters M68 (NGC 4590 = C1236-264), NGC 6397 (C1736-536), and M92 (NGC 6341 = C1715+432), Stetson (1991a) found an excess of stars on the subgiant branch (SGB) just above the main-sequence turnoff (MSTO). Bolte (1994) and Bergbusch (1996) both observed M30 using a mosaic of small-field CCD images and found an excess of SGB stars. (The SGB is defined here as the transitional region between the main-sequence turnoff and the base of the red giant branch at the point of maximum curvature.)
Another unexpected observation involving LFs is a mismatch between theoretical predictions and the observed size of the "jump" dividing the main sequence (MS) and the red giant branch (RGB). When normalized to the MS, there is an excess of observed RGB stars compared to models (Stetson 1991a, Bergbusch & VandenBerg 1992, Bolte 1994, Bergbusch 1996, although this has been disputed by Degl'Innocenti, Weiss, & Leone (1997). These results might be explained by the action of core rotation (VandenBerg, Larson, & DePropris 1998), or perhaps (as discussed later) we are witnessing the results of deep mixing and the delivery of fresh fuel into the hydrogen shell-burning regions. Langer & Hoffman (1995) suggested that, if the abundance patterns of light elements seen in bright cluster giants (e.g. Kraft 1994) are due to deep mixing, hydrogen-rich envelope material is almost certainly mixed into the hydrogen burning shell (prolonging the giant phase of evolution), and some of the helium produced is returned to the envelope. Because of the potential importance of such non-standard physics in stars, and because of the caveats associated with earlier LF studies, the most productive next step is to derive better LFs in a number of Galactic globular clusters (GGCs).
In the next section, we describe our observations of the cluster. In §3, we discuss the features observed in the color-magnitude diagram, describe the method of computing the luminosity functions, and present the results of artificial star experiments. In §4, we discuss the constraints that can be put on the global parameters of the cluster -metallicity, distance, and age. The method of data reduction is described in Appendix A.
Observations
The data used in deriving the V -and I-band LFs of M30 were taken on July 7/8, 1994 at the Cerro Tololo Inter-American Observatory (CTIO) 4 m telescope. In all, six exposures of 120 s, one exposure of 60 s and two exposures of 10 s were made in V , and six exposures of 120 s, one exposure of 60 s, and one exposure of 10 s were made in I. All frames were taken using the 2048 × 2048 pixel "Tek #4" CCD chip, which has a sampling of about 0 ′′ .44 per pixel, and a field 15 ′ on a side. These exposures were reduced individually for the purpose of constructing the color-magnitude diagram. In performing artificial star experiments and deriving the LF, the three best-seeing images in both V -and I-bands were combined into master long-exposure images. The frames were centered approximately 2 ′ east of the cluster center, in order to avoid a bright field star nearby.
The night of the 4 m observations was not photometric. In order to set the observations on a standard photometric system, we used observations made at the CTIO 1.5 m telescope on one photometric night (October 18/19, 1996). The detector used was the "Tek #5" 2048 × 2048 CCD, having a field of about 14 ′ .8 on a side. Landolt (1992) standard star observations were used to calibrate a secondary field that overlapped the 4 m field. On that night, 10 s and 120 s exposures were taken in each band, along with exposures of 27 standards in 7 Landolt fields. A sample of 118 stars having 12.9 < V < 1.5 and −0.03 < (V − I) < 1.31 was calibrated as secondary standards in this way. The field was centered approximately 5 ′ south of the cluster center.
During the same run on the 1.5 m telescope, frames were taken of M30 on the non-photometric night of October 16/17. Five additional exposures were taken in each band (20 s, 200s, and 3×600s in V, and 15 s, 180 s, and 3×600s in I). The details of the data reduction and calibration are described in Appendix A.
The Color-Magnitude Diagram (CMD) and Luminosity Functions (LFs)
3.1. The CMD
In Figure 1, we plot the total V I sample of 25279 stars (upper panel) and a sample that has been restricted in projected radius to 110 ′′ < r < 10 ′ from the cluster center (lower panel). The inner radius was chosen in accord with the restriction placed on stars to be used later in the LFs, while the outer radius restriction was chosen to exclude regions that were affected by field star contamination.
Fiducial points for the MS and lower RGB of the clusters were determined by finding the mode of the color distribution of the points in magnitude bins. The fiducial line on the upper RGB was determined by finding the mean color of the stars in magnitude bins. Once a mean was determined, stars falling more than 7σ from the fiducial point were discarded (so as to eliminate AGB and HB stars, as well as blends and poorly measured stars), and the mean redetermined. This procedure was iterated until the star list did not change between iterations. At the tip of the RGB and on the AGB, the positions of individual stars were included as fiducial points if they appeared to be continuations of the mean fiducial line. The fiducial line for the HB was obtained by determining mean points in magnitude bins for the blue tail, and in color bins for the horizontal part of the branch. No smoothing has been applied. Table 1 lists the fiducial lines for our samples, as well as the number of stars used in computing each point.
The LFs and Incompleteness
The procedure used to correct the "observed" LF back to the "true" LF is described in detail in Sandquist et al. (1996). As in that paper, we carried out artificial star tests on only four frames: a long exposure frame (composed of the average of the three best-seeing images) and a short exposure frame in both V and I band. In 11 runs, 20965 artificial stars were processed. ALLFRAME's coordinate transformations were found to be unable to follow the nonuniform spatial distortions introduced by the 4 m field corrector. To avoid this problem, we reduced all of the frames through ALLSTAR as usual, derived a master detected star list for each filter, and rereduced the frames in ALLSTAR with the improved positions. This procedure improved the overall quality of the photometry (as judged by the scatter around the fiducial lines of the cluster), as ALLFRAME normally does.
In Figures 2 -4 we plot our computed values for median magnitude biases δ V , median external error σ ext (V ), and completeness probability f (V ) as a function of magnitude and radius. There is little variation in most of the quantities until the innermost radial region (r < 2 ′ .0, where crowding of stellar images is worst) is reached.
One change we have made since our first study was in the error estimation. The uncertainty in the incompleteness factor f was previously found by simultaneously varying F, σ, and δ in such a way as to cause the maximum change in f away from our best value. The magnitude of this change was used as the error estimate. We have improved this, following a suggestion by Bergbusch (1996), by estimating the error by varying F, σ, and δ individually and adding the resulting error estimates in quadrature.
Using this information, we eliminated stars from consideration for the LF if they fell far enough away from the nearest point on the fiducial line of the cluster. The "distance" was defined in terms of difference in magnitude and color divided by their respective external errors, and then added in quadrature. So, stars were eliminated if (∆V /σ ext (V )) 2 + (∆(V − I)/σ ext (V − I)) 2 > 25. On the upper RGB, where contamination by the AGB could be a factor, we adjusted the error cutoff by hand until we were sure the AGB stars were being eliminated, but not at the expense of the RGB stars.
The luminosity functions are listed in Tables 2 and 3. Totals of 14772 and 14507 stars were used in creating the V -and I-band LFs. Stars on the MS and SGB were only included if they fell more than 2 ′ .0 from the center of the cluster. This reduced the significance of crowding effects on the photometry. Stars with magnitudes V < 16.9 (or I < 17.0) were included in the determination of the LF to within 30 ′′ of the center of the cluster. This was done to get a better indication of the "global" LF for the red giants since mass segregation is expected to occur in M30, which would affect the RGB star counts taken from a small range of radii.
Discussion
Reddening and Metallicity
Most previous studies have made estimates of E(B − V ). We will assume E(V − I) = 1.25E(B − V ) (Cardelli, Clayton, & Mathis 1989) for the rest of the discussion and refer to the reddening in E(V − I). M30 is situated well out of the Galactic plane (b II = −47) and the reddening is likely small (E(V − I) < 0.1). Reddening maps of Burstein & Heiles (1982) indicate that E(V − I) < 0.04. Reed, Hesser & Shawl (1988) derive a negative reddening based on the comparison of M30's integrated color and spectral type. Zinn (1980) gets a value of 0.01 from integrated-light measurements. Reed, Hesser, & Shawl's data indicates that M30 appears to have the bluest intrinsic colors of all of the globular clusters they examined. However, M30 is the cluster that shows the strongest color gradient of any GGC and reddening values measured from integrated colors or spectra will depend on the range of radii over which the observations are made. Given the sense of the gradient, bluer towards the center, it seems likely that these reddening measurements will be biased to lower values. Dickens (1972) and Richer, Fahlman, & VandenBerg (1988;hereafter RFV) independently derived significantly higher values, E(V − I) ∼ 0.08, based on the UBV colors of blue HB stars in M30. On the other hand, U-band photometry is notoriously difficult to calibrate and the precision of reddening estimates from color-color plots is ∼ 0.04 mag even in the best cases. Neither the Dickens nor the RFV study appears to have had a photometric calibration good enough to warrant error bars less than this. Differential CMD comparisons with M92 (Vandenberg, Bolte, & Stetson 1990; hereafter VBS) and M68 ( Figure 7) imply E(V − I) ∼ 0.05 to 0.06. The recent reddening maps based on IRAS and COBE measurements of far-IR flux from infrared cirrus suggest E(V − I) ∼ 0.063.
We can make our own estimate using using Sarajedini's (1994) simultaneous reddening and metallicity method (for V HB = 15.04 ± 0.08 and (V − I) g = 0.97 ± 0.02, where the quoted errors allow some room for calibration errors). We find E(V − I) = 0.065 ± 0.003. The errors were derived from Monte Carlo tests with the quoted errors on V HB and (V − I) g . Most of the reddening estimates we have thus far indicate a relatively high value E(V − I) = 0.06 ± 0.02.
The compilation of Zinn & West (1984) has [Fe/H] = −2.13 ± 0.13 and numerous studies since have determined values between −1.9 and −2.3 (Geisler, Minniti, & Clariá 1992;Minniti et al. 1993;. From the simultaneous reddening and metallicity method above, we find [Fe/H]= −2.01 ± 0.09. Anticipating later discussion of the level of the HB, we examine the effects of an anomalously high M V (HB) on the simultaneous reddening and metallicity method. A high M V (HB) value can come about due to a high helium abundance, whether primordial or the result of a "deep mixing" scenario (Langer & Hoffman 1995). If the "true" M V (HB) value (in the absence of helium enrichment) is fainter by 0.10 mag, then we would have V HB = 15.14, and calculate [Fe/H] = −2.11 ± 0.07 and E(V − I) = 0.068 ± 0.002.
Distance Modulus
Recently Reid (1997), Gratton et al. (1997) and Pont et al. (1998) have used subdwarf parallaxes measured with the Hipparcos satellite (ESA 1997) to redetermine the distance moduli to several of the best observed clusters. The general result of these studies is to increase cluster distance moduli by 0.2 to 0.4 mag, implying high luminosities for the horizontal branches (as bright as M V ∼ 0.1 mag for the [Fe/H]∼ −2 clusters). For M30 specifically, Gratton et al. find (m-M)= 14.96 ± 0.10 (for E(V − I) = 0.05) although this is based on only three subdwarfs. We repeat this exercise for M30 with our data and a larger set of subdwarfs. This method is sensitive to uncertainties in the color of the unevolved main-sequence, which result from zero-point errors in the photometric calibration, the reddening uncertainties already mentioned, and uncertainties in the placement of the main-sequence fiducial line. Our data for M30 are not optimum for using subdwarf fitting to measure the distance, primarily because of the uncertainty in the reddening, but also because we suffer from each of the other problems to a small degree.
Nevertheless, we selected subdwarfs that satisfy the following criteria: parallaxes from the Hipparcos mission having relative errors σ π /π < 0.15, metal abundances from the study of Gratton, Carretta, & Castelli (1996), and V I photometry. The restriction on parallax error was chosen to minimize the effect of bias corrections. (As a result, Lutz-Kelker corrections only change our derived distance moduli by 0.01 mag.) The Gratton et al. metallicity scale was chosen for its homogeneity and because it minimizes the possibility of systematic abundance errors with respect to Carretta & Gratton globular cluster metallicities. When available, we used V I photometry for the subdwarfs tabulated in Mandushev et al. (1996). In the remaining cases, we followed their procedure of combining literature values from the following sources: Carney & Aaronson (1979), Carney (1980Carney ( , 1983b, and Ryan (1989Ryan ( , 1992. The studies involving Carney all used the Johnson I filter, so we applied the transformation from Carney (1983a) to convert them to the Cousins system. Known spectroscopic binaries were excluded, and, for the cases where it has been measured, any reddening of the subdwarfs (at most a very small amount) has been subtracted. Our sample of subdwarfs and metal-poor subgiants is shown in Table 4. We used the subdwarfs to estimate the distance to M30 in two different ways.
First, to simultaneously estimate the distance and E(V − I) of M30 we created a grid of chi-square-like sums. The M30 main sequence between 19 < V < 22.8 was represented by a third-order polynomial. For ranges of m-M and E(V − I), the minimum distance of each subdwarf from the main-sequence polynomial was calculated. This distance was normalized by the combined errors in the subdwarfs' colors and magnitudes and the main-sequence fiducial uncertainties in color and magnitude (assumed to be 0.04 mag in color and 0.05 mag in V magnitude). Our "χ 2 " sum is:
χ 2 = i (V i − V fiducial ) 2 + ((V − I) i − (V − I) fiducial ) 2 σ 2
We will refer to this as the chi-square sum although it does not match the usual definition of chi-square and it is not normalized in a way to give true confidence intervals. Because the slope of the main-sequence changes with V , there is a different weighting given to ∆V and ∆(V −I) for each subdwarf. The minimum chi-square values are for m−M∼ 14.6 and E(V − I) ∼ 0.02, although the reddening in particular is poorly constrained.
Our second approach was to fit the M30 main sequence to the subdwarfs using only the distance modulus as a variable for two E(V − I) values: 0.02 and 0.06. Table 5 shows the changes in this value for different subsets of our sample and for different input data. (The quoted errors include contributions from the scatter of values in the subdwarf fit, a cluster reddening error of 0.02 mag, and the absolute cluster metallicity error of 0.2 dex.) Clearly, reddening uncertainty is dominant in the total uncertainty. Two fits at different reddening are shown in Figure Considering the number of distance modulus measurements for M30 in the literature, it is best to try to compare using a common reddening of E(V − I) = 0.02 (and using ∂(m − M) V /∂E(V − I) ≈ 6 to correct values). From the two methods we presented here we have 14.6 and 14.65. From the pre-Hipparcos ground-based measurements of Bolte (1987) and RFV, we have 14.62 and 14.68 (the RFV value must also be corrected for their lower assumed metallicity for M30). For the Hipparcos-based distances of Reid (1997) and Gratton et al. (1997), we find 14.75 and 14.87 (although Gratton et al. use only three subdwarfs in their fit). It is clear that our distance moduli are consistent with the ground-based measurements, and over 0.1 mag smaller than the other Hipparcos-based measurements. Neverthless, the different distance measures are in good general agreement, at least for a fixed reddening value. Although it is not crucial for the conclusions that follow, we will adopt (m-M) V = 14.7 for the rest of the paper. From Figure 5, it is clear that the smaller reddening and distance modulus values are more consistent with the model isochrones and, for our preferred larger reddening value of E(V − I) = 0.06, the models do not match the shape of the M30 fiducial above the turnoff.
The R Method
The ratio R = N HB /N RGB , where N RGB is the number of RGB stars brighter than the luminosity level of the HB, is the traditional quantity used to estimate the helium abundance of stars in globular clusters. Dickens (1972) first noted that M30's value for "R" was unusually large and Alcaino & Wamsteker (1982) claimed a significant gradient in this ratio in the sense of a small value at the center of the cluster increasing to one of the largest measured in any cluster at large radii. We have sufficient numbers of stars on the red and blue sides of the instability strip to define the level of the HB. From five stars near the blue edge of the instability strip (with colors 0.25 < (V − I) < 0.35), we calculate an average magnitude V HB,b = 15.08 ± 0.06. From six stars on the red side of the instability strip, we find V HB,r = 14.90 ± 0.06. All stars used in these averages were found at radii greater than 1 ′ .0 from the center of the cluster, so that the photometry should be quite accurate.
Because the estimate of the HB magnitude is crucial in later arguments, we examine the topic further. Our blue side estimate is consistent with those of Bolte (1987) and Bergbusch (1996), even with the different calibrations of our respective datasets. Because it is possible that the red HB stars in M30 are evolved, it is wise to check this possibility before simply interpolating between the two sides. The number of stars (8 with projected radius r > 1 ′ .0, compared to 93 on the blue HB) at V ∼ 14.9 is roughly consistent with timescales for stars evolving from the blue HB, as the evolutionary tracks tend to parallel each other closely through this part of the CMD as they move toward the AGB. Thus, we have chosen to look at other clusters of similar metallicity with large, well-studied RR Lyrae populations to compute a magnitude correction to go from the red edge of the blue HB into the middle of the instability strip. For the clusters M15 (Bingham et al. 1984), M68 (Walker 1994), and M92 (Carney et al. 1992) we find agreement on a correction of −0.04 mag. Thus, we have V HB = 15.04 ± 0.06.
To define the RGB star sample for the R method, we need to establish the relative bolometric magnitudes of the HB and RGB stars. We have used the HB models of Dorman (1992) in conjunction with the isochrones of Bergbusch & VandenBerg (1992;hereafter BV92). The stellar models involved in these studies were computed with a consistent set of physics and compositions. Although the composition used is somewhat out-of-date, the differential bolometric corrections should be fine. The corrections as a function of [Fe/H] can be approximated by: 3), although for this range of metallicities, changes in metallicity have a very small effect on ∆V BC . Because of contamination and blending problems toward the center of the cluster, we restrict our samples of RGB and HB stars to r > 20 ′′ . With this choice, we find R = 1.45 ± 0.20. This makes M30's R the highest of any of the clusters examined. In Table 6, we present R values that have been derived from published photometry for the clusters similar in [Fe/H] to M30 -M68 (Walker 1994), M53 (Cuffey 1965), NGC 5053 (Sarajedini & Milone 1995), NGC 5466 (Buonanno et al. 1984), and M15 ) -along with several more metal-rich clusters.
∆V BC ≡ V RGB − V HB = 0.709 + 0.548[M/H] + 0.229[M/H] 2 + 0.034[M/H] 3 .
Photometry exists for the central 25 ′′ of M30 from the Hubble Space Telescope (Yanny et al. 1994;hereafter YGSB). By merging their list with ours and eliminating common stars, we have created a master list of HB and RGB stars that completely covers the cluster out to 7 ′ .0, with portions included out to about 12 ′ . The data for the full sample is presented in Table 7. Using this sample, we find a global value for R of 1.49 ± 0.18. (Even if our value of V HB is too bright, and if we use the magnitude of the red edge of the blue HB, we find R = 1.36 ± 0.21 -still a high value in a relative sense.)
This global R value for M30 is on firm ground, because there is no place for the bright stars to hide. The photometry is easily good enough to distinguish between the AGB and RGB star samples, so this is not a source of uncertainty. (In fact, M30 may be deficient in AGB stars as well as RGB stars.) The use of the lower Buzzoni et al. (1983) value ∆V BC = 0.15 for the differential bolometric correction would increase the high R value. Approximately 30 additional RGB stars would have to be included to bring M30's value in line with that of other clusters -a 24% change in the sample size.
There are a few potential explanations for a global depletion of bright giants in this cluster. First, because the ratio R is a helium abundance indicator, the abnormally high value could indicate a higher-than-average helium abundance in M30 stars more luminous than the HB. This does not necessarily imply high Y for lower luminosity stars, since a deep-mixing mechanism would also be expected to dredge up freshly produced helium (Sweigart 1997). Second, the environment within the cluster may affect the stellar populations by a mechanism that truncates RGB evolution and/or produces additional HB stars. We now consider the arguments for the two sides.
Helium Abundance
Using the Buzzoni et al. (1983) calibration and our M30 R value, we find Y = 0.24 ± 0.02. The value derived using V HB of the blue edge of the instability strip gives Y = 0.23 ± 0.02. That V HB is definitely a faint limit, making Y = 0.23 a lower limit. In any case, the R value for M30 is significantly higher than those for other clusters when the revised differential bolometric corrections are used. [The low value of Y (R) for the other clusters is discussed in Sandquist (1998).]
One check we can make is to examine other helium indicators to see if they also indicate a high abundance relative to other clusters. Caputo, Cayrel, & Cayrel de Strobel (1983) introduced two indicators: A (the mass-luminosity relation for RR Lyrae stars of ab type) and ∆ (the magnitude difference between the HB and the point on the MS where the dereddened B − V color is 0.7). While M30's RRab variables have not been studied to the extent necessary to compute A, we can compute a ∆ value from the BV photometry of Bolte (1987) and RFV. Assuming for both that E(V − I) = 0.06 ± 0.02, we find 6.42 ± 0.12 and 6.28 ± 0.13 respectively, where the primary contributions to the error are the uncertainty in the reddening and the small number of stars used to define the HB magnitude. (The lower RFV value can be traced to a fainter HB magnitude relative to Bolte's data.) We have computed comparison values for the clusters M68 (McClure et al. 1987, Walker 1994, and M15 (Durrell & Harris 1993), as summarized in Table 6 We can directly compare our data with V I for other clusters if we redefine ∆ (hereafter, ∆ B−V ) by choosing the MS point to have (V − I) 0 = 0.85. From isochrones this is approximately equivalent to (B − V ) 0 = 0.7. From our fiducial line, we find ∆ V −I = 6.36 ± 0.10 for E(V − I) = 0.06 ± 0.02. V I data exist for M92 (Johnson & Bolte 1998) and M68 (Walker 1994). We find that ∆ V −I = 6.25 ± 0.12 (E(V − I) = 0.025 ± 0.0125) and 6.11 ± 0.07 (E(V − I) = 0.0875 ± 0.0125), respectively. The M92 value relies essentially on one star for the HB magnitude, so the ∆ V −I value is uncertain.
For ∆ B−V and ∆ V −I we see evidence (though not overwhelming) that the M30 value is high compared to other clusters. The values above indicate that M30's helium abundance is high by about 0.02 for [M/H]= −1.7 if the helium is primordial, or 0.03 if the helium enrichment only affects the level of the HB, as in the deep mixing scenario. (Note that a lower value for the reddening would bring the ∆ values into consistency with the other clusters.) A high helium abundance would tend to make the HB distribution bluer on the whole. According to Fusi Pecci et al. (1993), the color of the peak of the HB star distribution in M30 is one of the bluest, but other clusters are rather close (M53 is bluer, M15 has approximately the same peak color, and M92 and NGC 5466 are slightly redder).
Environmental Effects
Because M30 has a core of high stellar density, we consider the possibility that this environment has influenced the populations of evolved stars in the cluster. M30 has one of the most robustly determined color gradients (approximately linear in log r: ∼ +0.20 mag dex −1 ; Piotto, King, & Djorgovski 1988) of all the globular clusters in the Galaxy. The sense of the color gradient is such that the integrated colors become bluer towards the cluster center. In M30 and other post-core-collapse clusters it has been suggested that the color gradient is due to a decrease in the ratio of RGB-to-BHB stars resulting from stellar interactions in the dense cluster cores. Although claim the color gradient measured in M30 is due to a deficit of RGB stars in the inner few tens of arcseconds, Burgarella & Buat (1996) show that the gradient (which extends to radii > 2 ′ ) is not due to differences in the spatial distribution in the evolved-star populations or in the blue stragglers. Our results are consistent with this latter claim. Buonanno et al. (1988) claimed to have detected a radial variation in the ratio R = N HB /N RGB on the basis of a smaller sample of stars. From our sample, we find R ranges from about 1.30 ± 0.26 in the inner 30 ′′ to 1.35 ± 0.27 for 30 ′′ < r < 100 ′′ to 1.83 ± 0.36 for stars with projected radius r > 100 ′′ . There may be marginal evidence for a difference between the outer annulus, and the inner two, but over the range of radii for which the bluer-inward color gradient has been observed, there is no evidence of a trend in the bright populations. The sense of the difference between the outer and inner populations is in any case opposite to that required to make the color gradient. The cumulative radial distribution ( Figure 6) also shows no strong trends in radius, contrary to claims in other studies with smaller samples (Buonanno et al. 1988, Piotto et al. 1988), but in agreement with studies of the core (Yanny et al. 1994, Burgarella & Buat 1996. A Kolmogorov-Smirnoff test indicates a 33% chance that the two samples are drawn from the same distribution. The inhomogeneity of the RGB sample seems to be responsible for this noncommittal probability.
We conclude that despite the color gradient in M30, and the apparently ripe conditions for interactions to alter the stellar populations, environment-based processes are not responsible for the high R value we measure. Based on the color-difference method, VBS claimed that the most metal-poor clusters, including M30, are coeval at the level of 1 Gyr. Our comparison (Figure 7) of the fiducial lines of M30 and M68 (Walker 1994) in the neighborhood of the SGB indicates that the ages of M30 and M68 (assuming similar main-sequence Y and [α/Fe]) are nearly identical. With our uniform calibration of MSTO and evolved stars in M30, we can determine with good precision the other commonly applied age estimator ∆V HB T O . In computing the values presented in Table 6, we have attempted to use the studies with the largest samples having uniform photometry from the level of the HB to below the TO. We were able to derive values for the clusters M68 (Walker 1994), M53 and NGC 5053 (Heasley & Christian 1991), M92 (Bolte & Roman 1998), and M15 (Durrell & Harris 1993). Although the clusters all have blue HB morphologies, it has not been necessary to make corrections to find the "true" HB level: either the HB is populated on both sides of the instability strip (M53, NGC 5053) or there are a number of well-measured RR Lyrae stars (M15, M68, M92).
From our photometry of M30, we find V T O = 18.63 ± 0.05, and ∆V HB T O = 3.59 ± 0.06. While M15, M53, M68, M92, and NGC 5053 have ∆V HB
T O values that agree to within the errors (and also agree with the values derived for clusters of higher metallicity), M30 has a value about 0.15 mag higher. This is in disagreement with values given in the extensive tabulation of Chaboyer, Demarque, & Sarajedini (1996). The values for M30 and M92 in particular have been put on firmer ground here since consistent photometry exists from the HB to the TO. Using the more robust V(BTO) (the apparent magnitude of a point 0.05 mag redder than the turnoff; Chaboyer et al 1996), we can compare ∆V HB BT O values for M30 and M68, which also has V I data. We find 3.16 ± 0.06 for M30, and 2.96 ± 0.02 for M68.
There are two plausible ways to explain the apparent 0.15 mag excess in ∆V HB T O for M30 relative to other clusters. First, M30 could be older by ∼ 2 Gyr. This conclusion would be in conflict with that inferred by VBS based on the color-difference method. (This is perhaps the first case for which the relative age indicators ∆V HB T O and the subgiant-branch color extent give significantly different answers.) Alternatively, M30 stars could have a larger initial helium abundance by approximately 0.027. If the higher helium abundance is restricted to the cluster HB stars, as would be the case in a deep-mixing scenario, the increase required is approximately 0.045. (The agreement between the M30 and M68 CMDs everywhere but on the HB would argue against a difference in the initial helium abundances in the two clusters, as would beliefs about Galactic chemical evolution.) The size of the potential helium enhancement is close to what was inferred earlier from the helium indicators ∆ and R for M30.
Luminosity Functions
In the following, we will be using a combination of oxygen-enhanced (BV92) and α-element enhanced (VandenBerg 1997) theoretical LFs to interpret the data. The current state of knowledge indicates that all of the α elements have enhancements (Pilachowski, Olszewski, & Odell 1983;Gratton, Quarta, & Ortolani 1986;Sneden et al. 1992). The available evidence also suggests that the oxygen enhancement remains constant, at least for [Fe/H] −1.4 (e.g. Suntzeff 1993, Carney 1996.
On the RGB, stellar evolution is insensitive to the oxygen abundance because the luminosity evolution is driven almost entirely by the helium core mass (Refsdal & Weigert 1970), while the color is primarily determined by H − opacity. Oxygen has a relatively high ionization potential, and hence does not contribute electrons to the opacity. The fainter one goes on the MS, the more insensitive the evolution is to the oxygen abundance because of the same opacity effect, and because p − p chain reactions are dominant over CNO cycle reactions in influencing the luminosity. As a result, the different distribution of heavy elements causes negligible differences in the theoretical LFs on the RGB and lower MS (see Figure 17 of Sandquist et al. 1996).
It is primarily the turnoff region that is affected by changes in the oxygen abundance, because CNO cycle reactions begin to become important, and because oxygen ionization regions are close enough to the surface to influence surface temperatures. Increased oxygen abundance increases the envelope opacity, creating redder models. Increased CNO cycle activity causes a star to adjust to accommodate the increased luminosity by reducing the temperature and density of the hydrogen burning regions, which results in a net decrease in the luminosity of the turnoff and SGB relative to solar-ratio models. Thus, the SGB "jump" moves in magnitude in the LF. In the CMD, it also changes slightly in slope, but for metal-poor clusters like M30, this does not cause a significant change in the shape of the SGB jump in the LF.
An examination of BV92 models indicates that the V -band LF is not very age-sensitive for this range of metallicities. It is most sensitive on the SGB and then, as found in Sandquist et al. (1996), only when the SGB is nearly horizontal in the CMD. In V band for a cluster as metal-poor as M30, the SGB has a relatively large slope, and so only a large systematic age error will influence the fit. In light of the Hipparcos parallax data, this possibility should be considered, since derived distance moduli indicate brighter TO magnitudes (and thus, younger ages) for metal-poor clusters. Figure 8 shows a comparison of the V -band LF with theoretical LFs for different ages, using an apparent distance modulus of (m − M) V = 14.87, as derived from one fit to the Hipparcos subdwarf sample.
Previous studies of M30's V -band LF (Piotto et al. 1987, Bolte 1994, Bergbusch 1996 uncovered two unusual features in comparisons with theoretical models: an excess of faint red giants relative to main-sequence stars, and an excess of subgiant stars. Our photometry goes fainter on the MS, allowing us to verify that the normalization of the theoretical models has not been made in an "abnormal" section, while our wide field allows us to measure the largest sample of red giants in the cluster to date. Figure 9 shows a comparison of the studies, with magnitude shifts according to measured zero-point differences. In large part there is excellent agreement. Our LF is significantly below Bergbusch's at his faint end, most likely due to underestimated incompleteness corrections in his study. At the bright end of the RGB (V < 15), our LF points are also below most of Bergbusch's. However, we observed a larger number of giants, and our bins are larger, making our points more significant statistically.
In the following subsections, we discuss the main features in the LFs.
Red Giant Branch Excesses
There is a apparently a considerable excess of stars on the RGB for 15 < V < 17 (we will refer to this as the "lower RGB") when compared to the models normalized to the unevolved main sequence. To judge the reality of the excess, we need to accurately normalize the theoretical LFs in the horizontal and vertical directions, and choose the photometry subsample to maximize the statistical significance. The horizontal normalization can be accomplished by shifting the theoretical LF in magnitude so that the TO matches that of the observational LF (Stetson 1991a). In the vertical direction, we have normalized to the MS in a range of magnitudes where there are large numbers of measured stars, and where incompleteness is a relatively small consideration.
The mass function controls how well the normalized theoretical LF is fit to the MS portion in the present example. As shown in Figure 10, fits using small values for the power-law mass function exponent x indicate that the relative numbers of stars on the RGB and lower MS can be matched by canonical stellar evolution models. With such a choice though, bins with 18.3 < V < 20.1 are not well-modeled. We find that the LF can be modeled from near the faint limit of our survey to the base of the RGB if we use a higher value for x. This alleviates the depression in the star counts in this magnitude range seen by Bolte (1994). However, we are still left with an excess of RGB stars relative to MS stars in the range 15.1 < V < 16.6.
The effects of mass segregation have been previously observed within M30 in the form of a variation of the local mass function exponent x local with radius (RFV, Bolte 1989, Piotto et al. 1990, Sosin 1997. As a result, the best comparison that can be made would be between theory and a faint sample restricted to the outskirts of the cluster. The models of Pryor, Smith, & McClure (1986), as well as observational studies, indicate that restricting the sample to stars more than 20 core radii from the center should minimize the effects of mass segregation on faint end of the LF. Figure 11 shows the LF we computed for this purpose. The presence of the RGB-MS discrepancy in this case suggests that the problem is not related to the dynamical effects on the mass function, at least in the outskirts. We can get good overall agreement with the shape of the LF on the MS, but there is a relative excess of RGB stars, and the SGB region is not well fit. The SGB comparison is insensitive to age and metallicity using this method of matching the MSTO. Helium abundance, however, has a larger effect (Stetson 1991a). Because the RGB stars in M30 become more populous relative to HB stars (and presumably MS stars) towards the center, we expect that the cluster core would show a larger discrepancy.
As with the SGB excess, this effect has only been observed in metal-poor clusters (in other words, not in the LFs of NGC 288 or M5). To add stars to the canonical number at a point in the red giant LF, one must either increase the hydrogen content of the mass being fed into the hydrogen-burning shell, or reduce the density or temperature of the burning shell. One possibility for the excess stars on the lower RGB is that we are seeing the effects of deep mixing, which brings hydrogen-rich envelope material into the energy generating shell. If this kind of mixing occurred on the lower RGB, it could eliminate the RGB bump by erasing the chemical discontinuity left by a surface convection zone.
Alternately, VandenBerg, Larson, & DePropris (1998) have examined the effects of rotation on RGB evolution. They found that core rotation can expand the outer portions of the stellar core enough to cause a reduction of the shell temperature. This results in a decrease in the rate of evolution for RGB stars, and hence leads to an increase in the number of stars per luminosity bin. This is in the correct direction to explain the RGB excess. This rotation could be related to deep mixing scenarios that are required to explain abundance anomalies in RGB stars (e.g. Shetrone 1996) -most notably a decline in the 12 C/ 13 C ratio relative to theoretical predictions, and Na-O and Al-O anti-correlations (as surface material is mixed into regions where O is being converted to N in the CNO cycle).
[Note that rotation cannot explain subgiant branch excesses because the burning region in core-burning stages is too small to contain a significant amount of angular momentum (VandenBerg 1995). Even if rotation does affect the structure of the star outside the core, and thereby changes the core temperature, this would not produce isothermalization that would lead to SGB excesses.]
The rotation and mixing pictures (with the assumption that mixing is somehow based on internal rotation) receive some support from observations of rotation in HB stars of some clusters. There is definite evidence of stellar rotation in blue HB stars in NGC 288, M3, and M13 (Peterson, Rood, & Crocker 1995). M13 has the fastest rotators, with stars falling into two groups: some with v ≈ 15 km s −1 , and some with v ≈ 38 km s −1 . M3 has a v sin i distribution consistent with v = 13 ± 2 km s −1 , while NGC 288's stars are consistent with v = 9 ± 2 km s −1 . Cohen & McCarthy (1997) also found projected rotation rates between 15 and 40 km s −1 for five blue HB stars in M92. The presence of stellar rotation on the HB implies that angular momentum may have been stored during the RGB phase in a rapidly rotating core, avoiding loss of angular momentum through the stellar wind. (Such mass loss is needed to be able to create HB stars of appropriate masses to match observed cluster HB morphologies.) If this is true, it would be particularly interesting to compare LFs for M3 and M13 to look for the effects of rotation, and perhaps even different levels of rotation. Further stellar rotation measurements for M30, M68, and M92 would also be helpful in examining rotation as a cause of MS-RGB discrepancy in the combined LF. Figure 12 presents the cumulative LF (CLF) for the cluster. In this graph we have included RGB stars from 1 ′ to 6 ′ from the center of the cluster. The RGB bump is typically identified from a break in slope in the cumulative LF. At this point, the shell-burning source begins consuming material of constant, lower helium content (in other words, the shell reaches what was formerly the base of the convection zone at its maximum extent -Fusi Pecci et al. 1990). Fusi Pecci et al. examined clusters over a range of metallicities, and found a linear relation between ∆V HB bump = V bump − V HB and [Fe/H], as predicted by theory. By combining CMDs for three of the most metal-poor clusters (M15, M92, and NGC 5466), they found ∆V HB bump = −0.51 ± 0.05. In addition, for NGC 6397, the most metal-poor cluster for which they were able to find the bump, they found ∆V HB bump = −0.40 ± 0.16. As shown in Figure 12, we have examined data for M68 (Walker 1994), a cluster of nearly the same metallicity as M30, in order to get a better idea of where the bump should be. There is a clear indication of a slope break for M68: ∆V HB bump = −0.46 ± 0.03, or V T O − V = 3.87, for M68. That result shows that the continuation of the Fusi Pecci et al. relation to lower metallicity appears correct. We have chosen to shift M30 and M68 so that their MSTOs align because of the evidence that M30's HB may be anomalously bright (see § 4.2). The comparison reveals that there may be a feature at the same position as in M68, although we do not see significant signs of slope change in the CLF at the position of the feature.
The RGB Bump
Subgiant Branch Excesses
We find that a few bins on the SGB (V ≈ 18.17) show an excess of stars relative to the theoretical predictions for the best fitting models, confirming the result of Bolte (1994). In Figure 13, we plot the LF with a radius cut closer to the cluster center so as to get better statistics on the SGB. As Figure 14 shows, there is little scatter in the vicinity of the SGB in the CMD that would tend to wash out or contribute to the observed excess at V ≈ 18.2. The excess is based on a single point having a significance of 2.7σ, where the error in almost entirely due to Poisson statistics. Bolte (1994) states the significance of the bump as 4.8σ, and it appears to occupy two LF bins in his Figure 7. The significance of his result is probably smaller than that because of the difficulty in determining the position of the "jump" (≈ 0.8 mag brighter than the MSTO in V ) in his LF.
An examination of the I-band LF in Figure 15 shows the presence of a deviation at the same position in the CMD. This is important because the slope of the SGB is steeper in an (I, V − I) CMD than in a (V, V − I) CMD. As a result, the bump can no longer be ascribed to a feature caused by the exact slope: it must be the result of an increase in the number of stars congregating near a point on the cluster's fiducial line. At the analogous position in the I-band LF, there are two bins with excesses of 1.7σ and 2.8σ compared to theory, for a combined significance of 3.3σ. The appearance of the subgiant branch excess in both the V -and I-band LFs indicates that the cause must be due to an excess of stars (rather than being caused by the exact slope of the SGB -thus eliminating the exact metallicity, helium content, and oxygen abundance as causes).
So, the SGB excess has marginal significance in our LFs. In order to more definitively determine the reality of the feature, photometry reaching into the center of this cluster will be needed. The observation of this feature in the combined LF of M68, M92, and NGC 6397 (Stetson 1991a) lends more credence to the phenomenon, but more investigation is necessary. Sandquist et al. (1996) found that there was no evidence for an SGB excess in the LF of the more metal rich cluster M5 (which has a good I-band LF for easy comparison with Figure 15). Bergbusch (1993) saw no evidence of an excess in his V -band LF of NGC 288. These pieces of evidence seem to indicate that any cause must only be effective at low metallicities. There is, however, a general lack of useful LF data covering the SGB for globular clusters with metallicities between M5 and M30, or more metal rich than M5.
If the feature is real, there are at least two potential means of creating such an excess: a fluctuation in the initial mass function, and an unknown physical process isothermalizing the stellar core of turnoff mass stars. The excess in Stetson's M68-M92-NGC 6397 LF makes mass function fluctuations less likely. A star can be forced to pause on the SGB, but still burning hydrogen in its core, if isothermality is imposed on a large portion of the core (Faulkner & Swenson 1993). If such a process occurred in a large enough fraction of the stars in a cluster, a SGB excess could be created in the LF. A way to create such an excess is to invoke a process that increases the efficiency of energy transfer over a large portion of the core. For this to happen, the mean free path of the transporting particle must be large. No such particle has been identified to date.
Conclusions
1. Determinations of the reddening for M30 disagree at a ±0.03 mag level and cases can be made for values ranging from 0.03 to 0.07 in E(V − I). This uncertainty is the main factor preventing a more accurate determination of the distance modulus. By fitting subdwarfs with Hipparcos parallax data to the V I fiducial line, we find satisfactory fits for (m − M) V ; E(V − I) pairs ranging from 14.87; 0.06 to 14.65; 0.02 with the statistical errors of around 0.12 mag (all for the case [Fe/H] = −1.91). When shifted to a common reddening, we find our distance modulus is consistent with ground-based estimates, and at least 0.1 mag smaller than other Hipparcos-based estimates.
2. M30 has a larger R value [N HB /N RGB ] than any of the other metal-poor clusters for which this quantity has been measured. This quantity is usually used as a helium indicator and our measured R value suggests a helium abundance ∼ 0.03 − 0.04 larger than the mean of the other metal-poor clusters. M30's value for the helium indicator ∆ is also relatively high although for the case of E(V − I) < 0.02, it is consistent with the other metal-poor clusters. If there is a helium abundance enhancement in M30, it is probably not an initial abundance difference since Galactic chemical evolution and the similarity of the M30 and M68 fiducial lines (see next point) argue against it.
3. The ∆V HB T O value for M30 is demonstrably large relative to clusters of similar metallicity. The M30 fiducial line (except for the HB) overlies that of M68 (see Figure 7) and M92 (VBS) very closely, indicating that M30 probably has the same age as these two clusters. We suggest that the HB luminosity in M30 is high due to a larger-than-average Y for the M30 HB stars.
4. The LFs of the cluster show definite evidence for an excess of RGB stars relative to MS stars, and marginally significant (≈ 3σ) evidence for an excess of SGB stars, as compared with theory. The SGB feature has slightly higher significance in the I band. The possibility remains that these anomalies are only present in low-metallicity clusters.
Stellar rotation is a possible explanation for the excess number of RGB stars relative to MS stars. Alternatively, the excess giants could be a signpost for mixing events on the lower RGB in which fresh hydrogen is mixing into the energy generation region. This could also be identified as the source of the envelope Y-enrichment we infer from the HB and brighter RGB stars. 5. We do not find an obvious RGB bump in M30, in spite of the size of our RGB sample. Using the cumulative LF, we have detected the bump in the metal-poor cluster M68 with a ∆V HB bump value that agrees with the linear trend with [Fe/H] found by Fusi Pecci et al. (1990).
It is possible that points 2 -5 are all related to the deep-mixing events inferred for some globulars based on the surface abundances of elements that participate in the energy generation cycles. The hypothesis that we are seeing the effects of the mixing of hydrogen-rich material into the energy-generation regions and helium-rich material out into the stellar envelope can qualitatively explain all of these (2 through 5) observations. If this hypothesis is correct then we predict that detailed abundance studies of the bright giants in M30 should show the characteristic patterns of deep mixing -low oxygen and carbon abundances accompanied by high nitrogen, aluminum, and sodium.
We would especially like to thank D. VandenBerg for providing us with theoretical α-enhanced isochrones and luminosity functions prior to publication and P. Stetson for the use of his excellent software. It is a pleasure to thank P. Guhathakurta, Z. Webster, and R. Rood for useful conversations. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. M.B. is happy to acknowledge support from NSF grant AST 94-20204.
Electronic copies of the listing of the photometry are available on request to the first author.
A. Data Reduction
A.1. Primary Standard Calibration Fields
A.1.1. Aperture Photometry Aperture photometry was performed using the program DAOPHOT II (Stetson 1987). Using these data, growth curves were constructed for each frame using DAOGROW (Stetson 1990) in order to extrapolate from the flux measurements over a circular area of finite radius to the total flux observable for the star. The aperture magnitudes and the known standard system magnitudes of Landolt (1992) were then used to derive coefficients for the transformation equations:
v = V + a 0 + a 1 · (X − 1.25) + a 2 · (V − I) + a 3 · (V − I) 2 + a 4 · (V − I) 3 i = I + b 0 + b 1 · (X − 1.25) + b 2 · (V − I),
where v and i are observed aperture photometry magnitudes, V and I are the standard system magnitudes, and X is the airmass. The primary standard stars covered a color range −0.35 < (V − I) < 1.67, completely encompassing the color range of the cluster sample. The coefficients for the transformation equations are given in Table A1. The residuals for the sample of 25 stars are shown in Figure 16, and the average residuals are given in Table A2. (In this and all subsequent comparisons, the residuals are calculated in the sense of ours -theirs.)
A.2. Secondary Standard Calibration
We chose 118 relatively bright and isolated stars in the M30 field observed with the 1.5 m telescope during the photometric night to be "secondary standards". The stars selected were required to be unsaturated, brighter than the turnoff, in relatively uncrowded regions of the images, and close to the apparent fiducial line of the cluster (since this acts as an additional check on the accuracy of the photometry). Once the list was finalized, all other stars were subtracted from the frames and aperture photometry was obtained. The colors for these standards cover the range 12.82 < V < 16.54 and −0.069 < (V − I) < 1.280
A.3. Object Frames
A.3.1. Profile Fitting Photometry
Both the CTIO 4 m and 1.5 m data for M30 were reduced using the standard suite of programs developed by Peter Stetson (DAOPHOT/ALLSTAR;Stetson 1987Stetson , 1989, and following the procedures in Sandquist et al. (1996).
A.3.2. Calibration
We used the secondary standards established with the 1.5 m observations on the single photometric night to determine the coefficients in the transformation equations for all of the 1.5 m profile fitting photometry:
v = V + a 0,k + a 1 · (V − I) i = I + b 0,k + b 1 · (V − I),
where v and i are the instrumental magnitudes from the profile fitting, V and I are the standard values from the aperture photometry, and k is an index referring to individual frames. The coefficients of the color terms are given in Table A1, the average residuals and standard deviation of the residuals for the comparison of the profile fitting and aperture photometry are given in Table A2, and individual star residuals are shown in Figure 17.
In the next step, we chose to calibrate the 4 m profile fitting photometry to the 1.5 m profile fitting photometry rather than the aperture photometry of the secondary standards. This was done primarily to ensure that all of our profile fitting was on the same system over as large a range of magnitudes as possible. The M30 frames taken at the 1.5 m telescope on the one photometric night did not go particularly deep, while the 4 m photometry had few unsaturated observations of the brighter stars in the cluster. The data taken on the non-photometric nights at the 1.5 m telescope did, however, cover a range of magnitudes similar to that of the 4 m data.
We selected a sample of 248 stars found in both fields at least 300 pixels away from the cluster center. These stars were used to determine the transformation coefficients for the equations:
v = V + a 0,k + a 1 · (V − I) + a 2 · (V − I) 2 i = I + b 0,k + b 1 · (V − I) + b 2 · (V − I) 2 ,
where v and i are the instrumental magnitudes from the 4 m observations, and V and I are the standard values from the 1.5 m observations. The coefficients of the color terms are given in Table A1, while the residuals of the comparison of the photometry for the 4 m and 1.5 m measurements of the secondary standards are shown in Figure 18.
For the final calibration, we used the transformation equations for the 1.5 m and 4 m profile-fitting data. All of the profile-fitting photometry from both telescopes was combined with weights equal to the inverse square of the internal measurement errors in order to determine our standard-system magnitude and color values.
Because of the large sky coverage of the CTIO frames, most other surveys of M30 overlap the program area at least partially. Table A2 provides a summary of the zero-point offsets for comparisons with these studies. We would particularly like to point out that there is considerable difference among them, highlighting the importance of the calibration. The fields used by Bolte (1987), RFV, and Samus et al. (1995) are completely included on all frames. A comparison with the photometry of Bolte is given in Figure 19. In Figure 20, we show the comparison with the study of M30 by Samus et al. (1995). We do this partly because it involves the same filter bands, and partly because the residuals are the lowest on average (although the scatter in star-to-star residuals is large). As a note, comparisons with the most recent study (Bergbusch 1996) show no signs of color trends in the residuals except within about a magnitude of the tip of the giant branch. Fig. 11.-The V -band luminosity function for M30, restricted to r > 4 ′ ≈ 30r c with theoretical α-enhanced LFs for [Fe/H] = −2.01, and x = 2. The four curves are for ages 10, 12, 14, and 16 Gyr, and they have been shifted so that the theoretical and observational TO magnitudes match, and then normalized to the observed LF in a 0.5 mag bin 2 magnitudes below the turnoff. The implied distance moduli from the isochrone TOs are 14.98, 14.77, 14.61, and 14.47 respectively. Fig. 12.-The cumulative luminosity functions for M30 and M68 (Walker 1994), shifted so that the main sequence turnoffs have the same magnitude. The dotted lines indicate linear fits to the functions above and below the slope break, which is an indication of the position of the RGB bump. Bolte (1987) and our dataset. The data for Bolte's short exposure frames are in the plots on the left, and his long exposure data is used on the right. The residuals are in the sense (ours -Bolte's). a Lutz-Kelker corrections calculated using ∆M LK = −7.60(σ π /π) 2 − 47.23(σ π /π) 4
V V − I N V V − I N
5. If we use the value of metal content given by Zinn & West ([Fe/H] = −2.13) along with the Carney et al. (1994) abundance scale for the subdwarfs (which has roughly the same zero point as Zinn & West), the distance modulus is increased by only a few hundredths of a magnitude. Restricting the sample to only metal-poor ([Fe/H]< −1.3) subdwarfs also does not significantly change the distance modulus.
Because α-element enhancements influence the position of the HB and RGB in the CMD like a change in [Fe/H] (Salaris, Chieffi, & Straniero 1993), they must be taken into account when computing [M/H]. For M30, we find that ∆V BC = 0.27. We have used [M/H] = −1.70 (correcting [Fe/H] by 0.21 dex for [α/Fe]= +0.
. The ∆ values for the other clusters agree with the theoretical value of 6.30 for [M/H] = −1.82 ([Fe/H]= −2.03).
Age Indicators: ∆V HB T O and ∆(V − I)
Fig. 1 .
1-The M30 color-magnitude diagram for a) all measured stars, and b) the sample restricted to stars having 110 ′′ < r < 10 ′ .Fig. 2.-Magnitude bias versus V magnitude. Fig. 3.-External magnitude errors versus V magnitude.
Fig. 4 .
4-Completeness fraction versus V magnitude.
Fig. 5 .
5-Subdwarf fits to the V I main sequence fiducial for two different reddenings, assuming [Fe/H] M 30 = −1.91 (Carretta & Gratton 1996). The M30 fiducial line is plotted as open boxes. The solid boxes with error bars are local subdwarfs and giants. Only those subdwarfs with M V > 4.5 and [Fe/H]< −1.2 were used in the main-sequence fitting. The isochrones (plotted as solid lines) are preliminary α-enhanced versions (VandenBerg 1997) with [Fe/H] = −2.01 and ages 10, 12, 14, and 16 Gyr (from top to bottom). The two reddening values correspond to E(B − V ) = 0.016 and 0.048.
Fig. 6 .
6-The M30 cumulative radial distributions for RGB stars (solid line) HB stars (dotted line), and AGB stars (dashed line), using wide-field data from this paper, and HST data for the core (YGSB).
Fig. 7 .
7-A comparison of the (V, V −I) fiducial lines for M30 (thick line) with our computed fiducials from M68 (thin line; Walker 1994 data). The M68 fiducial has been shifted 0.45 mag brighter in V and 0.03 mag bluer in color.
Fig. 8 .
8-The V -band luminosity function for M30 compared with theoretical α-enhanced LFs for [Fe/H] = −2.01 and a distance modulus (m − M) V = 14.87 for ages (from left to right 10, 12, 14, and 16 Gyr. The theoretical luminosity functions have been normalized to the range 20.4 < V < 20.9.
Fig. 9 .
9-The V -band luminosity function for 14772 stars in M30 as derived here (• with error bars), compared with LFs from Bergbusch (1996; △) and Bolte (1994; ✷), as well as a theoretical α-enhanced LFs for [Fe/H] = −2.01 and x = 2.0 with age 12 Gyr and (m − M) V = 14.70 (solid line) and age 10 Gyr and (m − M) V = 14.87 (dotted line) for comparison purposes. All of the luminosity functions have been normalized to the range 18.5 < V < 19.
Fig. 10 .
10-The V -band luminosity function in M30, with theoretical α-enhanced LFs for [Fe/H] = −2.01 and age 12 Gyr. The curves are for mass function exponents x = 1.5 (solid line), x = 2 (dotted line), and x = 2.5 (dashed line). The theoretical LFs have been shifted in magnitude using an apparent distance modulus (m − M) V = 14.70.
Fig. 13 .
13-The V -band luminosity function for M30 including stars down to a radius 150 pixels (66 ′′ ) from the cluster center. The theoretical α-enhanced LFs have [Fe/H] = −2.01 and x = 2.5, but have the same combinations of age and distance modulus asFigure 9. The luminosity function has been normalized to the range 19.4 < V < 19.9.
Fig. 14 .
14-The M30 color-magnitude diagram for all the stars in the V I sample used in computing the V -band luminosity function inFigure 13.
Fig. 15 .
15-The I-band luminosity function for 14507 stars, with [Fe/H] = −2.01, x = 2.5 theoretical α-enhanced LFs for age 12 Gyr and (m − M) I = 14.67 (solid line), and age 10 Gyr and (m − M) I = 14.81 (dotted line).
Fig. 16 .
16-Final residuals for the comparison of standard and measured values forLandolt (1992) primary standard stars observed at the CTIO 1.5 m telescope.
Fig. 17 .
17-Residuals for the comparison of 118 M30 secondary standard stars measured using profile fitting and aperture photometry in the CTIO 1.5 m frames. The residuals are in the sense (profile fitting -aperture).
Fig. 18 .
18-Final residuals for the comparison of 248 M30 stars used in the calibration of the CTIO 4 m frames. The residuals are in the sense (4 m -1.5 m).
Fig. 19 .
19-Residuals for the comparison between the M30 CCD photometry of
Fig. 20 .
20-Residuals for the comparison between the M30 CCD photometry ofSamus et al. (1995) and our dataset. The residuals are in the sense (ours -Samus').
Table 1 .
1M30 [V, (V − I)] Fiducial Points
Table 1 -Table 4 .
14Continued Table 2. V -Band Luminosity Function for M30Table 3. I-Band Luminosity Function for M30 Metal-Poor Field Subdwarfs and SubgiantsV
V − I
N
V
V − I N
17.800 0.764 101 14.354 0.924 1
17.575 0.793
57 14.483 0.914 1
17.425 0.809
44 14.564 0.894 1
Table 5 .
5Measured Distance Moduli [Fe/H] E(V − I) 69 ± 0.13 14.68 ± 0.13 14.66 ± 0.12 14.71 ± 0.13 Table 6. Characteristics of Metal-Poor Globular ClustersSubdwarf Sample
All
M V > 4.25
M V > 5
[Fe/H]< −1.3
−1.91
0.06
14.93 ± 0.12 14.87 ± 0.12 14.86 ± 0.11 14.92 ± 0.11
−1.91
0.02
14.70 ± 0.12 14.65 ± 0.12 14.64 ± 0.11 14.76 ± 0.12
−2.13
0.06
14.93 ± 0.12 14.91 ± 0.12 14.87 ± 0.11 14.93 ± 0.11
−2.13
0.02
14.ID
[Fe/H] ZW
Y Indicators
∆V HB
T O
R HB
R
∆ B−V
NGC 104 (47 Tuc)
−0.71
1.21 ± 0.13 5.32 ± 0.08 3.59 ± 0.10
−1.00
NGC 5904 (M5)
−1.40
1.08 ± 0.09 5.78 ± 0.04 3.47 ± 0.06
0.39
NGC 5272 (M3)
−1.66
1.19 ± 0.10 5.84 ± 0.04 3.52 ± 0.09
0.07
NGC 4590 (M68)
−2.09
0.91 ± 0.17 6.34 ± 0.05 3.41 ± 0.05
0.44
NGC 5024 (M53)
−2.04
1.18 ± 0.18
· · ·
3.46 ± 0.08
0.76
NGC 5053
−2.41
0.95 ± 0.25
· · ·
3.44 ± 0.08
0.61
NGC 5466
−2.22
1.21 ± 0.27
· · ·
· · ·
0.51
NGC 6341 (M92)
−2.24
1.26 ± 0.18
· · ·
3.44 ± 0.06
0.88
NGC 6397
−1.91
1.15 ± 0.17
· · ·
3.6 ± 0.14
0.93 (0.69)
NGC 7078 (M15)
−2.17
1.23 ± 0.21 6.32 ± 0.11 3.46 ± 0.10
0.72
NGC 7099 (M30)
−2.13
1.49 ± 0.18 6.42 ± 0.13 3.59 ± 0.06
0.84
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| []
|
[
"Electron-hole correlations govern Auger recombination in nanostructures",
"Electron-hole correlations govern Auger recombination in nanostructures"
]
| [
"John P Philbin \nDepartment of Chemistry\nUniversity of California\n94720BerkeleyCaliforniaUnited States\n",
"Eran Rabani \nDepartment of Chemistry\nUniversity of California\n94720BerkeleyCaliforniaUnited States\n\nMaterials Science Division\nLawrence Berkeley National Laboratory\n94720BerkeleyCaliforniaUnited States\n\nThe Sackler Center for Computational Molecular and Materials Science\nTel Aviv University\n69978Tel AvivIsrael\n"
]
| [
"Department of Chemistry\nUniversity of California\n94720BerkeleyCaliforniaUnited States",
"Department of Chemistry\nUniversity of California\n94720BerkeleyCaliforniaUnited States",
"Materials Science Division\nLawrence Berkeley National Laboratory\n94720BerkeleyCaliforniaUnited States",
"The Sackler Center for Computational Molecular and Materials Science\nTel Aviv University\n69978Tel AvivIsrael"
]
| []
| The fast nonradiative decay of multiexcitonic states via Auger recombination is a fundamental process affecting a variety of applications based on semiconductor nanostructures. From a theoretical perspective, the description of Auger recombination in confined semiconductor nanostructures is a challenging task due to the large number of valance electrons and exponentially growing number of excited excitonic and biexcitonic states that are coupled by the Coulomb interaction. These challenges have restricted the treatment of Auger recombination to simple, noninteracting electronhole models. Herein we present a novel approach for calculating Auger recombination lifetimes in confined nanostructures having thousands to tens of thousands of electrons, explicitly including electron-hole interactions. We demonstrate that the inclusion of electron-hole correlations are imperative to capture the correct scaling of the Auger recombination lifetime with the size and shape of the nanostructure. In addition, correlation effects are required to obtain quantitatively accurate lifetimes even for systems smaller than the exciton Bohr radius. Neglecting such correlations can result in lifetimes that are 2 orders of magnitude too long. We establish the utility of the new approach for CdSe quantum dots of varying sizes and for CdSe nanorods of varying diameters and 1 arXiv:1808.10404v1 [cond-mat.mes-hall] 30 Aug 2018 lengths. Our new approach is the first theoretical method to postdict the experimentally known "universal volume scaling law" for quantum dots and makes novel predictions for the scaling of the Auger recombination lifetimes in nanorods.The fast nonradiative decay of multiexcitonic states is a central process to many nanocrystal-based applications. 1, 2 This nonradiative decay occurs primarily via Auger recombination (AR) in which one electron-hole pair recombines by transferring its energy to an additional charge carrier(Fig. (1)). In some cases, such as light harvesting devices, AR can limit performance by rapidly quenching the photoluminescence 1, 3-6 and destroying the population inversion required for nanocrystal based lasers, 7 while in other cases, such as photodetectors, 8 single photon sources 9 and even for photocatalysis, 10 it can improve performance by providing a source of hot electrons. Therefore, developing a unified framework to describe AR is important from both fundamental and applied perspectives.In recent years, much effort has been put into -and much success obtained in -the development of synthetic techniques and principles that result in nanocrystals (NCs) with rationally designed AR lifetimes. 2 Synthesizing giant NCs offers the simplest and most well-known approach to increase the AR lifetime. This approach works well because the AR lifetime, τ AR , in single-material quantum dots (QDs) obeys the "universal volume scaling law" (i.e., τ AR,QD ∝ V in QDs). 1, 11-13 However, current theories predict a steeper scaling with the QD volume, 14-16 signifying only a partial understanding of the AR process even in spherical, 0D NCs. In addition to controlling the AR lifetime by changing the system size, many reports have found that an intelligent design of core/shell NCs with sharp or gradual interfaces allows for the AR lifetimes in NCs to be tuned.[16][17][18][19][20][21][22]The situation is somewhat more confusing for non-spherical NCs.[23][24][25][26][27][28][29][30][31]The AR lifetime in 1D nanorod (NR) structures was reported to scale linearly with the length (L) of the NRs (i.e., τ AR,NR ∝ L), | 10.1021/acs.nanolett.8b03715 | [
"https://arxiv.org/pdf/1808.10404v1.pdf"
]
| 53,207,164 | 1808.10404 | 4bd1ecbd671746129f54f33a5e61a4dd11b1d557 |
Electron-hole correlations govern Auger recombination in nanostructures
30 Aug 2018
John P Philbin
Department of Chemistry
University of California
94720BerkeleyCaliforniaUnited States
Eran Rabani
Department of Chemistry
University of California
94720BerkeleyCaliforniaUnited States
Materials Science Division
Lawrence Berkeley National Laboratory
94720BerkeleyCaliforniaUnited States
The Sackler Center for Computational Molecular and Materials Science
Tel Aviv University
69978Tel AvivIsrael
Electron-hole correlations govern Auger recombination in nanostructures
30 Aug 2018
The fast nonradiative decay of multiexcitonic states via Auger recombination is a fundamental process affecting a variety of applications based on semiconductor nanostructures. From a theoretical perspective, the description of Auger recombination in confined semiconductor nanostructures is a challenging task due to the large number of valance electrons and exponentially growing number of excited excitonic and biexcitonic states that are coupled by the Coulomb interaction. These challenges have restricted the treatment of Auger recombination to simple, noninteracting electronhole models. Herein we present a novel approach for calculating Auger recombination lifetimes in confined nanostructures having thousands to tens of thousands of electrons, explicitly including electron-hole interactions. We demonstrate that the inclusion of electron-hole correlations are imperative to capture the correct scaling of the Auger recombination lifetime with the size and shape of the nanostructure. In addition, correlation effects are required to obtain quantitatively accurate lifetimes even for systems smaller than the exciton Bohr radius. Neglecting such correlations can result in lifetimes that are 2 orders of magnitude too long. We establish the utility of the new approach for CdSe quantum dots of varying sizes and for CdSe nanorods of varying diameters and 1 arXiv:1808.10404v1 [cond-mat.mes-hall] 30 Aug 2018 lengths. Our new approach is the first theoretical method to postdict the experimentally known "universal volume scaling law" for quantum dots and makes novel predictions for the scaling of the Auger recombination lifetimes in nanorods.The fast nonradiative decay of multiexcitonic states is a central process to many nanocrystal-based applications. 1, 2 This nonradiative decay occurs primarily via Auger recombination (AR) in which one electron-hole pair recombines by transferring its energy to an additional charge carrier(Fig. (1)). In some cases, such as light harvesting devices, AR can limit performance by rapidly quenching the photoluminescence 1, 3-6 and destroying the population inversion required for nanocrystal based lasers, 7 while in other cases, such as photodetectors, 8 single photon sources 9 and even for photocatalysis, 10 it can improve performance by providing a source of hot electrons. Therefore, developing a unified framework to describe AR is important from both fundamental and applied perspectives.In recent years, much effort has been put into -and much success obtained in -the development of synthetic techniques and principles that result in nanocrystals (NCs) with rationally designed AR lifetimes. 2 Synthesizing giant NCs offers the simplest and most well-known approach to increase the AR lifetime. This approach works well because the AR lifetime, τ AR , in single-material quantum dots (QDs) obeys the "universal volume scaling law" (i.e., τ AR,QD ∝ V in QDs). 1, 11-13 However, current theories predict a steeper scaling with the QD volume, 14-16 signifying only a partial understanding of the AR process even in spherical, 0D NCs. In addition to controlling the AR lifetime by changing the system size, many reports have found that an intelligent design of core/shell NCs with sharp or gradual interfaces allows for the AR lifetimes in NCs to be tuned.[16][17][18][19][20][21][22]The situation is somewhat more confusing for non-spherical NCs.[23][24][25][26][27][28][29][30][31]The AR lifetime in 1D nanorod (NR) structures was reported to scale linearly with the length (L) of the NRs (i.e., τ AR,NR ∝ L),
but this observation has not been derived from first principles. Recently, it was argued that the AR decay in PbSe NRs has a crossover from cubic to bimolecular scattering as the length of the NR is increased, 28 calling into question the monotonic length dependence. Further complications arise from the difficulty to measure precisely the AR lifetimes 24 and also to independently control the dimensions of NRs by current synthetic techniques. In fact, it was shown that NRs of equal volume (but differing diameters and lengths) can have AR lifetimes that differ by more than a factor of 2, 27 29 while recently it was argued to scale linearly with A, attributed to collisions of excitons limited by their spatial diffusion. 31 The scaling of the AR lifetime as a function of the number of monolayers (ML) was reported to obey a seventh power dependence, τ AR,NPL ∝ (ML) 7 , in CdSe NPLs. 31 This was rationalized by a simple noninteracting effective mass model. 31 In order to simplify and better understand the size and dimensionality dependence of AR lifetimes in NCs, a unified theoretical framework for calculating AR lifetimes in 0D, 1D and 2D nanostructures must be developed. Such a development has been hampered by various factors, including limitations resulting from the enormous number of excitonic and biexcitonic states in NCs as well as the difficulties in including electron-hole correlation effects. Indeed, previous theoretical works have relied on a non-atomistic model 14,32 or a noninteracting electron-hole picture, thought to be suitable for strongly confined systems. 14-17, 33, 34 However, this approach fails to handle the continuous transition from strong to weak confinement regimes as well as nanostructures that have both strong and weak confinement along different dimensions (e.g., weakly confined along the NR axis and strongly confined in the others).
In this Letter, we develop a unified approach for calculating AR lifetimes that is applicable to all degrees of confinement. The approach is based on Fermi's golden rule to couple excitonic with biexcitonic states. Electron-hole correlations are explicitly included in the initial biexcitonic states by solving the Bethe-Salpeter equation (BSE) to obtain correlated electron-hole states which are then used to form the initial biexcitonic states. This procedure captures most of the electron-hole correlation as the exciton binding energy is typically an order of magnitude larger than the biexciton binding energy. 35 Through a study of CdSe QDs and NRs of varying dimensions, we show that our approach predicts AR lifetimes in quantitative agreement with experiments whereas the noninteracting formalism often overestimates the AR lifetimes by 1 − 2 orders of magnitude. The shorter AR lifetimes are a consequence of electron-hole pair localization which increases the Coulomb coupling and thereby the AR rate in the interacting formalism. By comparing the interacting and noninteracting formalisms (Fig. (1)), we also make evident the importance of including electron-hole correlations for the first theoretical postdiction of the observed volume scaling of the AR lifetime in QDs. Interestingly, the transition to the regime where excitonic effects must be included for an accurate AR lifetime calculation occurs at a surprisingly small diameter in CdSe QDs, below the exciton Bohr radius of CdSe. Additionally, we explain the AR lifetime scaling behavior in terms of the scaling of the Coulomb matrix elements and the density of final states in QDs and NRs. The method presented in this Letter is generally applicable to 0D, 1D, 2D and NC heterostructures.
AR involves the coupling of an initial biexcitonic state (|B ) of energy E B to a final excitonic state (|S ) of energy E S via the Coulomb interaction (V ). We utilize Fermi's golden rule to calculate the AR lifetime (τ AR ) where we average over thermally distributed initial biexcitonic states and sum over all final decay channels into single excitonic states:
τ −1 AR = B e −βE B Z B 2π S | B |V | S | 2 δ (E B − E S ) .
(1)
In the above, the delta function δ (E B − E S ) enforces energy conservation between the initial and final states and Z B is the partition function for biexcitonic states. Note that later when we compare to experimental values, we use a room temperature β for this Boltzmann weighted average, but we do not include temperature fluctuations in our NC configurations. 36 A brute force application of equation (1)
|S (0) = a † a a i |0 ⊗ |χ S(2)|B (0) = a † b a j a † c a k |0 ⊗ |χ B ,(3)
where the superscript "(0)" signifies a noninteracting picture is used. In the above, a † a and a i are electron creation and annihilation operators in quasiparticle state "a" and "i", respectively. The indexes a, b, c...
B = ε b − ε j + ε c − ε k , respectively.
The AR lifetime takes an explicit form (see the Methods section for a detailed derivation and discussion of the spin states studied herein) given by:
τ (0) AR −1 = 2π Z (0) B bckj e −β(ε b −ε j +εc−ε k ) a |V bacj | 2 δ (ε b + ε c − ε j − ε a ) + 2π Z (0) B bckj e −β(ε b −ε j +εc−ε k ) i |V ijbk | 2 δ (ε b − ε j − ε k + ε i ) .(4)
The first term on the right hand side (rhs) of equation (4) describes the decay of a negative trion of energy ε b + ε c − ε j into an electron of energy ε a while one of the holes remains a spectator (we refer to this as the "electron channel" and it is shown pictorially on the left side of Fig. 1), and the second term on the rhs of equation (4) describes the decay of a positive trion of energy ε b − ε j − ε k into a hole of energy ε i while one of the electrons remains a spectator (we refer to this as the "hole channel"). The explicit form of the Coulomb coupling is then given by:
V rsut = φ r (r) φ s (r) φ u (r ) φ t (r ) |r − r | d 3 r d 3 r ,(5)
where φ s (r) are the quasiparticle states for electrons (s ∈ a) or holes (s ∈ i) and there is no screeningconsistent with Ref. 33 and Ref. 37.
As discussed in the introduction, the noninteracting approach is suitable for nanostructures in the very strong confinement regime, where the kinetic energy is large compared to electron-hole interactions.
This approach fails, as shown below, for system sizes in the moderate to weak confinement regimes.
The inclusion of electron-hole correlations is mainly of significance in the description of the initial biexcitonic states while for the final excitonic states, the noninteracting framework seems suitable even for weakly confined structures, since the final state describes a highly excited electron-hole pair, above their ionization energy. Therefore, we use a noninteracting description for |S given by equation (2), but include electron-hole correlations in the description of the initial biexcitonic state. Motivated by the work of Refaely-Abramson et al., 37 we express the biexcitonic state as two spatially noninteracting but spincorrelated excitons. This is justified since electron-hole correlations are most significant within excitons as reflected by the larger exciton binding energy compared to that of biexcitons. 35 In our interacting approach the biexcitonic states take the form:
|B = b,j c,k c B b,j c B c,k a † b a j a † c a k |0 ⊗ |χ B ,(6)
where the coefficients c B b,j are determined by solving the Bethe-Salpeter equation (BSE), 38 as detailed in
Ref. 39. The excitonic energy is given by the noninteracting expression, while the biexcitonic energy is now a sum of the exciton energies, each obtained from the BSE. Within the interacting framework, the AR lifetime is given as a sum of electron-dominated (shown pictorially on the right side of Fig. 1) and
hole-dominated contributions:
τ −1 AR = 2π Z B B e −βE B a,i b,c,j c B b,i c B c,j V bacj 2 δ (E B − ε a + ε i ) + 2π Z B B e −βE B a,i j,b,k c B a,j c B b,k V ijbk 2 δ (E B − ε a + ε i ) ,(7)
where there are coherent sums of the Coulomb matrix elements multiplied with the coefficients that were For the implementation of the above frameworks, we chose the semi-empirical pseudopotential method to model the quasiparticle states. [42][43][44][45] And because we only need quasiparticle states in specific energy ranges (near the band-edge for the initial biexcitonic states and those that satisfy energy conservation for the final excitonic states), we utilize the filter-diagonalization technique 46,47 to obtain only the required electron and hole eigenstates. 47 Fig. 2. The difference in the band and optical gap is the exciton binding energy and is in good agreement with previous studies. 40,41 This suggests that (a) our model is accurate enough to reproduce single-(fundamental gap) and twoparticle (optical gap) properties with the simplification of a uniform dielectric screening and (b) that our computational machinery shows mild scaling with the system size, allowing a direct comparison with experiments for realistic NC sizes. (4)) and interacting (equation (7)) formalisms along with experimental 1, 23, 24 measurements of the AR lifetimes.
It is clear that neglecting electron-hole correlations in the initial biexcitonic state is only reasonable in the very strong confinement limit, where R QD a B (where a B = 5.6 nm is the exciton Bohr radius of CdSe). 48 The noninteracting-based AR lifetimes increase too rapidly as the volume of the QD increases compared to both the interacting formalism and experimentally measured AR lifetimes. Quantitatively, the computed scaling of the AR lifetime by the noninteracting formalism is τ
AR,QD ∝ V 1.69 , which is in contrast to the known volume scaling of the AR lifetime in single material QDs. 1 On the other hand, the volume scaling is accurately captured by the interacting formalism (τ AR,QD ∝ V 0.99 ), and the overall agreement with the experiments is remarkable. Recall that the previous theoretical studies using a noninteracting formalism for the AR lifetime either studied QDs small enough that the noninteracting formalism was able to relatively accurately predict the volume scaling of the AR lifetime 33 or the theories predicted a stronger dependence on the volume (∝ V 5/3 to V 2 ). 14,15 To understand the origin of the volume scaling of the AR lifetimes for QDs, we start with Fermi's golden rule and, for simplicity, focus on the rate of decay to hot electrons via the electron channel (similar arguments also hold for the hole channel) at zero temperature (b = c ≡ = LUMO and j ≡ h = HOMO) in the noninteracting approach:
τ (0) AR,e −1 = 2π a |V a h | 2 δ (ε + ε − ε h − ε a ) ,(8)
where ε + ε − ε h = 2E g equals two times the fundamental gap, E g . The scaling of the AR lifetime depends on the scaling of the final density of state and the Coulomb coupling. The former scales linearly with the volume of the NC. 49,50 Determining the scaling of the latter is more involved. Naively, one would predict it to scale with R −1 QD due to the Coulomb potential. However, because the final hot electron state is highly oscillatory, reflecting the high kinetic energy of the hot electron, and the initial biexcitonic state is slowly varying, the leading term that scales as R −1 QD vanishes. The next term, which can be obtained by invoking the stationary phase approximation, scales as R −3 QD . 14 Altogether, these arguments predict an
Auger lifetime that is proportional to the volume: τ −1 AR,e ∝ R −3 Surprisingly, the noninteracting formalism shows pronounced deviations from the interacting formalism for CdSe QDs with diameters as small as ∼ 2.5 nm, much smaller than the exciton Bohr radius (a B = 5.6 nm for CdSe). 48 This was a rather surprising result as all QDs studied here have R QD < a B ,
where electron-hole interactions are rather small compared to the confinement kinetic energy (see inset in Fig. 2).
The deviations in AR lifetimes predicted by the two formalisms are even larger for CdSe NRs. In Interestingly, more recent experimental measurements show nearly no volume effect on the AR lifetimes in CdSe NRs (striped blue square), 24 however, the same authors reported on the inconsistencies between transient absorption and time-resolved photoluminescence measurements (for the largest system studied, the two measurements differ by a factor of ≈ 3). Similar inconsistencies for NRs were reported for the reverse process, by which a hot exciton decays into a biexcitonic state by impact excitation, leading to multiexciton generation (MEG). Preliminary measurements reported a notable volume dependence of the impact excitation rate, 51, 52 while more recent theoretical work, 53 followed by experimental validation, 27 argued that impact excitation rates are volume independent. This suggests that different experimental setups (synthesis and optical measurements) may lead to different scaling behavior. A similar reasoning may also explain the discrepancy between the two sets of experimental results on AR lifetimes shown in Fig. 5. However, more experimental work is needed to fully understand the diversity of experimental outcomes, in particular, given that our new theoretical predictions are consistent with one set of measurements but not the other.
Returning to the AR lifetime scaling with volume in NRs, the noninteracting formalism behaves as
τ (0)
AR,NR ∝ V 2.02 . This is expected based on the scaling of the Coulomb matrix elements with the diameter and length of the NR, 53 but is in contrast to the scaling observed both experimentally 23 and theoretically using the interacting formalism. Thus, including electron-hole correlations is needed for both a quantitatively and qualitatively accurate description of the AR lifetime calculation in NRs. Intuitively, this result makes sense due to both the lack of confinement along the NR axis and the large electron-hole binding energy in CdSe NRs (∼ 200 meV) 48 contributing to making the noninteracting carrier approximation invalid in NRs.
As mentioned above, it is experimentally difficult to independently control the NR diameter and length; however, it is trivial to do computationally, so we analyzed the AR lifetime scaling separately for the NR diameter and length. We found that the AR lifetime scales approximately quadratically-cubically with the length of the NR in the noninteracting formalism, while it scales nearly linearly in the interacting formalism (Fig. 6), in agreement with previous experimental measurements. 23,[26][27][28]30 However, the scaling with the length of the NR depends slightly on the diameter. We also observed an approximate D 3 scaling in the interacting formalism, which still awaits experimental validation.
Our finding that the noninteracting formalism is inaccurate for NRs whereas the interacting formal-ism is accurate further corroborates previous kinetic models and experiments that argued that the total AR rate in NRs increases quadratically with the number of excitons, n (k AR (n) ∝ n (n − 1) /2). 10,25,28,54 In other words, kinetic models of AR in NRs should model AR as a bimolecular collision of two excitons;
in opposition to the combinatorial scaling of n 2 (n − 1) /2 if modeling AR as a three particle collision between free, noninteracting electrons and holes. Overall, these results on CdSe NRs add to the body of work that electrons and holes form bound 1D Wannier excitons in 1D systems such as semiconductor NRs and carbon nanotubes. [55][56][57][58] In conclusion, the interacting approach developed here for calculating AR lifetimes in NCs provides a framework that is able to predict quantitatively accurate AR lifetimes in both QDs and NRs. Our interacting formalism is the first to postdict the experimentally observed linear volume dependence of the AR lifetime in QDs as well as the correct scaling of the AR lifetimes in NRs with respect to the length and volume. This result was rationalized by noting that the matrix elements in AR lifetime calculations involve a product of the initial electron and hole states; thus, taking into account electron-hole correlations will have a large impact in regimes where the confinement energy is comparable or smaller than the exciton binding energy. Electron-hole correlations result in a localization of the pair, thereby, increasing the Coulomb coupling between the initial and final states. This is especially true in NRs where the lack of confinement along the NR axis makes the electron-hole attraction even more important. The resulting localization of the electron-hole pair leads to dramatic decreases in the AR lifetimes, as large as 2 orders of magnitude, when including such correlations.
Altogether, the interacting formalism outlined in this Letter constitutes a large step in bringing theoretical studies up to speed with ability of experimentalists to measure AR lifetimes and, in general, multiexciton dynamics. Our approach allows for direct comparisons and joint investigations between theorists and experimentalists as it permits accurate theoretical calculations of AR lifetimes for experimentally relevant nanostructures of any dimensionality and composition. It should be noted that our framework assumes the excitons scatter coherently; thus, systems in which exciton diffusion is the rate limiting step are currently outside the scope of our approach. In future work we plan to apply our formalism to study AR in CdSe NPLs and extend it to also include exciton diffusion processes, to resolve another experimentally controversy where two different methods provide significantly different scaling behaviors in 2D NPLs. 29,31 Methods A detailed derivation of the equations along with additional information and discussion on the implementation of the theory using the semi-empirical pseudopotential method, filter-diagonalization technique, Power law fits, τ AR = a × V b , are also shown for each of the three sets of AR lifetimes.
but whether this indicates a deviation from the volume scaling observed in QDs remains an open question. Nanoplatelet (NPL) structures appear to provide an example of the breakdown of the volume scaling of AR lifetimes. Contradictory results have been reported for the scaling of AR lifetimes with the lateral area (A). She et al. showed that the AR lifetimes are independent of A,
refer to the quasiparticle electron (unoccupied) states and i, j, k... refer to quasiparticle hole (occupied) states, with corresponding quasiparticle energies ε a and ε i . In equation(3), |0 is the ground state and |χ S and |χ B are the spin parts of the wavefunctions for excitons and biexcitons, respectively. Within the noninteracting formalism, the excitonic and biexcitonic energies are given by E(0) S = ε a − ε i and E (0)
obtained by diagonalizing the Bethe-Salpeter Hamiltonian matrix. Due to the presence of electron-hole interactions, all particles are involved in the AR process in the interacting formalism. For further details regarding the theory and the derivations of the above equations, please consult the Methods section.
Fig. 3
3displays the AR lifetimes obtained by using both the noninteracting (equation
QD2 R 3
3QD ∝ R −3 QD . Similar arguments hold for the scaling of the Auger lifetime in the interacting formalism. We find, as predicted, that the density of hot electrons and holes scales linearly with the volume of the NCs (top panel, Fig. 4) in both formalisms. However, the scaling of the average Coulomb coupling squared shows significant deviations from the expected V −2 stationary phase result in the noninteracting formalism (∝ V −2.74 ), while in the interacting formalism it scales as expected, ∝ V −1.99 . These different scalings can be rationalized by a more localized electron-hole wavefunction in the interacting case, due to the screened Coulomb electron-hole attraction term in the BSE, leading to more overlap with the wavefunction of the hot electron.
Fig. 5
5we show the calculated and measured[23][24][25] AR lifetimes for a series of CdSe NRs of different volumes. It is immediately evident that the noninteracting formalism is quantitatively incorrect for all NRs studied. The noninteracting-based AR lifetimes are also too long by approximately 1 − 2 orders of magnitude! This result arises from an underestimation of the Coulomb coupling due to the electron-hole wavefunctions being delocalized over the entire NR in the noninteracting formalism; there is no electronhole attraction to localize the electron-hole pair to form a bound Wannier exciton in the noninteracting formalism. In contrast, the interacting formalism predicts the scaling (nearly linearly with volume) as well as the magnitude of the AR lifetimes quiet accurately in comparison with the experimental results depicted by the solid blue squares.23 Based on the results reported for spherical QDs, this is to be expected and further signifies the importance of electron-hole correlations in the AR process in confined nanostructures.
Figure 1 :
1Pictorial representations are shown for the electron channel of an Auger recombination (AR) event in the noninteracting (left) and interacting (right) formalisms. The black horizontal lines represent the discrete quasiparticle states of the semiconductor nanostructures. The gray box in the interacting formalism represents the fact that the excitons (correlated electron-hole pairs) are a linear combination of the quasiparticle states within the box that were included in the BSE. E g is the fundamental gap and E opt is the optical gap. |B (0) is the initial state in the noninteracting formalism (note that one of the holes is a spectator and the AR process describes a negative trion, t − , decaying to an excited quasielectron state). |B is the initial state in the interacting formalism composed of two excitons and all 4 particles are involved in the AR process. The final states in both formalisms are given by |S (0) . The dashed line represents the Coulomb interaction.
Figure 2 :
2Energy gaps (in eV) for the seventeen CdSe QDs. The fundamental gap is shown in blue solid squares and the optical gap is shown in red solid circles. The inset shows the exciton binding energy (the energy difference between the fundamental and optical gaps) which ranges from ∼ 500 meV for the smallest QDs to ∼ 150 meV for the largest QDs studied here. For comparison, we also show the measured exciton binding energy (green stars, Ref. 40) and calculations based on a semi-empirical pseudopotential model using a perturbative scheme (maroon circles, Ref. 41).
Figure 3 :
3AR lifetimes, τ AR , for CdSe QDs as a function of the volume of the QD. Good agreement is observed between the interacting formalism (green circles) and experimental (blue squares: solid, 1 vertical lines23 and horizontal lines 24 ) AR lifetimes for all sizes. On the other hand, the noninteracting formalism (red triangles) deviates from the experimental values for QD volumes > 10 nm 3 . Power law fits, τ AR = a × V b , are also shown for each of the three sets of AR lifetimes.
Figure 4 :
4The top half shows the density of states at the energy of the hot electron and holes satisfying energy conservation for CdSe QDs as a function of the volume of the QD. The hot electrons (holes) have energies approximately E g above (below) the HOMO (LUMO) in the noninteracting case and in the interacting formalism the hot electrons (holes) have energies approximately E opt above (below) the HOMO (LUMO). The bottom half shows the average of the Coulomb couplings, W 2 , squared to the final states. The noninteracting formalism results are shown as red triangles and the interacting formalism results are shown as green circles. Power law fits, f (V ) = a × V b , are also shown for all sets.
Figure 5 :
5Auger recombination lifetimes for CdSe NRs as a function of the volume of the NRs predicted by the interacting (green circles), the noninteracting (red triangles) formalisms along with experimentally measured (blue squares: solid23 , vertical24 and horizontal 25 lines) AR lifetimes. The three different sizes used correspond to the three different diameters (1.53 nm, 2.14 nm and 2.89 nm) studied computationally.
Figure 6 :
6Interacting formalism based Auger recombination lifetimes for CdSe NRs as a function of the length (left) and diameter (right) of the NR. Power law fits, τ AR = a × D b and τ AR = a × L b , are also shown for each NR set.
for nanostructures is prohibitive for several reasons. First, there is currently no tractable electronic structure method for a fully-correlated biexcitonic state and for excitonic states at high energies. Second, the number of initial and final states that satisfy energy conservation increases rapidly with the system size. For these reasons, computational and theoretical studies of AR in confined nanostructures have relied on a noninteracting formalism to describe |S and|B : 14-17, 32-34
Electron-hole correlations were included in the interacting formalism by solving the BSE within the static screening approximation, where the dielectric constant was taken from the work of Wang & Zunger. 43 For QDs, we calculated the AR lifetimes for seventeen wurtzite CdSe QDs with diameters ranging from D QD = 2R QD = 1.2 nm (Cd 20 Se 19 ) to D QD = 2R QD = 5.3 nm (Cd 1358 Se 1360 ). For completeness, we also calculated the fundamental and optical gaps for the CdSe QDs, shown in
Bethe-Salpeter equation, Fermi's golden rule in the AR lifetime calculations presented in this Letter and the procedure used to construct the CdSe QDs and NRs is also outlined. This material is available athttps://doi.org/.
58. Pal, S., Casanova, D. & Prezhdo, O. V. Effect of Aspect Ratio on Multiparticle Auger Recombination
in Single-Walled Carbon Nanotubes: Time Domain Atomistic Simulation. Nano Lett. 18, 58-63
(2018).
Supplementary Information is available for this paper at https://doi.org/.
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Correspondence Correspondence and requests for materials should be addressed to J.P.P. (email: [email protected]) or to E.R. (email: [email protected]).
Acknowledgements The authors thank Mr. Devan Skubitz for preliminary tests of the developed code and Dr. Author Contributions J.P.P and E.R. developed the theoretical framework, computer code, performed the calculations and co-wrote the paper.Competing InterestsThe authors declare that they have no competing financial interests.
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"SCREENING RULES FOR OVERLAPPING GROUP LASSO",
"SCREENING RULES FOR OVERLAPPING GROUP LASSO"
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"Seunghak Lee \nCarnegie Mellon University\n\n",
"Eric P Xing \nCarnegie Mellon University\n\n",
"Seunghak Lee \nCarnegie Mellon University\n\n",
"Eric P Xing \nCarnegie Mellon University\n\n"
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"Carnegie Mellon University\n",
"Carnegie Mellon University\n",
"Carnegie Mellon University\n",
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| Recently, to solve large-scale lasso and group lasso problems, screening rules have been developed, the goal of which is to reduce the problem size by efficiently discarding zero coefficients using simple rules independently of the others. However, screening for overlapping group lasso remains an open challenge because the overlaps between groups make it infeasible to test each group independently. In this paper, we develop screening rules for overlapping group lasso. To address the challenge arising from groups with overlaps, we take into account overlapping groups only if they are inclusive of the group being tested, and then we derive screening rules, adopting the dual polytope projection approach. This strategy allows us to screen each group independently of each other. In our experiments, we demonstrate the efficiency of our screening rules on various datasets. | null | [
"https://arxiv.org/pdf/1410.6880v1.pdf"
]
| 15,256,251 | 1410.6880 | 20a16efb03c366fa4180659c2b2a0c5024c679da |
SCREENING RULES FOR OVERLAPPING GROUP LASSO
Seunghak Lee
Carnegie Mellon University
Eric P Xing
Carnegie Mellon University
SCREENING RULES FOR OVERLAPPING GROUP LASSO
Recently, to solve large-scale lasso and group lasso problems, screening rules have been developed, the goal of which is to reduce the problem size by efficiently discarding zero coefficients using simple rules independently of the others. However, screening for overlapping group lasso remains an open challenge because the overlaps between groups make it infeasible to test each group independently. In this paper, we develop screening rules for overlapping group lasso. To address the challenge arising from groups with overlaps, we take into account overlapping groups only if they are inclusive of the group being tested, and then we derive screening rules, adopting the dual polytope projection approach. This strategy allows us to screen each group independently of each other. In our experiments, we demonstrate the efficiency of our screening rules on various datasets.
1. Introduction. We propose efficient screening rules for regression with the overlapping group lasso penalty. Our goal is to develop simple rules to discard groups with zero coefficients in the optimization problem with the following form:
(1.1) min β 1 2 y − Xβ 2 2 + λ g∈G √ n g β g 2 , where X ∈ R N ×J is the input data for J inputs and N samples, y ∈ R N ×1 is the output vector, β ∈ R J×1 is the vector of regression coefficients, n g is the size of group g, and λ is a regularization parameter that determines the sparsity of β. In this setting, G represents a set of groups of coefficients, defined a priori, and we allow arbitrary overlap between different groups, hence "overlapping" group lasso. Overlapping group lasso is a general model that subsumes lasso (Tibshirani, 1996), group lasso (Yuan and Lin, 2006), sparse group lasso (Simon et al., 2013), composite absolute penalties (Zhao, Rocha and Yu, 2009), and tree lasso (Zhao, Rocha and Yu, 2009;Kim et al., 2012) with 1 / 2 penalty because they are a specific form of overlapping group lasso.
In this paper, we do not consider the latent group lasso proposed by Jacob et al. (Jacob, Obozinski and Vert, 2009), where support is defined by the union of groups with nonzero coefficients. Instead, we consider the For screening test on group g2, we consider g3 but disregard g1 and g4 to enable the independent screening test.
overlapping group lasso in the formulation of (1.1), where support is defined by the complement of the union of groups with zero coefficients (Jenatton, Audibert and Bach, 2011;Yuan, Liu and Ye, 2011). Therefore, unlike the latent group lasso, simple conversion from (1.1) to nonoverlapping group lasso problems by duplicating features overlapped between different groups is infeasible. Recently, to solve (1.1) efficiently, fast algorithms have been developed (Yuan, Liu and Ye, 2011;Chen et al., 2012;Deng, Yin and Zhang, 2013) (we refer readers to (Bach et al., 2012) for review of optimization with sparsity-inducing penalties); however, in many applications such as genomewide association studies (Lee and Xing, 2012;Yang et al., 2010), the number of features (or number of groups) can be very large. In such cases, fast optimization of (1.1) is challenging because it requires us to sweep over all coefficients/groups of coefficients many times until the objective converges. Furthermore, parallelization of existing sequential algorithms for speedup is nontrivial.
The past years have seen the emergence of screening techniques that can discard zero coefficients using simple rules in a single sweep over all coefficients. Examples include Sasvi rules (Liu et al., 2013), dual polytope projection (DPP) rules , dome tests (Xiang and Ramadge, 2012), sphere tests (Xiang, Xu and Ramadge, 2011), SAFE rules (Ghaoui, Viallon and Rabbani, 2012), and strong rules (Tibshirani et al., 2012). Among these, strong rules are not exact (true nonzero coefficients can be mistakenly discarded), whereas the other tests are exact. Furthermore, Bonnefoy et al. (Bonnefoy et al., 2014) recently developed an approach that merges screening approach with first-order optimization algorithms for lasso; however, to the best of our knowledge, none of the existing screening methods can be applied to (1.1) with overlapping groups. DPP and strong rules can be used for nonoverlapping group lasso, and the others are developed only for lasso.
In this paper, we develop exact screening rules for overlapping group lasso. The proposed screening rules can efficiently discard groups with zero coefficients by looking at each group independently. As a result, after screening, the number of groups that potentially include nonzero coefficients is small, and thus we can reduce the size of (1.1) by reformulating it using only the groups that survived. We then employ an optimization technique to solve the reduced problem. The resultant solution is optimal because the screening rules are exact in the sense that nonzero coefficients in a global optimal solution are never mistakenly discarded. The key idea behind our approach is to consider only groups that are inclusive of tested groups, while ignoring the other overlapping groups to perform independent screening tests. For example, in Figure 1, when performing a screening test on group g 2 , we take into account only g 3 because g 3 is a subset of g 2 , whereas g 1 and g 4 are non-inclusive of g 2 . The contributions of this paper are as follows:
1. We develop novel overlapping group lasso screening (OLS) and sparse overlapping group lasso screening (SOLS) rules. Sparse overlapping group lasso is a special case of overlapping group lasso, as formulated in (3.23). 2. We show that the screening rule for nonoverlapping group lasso via DPP (group DPP or GDPP) is also exact when applied to overlapping group lasso.
In our experiments, we demonstrate that OLS, SOLS, and GDPP give us significant speed-up against a solver without screening. For example, OLS and SOLS achieved a 3.7× and 3× speed-up on PIE image dataset, compared to an overlapping group lasso solver without screening. Furthermore, OLS and SOLS are substantially more efficient to discard features with zero coefficients than GDPP under various experimental settings, confirming that the proposed algorithms are capable of using overlapping groups for screening.
Notation. We refer matrices to boldface and uppercase letters; vectors to boldface and lowercase letters; and scalars to regular lowercase letters. Columns are indexed by subscripts (e.g., x j is the j-th column vector of the matrix X). We refer g or h to a group of coefficients, and w g represents a sub-vector of w, indexed by g. . The DPP screening rules are derived as follows: first, we find a dual form of group lasso and its Karush-Kuhn-Tucker (KKT) conditions. Then, using the KKT conditions and the relationship between primal and dual solutions, we find screening rules to discard groups with zero coefficients; however, such screening rules involve a dual optimal solution, which is unknown. Thus, the DPP approach finds a range of vectors that includes a dual optimal solution, which is easy to obtain, and uses it instead of a dual optimal solution for screening rules. Here we review the DPP screening rule for nonoverlapping group lasso because it gives us with a vehicle to derive screening rules for overlapping group lasso.
For the nonoverlapping group lasso, screening rules via DPP can be obtained by the following procedure: θ * is unknown. To address this problem, we estimate a range of vectors, denoted by Θ, that contains θ * based on θ * (λ 0 ), where λ 0 = λ. Specifically, to estimate Θ, we use the fact that P F is continuous and nonexpansive. Finally, the DPP screening rule for nonoverlapping group lasso is formulated as follows: if sup θ∈Θ X T g θ 2 < √ n g , then β * g = 0. By finding a closed-form solution for the left-hand side of the rule, i.e., sup θ∈Θ X T g θ 2 , we can obtain screening rules for nonoverlapping group lasso.
3. Overlapping Group Lasso Screening. Now we develop overlapping group lasso screening rules. The first challenge is that the groups are not separable, making independent tests infeasible. Second, it is also unclear how to make use of overlapping groups in screening rules. Intuitively, coefficients are more likely to be zero if they are involved in more groups; thus incorporating overlapping groups into screening rules can help discard more features.
We address the first challenge by considering only a set of groups that is a subset of the tested group. Consider the example in Figure 1. Suppose we want to test if β g 2 = 0, g 2 = {2, 3, 4} given three other groups: g 1 = {1, 2, 3}, g 3 = {3, 4}, and g 4 = {4, 5}. In such a case, we consider only g 3 for the screening test on g 2 because g 3 is included in g 2 , allowing us to test g 2 independent of other groups; we ignore g 1 and g 4 in testing g 2 because they involve coefficients not included in g 2 , preventing us from testing the groups independently. For the second challenge, we derive new screening rules that can exploit the overlapping groups for minimizing the left-hand side of screening rules. In other words, they discard more features as the number of overlapping groups, inclusive of a tested group, increases.
In §3.1, we start with a condition for β * g = 0 for overlapping group lasso that contains a dual optimal. Based on this condition, we derive a screening rule by replacing a dual optimal with a range that includes the dual optimal in §3.2. Finally, in §3.3, we present efficient algorithms for overlapping group lasso screening based on the screening rules obtained in §3.2.
3.1. Screening Condition for Overlapping Group Lasso. Let us start with the dual form of overlapping group lasso (see appendix for derivation of the dual form):
sup θ 1 2 y 2 2 − λ 2 2 θ − y λ 2 2 (3.1) subject to X T θ = v,
where θ is a vector of dual variables, and v is a subgradient of g∈G √ n g β g 2
with respect to β. A dual optimal θ * can be obtained by projecting y λ onto F ≡ X T θ = v , denoted by P F y λ . We will use the "non-expansiveness" property of this projection operator to derive screening rules in §3.2.
Next we derive the KKT conditions for overlapping group lasso. Introducing z = y − Xβ, (1.1) can be written as
min β 1 2 z 2 2 + λ g∈G √ n g β g 2 (3.2) subject to z = y − Xβ.
Then, a Lagrangian of (3.2) is
(3.3) L(β, z, θ) = 1 2 z 2 2 + λ g∈G √ n g β g 2 + λθ T (y − Xβ − z) ,
and the KKT conditions of (3.3) are as follows:
0 ∈ ∂L(β * , z * , θ * ) ∂β g = −λX T g θ * + λv g , (3.4) 0 = ∇ z L(β * , z * , θ * ) = z * − λθ * , (3.5) 0 = ∇ θ L(β * , z * , θ * ) = λ (y − Xβ * − z * ) , (3.6)
where v g is a subgradient of g∈G √ n g β g 2 with respect to β g . From (3.5) and (3.6), we obtain a bridge between the primal and dual solutions:
(3.7) λθ * = y − Xβ * .
Let us define two sets of groups that overlap with group g as follows:
G 1 = {h : h ∈ G − g, h ⊆ g, h ∩ g = ∅}, (3.8)Ḡ
whereḠ 1 andḠ 2 are sets of groups overlapping with g, andḠ 1 includes the groups that are inclusive of g. Then, we denote
v g = √ n g [γ 1 , . . . , γ ng ] T ,
where γ j is given by
(3.10) γ j = u j + j∈h,h∈Ḡ 1 w j + j∈h,h∈Ḡ 2 s j ,
where u j is a subgradient of β g 2 with respect to β j ; w j and s j are subgradients of β h 2 with respect to β j , where h belongs toḠ 1 andḠ 2 , respectively. The definition of 2 norm subgradient, i.e., u g 2 ≤ 1, w h 2 ≤ 1, and s h 2 ≤ 1, gives us
(3.11) j∈g u 2 j = j∈g γ j − j∈h,h∈Ḡ 1 w j − j∈h,h∈Ḡ 2 s j 2 ≤ 1,
where the equality holds when β * g = 0. Plugging (3.4) into (3.11), a sufficient condition for β * g = 0 is given by
(3.12) min w,s: w h 2 ≤1, s h 2 ≤1 j∈g x T j θ * − j∈h,h∈Ḡ 1 √ n h w j − j∈h,h∈Ḡ 2 √ n h s j 2 < √ ng.
To screen each group independently (i.e., test using only the coefficients in group g), we set s h = 0, for all h ∈Ḡ 2 . This is a valid subgradient because it always satisfies s h 2 ≤ 1. Given subgradients of the groups inclusive of g, we have the following screening condition for β
* g = 0: if b g < √ n g , then β * g = 0, where b g is defined by b g ≡ min w: w h 2 ≤1 j∈g x T j θ * − j∈h,h∈Ḡ 1 √ n h w j 2 . (3.13)
Note that b g is an upper bound on the left-hand side of (3.12) due to the fixed s h = 0, for all h ∈Ḡ 2 . If b g < √ n g , then (3.12) holds, and thus β * g = 0.
3.2. Screening Rules for Overlapping Group Lasso. So far, we have derived a condition for β * g = 0; however, it is not yet usable for screening because θ * is unknown. Thus, by following the DPP approach by Wang et al. , we first find a region Θ that contains θ * .
In (3.1), an optimal θ * is the projection of y λ onto the constraint F:
θ * = P F y λ = argmin θ∈F θ − y λ 2 ,
where P F is the projection operator with F which is a nonempty, closed convex subset of a Hilbert space (F is nonempty because 0 ∈ F, and closed convex because it is an intersection of closed half-spaces). Thus, we can use the "non-expansiveness" property of P F (Bertsekas et al., 2003), given by
(3.14) P F y λ − P F y λ 0 2 = θ * (λ) − θ * (λ 0 ) 2 ≤ y λ − y λ 0 2 ,
where λ 0 is a tuning parameter (λ 0 > λ), and θ * (λ) ≡ θ * , and θ * (λ 0 ) is a dual optimal solution given λ 0 . Here (3.14) shows that θ * (λ) lies within a sphere Θ centered at θ * (λ 0 ) with a radius of ρ = y λ − y λ 0 2
. Based on this, we can represent θ * (λ) = θ * (λ 0 ) + r, where r 2 ≤ ρ. By plugging it into (3.13) and maximizing the objective over r, we have the following screening
rule: if b g < √ n g , then β * = 0, where b g is defined by b g ≡ sup r: r 2 ≤ρ min w: w h 2 ≤1 j∈g x T j {θ * (λ0) + r} − j∈h,h∈Ḡ 1 √ n h wj 2 . (3.15) Notice that b g is an upper bound on b g , and thus b g < √ n g ⇒ b g < √ n g ⇒ β * = 0.
With a little bit of algebra, we get our screening rule for overlapping group lasso.
Theorem 1. For the overlapping lasso problem, suppose that we are given an optimal dual solution θ * (λ 0 ). Then for λ < λ
0 , β * g (λ) = 0 if (3.16) min w h : w h 2 ≤1 j∈g x T j θ * (λ 0 ) − j∈h,h∈Ḡ 1 √ n h w j 2 < √ ng − Xg F y 2 1 λ − 1 λ 0 . Proof. See Appendix B.
Using the bridge between the primal and dual in (3.7), we can also obtain a screening rule in a primal form.
Theorem 2. For the overlapping lasso problem, suppose that we are given an optimal solution β * (λ 0 ). Then for λ < λ 0 , β *
g (λ) = 0 if (3.17) min w h : w h 2 ≤1 j∈g x T j y − Xβ * (λ 0 ) λ 0 − j∈h,h∈Ḡ 1 √ n h w j 2 < √ ng − Xg F y 2 1 λ − 1 λ 0 .
We also note that Theorem 2 can be employed to solve lasso problems following a λ path {λ 1 , λ 2 , . . . , λ T } in a descending order. The λ path is determined a priori, and one may choose linearly, geometrically, or logarithmically spaced λ values. In the sequential version of screening, we first perform screening with λ 1 using λ with β * (λ ) = 0. We then run a solver using the remaining coefficients with their corresponding groups; its results become β * (λ 1 ). Now, using λ 1 with β * (λ 1 ), we perform screening for λ 2 . We repeat the above procedure for all remaining λ parameters. The following theorem shows sequential screening rule, where a screening rule for λ t is constructed based on β * (λ t−1 ), t ≥ 2.
Theorem 3. For the overlapping lasso problem with a λ path {λ 1 , . . . , λ T }, λ t−1 > λ t , t = 2, . . . , T , suppose that we are given an optimal solution β * (λ t−1 ).
Then, β * g (λ t ) = 0 if (3.18) min w h : w h 2 ≤1 j∈g x T j y − Xβ * (λ t−1 ) λ t−1 − j∈h,h∈Ḡ 1 √ n h w j 2 < √ ng − Xg F y 2 1 λ t − 1 λ t−1 .
We omit the proofs for Theorem 2 and Theorem 3 because it is straightforward to derive them from Theorem 1.
3.3. Screening Algorithms for Overlapping Group Lasso. To use Theorems 1, 2, 3 we need an efficient way to obtain the left-hand side. Instead of solving the left-hand side directly, we minimize an upper bound on the left-hand side because it can be quickly solved. Any upper bounds give us a valid screening rule, but the tighter the bound, the better the screening efficiency (it discards more features). Note that the goal of screening is to speed up optimization, and thus we intend to present a simple yet efficient algorithm.
We first present our algorithm and then verify that it minimizes an upper bound on the left-hand side. We adopt a simple coordinate descent-type approach, where each group is used for minimization one at a time. Suppose that we perform screening on group g. We start with making two variables: l ← 0, a ← g, and a set of overlapping groups {h 1 , . . . , h K : h k ∈Ḡ 1 , k = 1, . . . , K} (any ordering works for our purpose). For each group h k , we take the intersection between h k and a, i.e., h k ← h k ∩ a. If h k = ∅, we skip h k and proceed to the next h k+1 . If h k = ∅, we take the following procedure.
If X h k θ * (λ 0 ) 2 ≤ √ n h k , we set l ← l; otherwise l ← l + z, where z = j∈h k x T j θ * (λ 0 ) − √ n h k w j 2
, and {w j : j ∈ h k } is determined by the following algorithm.
1. Set d = 1. 2. For each j ∈ h k , we compute (3.19) w j = x T j θ * (λ 0 ) √ n h k , if x T j θ * (λ 0 ) √ n h k ≤ √ d sign x T j θ * (λ 0 ) √ d , otherwise, and update d ← d − w 2 j .
Algorithm 1: Screening for overlapping group Lasso
Input: X, y, λ t−1 , λt (λ t−1 > λt), G, θ * (λ t−1 ) = y−Xβ * (λ t−1 ) λ t−1
Output: T (a set of groups with potential nonzero coefficients)
1 T ← ∅; 2 for g ∈ G do 3Ḡ 1 = {h : h ∈ G − g, h ⊆ g, h ∩ g = ∅}; 4 a ← g; 5 l ← 0; 6 for h ∈Ḡ 1 do 7 h ← h ∩ a; 8 if X T h θ * (λ t−1 ) 2 > √ n h then 9 d ← 1; 10 for j ∈ h do 11 if x T j θ * (λ t−1 ) ≤ √ dn h then 12 d ← d − x T j θ * (λ t−1 ) √ n h 2 ; 13 else 14 l ← l + x T j θ * (λ t−1 ) − √ dn h sign x T j θ * (λ t−1 ) 2 ; 15 break; 16 a ← a − h ; 17 if l + X T a θ * (λ t−1 ) 2 2 ≥ √ ng − Xg F y 2 1 λ t − 1 λ t−1 then 18 T ← T ∪ gi;
We then set a ← a − h k , and iterate this procedure over all groups inḠ 1 .
Finally, a minimized left-hand side for the screening rules is l + x T a θ * (λ 0 ) 2 2 . This algorithm in a sequential setting for overlapping group lasso (screening for λ t given λ t−1 , and λ t ∈ {λ 1 , . . . , λ T }) is summarized in Algorithm 1. Now, we show that this algorithm minimizes an upper bound on the lefthand side of Theorem 1. The key idea is that, at the (k + 1)-th iteration, we minimize an upper bound on the bound obtained at the k-th iteration. We denote a by the set of coefficients to be processed, and k by the iteration counter, initialized by a ← g and k = 1. At the k-th iteration, the squared left-hand side is bounded as follows:
min w h ,∀h∈Ḡ 1 j∈g x T j θ * (λ0) − j∈h,h∈Ḡ 1 √ n h wj 2 (3.20) ≤ min w h ,∀h∈Ḡ 1 −h k j∈g−h k x T j θ * (λ0) − j∈h,h∈Ḡ 1 −h k √ n h wj 2 (3.21) + min w h k X T h k θ * (λ0) − √ n h k w h k 2 2 .
To obtain the upper bound in (3.21), we set w j = 0 for all {j : j ∈ h∩h k , h ∈ G 1 − h k }. Let us fix {w h : h ∈Ḡ 1 − h k } and then find the bound in (3.21). Since the first term is a constant due to fixed {w h }, we find z that bounds the second term:
min w h k X T h k θ * (λ 0 ) − √ n h k w h k 2 2 ≤ z. Based on the subgradient condition w h k 2 ≤ 1, if X T h k θ * (λ 0 ) 2 ≤ √ n h k , we set X T h k θ * (λ 0 ) − √ n h k w h k 2 2
= 0 = z; otherwise, we find an upper bound z using the simple coordinate descent-type procedure with (3.19).
It is easy to see that the procedure with (3.19) satisfies the subgradient condition w h k 2 ≤ 1, and thus z is a valid upper bound. Then, we set a ← a − h k because h k is processed, and set l ← z. Subsequently, we get an upper bound on (3.21):
min w h ,∀h∈Ḡ 1 −h k j∈g−h k x T j θ * (λ 0 ) − j∈h,h∈Ḡ 1 −h k √ n h w j 2 + l ≤ min w h ,∀h∈Ḡ 1 −h k −h k+1 j∈g−h k −h k+1 x T j θ * (λ 0 ) − j∈h,h∈Ḡ 1 −h k −h k+1 √ n h w j 2 (3.22) + min w h k+1 X T h k+1 θ * (λ 0 ) − n h k+1 w h k+1 2 2 + l, where h k+1 = h k+1 ∩ a. Fixing {w h : h ∈Ḡ 1 − h k − h k+1 }, and setting w j = 0, ∀j ∈ h ∩ h k , h ∈Ḡ 1 − h k − h k+1 , we minimize an upper bound z on X T h k+1 θ * (λ 0 ) − √ n h k+1 w h k+1 2 2
using the procedure with (3.19), and l ← l + z. We iterate this procedure over all groups inḠ 1 , i.e., {h ∈Ḡ 1 }, resulting in an upper bound on the left-hand side as follows:
min w h ,∀h∈Ḡ 1 −h 1 −...−h K j∈g−h 1 −...−h K x T j θ * (λ0) − j∈h,h∈Ḡ 1 −h 1 −...−h K √ n h wj 2 + l. By setting w j = 0 for all j ∈ h ∩ h k , h ∈Ḡ 1 − h 1 − . . . − h K ,
we get an upper bound on the squared left-hand side, i.e., l + x T a θ * (λ 0 ) 2 2 ; taking the square root on it, we obtain the left-hand side of Theorem 1.
We note that the DPP screening rule for nonoverlapping group lasso (GDPP) ) is a special case of the proposed screening rules for overlapping group lasso. In Theorem 1, by setting w j = 0 for all j ∈ h, we obtain GDPP, where its left-hand side is an upper bound on that of our screening rules. This implies that GDPP is also an exact screening Algorithm 2: Screening for sparse overlapping group lasso
Input: X, y, λ t−1 , λt (λ t−1 > λt), G, θ * (λ t−1 ) = y−Xβ * (λ t−1 ) λ t−1
Output: T (a set of groups with potential nonzero coefficients)
1 T ← ∅; 2 for g ∈ G do 3 if |G| ≥ 2 then 4 l ← 0; 5 for j ∈ G do 6 if x T j θ * (λ t−1 ) > 1 then 7 l ← l + x T j θ * (λ t−1 ) − sign x T j θ * (λ t−1 ) 2 ; 8 else 9 l ← x T j θ * (λ t−1 ) 2 ; 10 if √ l ≥ √ ng − Xg F y 2 1 λ t − 1 λ t−1 then 11 T ← T ∪ gi;
rule for overlapping group lasso; however, GDPP would not be as efficient as Theorem 1 due to the lack of degrees of freedom to decrease its left-hand side. It is surprising that GDPP is applicable to overlapping group lasso because GDPP is derived under the assumption that groups do not overlap. In practice, findingḠ 1 can be an algorithmic bottleneck (line 3 in Algorithm 1). To search forḠ 1 efficiently, we used a simple algorithm. We first sort the groups based on the smallest index of each group, resulting in G = {g (1) , . . . , g (M ) }. To perform a screening test on g (m) , we find itsḠ 1 by searching for the groups between g (m+1) and g (m+W ) , where W is the user-defined window size. With larger W , we may discard more features, but the computational complexity for screening increases linearly in W .
3.3.1. Screening Algorithm for Sparse Overlapping Group Lasso. Sparse overlapping group lasso is a special case of overlapping group lasso that includes 1 penalty, defined by
(3.23) min β 1 2 y − Xβ 2 2 + λ 1 β 1 + λ 2 g∈G √ n g β g 2 .
For this model, we provide a simple and fast algorithm by considering only the individual coefficients in 1 penalty forḠ 1 . Note that individual features can be considered as groups of size one, inclusive of other groups. Substitut-ingḠ 1 in line 3 in Algorithm 1 byḠ 1 = {j : j ∈ g}, we obtain a screening algorithm for sparse overlapping group lasso, summarized in Algorithm 2.
Algorithm 3: Finding a small λ that sets all coefficients to zero
Input: X, y, G, r (0 < r < 1 is a common ratio for a geometric series) Output: λ that discards all features
1 λ 1 = max g∈G 1 √ ng X T g y 2 ; 2 for t = 2 to T do 3 λt = λ t−1 r; 4 θ * (λ t−1 ) = y λ t−1 ; 5 T ← Algorithm1(X, y, λt, λ t−1 , G, θ * (λ t−1 )); 6 if T = ∅ then 7 λ = λ t−1 ; 8 break;
It is worthwhile to mention that Algorithm 2 is significantly faster than Algorithm 1 due to the lack of set operations in Algorithm 1. Furthermore, in our experiments, we observed that Algorithm 2 discards similar numbers of features to Algorithm 1 on various datasets. Thus, if the model in (3.23) is utilized in an application, Algorithm 2 would be appealing in terms of both its screening rejection power and speed.
Remark. For nonoverlapping group lasso, we find λ max (the smallest λ that sets β * = 0) as follows: λ max = max g∈G 1 √ ng X T g y 2 . However, this is not necessarily the smallest λ for zero solutions for overlapping group lasso. In fact, it is nontrivial to compute λ max for overlapping group lasso because different groups are coupled through overlaps, preventing us from using the simple equation above. Instead, using a screening algorithm, we can find a small λ that sets all coefficients to zero, denoted by λ . The key idea is that we decrease λ following a sequence until all coefficients are discarded by a screening algorithm. We denote λ by the smallest λ that sets β * = 0 in our regularization path. This technique is summarized in Algorithm 3 with a geometric sequence of λ parameters.
Experiments.
We demonstrate the efficiency of the proposed screening algorithms in terms of the screening rejection ratio and the speed. The rejection ratio is defined by the ratio of discarded coefficients to true zero coefficients, obtained by an overlapping group lasso solver without screening. Here we refer Algorithm 1 to OLS and Algorithm 2 to SOLS.
To the best of our knowledge, there are no existing screening rules for overlapping group lasso; however, we showed that GDPP can also be used for overlapping group lasso as an exact screening rule. Therefore, as a baseline screening algorithm, we used GDPP. Comparing OLS 2 and SOLS against GDPP, we investigate the benefits of using overlapping groups for screening because GDPP makes no use of overlapping groups.
We used the following three image datasets 3 and one genome dataset for our experiments, chosen to cover the cases where N >> J, N << J, and N ≈ J, and a real-world application in genetics. a) The PIE image database (Sim, Baker and Bsat, 2002), which contains 11,554 face images of 68 people under different poses, illumination conditions, and expressions. The images are represented by D ∈ R 11554×1024 . We generated the response vector y ∈ R 11554×1 by randomly choosing a feature in D, and the rest of the features are concatenated to be the design matrix X ∈ R 11554×1023 . b) The Alzheimer's disease (AD) dataset , which contains 541 AD individuals, represented by 511,997 single nucleotide polymorphisms (the most common genetic variants). For the same individuals, the AD dataset contains 40,638 expression levels for known and predicted genes, splice variants, miRNAs, and non-coding RNA sequences in the brain region of the cerebellum. We used the genetic information for X ∈ R 541×511997 , and randomly choose a gene expression for y ∈ R 541×1 . This is a typical experimental setting for expression quantitative mapping (Lee and Xing, 2012), where the goal is to identify genetic variants that affect gene expression levels. c) The COIL database (Nayar, Nene and Murase, 1996) contains color images of 100 objects. For each object, 72 images are taken with different angles. We selected object 10, whose image data are represented by D ∈ R 72×49152 . X and y are generated in the same way as the PIE dataset. d) The digit recognition data (GISETTE) (Guyon et al., 2004), which contains 6,000 digit images of four and nine. Each sample contains 5,000 features, where 70% of features are constructed from MNIST dataset (LeCun et al., 1998), and 30% of them are artificially generated following a distribution of true features, resulting in X ∈ R 6000×5000 . The response vector y ∈ R 6000×1 consists of binary labels, indicating the class (either four or nine) of each sample.
Furthermore, we test screening algorithms on the sparse overlapping group lasso in (3.23) with λ 1 = λ 2 under different group structures. We chose this model because it induces individual level sparsity, allowing us to capture complex patterns of nonzero coefficients. Moreover, we generated group structures based on feature locality because features located nearly are often associated with an output vector jointly in image or genome datasets. The group structures used in our experiments are as follows:
1. 1 + nonoverlap groups: G contains nonoverlapping groups, where 3 from http://www.csie.ntu.edu.tw/∼cjlin/libsvmtools/datasets/ each group consists of 20 consecutive features. 2. 1 + tree structure groups: G contains tree structured groups with four levels, where from the root to the leaves, the groups consist of 20/15/10/5 consecutive features. A parent group is always super set of their children groups. 3. 1 + overlap groups: G contains overlapping groups, in which each group contains 20 features and consecutive groups overlap by 5 features. 4. 1 + overlap groups guided by prior knowledge: This group structure is constructed only for AD dataset. G contains 26,222 groups, in which each group contains single nucleotide polymorphisms (i.e., features in AD dataset) located on the same gene region, defined by the interval between the gene's transcription start and end site. Note that groups may overlap due to overlapping genes.
We solved the overlapping group lasso problems with OLS, SOLS, and GDPP on a sequence of {0.9 1 λ , 0.9 2 λ , . . . , 0.9 30 λ }, where λ was obtained by Algorithm 3. In the sequential screening, we stop using screening from λ t+1 if no features are discarded at λ t (1 ≤ t ≤ 29) because it is likely that screening with λ < λ t discards a few features, if at all. All the experiments except GISETTE (which contains a single y) were repeated 10 times with a different y for each run, and we report the average performance. As an overlapping group lasso solver, we used FoGLasso (Yuan, Liu and Ye, 2011), a state-of-the-art solver in the SLEP package (Liu, Ji and Ye, 2009), and all screening rules were implemented in Matlab. Below, we our demonstrate empirical results under different datasets, different group structures, and different group sizes to study how the screening efficiency changes under various scenarios. 4.1. Different Datasets. We first investigate the effectiveness of screening algorithms in terms of the screening rejection ratio under the four different datasets. For this experiment, we used the 1 + tree structure groups. Figures 2 (a-d) show the rejection ratio of OLS, SOLS, and GDPP on the four datasets as we change λ parameter. Figures 2 (e-h) demonstrate the differences between screening rejection ratios of the OLS and those of GDPP; each dot represents a result for a single λ, and the distance between dots and the diagonal line denotes the increased rejection ratio by OLS's use of overlapping groups.
For all datasets except AD dataset, OLS and SOLS reject significantly more features than GDPP, taking advantage of overlapping groups to decrease the left-hand side of screening rules. However, given the AD dataset, the performance gap between OLS (or SOLS) and GDPP was the smallest. This phenomenon is due to the small groups used for the AD dataset. The median of AD group sizes was 8; in contrast, the other datasets used the groups, sizes of 20/15/10. In the experiments with different group sizes in §4.3, we confirm that the performance gap between OLS (or SOLS) and GDPP increases as the group size increases. This is not surprising because as group size increases, the number of free variables to minimize the lefthand side of screening rules increases, resulting in a better screening rejection ratio. Note that OLS and SOLS outperform GDPP for all N >> J (PIE), N << J (COIL), and N ≈ J (GISETTE) settings. This observation suggests to us that OLS and SOLS would be useful for all large datasets regardless of their sample or feature sizes. Furthermore, OLS and SOLS show similar rejection ratios for all datasets. Therefore, in a single-machine setting, if 1 norm is included in the regularization, SOLS would be preferred over OLS due to SOLS's low computational complexity as well as its rejection ratio comparable to that of OLS. We then measured the speed gain achieved by the screening rules on the four different datasets. In Figure 3, we compare the running times of solving overlapping group lasso problems given a sequence of λ parameters among the solver without screening, the solver with OLS, the solver with SOLS, and the solver with OGDPP. For the results with screening, we also illustrate the portion of running times consumed by the solver and by screening. For the PIE, AD, COIL, and GESETTE datasets, we observed 3×, 2.2×, 2.7×, and 1.8× speed-up by SOLS in comparison to the solver without screening. One interesting observation is that for N >> J setting such as in the PIE dataset, running times of screening with OLS and SOLS are similar; but for large J such as in the COIL dataset, OLS is significantly more expensive than SOLS. As a result, the solver with OLS was slower than the solver with SOLS, even though OLS discarded more features than SOLS. However, we note that screening rules are embarrassingly parallel. Thus, in a distributed setting, OLS would be appealing because the screening portion of running times can be reduced proportionally to the number of cores. Hereafter, we show only the screening rejection ratio by different screening algorithms because the similar patterns of running times are observed under the other experimental settings.
4.2. Different Group Structures. Next, we run the screening algorithms on the COIL dataset under three different group structures, including 1 + nonoverlap groups, 1 + tree structure groups, and 1 + overlap groups. Figures 4(a-c) show the rejection ratios of different algorithms as λ changes given different group structures; we compare the rejection ratio of SOLS with that of GDPP below each corresponding plot. Overall, we can see that OLS and SOLS maintain high rejection ratios over a wide range of λ parameters, but GDPP's rejection ratio drops quickly as λ decreases. The rejection ratios by OLS and SOLS were identical under 1 + nonoverlap groups and 1 + overlap groups because only individual coefficients due to 1 penalty are inclusive groups for both screening methods. Interestingly, even under 1 + tree structure groups, OLS and SOLS show similar rejection ratios, which indicates that use of 1 penalty forḠ 1 is effective.
Different Group Sizes.
We also observed the effects of group sizes on the screening rejection ratio. For this experiment, we used the COIL dataset and 1 + overlap groups, where the size of groups is changed from 10 to 160. Figure 5 shows the screening efficiency of GDPP and OGDPP given different group sizes. Clearly, the rejection ratio of GDPP decreases as the group size increases. However, SOLS was not affected by the increased group sizes because SOLS can take advantage of the increased number of overlapping groups,Ḡ 1 = {j : j ∈ g}, within each tested group g. As the number of overlapping groups increases, the number of free variables to optimize in the left-hand side of SOLS rule also increases. Therefore, SOLS kept the high rejection ratios given all group sizes. The rejection ratio of OLS is decreased for group sizes ≥ 80 because we fixed the window size to search forḠ 1 by 50. Thus, OLS's rejection ratio started to decline from the group size of 80 due to the fixed number of overlapping groups inḠ 1 .
Conclusion.
In this paper, we developed screening rules including OLS and SOLS for overlapping group lasso. We make it possible to screen each group independently of each other by considering only groups inclusive of each tested group. Taking advantage of groups that overlap with tested groups, we showed that OLS and SOLS are efficient in terms of the screening rejection ratio. In addition, we verify that the GDPP screening rule ) is a special case of OLS, but it is less efficient than OLS because it lacks the capability to use overlapping groups. In OLS, there is a step to find all groups overlapped with the group of interest, which can be computationally heavy. Thus, as a special case of OLS, we also present SOLS for sparse overlapping group lasso that includes 1 penalty. In our experiments, SOLS was much faster than OLS for screening, while maintaining its rejection ratio comparable to that of OLS. Motivated by enhanced DPP , which uses a tight range of dual optimal solutions, developing more efficient OLS or SOLS would be an interesting research direction. Furthermore, extending OLS to various loss functions such as logistic loss or hinge loss is left for future research.
APPENDIX A: DUAL FORMULATION OF OVERLAPPING GROUP LASSO
Overlapping group lasso problem is defined by
(A.1) min β 1 2 y − Xβ 2 2 + λ g∈G √ n g β g 2 ,
where X ∈ R N ×J is the input data for J inputs and N samples, y ∈ R N ×1 is the output vector, β ∈ R J×1 is the vector of regression coefficients, n g is the size of group g, and λ is a regularization parameter that determines the sparsity of β. Here, we convert the primal overlapping group lasso problem in (A.1) to a dual problem.
Introducing z = y − Xβ, (A.1) can be written as min β 1 2 z 2 2 + λ g∈G √ n g β g 2 (A.2) subject to z = y − Xβ.
Lagrangian of (A.2) is (A.3) L(β, z, η) = 1 2 z 2 2 + λ g∈G √ n g β g 2 + η T (y − Xβ − z) .
Dual function g(η) of (A.3) is g(η) = inf β,z L(β, z, η)
= η T y + inf β −η T Xβ + λ g∈G √ ng β g 2 + infz 1 2 z 2 2 − η T z .
(A.4)
To obtain g(η), we solve the following two optimization problems:
(A.5) inf β −η T Xβ + λ g∈G √ n g β g 2
and (A.6) inf z 1 2 z 2 2 − η T z .
We first solve (A.5). Let us denote f 1 (β) ≡ −η T Xβ + λ g∈G √ n g β g 2 .
A subgradient of f 1 (β) with respect to β is
(A.7) ∂f 1 (β) ∂β = −X T η + λv,
where v is a subgradient of g∈G √ n g β g 2 with respect to β. The j-th element of v is given by
(A.8) v j = {g:j∈g,β g =0,g∈G} √ n g β j β g 2 + {g:j∈g,β g =0,g∈G} √ n g o j ,
where o j is a subgradient that satisfies o g 2 ≤ 1, where j ∈ g. To obtain an optimal β * for (A.5), we set ∂f 1 (β) ∂β = 0. Then, we have (A.9) X T η = λv.
Plugging it into f 1 (β), we get Here (3.10) is used for (A.12). Therefore, inf β f 1 (β) = 0. Now we solve the second problem f 2 (z) ≡ 1 2 z 2 2 − η T z. This result was presented in . However, here we show the derivation for self-containedness.
f 2 (z) = 1 2 z 2 2 − η T z = 1 2 z − η 2 2 − η 2 2 . (A.15)
Note that η = argmin z 1 2 z − η 2 2 − η 2 2 , and thus inf z f 2 (z) = − 1 2 η 2 2 . Based on these results for (A.5) and (A.6), the dual function g(η) in (A.4) is given by, (A.16) g(η) = η T y − 1 2 η 2 2 = 1 2 y 2 2 − 1 2 η − y 2 2 .
Finally, we denote θ = η λ . Combining (A.16) with (A.9), a dual formulation of overlapping group lasso is as follows:
sup θ 1 2 y 2 2 − λ 2 2 θ − y λ 2 2 (A.17)
subject to X T θ = v.
APPENDIX B: PROOF OF THEOREM 1
Theorem 1. For the overlapping lasso problem, suppose that we are given an optimal dual solution θ * (λ 0 ). Then for λ < λ 0 , β * g (λ) = 0 if
(B.1) min w h , w h 2 ≤1 j∈g x T j θ * (λ 0 ) − j∈h,h∈Ḡ 1 √ n h w j 2 < √ ng − Xg F y 2 1 λ − 1 λ 0 .
Proof. Based on (3.14), we have a sphere Θ that contains θ * (λ), which is centered at θ * (λ 0 ) with a radius of ρ = y λ − y λ 0 2
. Thus, we can represent θ * (λ) = θ * (λ 0 ) + r, where r 2 ≤ ρ. Plugging it into (3.13) we get bg ≤ min
w h : w h 2 ≤1 j∈g x T j θ * (λ 0 ) + x T j r − j∈h,h∈Ḡ 1 √ n h w j 2 (B.2) ≤ min w h : w h 2 ≤1 j∈g x T j θ * (λ 0 ) − j∈h,h∈Ḡ 1 √ n h w j 2 + j∈g x T j r 2 (B.3) ≤ min w h : w h 2 ≤1 j∈g x T j θ * (λ 0 ) − j∈h,h∈Ḡ 1 √ n h w j 2 + Xg 2 F y 2 2 1 λ − 1 λ 0 2 . (B.4)
We used Minkowski's inequality for (B.3), and Cauchy-Schwarz inequality and X g 2 2 ≤ X g 2 F for (B.4). From (3.13), if b g < √ n g , then β * g = 0, and the result follows.
Fig 1. For screening test on group g2, we consider g3 but disregard g1 and g4 to enable the independent screening test.
Fig 2 .
2Rejection ratio of OLS (overlapping group lasso screening), SOLS (sparse overlapping group lasso screening) and GDPP (group dual polytope projection) and their comparison on (a,e) PIE, (b,f ) COIL, (c,g) AD, and (d,h) GISETTE datasets for different λ/λmax parameters.
Fig 3 .
3Running times of overlapping group lasso solver without screening (Sol), solver with OSL screening (OSL), solver with SOSL screening (SOSL), and solver with GDPP screening (GDPP) on (a) PIE, (b) AD, (c) COIL, and (d) GISETTE datasets.
Fig 4 .
4Rejection ratio of OLS, SOLS, and GDPP with different group structures: (a,d) 1 + nonoverlap groups, (b,e) 1 + tree structure groups, and (c,f ) 1 + overlap groups.
Fig 5 .
5Rejection ratio of OGDPP and GPP with different sizes of overlapping groups. The sizes of groups are (a) 10 , (b) 40, (c) 80, and (d) 160 features.
2 .
2Background: Screening Rules via DPP. Recently, Wang et al. proposed screening rules via DPP for nonoverlapping group lasso
For simple notation, we denote θ * by a dual optimal solution given λ. We use notation θ * (λ) when we refer to a specific λ.
We set the window size to 50 to search for the overlapping groups inḠ1.
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| []
|
[
"Measuring Word Significance using Distributed Representations of Words",
"Measuring Word Significance using Distributed Representations of Words"
]
| [
"Adriaan M J Schakel [email protected] \nLateral GmbH\n\n",
"Benjamin J Wilson [email protected] \nLateral GmbH\n\n"
]
| [
"Lateral GmbH\n",
"Lateral GmbH\n"
]
| []
| Distributed representations of words as real-valued vectors in a relatively lowdimensional space aim at extracting syntactic and semantic features from large text corpora. A recently introduced neural network, named word2vec(Mikolov et al., 2013a;Mikolov et al., 2013b), was shown to encode semantic information in the direction of the word vectors. In this brief report, it is proposed to use the length of the vectors, together with the term frequency, as measure of word significance in a corpus. Experimental evidence using a domain-specific corpus of abstracts is presented to support this proposal. A useful visualization technique for text corpora emerges, where words are mapped onto a two-dimensional plane and automatically ranked by significance. | null | [
"https://arxiv.org/pdf/1508.02297v1.pdf"
]
| 17,841,938 | 1508.02297 | 958d165f8bb77838ec915d4f214a2310e3adde19 |
Measuring Word Significance using Distributed Representations of Words
Adriaan M J Schakel [email protected]
Lateral GmbH
Benjamin J Wilson [email protected]
Lateral GmbH
Measuring Word Significance using Distributed Representations of Words
Distributed representations of words as real-valued vectors in a relatively lowdimensional space aim at extracting syntactic and semantic features from large text corpora. A recently introduced neural network, named word2vec(Mikolov et al., 2013a;Mikolov et al., 2013b), was shown to encode semantic information in the direction of the word vectors. In this brief report, it is proposed to use the length of the vectors, together with the term frequency, as measure of word significance in a corpus. Experimental evidence using a domain-specific corpus of abstracts is presented to support this proposal. A useful visualization technique for text corpora emerges, where words are mapped onto a two-dimensional plane and automatically ranked by significance.
Introduction
Discovering the underlying topics or discourses in large text corpora is a challenging task in natural language processing (NLP). A statistical approach often starts by determining the frequency of occurrence of terms across the corpus, and using the term frequency as a criterion for word significance-a thesis put forward in a seminal paper by Luhn (Luhn, 1958). From the list of terms ranked by frequency, terms that are either too rare or too common are usually dropped, for they are of little use. For a domain-specific corpus, the top ranked terms in the trimmed list often nicely summarize the main topics of the corpus, as will be illustrated below.
For more detailed corpus analysis, such as discovering the subtopics covered by the documents in the corpus, the term frequency list by itself is, however, of limited use. The main problem is that within a given frequency range, function words, which primarily have an organizing function and carry little or no meaning, appear together with content words, which represent central features of texts and carry the meaning of the context. In other words, the rank of a term in the frequency list is by itself not indicative of meaning (Luhn, 1958).
This problem can be tackled by replacing the corpus-wide term frequency with a more refined weighting scheme based on document-specific term frequency (Aizawa, 2000). In such a scheme, a document is taken as the context in which a word appears. Since key words are typically repeated in a document, they tend to cluster and to be less evenly distributed across a text corpus than function words of the same frequency. The fraction of documents containing a given term can then be used to distinguish them. Much more elaborate statistical methods have been developed to further explore the distribution of terms in collections of documents, such as topic modeling (Blei et al., 2003) and spacing statistics (Ortuño et al., 2002).
An even more refined weighting scheme is obtained by reducing the context of a word from the document in which it appears to a window of just a few words. Such a scheme is suggested by Harris' distributional hypothesis (Harris, 1954) which states "that it is possible to define a linguistic structure solely in terms of the 'distributions' (= patterns of co-occurrences) of its elements", or as Firth famously put it (Firth, 1957) "a word is characterized by the company it keeps".
Word co-occurrence is at the heart of several machine learning algorithms, including the recently introduced word2vec by Mikolov and collaborators (Mikolov et al., 2013a;Mikolov et al., 2013b). Word2vec is a neural network with a single hidden layer that uses word co-occurrence for learning a relatively low-dimensional vector representation of each word in a corpus, a so-called distributed representation (Hinton, 1986). The di-mension is typically chosen of order 100 or 1000. This is easily orders of magnitude smaller than the size of a vocabulary, which would be the dimension when a one-hot representation of words is chosen instead. Given the words appearing in a context, the neural network learns by predicting (the representation of) the word in the middle, or vice versa. During training, words that appear in similar contexts are grouped together in the same direction by this unsupervised learning algorithm. The distributed representation thus ultimately captures semantic similarities between words. This has been impressively demonstrated by a series of experiments in the original word2vec papers, where semantic similarity was measured by the dot product between normalized vectors.
In this brief report, we consider the problem of identifying significant terms that give information about content in text corpora made up of short texts, such as abstracts of scientific papers, or news summaries. It is proposed to use the L 2 norm, or length of a word vector, in combination with the term frequency, as measure of word significance.
In a discussion forum dedicated to word2vec, 1 it has been argued by some that the length of a vector merely reflects the frequency with which a word appears in the corpus, while others argued that it in addition reflects the similarity of the contexts in which a word appears. According to this thesis, a word that is consistently used in a similar context will be represented by a longer vector than a word of the same frequency that is used in different contexts. Below, we provide experimental support for this thesis. It is this property that justifies measuring significance by word vector length, for words represented by long vectors refer to a distinctive context.
It is further proposed that the scatter plot of word vector length versus term frequency of all the words in the vocabulary provides a useful twodimensional visualization of a text corpus.
The paper is organized as follows. The next section introduces the language corpus used and gives a global characterization based on term frequency. Section 3 describes the experiments carried out using word2vec, and presents the main results. Section 4 concludes the paper with a short discussion.
Dataset
For our experiments we use a dataset from the arXiv, 2 a repository of scientific preprints. The dataset consists of about 29k papers from one single subject class in the arXiv, viz. the hep-th section on theoretical high-energy physics posted in the period from January 1992 to April 2003. Although full papers are available, we consider only title and abstract of the papers, which have about 100 word tokens on average. 3 LaTeX (or TeX) commands are removed from input through use of the detex program. 4 The input text is further converted to lowercase, and punctuation marks and special symbols are separated from words, as was done in the preprocessing step of a word2vec experiment by Mikolov on the IMDB dataset of movie reviews. 5
Term Frequency List
After removing stop words and punctuation marks, the list of 50 most frequently used words in the corpus reduces to the one given in Table 1. Deriving from a domain-specific dataset, this list indeed gives a succinct and fairly precise characterization of the hep-th corpus, which is primarily about "gauge theory", "quantum field theories", and "string theory". It correctly reveals the importance of "models" in this research area, as well as the importance of the concepts "space", "solutions", "action", "dimensions", "symmetry group", "equations", and "algebra". The term "black" refers to "black holes", which play a distinctive role in this corpus. The term "also" is not filtered out by the NLTK 6 stop word list we use. Finally, "show" (used exclusively as verb in this corpus, not as noun) appears mostly in the context "we show that" and reveals that a large portion of the corpus consists of research papers. Note that "show" is the only verb besides the stop word "be" that made it into the top 50 list.
Experiments
We next turn to word2vec. 7 For training the neural network, we use the same parameter settings as advertised for the IMDB dataset referred to above. 8 With these settings, the vector dimension is 100, the (maximum) context window size is 10, and the algorithm makes 20 passes through the dataset for learning. The total number of tokens processed by the algorithm is 3.2M. As is typical for a highly specific domain, the vocabulary is relatively small, containing about 44k terms, of which about half is used only once. 7 The code is available for download at https:// code.google.com/p/word2vec 8 Specifically, the parameters used are: word2vec -train $inputfile -output $outputfile -cbow 0 -size 100 -window 10 -negative 5 -hs 0 -sample 1e-4 -threads 40 -binary 0 -iter 20 -min-count 1
Similarity Distribution
During training, similar words are grouped together in the same direction by the learning algorithm, so that after training the vectors encode word semantics. One of the most popular measures of semantic similarity in NLP is the cosine similarity given by the dot product between two normalized vectors. Denoting the cosine of the angle between the two vectors, the cosine similarity can take values in the interval [−1, 1].
To analyze the hep-th corpus, we built a histogram of the cosine similarity between arbitrarily chosen pairs of word vectors. The words are randomly selected from the vocabulary irrespective their frequency. We have, however, discarded terms that appear only once. The result, given in Fig. 1, is a bell-shaped distribution. To our surprise, the distribution is not centered around zero, but around a positive value, 0.23. This means that word vectors in the hep-th corpus have on average a certain similarity. Closely related to this is that the average word vector is non-zero, having a small length, v = 1.37 (v = 1.51 when words that only appear once are excluded). This vector marks the center of the word cloud spanned in the word vector space by all the words in the vocabulary.
To see if this behavior is shared by general purpose corpora, we considered Wikipedia by way of example. 9 For that corpus, covering diverse topics, we found, as expected, the histogram to be centered around zero (and slightly right skewed). The non-zero value found for the hep-th corpus is therefore probably a sign of the homogeneity of this dataset.
The reason for excluding terms that only appear once, which after all make up half of the vocabu- lary, is that they have their own bell-shaped distribution, slightly more peaked than the one shown in Fig. 1 and centered at a higher cosine similarity of about 0.5. Note the outlier at zero cosine similarity in the distribution in Fig. 1. The reason for this outlier eludes us.
Vector Length as Significance Factor
To demonstrate that, besides depending on consistent use, the length of a word vector also depends on term frequency, we consider the months of the year, see Table 2. Apart from the word token "may", which in addition denotes a verbal auxiliary and is therefore used in many different contexts, these terms consistently appear in the abstracts of the hep-th corpus to indicate the time of a school or conference where the paper was presented. The data clearly show that for fixed context, the vector length increases with term frequency, see Fig. 2.
The three terms with the largest vector length are besides "june", "school" (v = 5.97, tf = 114) and "conference" (v = 5.89, tf = 93). If we take vector length as a measure of word significance, this finding surprisingly supports another thesis by Luhn (Luhn, 1958), which states that:
The more often certain words are found in each others company within a sentence, the more significance may be attributed to each of these words.
Here, the phrase "certain words" refers to words that a writer normally repeats [· · ·] as he advances or varies his arguments and as he elaborates on an aspect of a subject. Table 2 also nicely demonstrates that term frequency alone does not determine the length of a word vector. The term "may" has a much higher frequency than the other terms in the table, yet it is represented by the shortest vector. This is because it is used in the corpus mostly as a verbal auxiliary in opposing contexts. When a word appears in different contexts, its vector gets moved in different directions during updates. The final vector then represents some sort of weighted average over the various contexts. Averaging over vectors that point in different directions typically results in a vector that gets shorter with increasing number of different contexts in which the word appears. For words to be used in many different contexts, they must carry little meaning. Prime examples of such insignificant words are high-frequency stop words, which are indeed represented by short vectors despite their high term frequencies, see Table 3.
Vector Length vs. Term Frequency
To study to what extent term frequency and word vector length can serve as indicators of a word's significance, we represent all words in the vocabulary in a two-dimensional scatter plot using these variables as coordinates. Figure 3 gives the result for the hep-th corpus. For given term frequency, the vector length is seen to take values only in a narrow interval. That interval initially shifts upwards with increasing frequency. Around a frequency of about 30, that trend reverses and the interval shifts downwards. 2.03 28878 Table 3: Top 12 terms in the term frequency list of the hep-th corpus with their word vector length v and term frequency tf . In addition to punctuation marks, this list exclusively features stop words.
Both forces determining the length of a word vector are seen at work here. Small-frequency words tend to be used consistently, so that the more frequently such words appear, the longer their vectors. This tendency is reflected by the upwards trend in Fig. 3 at low frequencies. Highfrequency words, on the other hand, tend to be used in many different contexts, the more so, the more frequently they occur. The averaging over an increasing number of different contexts shortens the vectors representing such words. This tendency is clearly reflected by the downwards trend in Fig. 3 at high frequencies, culminating in punctuation marks and stop words with short vectors at the very end.
Words represented by the longest vectors in a given frequency bin often carry the content of distinctive contexts. Typically, these contexts are topic-wise not at the core of the corpus but more on the outskirts. For example, the words with the longest vector in the high-frequency ranges [2 k−1 , 2 k − 1] with k = 9, 10, 11, "inflation" (v = 4.64, tf = 571), "sitter" (v = 3.81, tf = 1490) as in "de Sitter", and "holes" (v = 3.41, tf = 2465) as in "black holes", all refer to general relativity. General relativity, having its own subject class (gr-qc) in the arXiv, is not one of the main subjects of the hep-th corpus. It takes a distinctive position in this corpus as it mostly appears in studies that aim at reconciling general relativity with the laws of quantum mechanics.
POS Tagging
To further assess the ability of word vector length to measure word significance, we assign part-ofspeech (POS) tags to each word in the corpus. For this task, we use the Stanford POS tagger 10 (Toutanova et al., 2003). The final tag assigned to a word in the vocabulary is decided by majority vote.
By the way that word2vec learns word representations, we expect nouns (excluding proper nouns) and adjectives to be similarly distributed in the vtf plane. This is indeed what we observe, see Fig. 4. Note that these word types pervade almost the entire region covered in the full plot in Fig. 3 by the complete vocabulary. We also find verbs and adverbs to be similarly distributed in the v-tf plane. Again this was to be expected given that word2vec learns word representations from word co-occurrences. Somewhat surprisingly, we observe in Fig. 5 that the distribution of verbs also overlaps with that of function words. 11 These word types no longer pervade the entire region covered in the full plot, but are confined to the bottom band, corresponding to short vectors. The fact that function words are represented by short vectors underscores the ability of vector length to measure word significance.
This proficiency is even more brought out by comparing the distribution of function words and proper nouns, which typically are indicative of distinctive contexts in the hep-th corpus. The plot in Fig. 6 shows a clear separation of proper nouns and function words for sufficiently large term frequencies.
Visualization
These results suggest an interesting technique for visualizing text corpora. By labeling the data points in the v-tf scatter plot with their terms, one obtains a two-dimensional visualization of all the words in the vocabulary. One advantage of a v-tf plot is that words are ranked by significance, thus allowing for effective exploration of a corpus. To deal with the large number of data points, an interactive visualization tool can be build that allows the user to navigate a mouse pointer over the plot, and that shows only the labels of the data points in 11 As function words we classify: prepositions (IN), pronouns (PRP, PRP$, WP, WP$), determiners (DT, PDT, WDT), conjunctions (CC), modal verbs (MD), and particles (RP). In brackets, we included here the tags used by the Stanford POS tagger. There exist other techniques for visualizing high-dimensional data, such as the popular tdistributed stochastic neighbor embedding (t-SNE) (van der Maaten and Hinton, 2008). That machine learning algorithm, being an example of multidimensional scaling, projects highdimensional data points onto a plane such that the distances, or similarities between them are preserved as well as possible. Words of similar meaning thus tend to be projected together by the t-SNE algorithm. Since the cosine similarity is independent of vector lengths, word significance is ignored when using this measure. The t-SNE algorithm therefore arranges the data points entirely differently from our proposal. Moreover, in contrast to the axes in the v-tf scatter plot, those in the t-SNE plot have no direct meaning.
Figure 1 :
1Cosine similarity between arbitrarily chosen pairs of word vectors with tf > 1.
Figure 2 :
2Word vector length as a function of frequency of appearance of the months of the year excluding "may". The line through the data points serves as a guide to the eye.
Figure 3 :Figure 4 :
34Word vector length v versus term frequency tf of all words in the hep-th vocabulary. Note the logarithmic scale used on the frequency axis. The dark symbols denote bin means with the kth bin containing the frequencies in the interval [2 k−1 , 2 k − 1] with k = 1, 2, 3, . . .. These means are included as a guide to the eye. The horizontal line indicates the length v = 1.37 of the Word vector length v versus term frequency tf of all words in the hep-th vocabulary labeled nouns (red dots) or adjectives (blue dots).
Figure 5 :
5Word vector length v versus term frequency tf of all words in the hep-th vocabulary labeled verbs (red dots) or function words (blue dots).
Figure 6 :
6Word vector length v versus term frequency tf of all words in the hep-th vocabulary labeled proper nouns (red dots) or function words (blue dots).the vicinity of the pointer.
Table 1: Top ranked words in the term frequency list of the hep-th corpus with their vector length v (included for later convenience) and term frequency tf . Punctuation marks and stop words are removed from the list.term
v
tf
theory
1.90 27702
field
2.00 17510
gauge
2.13 15536
string
2.33 13523
model
2.19 12389
quantum
2.14 12307
theories
2.22 10528
space
2.18
8035
also
1.67
7907
models
2.39
7313
two
2.03
7286
fields
2.11
7261
solutions
2.49
7129
show
1.87
7125
action
2.39
6602
one
1.82
6440
black
3.24
6011
dimensions 2.34
5953
symmetry
2.35
5792
group
2.46
5696
equations
2.54
5509
algebra
2.70
5461
Table 2 :
2The months of the year with their word vector length v and term frequency tf .
http://groups.google.com/forum/#!forum/word2vectoolkit
http://arxiv.org/ 3 The dataset is available from the KDD Cup 2003 homepage http://www.sigkdd.org/kdd-cup-2003-network-miningand-usage-log-analysis 4 http://www.cs.purdue.edu/homes/trinkle/detex/ 5 https://groups.google.com/forum/#!msg/word2vectoolkit/Q49FIrNOQRo/J6KG8mUj45sJ 6 www.nltk.org/
For a cleaned version of the Wikipedia corpus from October 2013, see https://blog.lateral.io/2015/06/the-unknownperils-of-mining-wikipedia/
Specifically, we use the english-caseless-left3wordsdistsim tagger. For details on this model and for downloading, see http://nlp.stanford.edu/software/tagger.shtml.
DiscussionMost applications of distributed representations of words obtained through word2vec so far centered around semantics. A host of experiments have demonstrated the extent to which the direction of word vectors captures semantics. In this brief report, it was pointed out that not only the direction, but also the length of word vectors carries important information. Specifically, it was shown that word vector length furnishes, in combination with term frequency, a useful measure of word significance. Also an alternative to the t-SNE algorithm for projecting word vectors onto a plane was introduced, where words are ordered by significance rather than similarity.We have restricted ourselves to unigrams in this exploratory study. For more extended experiments and applications, including important bi-and trigrams into the vocabulary will certainly improve results. We have also restricted ourselves to running word2vec using parameters that were recommended by the developers, and have not attempted to optimize them.Finally, the question arises whether word vectors produced by other highly scalable machine learning algorithms built on top of word cooccurrences, such as the log bilinear model(Mnih and Kavukcuoglu, 2013)and GloVe(Pennington et al., 2014), also encode word significance in their length.
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Feature-rich part-of-speech tagging with a cyclic dependency network. [ Toutanova, Proceedings of the 2003 Conference of the North American Chapter of the Association for Computational Linguistics on Human Language Technology. the 2003 Conference of the North American Chapter of the Association for Computational Linguistics on Human Language TechnologyStroudsburg, PA. ACL1[Toutanova et al.2003] Kristina Toutanova, Dan Klein, Christopher D. Manning, and Yoram Singer. 2003. Feature-rich part-of-speech tagging with a cyclic de- pendency network. In Proceedings of the 2003 Con- ference of the North American Chapter of the As- sociation for Computational Linguistics on Human Language Technology, volume 1, pages 173-180, Stroudsburg, PA. ACL.
| []
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[
"Theory of Fano resonance in single molecule electroluminescence induced by a scanning tunneling microscope",
"Theory of Fano resonance in single molecule electroluminescence induced by a scanning tunneling microscope",
"Theory of Fano resonance in single molecule electroluminescence induced by a scanning tunneling microscope",
"Theory of Fano resonance in single molecule electroluminescence induced by a scanning tunneling microscope"
]
| [
"Lei-Lei Nian \nSchool of Physics\nWuhan National High Magnetic Field Center\nHuazhong University of Science and Technology\n430074WuhanP. R. China\n",
"Jing-Tao Lü \nSchool of Physics\nWuhan National High Magnetic Field Center\nHuazhong University of Science and Technology\n430074WuhanP. R. China\n",
"Lei-Lei Nian \nSchool of Physics\nWuhan National High Magnetic Field Center\nHuazhong University of Science and Technology\n430074WuhanP. R. China\n",
"Jing-Tao Lü \nSchool of Physics\nWuhan National High Magnetic Field Center\nHuazhong University of Science and Technology\n430074WuhanP. R. China\n"
]
| [
"School of Physics\nWuhan National High Magnetic Field Center\nHuazhong University of Science and Technology\n430074WuhanP. R. China",
"School of Physics\nWuhan National High Magnetic Field Center\nHuazhong University of Science and Technology\n430074WuhanP. R. China",
"School of Physics\nWuhan National High Magnetic Field Center\nHuazhong University of Science and Technology\n430074WuhanP. R. China",
"School of Physics\nWuhan National High Magnetic Field Center\nHuazhong University of Science and Technology\n430074WuhanP. R. China"
]
| []
| The coupling between molecular exciton and gap plasmons plays a key role in single molecular electroluminescence induced by a scanning tunneling microscope (STM). But it has been difficult to clarify the complex experimental phenomena. By employing the nonequilibrium Green's function method, we propose a general theoretical model to understand the light emission spectrum from single molecule and gap plasmons from an energy transport point of view. The coherent interaction between gap plasmons and molecular exciton leads to a prominent Fano resonance in the emission spectrum. We analyze the dependence of the Fano line shape on the system parameters, based on which we provide a unified account of several recent experimental observations. Moreover, we highlight the effect of the tip-molecule electronic coupling on the spectrum, which has hitherto not been considered. | 10.1021/acs.nanolett.8b02706 | [
"https://arxiv.org/pdf/1803.01189v1.pdf"
]
| 53,012,168 | 1803.01189 | dd83f2c10f232c6d650caa7933f23e60fd0726d4 |
Theory of Fano resonance in single molecule electroluminescence induced by a scanning tunneling microscope
3 Mar 2018
Lei-Lei Nian
School of Physics
Wuhan National High Magnetic Field Center
Huazhong University of Science and Technology
430074WuhanP. R. China
Jing-Tao Lü
School of Physics
Wuhan National High Magnetic Field Center
Huazhong University of Science and Technology
430074WuhanP. R. China
Theory of Fano resonance in single molecule electroluminescence induced by a scanning tunneling microscope
3 Mar 2018
The coupling between molecular exciton and gap plasmons plays a key role in single molecular electroluminescence induced by a scanning tunneling microscope (STM). But it has been difficult to clarify the complex experimental phenomena. By employing the nonequilibrium Green's function method, we propose a general theoretical model to understand the light emission spectrum from single molecule and gap plasmons from an energy transport point of view. The coherent interaction between gap plasmons and molecular exciton leads to a prominent Fano resonance in the emission spectrum. We analyze the dependence of the Fano line shape on the system parameters, based on which we provide a unified account of several recent experimental observations. Moreover, we highlight the effect of the tip-molecule electronic coupling on the spectrum, which has hitherto not been considered.
I. INTRODUCTION
Recently, single molecular electroluminescence (EL) induced by the inelastic electron tunneling from a scanning tunneling microscope (STM) has attracted a lot of attention, yielding many fascinating physics and potential applications [1][2][3][4][5][6][7][8]. In such STM-induced luminescence (STML) experiments, light emission from gap plasmon modes is a common process [9][10][11][12][13][14][15], which in turn can dominate, accompany, or influence the luminescence of single molecules positioned nearby STM tip [3,4,16,17]. The resulting coupling between the molecular exciton and gap plasmons is of interest because it contributes to the study of fundamental quantum phenomena, including coherent energy transfer, cavity quantum electrodynamics, and entanglement [18].
The importance of coherent interaction between molecular exciton and gap plasmons in STML has been revealed in recent experiments [19][20][21][22][23][24][25][26][27][28][29][30][31][32], through, for example, prominent Fano [33] line shapes in the emission spectrum. Its possible applications in single molecule detection [29][30][31], single photon generation [34] have been envisioned. Although these experiments show the important role played by the coherent interaction between molecular exciton and gap plasmons, a systematic theoretical model to account for all these experimental results is so far lacking. Revealing the connection between the line shape and the system parameters is important for further development and application of this technique.
Here, we propose a general theoretical model that is able to account for all these experimental results. We first demonstrate the coherent optical coupling between molecular exciton and gap plasmons leads to a pronounced Fano resonance, whose line shape depends sensitively on the system parameters. Using experimentally based parameters, the simulated spectrum shows quantitative agreement with the experimental results. This is an essential step to predict or control the dynamic energy transfer process in STML experiments.
II. MODEL AND THEORY
We consider a model system schematically shown in Fig. 1 (a). The voltage bias applied between the tip and the substrate generates a flowing electrical current between them, which is used to excite the localized gap plasmons. A single molecule is represented by two electronic states l and h, representing the lowest unoccupied molecular orbital (LUMO) and highest occupied molecular orbital (HOMO), respectively. If the molecule is present underneath the STM tip, under certain bias, a molecular exciton may also be created by the electrical current, i.e., injecting an electron to the LUMO and a hole to the HOMO orbital. If the molecule is not far away from the tip, the molecular exciton can be created by the gap plasmons, given its much wider frequency and larger spatial distribution. This requires a direct coupling between the gap plasmons and the molecular exciton.
We model the gap plasmon using a photon field with angular frequency ω p . In reality, there could be several modes with similar frequencies. Similarly, we model the molecular exciton using a photon field with angular frequency ω 0 . The two photon fields couple to each other through the parameter t p (x), which depends on the tipmolecule distance x.
To study the energy transfer between the electron and photon fields, we use an effective model shown in Fig. 1 (b). The biased electronic system acts as an effective nonequilibrium energy bath, which supplies energy to the gap plasmons and the molecular exciton. The energy absorbed by the photon fields is either dissipated into the environment or radiated to the free space. The radiation then goes to the detector.
Energy transport for this effective model can be studied using the nonequilibrium Green's function (NEGF) ) ( Effective model to study the energy transfer in the experimental setup in (a). The nonequilibrium electronic bath includes the STM tip, the substrate and the single molecule under certain voltage bias. The gap plasmon is represented by a photon mode with angular frequency ωp, while the molecular exciton by a mode with frequency ω0. There is a direct coupling whose magnitude depends on the relative position of the STM tip and the molecule x (see (a)). The emitted light is collected by the photon detector. There are also non-radiative channels into which the two photon modes can dissipate their energy, represented by the environment. method [35][36][37][38]. The frequency-resolved energy flux going into bath α is written as
I α (ω) = ω 2π Tr Π < α (ω)D > (ω) − Π > α (ω)D < (ω) . (1)
Here, D > (D < ) is the greater (less) Green's function of the photon fields, Π > α (Π < α ) is the corresponding selfenergy due to coupling to bath α. We have considered three kinds of baths: (1) the nonequilibrium electronic system which supplies the energy, thus I el < 0;
(2) the photon detector which collects the radiation and corresponds to the measured photon flux; (3) the non-radiative environment into which the non-radiative energy goes. The two terms in Eq. (1) correspond to energy flowing into and out of the bath, respectively. The Green's functions and self-energies in Eq. (1) are solved within the self-consistent Born approximation (SCBA) [37,[39][40][41]. The photon flux can then be calculated from them. The details of the method can be found in the Appendices A and B.
III. RESULTS AND DISCUSSIONS
A. Important parameters
The advantage of the effective model is that, we separate the electronic part of the whole system from the photonic part. All the electronic part, including the STM tip, molecule and substrate, is modeled as a nonequilibrium energy bath. The most important feature of the nonequilibrium bath is that, the width of its energy spectrum is determined by the applied bias |eV |, i.e., the bath can not excite photon mode whose energy is larger than the applied bias ω > |eV |. It enters into our theory through the self-energy Π el , on which the Green's functions D > and D < in Eq. (1) depend. Meanwhile, the line shape of the spectrum is mainly determined by the two parameters describing the photon modes, which we consider in the following.
The first important parameter that determines the line shape is the detuning ∆ p0 = ω p − ω 0 . In the experiment, the resonant frequency of gap plasmon ω p can be tuned by adjusting the tip shape, or modifying the dielectric properties of the substrate, i.e., introducing dielectric layers. Figure 2 displays the evolution of the spectrum with different values of energy detuning ∆ p0 . We note that the strong energy detuning dependence of the Fano line shape is in agreement with experimental findings [29,30,42] the detuning q ∝ −∆ p0 . The second parameter is the coupling between the gap plasmon and the molecular exciton t p (x), determined by the relative position of the tip and the molecule (x), according to which, we can define three regimes. They correspond to the tip apart from (I), slight aside from (II) and located on top of the molecule (III), respectively. The energy flux spectrum of different situations is plotted in Fig. 3 [43]. In case I, a broadband emission in the STML spectra can be observed in Fig. 3 (I). This is from the radiative decay of the gap plasmon, while the molecular exciton does not participate to the transport. The other two cases are more interesting, which we focus in the following.
In case II, the interaction between the molecular exciton and the gap plasmon occurs, which results in coherent energy transfer between them. This interaction generates a sharp dip in the broadband emission spectra, as shown in Fig. 3 (II). The resulting asymmetric lineshape is a signature of the Fano resonance. Essentially, the single molecule only couples to the substrate in this case, no tunneling electrons excite the single molecule directly. But it can be excited by the gap plasmon indirectly. Also shown in the figure are the separate contributions of the flux from the gap plasmon and the molecular exciton. The spectrum of the molecular exciton is a normal Lorentzian-like peak, while that of the plasmon shows the typical Fano line shape and contributes dominantly to the total spectrum. In case III, the molecule is underneath the STM tip. Both optical fields can be excited directly by the tunneling electrons. Their coherent interaction results in a Fano-shaped emission spectrum shown in Fig. 3 (III). In this case, a sharp peak instead of dip is observed. The signature of the molecular exciton becomes dominant, in contrast to case (II). In Appendix C, through a simple model, we show that the asymmetric line shapes originate from the Fano interference between the two photon fields.
We now apply our theory to consider three recent ex-periments. We show that they fall into one of the above discussed three regimes. In the first experiment, the molecule is attached to the metallic electrodes (tip and substrate) through molecular linkers [42], corresponding to case III. In the other two, the molecule lies on thin insulating layer deposited on the metal substrate [29,30]. The relative position of the tip and molecule can be adjusted to cover all the three regimes.
B. A suspended molecular wire
STM-induced narrow-line emission from a single molecular emitter (H2P) connected to the tip and the substrate through oligothiophene linkers was reported in Ref. 42. The light spectra exhibits an asymmetric line shape in broad background, with the peak position closely associated with the emission energy of the fused H2P molecule. The oligothiophene wires decouple the H2P emitter from the substrate and the tip. The length of the linker can be adjusted by lifting the STM tip away from the substrate. Therefore, the distance between tip and substrate plays a key role in achieving molecular luminescence. Here we simulate the evolution of tip-substrate distance by adjusting the non-radiative decay parameter γ 0e of the photon field ω 0 while keeping all other parameters fixed. Figure 4(a) plots the emission spectrum (photon energy flux versus frequency/energy ω) for several values of γ 0e (proportional to the lifetime of the molecular excited state-LMES). For the short distance case, i.e., the most part of the molecular linker is adsorbed on the substrate, it is difficult to observe a well-defined fluorescence from the molecular emitter because of the quenching of molecular luminescence, i.e., the LMES is very short. In this case, the spectrum observed is similar to most STMinduced light emission experiments, showing a broad gap plasmon spectra [see γ 0e = 100 meV in Fig. 4(a)]. With increased tip-substrate distance, the non-radiative decay becomes smaller. This results in an increased LMES. EL from the molecule can then be observed as a peak in the spectrum. The intensity of the peak becomes stronger with further decoupling from the substrate. This is similar to the case (III) in Fig. 3. The other parameters are the same as in Fig. 3 (II).
C. Single molecule on insulating layer
In Refs. 29 and 30, STM-induced light emission from a single molecule decoupled from substrate by insulating NaCl layer was studied. It was found that the relative tip-molecule position [t p (x)] modifies the emission spectrum significantly. Figure 4 (b) and (c) show the tip position-dependent flux obtained from our theory, corresponding results from Refs. [29,30]. In the case the molecule and the tip are far apart [t p (x) = 0], the two optical fields do not couple directly. Since the molecule does not participate the electron tunneling process, the molecular luminescence is not observed. Approaching the tip to the molecule generates a nonzero t p (x) = 0, the coupling of the molecular exciton and gap plasmon mode opens the energy transfer channel between them. A sharp dip [(b)] or peak [(c)] develops due to the Fano interference. The interaction between the molecule exciton and the gap plasmons can be tuned by varying the tip position near molecule. This allows one to control the hybridization of the two states. To provide a quantitative description that can be compared with the experimental data, we simulate the tip distance-dependent light spectra by adjusting the t p (x) ranging from 0 to 15 meV, all the main features of the experimental results are reproduced by our theory, e.g., the dip/peak structure becomes more and more prominent as t p (x) increases. We note that, although we consider only one exciton mode in Fig. 4 (c), in the experiment, two peaks are observed corresponding to transition dipoles along the two ligands axes of H2Pc. The two dipoles are not degenerate due to the breaking of four-fold rotational symmetry in H2Pc.
Encouraged by the good agreement between our results and the experimental data, we go one step further. We study here the effect of the electronic coupling between the tip and the molecule on the spectrum. The blue lines in Fig. 4 (b) and (c) show the evolution of the spectrum with increasing electronic coupling while fixing the other parameters. We can see that the contribution from the molecular exciton becomes larger with stronger tip-molecule coupling. The Fano dip in Fig. 4 (b) gradually develops into an asymmetric peak, resembling the case in Fig. 4 (a). These results show the importance of electronic subsystem on the photon emission spectrum. This has hitherto not been considered, and is beyond the simple model in the SI. This prediction can be verified in experiment by changing the vertical tip-molecule distance, which has used in related studies [13,14].
IV. SUMMARY
In summary, we have developed a general theoretical model based on NEGF to investigate the single moleculemediated light emission from a STM junction inspired by the recent experiments. Three different regimes are highlighted to explain the experimental results. Our model provides a clear description of the evolution of the spectra line shapes with the STM tip position. Moreover, this approach can also be used to study the light emission from other molecules such as DNA and RNA molecules, as the mismatch of base-pairs can be distinguished by the emission spectra [44][45][46]. This provides a novel opportunity to detect the gene mutation.
ACKNOWLEDGMENTS
We are grateful to financial support from the National Natural Science Foundation of China (grant No.: 61371015).
Appendix A: Model and Hamiltonian
In our model as shown in Fig. 1(a), two extra 'agents' from the substrate (a 2 ) and from the tip (a 1 ) are introduced, which couple to the substrate and the STM tip, respectively. In reality, each of them is part of the tip or the substrate. They are introduced mainly to avoid direct coupling between the tip and substrate, which is convenient to apply the nonequilibrium Green's function (NEGF) theory. When the molecule is away from the tip, it couples only to the substrate. When it is underneath or very near the tip, it couples to the tip and the substrate through the two agents.
The model Hamiltonian, consisting of the electronic reservoir, two photon fields and electron-photon interaction terms, can be defined as
H = H el + H ph + H e−p ,(A1)
where
H el = H b + H a + H m ,(A2)H ph = H p0 + H p1 + H p01 ,(A3)H e−p = H ep0 + H ep1 .(A4)
The Hamiltonian of the tip (t) and substrate (s) electrons is written as
H b = kν=t,s ε kν c † kν c kν ,(A5)
where c † kν (c kν ) creates (annihilates) an electron in the ν (tip or substrate) reservoir with momentum k and energy ε kν . Hamiltonian of the agents, including coupling to s and t, is
H a = i=1,2 ε ai d † i d i + (t 12 d † 1 d 2 + h.c.) + kt t 1t d † 1 c kt + h.c. + ks t 2s d † 2 c ks + h.c. ,(A6)
where d † i (d i ) creates (annihilates) an electron on the agent i = 1, 2 with energy ε ai , t 12 is the tunnel coupling between two agents, and t 1t (t 2s ) is the agent-reservoir electron transfer coupling. Hamiltonian of the molecule is
H m = i=h,l ε i d † i d i + j=1,2 (t ij d † i d j + h.c.) + ks t is d † i c ks + h.c. ,(A7)
where d † i (d i ) creates (annihilates) an electron on the molecular orbital i = h, l with energy ε i , t ij is the tunnel coupling between molecule and agent, and t is is the molecule-reservoir (substrate) electron transfer coupling.
The Hamiltonian for two photon fields are
H p0 =hω 0 1 2 + a † 0 a 0 ,(A8)H p1 =hω p 1 2 + a † p a p ,(A9)
and
H p01 = t p (x)a † p a 0 + h.c.,(A10)
where a † 0 (a 0 ) and a † p (a p ) create (annihilate) photons in the two photon fields. The term H p01 is the coupling Hamiltonian between the two photon fields, and the coupling parameter t p (x) depends on the distance x between the tip and the molecule.
The interaction between the photon (plasmon) mode with the electronic system is described within the rotating wave approximation [47]
H ep0 = m 0 (d † h d l a † 0 + d † l d h a 0 ), (A11) H ep1 = m p (d † 2 d 1 a † p + d † 1 d 2 a p ),(A12)
where m 0 and m p are the coupling parameters of molecular exciton-photon and the agent-gap plasmon, respectively.
Appendix B: The NEGF method
The NEGF method [35][36][37][38] is a powerful tool to investigate the luminescence properties of the STM junction with consideration of the electron-photon coupling. We first define the photon Green's functions in the Keldysh contour with time on the contour as τ
D ij (τ, τ ′ ) = − ī h T t {a i (τ )a † j (τ ′ )} .(B1)
In real time, six different components of the Green's function can be defined as (seth = 1)
D t ij (t, t ′ ) = −iθ(t − t ′ ) a i (t)a † j (t ′ ) − iθ(t ′ − t) a † j (t ′ )a i (t) , Dt ij (t, t ′ ) = −iθ(t − t ′ ) a † j (t ′ )a i (t) − iθ(t ′ − t) a i (t)a † j (t ′ ) , D < ij (t, t ′ ) = −i a † j (t ′ )a i (t) , D > ij (t, t ′ ) = −i a i (t)a † j (t ′ ) , D r ij (t, t ′ ) = −iθ(t − t ′ ) [a i (t), a † j (t ′ )] , D a ij (t, t ′ ) = iθ(t ′ − t) [a i (t), a † j (t ′ )] .(B2)
For our purpose, the most suitable functions are the D <,> and D r,a . In general, D r is linked to the response function and D <,> is related to the light emission spectra, which can be obtained from the Dyson-Keldysh equations
D r (ω) = D r 0 (ω) + D r 0 (ω)Π r t (ω)D r (ω), D < (ω) = D r (ω)Π < t (ω)D a (ω).(B3)
Without electron-photon interaction, the Green's function D r 0 for the bare photon system can be solved exactly using equation of motion method. Π t = Π el + Π ev + Π d is the total photon self-energy, where Π el , Π ev , and Π d account for the interaction with electrons, non-radiative decay, and radiative decay, respectively. We consider wide-band environment and detector, such that Π ev and Π d can be expressed as
Π r ev (ω) = − i 2 diag{γ 0e , γ pe }, Π r d (ω) = − i 2 diag{γ d0 , γ dp },(B4)
and
Π < ev,de (ω) = f b (ω)[Π r ev,de (ω) − Π a ev,de (ω)].(B5)
Here, γ 0e (γ pe ) and γ d0 (γ dp ) are the radiative and non-radiative dissipation rate of molecular exciton (gap plasmon) due to coupling to the environment and the detector, respectively, f b (ω) = [e ω/kB T − 1] −1 is the Bose-Einstein distribution function with the temperature T . Based on standard SCBA, the self-energies due to electron-photon interaction are given by
Π r el,mn (ω) = −i ijkl M m ij dε 2π [G r li (ε)G < jk (ε − ω) + G < li (ε)G a jk (ε − ω)]M n kl , Π < el,mn (ω) = −i ijkl M m ij dε 2π G < li (ε)G > jk (ε − ω)M n kl .(B6)
Similarly, we can define the Green's function for electrons
G ij (τ, τ ′ ) = −i T t {d i (τ )d † j (τ ′ )} . (B7)
In the energy space, the retarded and lesser Green's functions can be calculated from Dyson-Keldysh equations
G r (ε) = G r 0 (ε) + G r 0 (ε)Σ r ep (ε)G r (ε), G < (ε) = G r (ε)[Σ < ep (ε) + Σ < 0 (ε)]G a (ε),(B8)
where G 0 is the Green's function for electronic system without electron-photon interaction, and Σ 0 = Σ a−t + Σ a−s + Σ m−s is the electronic self-energy describing the coupling to the tip (Σ a−t ) and substrate (Σ a−s , Σ m−s ), respectively. Using the wide-band approximation for the tip and substrate electrodes, we have
Σ r a−t (ε) = − i 2 diag{Γ a1 , 0, 0, 0}, Σ r a−s (ε) = − i 2 diag{0, 0, 0, Γ a2 },
where Γ ai(=1,2) , and Γ i(=l,h)s are the linewidth functions, f e ν (ε) = [1 + e (ε−µν )/kB T ] −1 is the Fermi-Dirac distribution function for the electrode ν = t, s with the chemical potential µ ν and the temperature T , µ s − µ t = eV st is the tip-substrate voltage drop. The self-energies (Σ r,< ep ) due to electron-photon coupling are given within SCBA
Σ r ep,mn (ε) = −i ijkl M i mn D r 0,ij (ω = 0)M j kl dε 2π G < lk (ε) + i ijkl M k mi dω 2π [G r ij (ε − ω)D < kl + G < ij (ε − ω)D r kl + G r ij (ε − ω)D r kl ]M l jn , Σ < ep,mn (ε) = i ijkl M k mi dω 2π G < ij (ε − ω)D < kl (ω)M l jn .(B11)
The interaction matrix M describes molecule (agent)-photon field coupling, which can be divided into two types of contributions: (a) m 0 d † i d j a † 0 (i, j = h, l, i = j) in Eq. (A11) describe excitation and de-excitation between two molecular orbits, and (b) m p d † i d j a † p (i, j = 1, 2, i = j) in Eq. (A12) describe transitions between two electronic states of the agents that couple to the plasmon field.
Following the standard procedure [37,38,48], the energy flux of photon can be expressed as
J α ph = dω 2π ωTr[Π < α (ω)D > (ω) − Π > α (ω)D < (ω)], α = el, ev, d.(B12)
As expected, the conservation of energy J el ph +J ev ph +J d ph = 0 is satisfied in steady state within SCBA. For characterizing the luminescence properties of a STM junction, we may define the flux probed by the detector by let α = d.
In the lowest order approximation to the electron-photon coupling, we can replacing D > and D < in Eq. (B12) using D > 0 and D < 0 . After the replacement, we can see that: (1) the energy spectrum goes into the photonic system is determined by both the electronic and the photonic system through Π e and D 0 , respectively. The Fano effect is reflected in the photonic system D 0 , especially the gap plasmons, as analyzed in Appendix C.
FIG. 1 .
1(a) Schematic picture of the experimental setup in STM induced light emission and single molecule luminescence. (b)
FIG. 2 .
2The energy detuning ∆p0 dependence of flux with tp(x) = 0.013 eV, the red lines represent the spectrum of gap plasmon mode. The other parameters are the same as inFig. 3(II), and keep fixed.
FIG. 3 .
3and can be fitted by a simple model detailed in Appendix C [Eq. (C10) or (C11)], where the magnitude of the Fano q factor is mainly determined by The photon energy flux calculated as the tip is: (I) apart from the molecule, (II) slight aside from the molecule so that there is a coupling between gap plasmon and the molecular exciton, but no direct electronic coupling between the tip and the molecule, (III) located on top of the molecule, so that there is direct electronic coupling between the tip and the molecule. The separate contributions from the plasmon and the molecular exciton are shown as dotted lines, while the sum of the two is shown as red solid line.
FIG
. 4. (a) The photon energy flux for different values of the non-radiative damping γ0e with other parameters fixed. The red line represents the spectrum of gap plasmon for Vst = −3 V. Other parameters are: εa1 = 0.16 eV, εa2 = 1.0 eV, ε h = 1.5 eV, ε l = 3.01 eV, t12 = 0.2 eV, Γae = 0.2 eV, Γms = 0.0, tma1 = 0.2 eV, tma2 = 0.2 eV, ω0 = 1.51 eV, ωp = 1.41 eV, m0 = 0.01 eV, mp = 0.003 eV, tp(x) = 0.022 eV, γpe = 0.25 eV, γ d0 = 1 × 10 −5 eV, γ dp = 0.008 eV, T = 4.5 K, Vst = 1.6 V. (b) The photon emission spectrum for different values of tp(x) without tip-molecule coupling (black lines), and for different values of tip-molecule coupling tma1 when tp(x) = 15 meV. Other parameters are the same with Fig. 3 (II). (c) Similar to (b), with different parameters ω0 = 1.81 eV, ωp = 2.0 eV, ε h = 0.48 eV, ε l = 2.29 eV, γ0p = 0.35 eV, γ dp = 0.02 eV, T = 4 K, Γms = 0.1 and Vst = −2.3 V.
Appendix C: The formula for Fano resonanceTo obtain a standard formula of Fano resonance, we consider a simple model to describe the coupling between the molecular exciton and the gap plasmon [seeFig. 1(b)], in which the photon transport due to molecular exciton is regarded as a scatter[49,50]. Then the correction to the Green's function for gap plasmon readswhereΠ r p−m represents retarded self-energy due to the coupling between gap plasmon and molecular exciton. For noninteracting photons, the expression for T-matrix obtained by equation of motion takes the formwhere D r m is the retarded Green's function of the molecular exciton.By using the definition of the spectral density of the gap plasmon with and without interaction: ρ p (ω) = −ℑD r p (ω)/π and ρ 0 p (ω) = −ℑD 0,r p (ω)/π, and taking the imaginary part of Eq. (C2)where we have definedHere, q is the Fano-factor, Q = t 2 p (x)ℜD 0,r p and γ = πt p (x) 2 ρ 0 p (ω). In the noninteracting case, the D r m can be written asγ dr represents the coupling with the electronic system, the environment and the detector. So, the T p−m takes the formWe introduce the ǫ = (ω − ω 0 − Q)/(γ + γ dr ), then T p−m can be expressed asSubstituting the expression of ℑD 0,r p and T p−m into ρ p (ω), we getwith γ t = πt p (x)t * p (x)/γ 0 , γ 0 = γ + γ dr , ǫ = (ω − ω t )/γ 0 and ω t = ω 0 + Q. Here, (q 2 + 2qǫ − 1)/(ǫ 2 + 1) indicates the spectral density of the gap plasmon has a Fano profile determined by the parameter q. Note that γ t characterizes the effective coupling strength of the gap plasmon with the molecular exciton.To demonstrate the Fano resonance is an universal phenomenon in the single molecule-based STML experiments, we can fit the numerical results of the spectral density of gap plasmon with the formula in Eq. (C10). The results of the fitting are shown inFig. C1(see the blue dotted lines). It is found that the Fano resonance can well describe the gap plasmon spectral density, which implies that the asymmetric line shape with a dip in the light spectra originates from the Fano resonance between the molecular exciton and gap plasmon with different lifetimes. Moreover, the Fano line shape becomes prominent as the t p (x) increases as well as the corresponding fitting parameter γ t . Therefore, one can use the Eq. (C10) to fit and predict the light emission spectra in the single molecule-based STML experiments. On the other hand, Eq. (C10) can be expressed as a more frequently used form to fit the experimental resultwhere F (ǫ) = (ǫ + q) 2 /(ǫ 2 + 1) is Fano function.
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In the calculations, we set t 1l,h = tma1, t 2l,h = tma2, Γa1,2 = Γae and Γ ls = Γ hs = Γms. The parameters producing the plots are the following: (I) Γms = 0. 5 eV, tma1 = 0, tma2 = 0, γ d0 = 0, γ0e = 0 and tp(x) = 0; (IIIn the calculations, we set t 1l,h = tma1, t 2l,h = tma2, Γa1,2 = Γae and Γ ls = Γ hs = Γms. The parameters pro- ducing the plots are the following: (I) Γms = 0.5 eV, tma1 = 0, tma2 = 0, γ d0 = 0, γ0e = 0 and tp(x) = 0; (II)
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| []
|
[
"Census of H ii regions in NGC 6754 derived with MUSE: Constraints on the metal mixing scale",
"Census of H ii regions in NGC 6754 derived with MUSE: Constraints on the metal mixing scale"
]
| [
"S F Sánchez \nInstituto de Astronomía\nUniversidad Nacional Autonóma de México\nA.P. 70-26404510MéxicoD.F\n",
"L Galbany \nMillennium Institute of Astrophysics\nUniversidad de Chile\nCasilla 36-DSantiagoChile\n\nDepartamento de Astronomía\nUniversidad de Chile\nCasilla 36-DSantiagoChile\n",
"E Pérez \nInstituto de Astrofísica de Andalucía (CSIC)\nGlorieta de la Astronomía s/nAptdo. 3004E-18080GranadaSpain\n",
"P Sánchez-Blázquez \nDepartamento de Física Teórica\nUniversidad Autónoma de Madrid\n28049MadridSpain\n",
"J Falcón-Barroso \nInstituto de Astrofísica de Canarias (IAC)\nE-38205La Laguna, TenerifeSpain\n\nDepto. Astrofísica\nUniversidad de La Laguna (ULL)\nE-38206La Laguna, TenerifeSpain\n",
"F F Rosales-Ortega \nInstituto Nacional de Astrofísica\nÓptica y Electrónica, Luis E. Erro 172840Tonantzintla, PueblaMéxico\n",
"L Sánchez-Menguiano \nInstituto de Astrofísica de Andalucía (CSIC)\nGlorieta de la Astronomía s/nAptdo. 3004E-18080GranadaSpain\n\nDpto. de Física Teórica y del Cosmos\nFacultad de Ciencias (Edificio Mecenas)\nUniversity of Granada\nE-18071GranadaSpain\n",
"R Marino \nDepartamento de Astrofísica y CC. de la Atmósfera\nFacultad de CC. Físicas\nCEI Campus Moncloa\nUCM-UPM\nUniversidad Complutense de Madrid\nAvda. Complutense s/nE-28040MadridSpain\n",
"H Kuncarayakti \nMillennium Institute of Astrophysics\nUniversidad de Chile\nCasilla 36-DSantiagoChile\n\nDepartamento de Astronomía\nUniversidad de Chile\nCasilla 36-DSantiagoChile\n",
"J P Anderson \nEuropean Southern Observatory\nAlonso de Cordova\n3107 Casilla 19001 -Vitacura -SantiagoChile\n",
"T Kruehler \nEuropean Southern Observatory\nAlonso de Cordova\n3107 Casilla 19001 -Vitacura -SantiagoChile\n",
"M Cano-Díaz \nInstituto de Astronomía\nUniversidad Nacional Autonóma de México\nA.P. 70-26404510MéxicoD.F\n",
"J K Barrera-Ballesteros \nInstituto de Astrofísica de Canarias (IAC)\nE-38205La Laguna, TenerifeSpain\n\nDepto. Astrofísica\nUniversidad de La Laguna (ULL)\nE-38206La Laguna, TenerifeSpain\n",
"J J González-González \nInstituto de Astronomía\nUniversidad Nacional Autonóma de México\nA.P. 70-26404510MéxicoD.F\n"
]
| [
"Instituto de Astronomía\nUniversidad Nacional Autonóma de México\nA.P. 70-26404510MéxicoD.F",
"Millennium Institute of Astrophysics\nUniversidad de Chile\nCasilla 36-DSantiagoChile",
"Departamento de Astronomía\nUniversidad de Chile\nCasilla 36-DSantiagoChile",
"Instituto de Astrofísica de Andalucía (CSIC)\nGlorieta de la Astronomía s/nAptdo. 3004E-18080GranadaSpain",
"Departamento de Física Teórica\nUniversidad Autónoma de Madrid\n28049MadridSpain",
"Instituto de Astrofísica de Canarias (IAC)\nE-38205La Laguna, TenerifeSpain",
"Depto. Astrofísica\nUniversidad de La Laguna (ULL)\nE-38206La Laguna, TenerifeSpain",
"Instituto Nacional de Astrofísica\nÓptica y Electrónica, Luis E. Erro 172840Tonantzintla, PueblaMéxico",
"Instituto de Astrofísica de Andalucía (CSIC)\nGlorieta de la Astronomía s/nAptdo. 3004E-18080GranadaSpain",
"Dpto. de Física Teórica y del Cosmos\nFacultad de Ciencias (Edificio Mecenas)\nUniversity of Granada\nE-18071GranadaSpain",
"Departamento de Astrofísica y CC. de la Atmósfera\nFacultad de CC. Físicas\nCEI Campus Moncloa\nUCM-UPM\nUniversidad Complutense de Madrid\nAvda. Complutense s/nE-28040MadridSpain",
"Millennium Institute of Astrophysics\nUniversidad de Chile\nCasilla 36-DSantiagoChile",
"Departamento de Astronomía\nUniversidad de Chile\nCasilla 36-DSantiagoChile",
"European Southern Observatory\nAlonso de Cordova\n3107 Casilla 19001 -Vitacura -SantiagoChile",
"European Southern Observatory\nAlonso de Cordova\n3107 Casilla 19001 -Vitacura -SantiagoChile",
"Instituto de Astronomía\nUniversidad Nacional Autonóma de México\nA.P. 70-26404510MéxicoD.F",
"Instituto de Astrofísica de Canarias (IAC)\nE-38205La Laguna, TenerifeSpain",
"Depto. Astrofísica\nUniversidad de La Laguna (ULL)\nE-38206La Laguna, TenerifeSpain",
"Instituto de Astronomía\nUniversidad Nacional Autonóma de México\nA.P. 70-26404510MéxicoD.F"
]
| []
| We present a study of the H ii regions in the galaxy NGC 6754 from a two pointing mosaic comprising 197,637 individual spectra, using Integral Field Spectrocopy (IFS) recently acquired with the MUSE instrument during its Science Verification program. The data cover the entire galaxy out to ∼2 effective radii (r e ), sampling its morphological structures with unprecedented spatial resolution for a wide-field IFU. A complete census of the H ii regions limited by the atmospheric seeing conditions was derived, comprising 396 individual ionized sources. This is one of the largest and most complete catalogue of H ii regions with spectroscopic information in a single galaxy. We use this catalogue to derive the radial abundance gradient in this SBb galaxy, finding a negative gradient with a slope consistent with the characteristic value for disk galaxies recently reported. The large number of H ii regions allow us to estimate the typical mixing scale-length (r mix ∼0.4 r e ), which sets strong constraints on the proposed mechanisms for metal mixing in disk galaxies , like radial movements associated with bars and spiral arms, when comparing with simulations. We found evidence for an azimuthal variation of the oxygen abundance, that may be related with the radial migration. These results illustrate the unique capabilities of MUSE for the study of the enrichment mechanisms in Local Universe galaxies. | 10.1051/0004-6361/201424950 | [
"https://arxiv.org/pdf/1411.4967v1.pdf"
]
| 53,117,899 | 1411.4967 | 5451c07c571456ff8c45037fff152f2587ec55d5 |
Census of H ii regions in NGC 6754 derived with MUSE: Constraints on the metal mixing scale
18 Nov 2014
S F Sánchez
Instituto de Astronomía
Universidad Nacional Autonóma de México
A.P. 70-26404510MéxicoD.F
L Galbany
Millennium Institute of Astrophysics
Universidad de Chile
Casilla 36-DSantiagoChile
Departamento de Astronomía
Universidad de Chile
Casilla 36-DSantiagoChile
E Pérez
Instituto de Astrofísica de Andalucía (CSIC)
Glorieta de la Astronomía s/nAptdo. 3004E-18080GranadaSpain
P Sánchez-Blázquez
Departamento de Física Teórica
Universidad Autónoma de Madrid
28049MadridSpain
J Falcón-Barroso
Instituto de Astrofísica de Canarias (IAC)
E-38205La Laguna, TenerifeSpain
Depto. Astrofísica
Universidad de La Laguna (ULL)
E-38206La Laguna, TenerifeSpain
F F Rosales-Ortega
Instituto Nacional de Astrofísica
Óptica y Electrónica, Luis E. Erro 172840Tonantzintla, PueblaMéxico
L Sánchez-Menguiano
Instituto de Astrofísica de Andalucía (CSIC)
Glorieta de la Astronomía s/nAptdo. 3004E-18080GranadaSpain
Dpto. de Física Teórica y del Cosmos
Facultad de Ciencias (Edificio Mecenas)
University of Granada
E-18071GranadaSpain
R Marino
Departamento de Astrofísica y CC. de la Atmósfera
Facultad de CC. Físicas
CEI Campus Moncloa
UCM-UPM
Universidad Complutense de Madrid
Avda. Complutense s/nE-28040MadridSpain
H Kuncarayakti
Millennium Institute of Astrophysics
Universidad de Chile
Casilla 36-DSantiagoChile
Departamento de Astronomía
Universidad de Chile
Casilla 36-DSantiagoChile
J P Anderson
European Southern Observatory
Alonso de Cordova
3107 Casilla 19001 -Vitacura -SantiagoChile
T Kruehler
European Southern Observatory
Alonso de Cordova
3107 Casilla 19001 -Vitacura -SantiagoChile
M Cano-Díaz
Instituto de Astronomía
Universidad Nacional Autonóma de México
A.P. 70-26404510MéxicoD.F
J K Barrera-Ballesteros
Instituto de Astrofísica de Canarias (IAC)
E-38205La Laguna, TenerifeSpain
Depto. Astrofísica
Universidad de La Laguna (ULL)
E-38206La Laguna, TenerifeSpain
J J González-González
Instituto de Astronomía
Universidad Nacional Autonóma de México
A.P. 70-26404510MéxicoD.F
Census of H ii regions in NGC 6754 derived with MUSE: Constraints on the metal mixing scale
18 Nov 2014Received --; accepted --Astronomy & Astrophysics manuscript no. draft c ESO 2014 November 19, 2014Galaxies: abundances -Galaxies: fundamental parameters -Galaxies: ISM -Galaxies: stellar content -Techniques: imaging spectroscopy -techniques: spectroscopic -stars: formation -galaxies: ISM -galaxies: stellar content
We present a study of the H ii regions in the galaxy NGC 6754 from a two pointing mosaic comprising 197,637 individual spectra, using Integral Field Spectrocopy (IFS) recently acquired with the MUSE instrument during its Science Verification program. The data cover the entire galaxy out to ∼2 effective radii (r e ), sampling its morphological structures with unprecedented spatial resolution for a wide-field IFU. A complete census of the H ii regions limited by the atmospheric seeing conditions was derived, comprising 396 individual ionized sources. This is one of the largest and most complete catalogue of H ii regions with spectroscopic information in a single galaxy. We use this catalogue to derive the radial abundance gradient in this SBb galaxy, finding a negative gradient with a slope consistent with the characteristic value for disk galaxies recently reported. The large number of H ii regions allow us to estimate the typical mixing scale-length (r mix ∼0.4 r e ), which sets strong constraints on the proposed mechanisms for metal mixing in disk galaxies , like radial movements associated with bars and spiral arms, when comparing with simulations. We found evidence for an azimuthal variation of the oxygen abundance, that may be related with the radial migration. These results illustrate the unique capabilities of MUSE for the study of the enrichment mechanisms in Local Universe galaxies.
Introduction
Nebular emission lines have been historically the main tool at our disposal for direct measurement of the gas-phase abundance at discrete spatial positions in low-redshift galaxies (e.g. Alloin et al. 1979). They trace the young, massive star component in galaxies, illuminating and ionizing cubic kiloparsecsized volumes of interstellar medium. Metals play a fundamental role in cooling mechanisms in the intergalactic and interstellar medium, and in processes of star-formation, stellar physics, and planet formation.
Previous spectroscopic studies have unveiled some aspects of the complex processes at play between the chemical abundances of galaxies and their physical properties. These studies have been successful in determining important relationships, scaling laws and systematic patterns (e.g. Lequeux et al. 1979;Diaz 1989;Zaritsky et al. 1994;Garnett 2002;Tremonti et al. 2004;Moustakas & Kennicutt 2006). However, these results are limited by statistics, either in the number of observed H ii regions or in the coverage of these regions across the galaxy surface.
The advent of multi-object spectrometers and IFS instruments with large fields of view (FoV) now offers the opportunity to undertake a new generation of emission-line surveys, based on samples of hundreds of H ii regions and full twodimensional (2D) coverage of the disks of nearby spiral galaxies (e.g. Rosales-Ortega et al. 2010). One of the most interesting results recently derived using IFS data is that the oxygen abundance gradient seems to present a common slope ∼−0.1 dex/r e for non-interacting galaxies (Sánchez et al. 2012b(Sánchez et al. , 2014b.
This result agrees with models based on the standard insideout scenario of disk formation, which predict a relatively quick self enrichment with oxygen and an almost universal negative metallicity gradient once it is normalized to the galaxy optical size (Boissier & Prantzos 1999. From the seminal works of Lacey & Fall (1985a), Guesten & Mezger (1982) and Clayton (1987), most numerical models of chemical evolution explain the existence of the radial gradient of abundances by the combined effects of a star-formation rate and an infall of gas, both varying with galactocentric radius (e.g., Mollá & Roy 1999).
Although there is a large number of studies focused on the analysis of the abundance gradient in galaxies, in contrast, little is known about the possible presence of azimuthal asymmetries in this distribution. Deviations from the radial abundance gradient are well known features in the Milky Way, based on the study of Cepheids and open clusters (e.g., Chiappini et al. 2001;Lépine et al. 2011). However, the situation in other spiral galaxies is less clear, and suffers from poor statistics, either for the low number of H ii regions sampled per galaxy or for the large errors of the abundance estimation. Recently, using wide-field IFS Rosales-Ortega et al. (2011) showed that the radial metallicity gradient of NGC 628 varies slighly for different quadrants, although the differences are comparable to the uncertainties introduced by the adopted estimators of the oxygen abundances. More recently, Li et al. (2013) found marginal evidence for the existence of moderate deviations from chemical abundance homogeneity in the intestellar medium of M101, using a combination of strong-line abundance indicators and direct estimations based on the detection of the [O iii]λ4363 auroral line.
Despite the advances of recent IFS-surveys in our understanding of the evolution of the chemical enrichment processes in galaxies, they present some limitations. The most important one is the lack of the spatial resolution required to propertly resolve individual small-scale morphological structures, in particular individual H ii regions. The IFS surveys with the best physical resolution, such as PINGS (Rosales-Ortega et al. 2010) or CALIFA (Sánchez et al. 2012a), have ∼5 times worst spatial resolution than the typical ground-based imaging surveys. This results in a bias in the detection of H ii regions, that are aggregated based on their spatial vicinity (decreasing their number by a factor three or more), and their spectra are polluted by diffuse gas emission (Mast et al. 2014).
In principle, the abundance scatter and azimuthal asymmetries of H ii regions around the average radial gradient can be used to constrain the spatial-scale of radial mixing (e.g. Scalo & Elmegreen 2004;Di Matteo et al. 2013). In the absence of radial mixing the only observed scatter around the abundance gradient should be produced by the errors in the individual measurements. Regardless of its origin (e.g., Athanassoula 1992), any radial mixing increases the scatter by moving regions of a certain abundance from a certain galactocentric distance to a different one. Therefore, the dispersion around the average slope is a constraint to the maximum radial mixing scale.
So far, for the reasons outlined above, the current IFS surveys lacked the required resolution to address this important key issue in the chemical evolution of galaxies. The Multi Unit Spectroscopic Explorer (MUSE Bacon et al. 2010) has changed dramatically the perspective for these studies. This instrument is a unique tool for the spectroscopic analysis of resolved structures in galaxies, particularly in the local universe. The combination of a large field-of-view (FoV) (∼60 ′′ ×60 ′′ ), unprecedented spatial sampling (0.2 ′′ /spaxel) for a wide-field IFU, which limits the spatial resolution to the atmospheric seeing, the spectral resolution and large wavelength coverage, and the large aperture of the VLT telescope, makes MUSE a well suited instrument to address these problems. Of course, there are other IFUs with similar or even larger FoVs, like PPAK (Kelz et al. 2006), VIMOS (Le Fèvre et al. 2003 or VIRUS-P (Hill et al. 2008), and also other IFUs operating in the optical range have similar or better spatial sampling, like GMOS (Allington-Smith et al. 2002) or OASIS 1 . However, MUSE is the first one that combines at the same time the large FoV and the image-like spatial sampling. 1 http://cral.univ-lyon1.fr/labo/oasis/present/ In this work we study the oxygen abundance gradient of the spiral galaxy NGC 6754 using the data recently observed by MUSE as part of the Science Verification programs (SV). NGC 6754 is a barred Sb galaxy mildly inclined (i ∼60 • ). Its brightness (B∼13 mag), projected size (r 25 ∼ 1 ′ ) and redshift (z =0.0108), similar to the footprint of the CALIFA galaxies (e.g. Walcher et al. 2014), makes it suitable to perform a census of the H ii regions using MUSE, to further understand the abundance distribution in this galaxy.
Data acquisition and reduction
NGC 6754 was observed on June 28th and 30th 2014 in the context of Program 60.A-9329 (PI: Galbany) of the MUSE SV run.
The observations were divided into two pointings covering the east and west parts of the galaxy, respectively. The final cube for each pointing is the result of 3 exposures of 900 seconds, where the second and third exposure were slightly shifted (2 arcsec NE and SW, respectively) and rotated 90 • from the first exposure, in order to provide a uniform coverage of the field and to limit systematic errors in the reduction.
The reduction of the raw data was performed with Reflex (Freudling et al. 2013) using version 0.18.2 of the MUSE pipeline (Weilbacher et al. 2014), including the standard procedures of bias subtraction, flat fielding, wavelength calibration, flux calibration, and the final cube reconstruction by the spatial arrangement of the individual slits of the image slicers.
The final dataset comprises two cubes of ∼100k individual spectra, each covering a FoV slightly larger than ∼1 . Each spectrum covers the wavelength range 4800-9300 Å, with a typical spectral resolution between 1800 and 3600 (from blue to red). The cubes are aligned east-west, with an overlapping area of ∼16 ′′ , where the galaxy center has been sampled twice. The final mosaiced datacube comprises almost 200k individual spectra, covering the entire galaxy up to 2 effective radii, with a FoV of ∼2 ′ ×1 ′ . For practical reasons, we analysed each cube separately and later combined the different data products. Figure 1 illustrates the power of the combined large FoV and high spatial resolution of MUSE, and the quality of the data. It shows a true color image created using a combination of a Vband image, and two continuum subtracted narrow-band images of 30Å width, centred in [O iii]λ5007 and Hα, at the redshift of the galaxy. The three maps were synthetized from each datacube, and combined to create a single image for each band. For each of the narrow-band images the continuum was estimated from the average of two additional narrow-band images of similar width (i.e., 30Å) extracted at a wavelength redshifted and blueshifted 100Å from the nominal wavelength of the considered emission line at the redshift of the object. Finally, the V-band image was synthetized by convolving the spectra at each spaxel by the nominal response curve of the Johnson V-band filter. Those images were used to illustrate the spatial resolution and image quality of the data. We should note here than the V-band image does not include only continuum emission, since it is contaminated by both Hβ and [O iii]λ5007, and the Hα intensity map is contaminated by the adjacent [N ii] doublet. However, they clearly trace the continuum and emission line distribution across the galaxy. The spatial distribution of the individual H ii /ionized regions can be easily recognized, tracing the star forming regions along the spiral arms of the galaxy. Different seeing conditions for the for the East (∼0.8 ′′ ) and for the West (∼1.8 ′′ ) pointings are also clear. Fig. 1. RGB color image of NGC 6754 created using the line intensity maps of [O iii]λ5007 (blue), V-band (green) and Hα (red) extracted from the datacubes. Each large tickmark corresponds to 10 ′′ (or 400 pc at the redshift of the galaxy). The green point-like sources are field stars. Each red structure corresponds to a single H ii region.
Right Ascension Declination
Analysis
The main goals of this study are to characterize the abundance gradient in NGC 6754 and to estimate the dispersion of the abundances of the individual H ii regions around the average gradient. In this section we describe briefly how we select the H ii regions, extract and analyze their individual spectra, derive the corresponding oxygen abundance, and analyze their radial gradient. More details on the procedure are described in Sánchez et al. (2012b), and references therein.
Detection of the ionized regions
The segregation of H ii regions and the extraction of the corresponding spectra is performed using a semi-automatic procedure named HIIexplorer 2 . The details of this program are given in Sánchez et al. (2012b), and a detailed description of the overall detection process, Sánchez et al. (2014b). HIIexplorer requires as input a map of emission line intensities or equivalent widths, a minimum threshold above which the peak intensity of the H ii region is detected, and three different convergence criteria: (i) the maximum fractional difference between the peak intensity and the adjacent ones to be agregated to a particular region, (ii) the minimum absolute intensity for a pixel to be agregated, and (iii) the maximum distance between the pixel considered and the peak intensity . In this particular case we use the map of the equivalent width of Hα, EW(Hα), derived from the narrow-band image described above. The use of the EW guarantees that the analysis is more homogeneous between the two pointings, since this parameter is less affected by possible spectrophotometric differences and it is less sensitive to seeing variations. The fact that the equivalent width may be contaminated or not by the adjacent [N ii] doublet is not relevant, since this map is used only to detect the emission line regions, and not in any further analysis along the article. Therefore, a possible contamination by [NII] may affect only the contrast, and only marginally, but not the detectability of the regions, since the average contamination by this line is about a 30% of the total flux. The output of HIIexplorer is a segmentation map and the integrated spectrum for each of the H ii regions detected.
We processed individually the two datacubes, fixing the input parameters to the optimal ones for the west-pointing, that was observed under worst seeing conditions. We selected a threshold in the peak EW(Hα) = 20Å, and minimum EW(Hα) = 8Å, a minimum fractional peak of 1%, and a maximum distance of 2 ′′ . Therefore, the convergence criteria restrict the detection of regions with at least EW(Hα) = 8Å in every spaxel and a maximum diameter of 4 ′′ . The selection of these parameters is based on our previous studies with other IFU data and different tests to optimize the results: (i) the minimum absolute EW(Hα) is selected to guarantee that all the pixels agregated to a particular region are above the boundary between retired and star-forming regions proposed by Cid Fernandes et al. (2010) and discussed in Sánchez et al. (2014b), even if a very conservative error of 25% is assumed for this parameter, and/or taking into account the contamination by [N ii]; (ii) the threshold in the peak EW(Hα) is selected to be more than twice the minimum, to guarantee that the region is actually clumpy/peaky, and not a diffuse ionized region; (iii) the maximum distance is fixed to the estimated size of an H ii region, that could have a diameter as large as ∼1 kpc (e.g., NGC 5471, Oey et al. 2003;García-Benito et al. 2011), thus, ∼4 ′′ at the redshift of the object. Using these parameters we detect a similar number of H ii regions in each pointing: 207 in the east pointing and 220 in the west one. The final catalogue was cleaned for double detections in the overlapping area by removing those H ii regions with coordinates that differ less than 3 ′′ . A total of 396 individual clumpy ionized regions are detected, a factor of 5-10 times larger than the number found with lower spatial resolution IFU data (e.g. Sánchez et al. 2013), as predicted by the simulations presented by Mast et al. (2014). Figure 2 shows the EW(Hα) map illustrating the result of this procedure. The location and relative size of the H ii regions detected are indicated with black circles. We note here that the HIIexplorer provides with a segmentation map, not with circular apertures. The current representation is therefore illustrative of the size of the H ii regions, but does not show the actual detailed shape of the associated segmented regions for which the spectra are extracted.
Measurement of the emission line intensities
In this analysis we follow the procedures described in Sánchez et al. (2014b), using the fitting package FIT3D 3 , (Sánchez et al. (2006b). We perform a Monte-Carlo fitting using two different single stellar population (SSP) libraries. In order to compare with previous results and provide with useful information of the underlying stellar population, we first use a library that comprises 156 templates to model and 3 http://www.caha.es/sanchez/FIT3D/ remove the underlying stellar population. This library comprises 39 stellar ages, from 1 Myr to 13 Gyr, and 4 metallicities (Z/Z ⊙ = 0.2, 0.4, 1, and 1.5), and it is described in detail in Cid Fernandes et al. (2013). These templates were extracted from a combination of the synthetic stellar spectra from the GRANADA library Martins et al. (2005) and the SSP library provided by the MILES project (Sánchez-Blázquez et al. 2006;Vazdekis et al. 2010;Falcón-Barroso et al. 2011). Therefore they are restricted to a wavelength range lower than 7000Å. Hence, they cannot be used to remove the underlying stellar population in the full spectral range covered by MUSE. For doing so, we used a more restricted library extracted from the MIUSCAT models (Vazdekis et al. 2012). It comprises 8 stellar ages, from 65 Myr to 17.7 Gyr, and 3 metallicities (Z/Z ⊙ = 0.4, 1, and 1.5). Our previous experience indicates that to decouple the underlying stellar population from the emission lines a restricted library like this one is enough (e.g. Sánchez et al. 2014a). We do not find large differences between the residual spectra for the wavelength range in common, and therefore we finally adopted the results from the second library for the analysis of the emission lines.
Dust attenuation and stellar kinematics were taken into account as part of the fitting process. The stellar kinematics was derived as a first step, fitting the underlying stellar population with a sub-set of the full stellar library changing the systemic velocity and the velocity dispersion at random within the range of allowed values. Then a first model for the stellar population is derived. This model is used to obtain the dust attenuation, allowed to change randomly within a pre-defined range. The extinction law by Cardelli et al. (1989) was assumed, with a specific dust attenuation of R V = 3.1. For each iteration over the dust attenuation values a new model of the underlying stellar population is derived. The best combination of the stellar velocity, velocity dispersion and dust attenuation is recovered based on the lowest reduced χ 2 provided. Finally, these parameters are fixed and the full stellar library is used to recover the underlying stellar population. Due to the Monte-Carlo fitting adopted, the inaccuracies in the derivation of the underlying stellar population are propagated to the error budget in the emission line fitting, and therefore, into the errors estimated for the emission line fluxes. Individual emission line fluxes were measured in the stellarpopulation subtracted spectra by fitting each of them with a single Gaussian function. For this particular dataset we extracted the flux intensity of the following emission lines: Hα, Hβ,
[O iii] λ4959, [O iii] λ5007, [N ii] λ6548, [N ii] λ6583, [S ii] λ6717, [S ii] λ6731, [Ar iii] λ7135 and [S iii] λ9069.
We may notice here that many other relevant emission lines are detected within the wavelength range covered by the spectra. Figure 3 shows a detail of the spectrum of a typical H ii region within the sample, together with the best fitted stellar population model and the resulting emission-line spectrum. In addition to the emission lines measured, it is possible to identify weaker lines like He I λ5876 and [O I] λ6300. The two components of the Na I absorption doublet at ∼5892Å are clearly resolved, with a contribution due to attenuation that cannot be reproduced by the stellar model, which was actully masked during the fitting process. The prominent CaTλ8542Å stellar absorptions in the near-infrared is also clearly detected. In addition to the emission line fluxes, we derive the emission line equivalent widths. For doing so, we divide the integrated flux of the line by the median flux density of the best fitted SSP-model in a window of 100Å centred in the wavelength of the emission line. Therefore, these equivalent widths are not contaminated by the contribution of any adjacent emission lines.
Two artifacts in the spectra are identified in Fig. 3, in particular in the residual emission line spectrum). They look like two bumps at ∼4750Å and ∼6550Å, just bluewards of Hβ and Hα.
Although they have been masked during the fitting process we prefer to show them in the plot, since they seem to be present in all the MUSE spectra we have analyzed. We are not sure if they are a product of the reduction process or a feature in the raw data. In any case, they do not affect the measurements of the emission lines. Two additional spectral features that are not well reproduced by the SSP models correspond to the Na iλ5890,5896 absorption feature, at ∼5950Å at the redshift of the object, and the CaTλ8542Å, at ∼8630Å at the redshift of the object. The Na i mismatch is a well known feature since this absorption line has two physical origins: (i) the absorption due to the presence of this element in the atmosphere of the stars, which is included in the SSP-templates, and (ii) the absorption due to the presence of this element in the inter-stellar medium, which produces an absorption proportional to the gas content (and dust attenuation). The CaTλ8542Å is often problematic in the current SSP-templates, that could be related to variations in the IMF, the abundance of Ca, or a non correct understanding of this absorption feature (although it is not the case in the particular example shown in Fig.3, for which we show the prediction from the fitted SSP-model). Due to these well known mismatches, both spectral regions were masked during the fitting process. Residuals from non perfect sky-subtraction of the strong OH-lines in the near-infrared are visible in the redder wavelength ranges, and a clear defect associated with a telluric absoption is shown at λ∼7600Å. None of the emission lines considered in this study are affected by this later effect.
Selection of H ii regions
Classical H ii regions are gas clouds ionized by short-lived hot OB stars, associated with ongoing star-formation. They are frequently selected on the basis of demarcation lines defined in the so-called diagnostic diagrams (e.g., Baldwin et al. 1981;Veilleux & Osterbrock 1987), which compare different line ratios. Figure 4 sample of H ii regions described above. This diagnostic diagram is frequently used since it uses very strong emission lines (e.g., Fig.3), and it is less affected by dust attenuation and imperfections in the spectrophotometric calibration. The classical demarcation lines described by Kauffmann et al. (2003) and Kewley et al. (2001) have been included. In this study we follow Sánchez et al. (2013) to select H ii regions as clumpy ionized regions with EW(Hα)>6Å and located below the Kewley et al. (2001) demarcation curve. This selection guarantees the exclusion of ionized regions possibly dominated by shocks (that are not clumpy in general), regions whose ionization is dominated by post-AGB stars, and AGN-dominated regions (e.g. Cid Fernandes et al. 2010). Following these criteria all the ionized regions selected by HIIexplorer have been classified as H ii regions.
The distribution of H ii regions across the BPT diagram follows the characteristic pattern in Sa/Sb early-type spirals Sánchez et al. (2014a). They are mostly located at the bottomright end of the classical location of H ii regions in this diagram, with a tail towards the so-called intermediate region between the two demarcation lines described above. Kennicutt et al. (1989) first recognized that H ii regions in the center of galaxies are spectroscopically different from those in the disk in their stronger low-ionization forbidden emission, that place them in the so-called intermediate region. More recently Sánchez et al. (2014b) found a similar behaviour for the H ii regions in earlytype disk galaxies, like NGC 6754. The location of the H ii re-gions change across the BPT diagram with the galactocentric distance (Fig. 4), as a consequence of the change of the ionization conditions and in particular the radial gradient in the oxygen abundance (e.g., Evans & Dopita 1985;Dopita & Evans 1986).
Physical conditions in the H ii regions
The location in the BPT diagram for a classical H ii region ionized by young stars of a certain age is defined well by three main parameters: (a) the ionization parameter or fraction of Lyman continuum photons with respect to total amount of gas, (b) the electron density of the gas, and (c) the metallicity or chemical abundance of the ionized gas. We derive here the first two of them. In addition, we estimate the dust attenuation to correct the emission line fluxes when required.
The dust attenuation, A V , was derived for each H ii region based on the Hα/Hβ Balmer line ratio. The extinction law by Cardelli et al. (1989) was assumed, with a specific dust attenuation of R V = 3.1, and the theoretical value for the unobscured line ratio for case B recombination of Hα/Hβ = 2.86, for T e =10,000 K and n e =100 cm −3 (Osterbrock 1989). For this study we have assumed that the intrinsic Hα/Hβ line ratio does not vary significantly, although it is known that it presents a dependence with the electron density and the temperature (e.g. Osterbrock 1989). Once derived the dust attenuation, all the considered emission line fluxes were corrected adopting the same extinction law, when needed.
The electron density, n e , was derived from the line ratio of the [S ii] doublet (e.g., Osterbrock 1989), by solving the equation,
I([S ii]λ6717) I([S ii]λ6731) = 1.49 1 + 3.77x 1 + 12.8x(1)
where x is the density parameter, defined as x = 10 −4 n e t −1/2 and t is the electron temperature in units of 10 4 K (McCall et al. 1985). For this calculation we assumed a typical electron temperature of T = 10 4 K, which is an average value that corresponds to the expected conditions in H ii regions (Osterbrock 1989). This equation reflects that the [S ii] doublet ratio is sensitive to changes in the electron density only for a limited range of values. For high and low values, it becomes asymptotic, and the value derived has to be treated with care and should not be used for quantitative statements. However, the value will still be valid to understand the possible dependences of the abundance gradient with this parameter. For the ionization parameter, u, we adopted the [S iii]λ9069,9532/[S ii]λ6717,6731 calibrator described by Kewley & Dopita (2002). Since [S iii]λ9532 is not covered by our wavelength range, we adopted a theoretical ratio of [S iii]λ9532/[S iii]λ9069 = 2.5 (Vilchez & Esteban 1996), fixed by atomic physics. Both emission lines were corrected for the dust attenuation prior to deriving the ionization parameter.
Oxygen abundance of H ii regions
Accurate abundance measurements for the ionized gas in galaxies require the determination of the electron temperature (T e ), usually obtained from the ratio of auroral to nebular line intensities (e.g. Osterbrock 1989). It is well known that this procedure is difficult to carry out for metal-rich galaxies, since as the metallicity increases the electron temperature decreases (as the cooling is via metal lines), and the auroral lines eventually become too faint to measure. Therefore, calibrators based on strong emission lines are used.
Strong-line indicators have the obvious advantage of using emission lines with higher signal-to-noise, detected in mostly all H ii regions, and with large dynamical ranges. However, they have the disadvantange that the line ratios considered do not trace only the oxygen abundance, but also depend on other properties of the ionized nebulae, like the electron density, and geometrical factors, and/or the shape of the ionizing radiation (normally parametrized by the ionization parameter, q, or u = q/c in its dimensionless form). It is well known that some of these parameters are correlated, like the trend between oxygen abundance and the ionization parameter, uncovered by the seminal studies by Evans & Dopita (1985); Dopita & Evans (1986), and recently revisited by Sánchez et al. (2014a).
There are two main schools in the derivation of oxygen abundance using strong-line indicators. One uses empirical calibrators based on the comparison of different of strong emission line ratios, with the corresponding abundance derived for a set of H ii regions for which T e is known. The line ratios in these methods are:
R23 = I([O iii] λλ5007, 4959) + I([O iii] λ3727) I(Hβ) (2) O3N2 = I([O iii] λ5007)/I(Hβ) I([N ii] λ6584)/I(Hα) (3) N2O2 = I([N ii] λ6584) I([O ii] λ3727) (4) N2 = I([N ii] λ6584) I(Hα) .(5)
This school adopts in most cases emission line ratios that present little or no dependence with the dust attenuation (i.e., not far in wavelength), that uses the strongest available emission lines, and calibrators that present either a monotonic or even a linear dependence with the abundance. For example, the O3N2 and the N2 indicators (Alloin et al. 1979;Pettini & Pagel 2004;Stasińska et al. 2006;Marino et al. 2013). In some cases they use a combination of all the available emission lines, like the counterpart-method described by Pilyugin et al. (2012), or more complex combinations of non-linear equations to derive the abundances (e.g. Pilyugin et al. 2010; Pérez-Montero 2014). Since they adopted empirical correlations, the intrinsic dependences with other parameters, like u, are subsummed in the calibrator by construction.
A second school prefers to use photoionization models to derive the dependence of the abundance and ionization strength with the different line ratios (e.g. Dopita et al. 2000;Kewley et al. 2001). A certain model for the ionizing stellar population is assumed, taking into account a certain burst of star formation, an initial mass function, a certain metallicity and, in some cases, the age of the cluster. Under certain conditions for the ionized nebulae (e.g., geometry, electron density), it is possible to derive trends and correlations between the abundance and the line ratios considered. For this school, the prefered line ratios are those that depend only of one of the parameters, or for which the correction on the other is well understood. The procedure/prescriptions described by Kewley & Dopita (2002) or López- Sánchez et al. (2012) on how to derive the abundances is a good example of this approach.
The main difference between the two schools is that for the first one the abundances derived are systematically lower. Dopita et al. (2014) studied a scenario in which the introduction of a κ distribution for the electron temperatures in the nebula allows to reconcile both estimates of the oxygen abundance, based on the early studies by Binette et al. (2009). However, most of the followers of the first school still consider that the T e or direct method is more representative of the physical conditions in the nebulae, and requires fewer assumptions or dependences on still not well understood physical properties (like the amount of ionizing photons of young stars, that differs among the different stellar evolution models). Another criticism is that in many cases the calibrations based on photoionization models assume tight or fixed correlations between the abundance of different elements (like the N/O ratio), that may affect their derived values (e.g. Pérez-Montero 2014).
We adopted the indicator based on the O3N2 ratio described before, in order to compare with previous results on the same field. This line ratio involves the stronger emission lines in the wavelength range, as clearly appreciated in Fig. 3, and therefore minimize the errors due to the inaccuracies in the measurement of the involved line ratios. By construction, it presents a weak dependence on dust attenuation, already noticed by pre-vious authors (e.g. Kewley & Dopita 2002). We adopted the recently updated calibration by Marino et al. (2013), that uses the largest sample of H ii regions with abundances derived using the direct, T e -based, method (M13 hereafter). This calibration corrects the one proposed by Pettini & Pagel (2004), that (due to the lack of H ii regions in the upper abundance range) combined direct measurements for the lower abundance range and values derived from photoionization models for the more metal rich ones. As demonstrated by Marino et al. (2013), it produces abundance values very similar to the ones estimated based on the Pilyugin et al. (2012) method, with an accuracy better than ±0.08 dex.
The wavelength range covered by MUSE at the redshift of the galaxy does not include the [O ii]λ3727 emission line. Therefore, all indicators that include it, like R23, N2O2, or the combination of any of them, can not be used. However, there are other less common abundance indicators covered in this wavelength range, such as:
S23 = I([S iii] λλ9069, 9532) + I([S ii] λ6717, 6731) I(Hβ) (6) S3O3 = I([S iii] λλ9069) I([O iii] λ5007) (7) Ar3O3 = I([Ar iii] λλ7135) I([O iii] λ5007) .(8)
The S23 indicator was first proposed by Díaz & Pérez-Montero (2000) as an alternative to the more widely used R23. Its main advantages are that the intensity of the lines are less affected by dust attenuation, that it presents a monotonic linear dependence with the abundance for a wide range of metallicities, and that it seems to be less dependent on the ionization parameter. This later statement was questioned by Kewley & Dopita (2002) on the basis of photoionization models. Oey & Shields (2000) already noticed that this indicator has a bi-valued behaivour with respect to the abundance, similar to R23, and restricted the use of the calibrator proposed by Díaz & Pérez-Montero (2000) to subsolar metallicities (Z < 0.5Z · ). This corresponds to an oxygen abundance of 12+log(O/H)<8.3, lower than the lowest abundances derived for the H ii regions discussed here based on the M13 calibrator.
There is no published calibration of the dependence of S23 with the oxygen abundance for the higher abundance branch. However, based on the photoionization models presented by Oey & Shields (2000), it is possible to derive an estimation of the abundance for that range:
12 + log(O/H) = 8.6 − 0.25log(S23)(9)
The accuracy of this calibrator has to be tested extensively, but based on the range of values covered by the described models we estimate to be not better than 0.15 dex.
The S3O3 and Ar3O3 indicators were proposed by Stasińska (2006) (S06 hereafter). She derived a non-linear correlation for both indicators and the oxygen abundance, that we adopt here. She estimated the accuracy of this calibrator to be of the order of ∼0.09 dex.
In contrast with the O3N2 abundance indicator, that is basically independent of dust attenuation, all these indicators involve ratios of emission lines widely separated in wavelength, and therefore the line intensities must be corrected for dust attenuation prior to derive the corresponding ratio and abundance. This introduces a new degree of uncertainty that we avoid by adopting the O3N2 calibrator. Figure 5 shows the comparison among the different estimators for the oxygen abundance discussed. We consider only the emission lines and line ratios detected above a 5σ detection limit. Hence, each panel shows a different number of H ii regions, ranging between 210 (for the panels involving the ArO3 indicator), and 360 (for the panels involving the O3N2, S23 and S3O3 indicators). This is because the [Ar iii] emission line is fainter than any of the other (Fig.3). There is very good agreement between the different estimators, despite the different ions involved in most of the calibrators, and the inhomogenous derivation of the calibrators. The largest differences are found in the calibrator involving the [Ar iii] emission line, for the reason indicated before: σ(X O3N2 − X Ar3O3 ) = 0.08 and σ(X S3O3 − X Ar3O3 ) = 0.07 dex, where σ is the standard deviation of the difference between the two estimations of the abundance, and X is the oxygen abundance, i.e., 12+log(O/H). The smallest differences are found between the O3N2 and S23 indicators, with σ(X O3N2 − X S23 ) = 0.04 dex, a value smaller than the expected accuracies of both calibrators. In summary, this comparison shows that our estimation of the oxygen abundance does not depend strongly on the adopted indicator, and that in average the accuracy of our estimation is of the order or better than ∼0.05 dex. This systematic error has been included in the error budget of the abundances derived for each individual H ii region.
Structural parameters of the galaxy
We derive the mean position angle, ellipticity, and effective radius of the disk, by a surface brightness and morphological analysis performed on the MUSE data using a V-band image of NGC 6754 synthetized from the IFS cube. The procedure is extensively described in Sánchez et al. (2014b). In summary, an isophotal analysis is performed using the ellipse_isophot_seg.pl tool included in the HIIexplorer package 4 . Unlike other tools, like ellipse included in IRAF, this tool does not assume a priori a certain parametric shape for the isophotal distributions. The following procedures were followed for the V-band image: (i) the peak intensity emission within a certain distance of a user defined center of the galaxy was derived. Then, any region around a peak emission above a certain percentage of the galaxy intensity peak is masked, which effectively masks the brightest foreground stars; (ii) once the peak intensity is derived, the image is segmented in consecutive levels following a logarithmic scale from this peak value; (iii) once the image is segmented in n levels isophotal regions, for each of them, a set of structural parameters was derived, including the mean flux intensity and the corresponding standard deviation, the semi-major and semi-minor axis lengths, the ellipticity, the position-angle, and the barycenter coordinates. The median values of the derived position angles and ellipticities along the different isophotes, once excluded those affected by the seeing in the very central regions, are adopted as the position angle and ellipticity of the galaxy. Their standard deviations are considered as an estimation of the error in the derivation of these parameters. We derive an ellipticity of e =0.88±0.07 and a position angle of PA =77±9 • . Assuming an intrinsic ellipticity for the galaxy of ∼0.13 (Giovanelli et al. 1995(Giovanelli et al. , 1997, the inclination is estimated to be i =64±6 • . Finally, we fit the surface brightness profile with a single exponential function to derive the disk scale-length, and the correponding disk effective radius, as de-
Results
Oxygen abundance gradient
We deproject the position of each H ii region using the morphological parameters described in the previous section. Then, we derive the galactocentric radial distribution of the oxygen abundance for NGC 6754, based on the abundances measured for each individual H ii region. For the 396 H ii regions detected, figure 6 shows the abundance gradient derived out to ∼2 r e along the galactocentric distance normalized to the effective radius.
The shape of the abundance gradient shown in this figure is totally consistent with the pattern found in many previous studies. The gradient shows an almost linear decrease between ∼0.3 and ∼1.7 effective radius, with a drop in the central regions and a flattenning and/or up-turn in the outer regions. The linear regime has been interpreted as evidence of inside-out growth in spiral galaxies, with a metal enrichment dominated by local processes (e.g., Sánchez et al. 2014b, and references therein). The drop in the inner region is found in a fraction of the spiral galaxies. In some cases (e.g., NGC 628) it has been associated with a circumnuclear ring of star formation at the expected location of the inner Lindblad resonance radius, where the gas is indeed expected to accumulate, due to non-circular motions exerted by a bar or spiral arms Rosales-Ortega et al. 2011). The nature of the flattening in the outer regions, that has also been observed in other galaxies (e.g., Marino et al. 2012), is still under debate. It could be an effect of the radial migration of stars that latter pollute the surrounding gas, or a consequence of a change in the star-formation efficiency.
The dashed-dotted black line in Fig. 6 shows the errorweighted linear fit to this radial distribution of the abundance. Following Sánchez et al. (2014b), the analysis is restricted to galactocentric distances 0.3< r/r e <2.1. We find a slope in the abundance gradient of α=−0.10±0.02 dex/r e , which is similar to the common abundance slope reported by Sánchez et al. (2014b) of ∼−0.1 dex/r e . If we used the same calibrator than the one adopted in that study, i.e., Pettini & Pagel (2004), instead of Marino et al. (2013), the slope would be slightly larger, α=−0.14±0.02 dex/r e . In any case, both slopes are totally compatible with the common gradient, since they are both within 1σ of the range of values described for this characteristic slope (Sánchez et al. 2014b).
The number of H ii regions detected for this galaxy is large enough to explore whether the abundance gradient depends on other properties of the nebular emission. The H ii regions in Fig. 6 have been color-coded according to the value of the EW(Hα). Adopting this scheme it is possible to distinguish between the regions with stronger specific star-formation rate, that trace mostly the spiral arms (Fig. 1), and those distributed more homogeneously across the entire disk. We repeat the fitting procedure splitting the sample in two, for H ii regions with equivalent width greater or smaller than 20Å. We find that the H ii regions with lower equivalent widths present a somewhat shallower gradient (α=−0.09±0.01) than those with higher equivalent widths (α=−0.12±0.02). The two gradients are shown in Fig. 6. However, the difference is rather small, and it may not be significant. In order to test it, we perform a Kolmogorov-Smirnov (KS) test to estimate how different the distributions of oxygen abundances in both cases are. We find that the probability that both distributions were not derived from the same sam- ple is just a 7.7%. Therefore, there is no significant difference between the abundance gradients for the regions with stronger or fainter specific star formation rates. We explore possible differences in the abundance gradients based on other properties of the ionized nebulae. First, we considered the ionization parameter, splitting the sample in two subsamples with log(u) greater or smaller than −3.6 (the median value for our sample). The differences in the slopes were even smaller, being α highlog(u) = −0.10±0.02 dex and α lowlog(u) = −0.08±0.03 dex. Then, we considered the electron density, splitting the sample in regions with n e greater or smaller than 75 cm −3 (the median value for our sample). In this case we find the same slope for both subsamples (α highn e = −0.10±0.03 dex).
Finally, we explore if the different spatial resolutions of the east and west pointings have an impact in the abundance distributions and gradients. We repeat the analysis restricting our sample to the regions detected in both pointings separately before joining them into a single catalog. We find very similar slopes for both subsamples: α east = −0.10±0.02 dex and α west = −0.12±0.03 dex. Even more, a KS-test indicates that the probability that both distributions were not derived from the sample sample is 0.02%.
Mixing scale-length
This sample of H ii regions is large enough to derive a estimation of the mixing scale-length. For doing so we compute the dispersion of galactocentric distances with respect to the linear regression. The average of this relative distance is zero (by construction), and the standard deviation is the typical mixing scale, i.e., how far a certain H ii region has moved from its expected location based on a pure inside-out chemical enrichment without radial mixing. We find a mixing scale-length r mix =0.43r e , that corresponds to 4.6 kpc at the redshift of this galaxy. We repeated the estimation for the different sub-samples of H ii regions discussed before, and found similar dispersions, covering a range of values of r mix =0.37−0.53r e . In particular, when taking into account the east and west pointing separately we derive a very similar radial mixing scale-length, slightly lower than the common one (r mix,east/west =0.35r e ). This indicates that (i) the different spatial resolution does not affect the result, and (ii) there seems to be an azimuthal variation of the oxygen abundances that increases the dispersion when not taken into account. Finally, we study if there is a dependence with galactocentric distance. We found that the mixing scale-length is slightly lower in the inner regions r mix (r/re < 0.9) =0.28r e than in the outer ones r mix (r/re > 0.9) =0.67r e .
To know how sensitive this dispersion is to the errors and uncertainties in the derived parameter, we performed a simple Monte-Carlo simulation, allowing each of the parameters (abundances, galactocentric distances, effective radius and inclination) to vary within the estimated errors. The standard deviation between the different estimated radial mixing scales is ∼0.15r e . This parameter puts a strong constraint on the metal mixing scale-length, independently of the mechanism required to produce the mixing. Obviously this is an upper limit to the mixing scale-length, in particular if the abundance gradient depends on the equivalent width of Hα.
Azimuthal variations of the oxygen abundance
In a pure inside-out scenario where the metal enrichment is dominated by local processes (the metal pollution by stars that dies at a certain location), the abundance gradient should not present any azimuthal variation. Different mechanisms proposed for the radial mixing predict different characteristic patterns in the azimuthal distribution of the oxygen abundance. The sample of H ii regions provided by our MUSE data is large enough to explore if there is an azimuthal variation in the distribution of oxygen abundances. Figure 7 shows the azimuthal distribution of the oxygen abundances for the H ii regions of our catalog, once removed the common radial gradient, for those regions within a ring of 0.3< r/r e <2.1 (i.e., the linear regime of the abundance gradient). There is a clear pattern, more clear when we derive the azimuthal average within a box of 25 • around each value. Indeed, when removing this average pattern the dispersion around the mean value is reduced by a 40%. The amplitud of the pattern does not follow a clear periodic sequence (like a sinusoidal structure), and there is no clear general dependence with the galactocentric distance.
However, in the strongest feature of this pattern, the wiggle between θ ∼90 • and θ ∼160 • with an amplitude of ∼0.05 dex, there seems to be a trend with distance, with the abundance decreasing at intermediate distances and increasing for both the inner and outer regions. This pattern corresponds to the spiral arm the south-east of the center of the galaxy. If real, this could be a hint of radial mixing.
The radial mixing scale defined in the previous section is reduced to 0.39r/re, or 4.1kpc, when the average azimuthal variation of the oxygen abundance is subtracted prior to derive the dispersion around the radial abundance gradient.
Discussion and Conclusions
In this study we analyse one of the first observations using MUSE on a spiral galaxy, NGC 6754. We detect and extract the spectroscopic information of a sample of 396 H ii regions, an order of magnitude larger than the average number observed in previous state-of-the-art IFU survey datasets (e.g. Sánchez et al. 2014b). This illustrates the capabilities of this unique instrument due to the combination of its large FoV and unprecedent spatial sampling/resolution. The abundance distribution derived has a negative gradient, with a slope consistent with the characteristic value reported by previous studies (Sánchez et al. 2014b), in the linear regime between 0.3 and 1.7 r e . The abundance decreases in the inner regions, and there is a hint of a flatenning in the outer parts. Both features have been already observed in previous studies in individual galaxies (e.g., Bresolin et al. 2009;Yoachim et al. 2010;Rosales-Ortega et al. 2011;Marino et al. 2012;Bresolin et al. 2012). The central drop is in many cases associated with a circumnuclear star-formation process (Sánchez et al. 2014b), and it could be related with the accumulation of gas due to non-circular motions exerted near the inner Lindblad resonance radius (e.g., Cepa & Beckman 1990).
The nature of the flatenning is still not clear. A detailed discussion on the different scenarios proposed was presented by Sánchez et al. (2014b). In summary, under the usually observed star-formation rates, the time required to enrich the ISM up to the observed abundances in these outer regions is of the order of the age of the Universe (Bresolin et al. 2012), and therefore it is unlikely that in situ star-formation could have enriched the interstellar medium to the values observed. Among the main mechanisms proposed to explain the flatenning we highlight the following ones: (i) angular momemtum transport that produces a radial mixing (e.g., Lacey & Fall 1985b;Goetz & Koeppen 1992;Portinari & Chiosi 2000;Schönrich & Binney 2009;Spitoni & Matteucci 2011); (ii) resonance scattering with transient spiral density waves (Sellwood & Binney 2002a); (iii) the overlap of spiral and bar resonances (Minchev et al. 2011); (iv) stellar radial migration (e.g., Roškar et al. 2008b,a); and (v) minor mergers and captures of satellite galaxies (Quillen et al. 2009;Bird et al. 2012).
We explore the possible dependence of the slope of the abundance gradient with different properties of the ionized gas. Significant differences are not expected if the metal enrichment is dominated by the inside-out growth of the galaxy. However, local processes, like outflows or metal raining induced by enhanced star-formation associated with the spiral arms may modify the chemical distribution locally. Under this assumption it would be expected that denser H ii regions in spiral arms, with greater EW(Hα) , and ionization strengths present a different distribution of oxygen abundances that those H ii regions located in the inter-arm regions. Our results indicate that local processes do not seem to be relevant enough to modify the galactocentric abundance gradient in this particular galaxy.
We define a parameter to estimate the amount of the redistribution of metals within the galaxy that we call the mixing scale-length, r mix . This parameter is defined as the dispersion around the abundance gradient along the galactocentric distance, and can be derived as the ratio between the dispersion in the abundance and the slope of the correlation. We estimate the typical r mix =0.43±0.15 r e , i.e., ∼4.6 kpc at the redshift of the galaxy. To our knowledge, this is the first time that this parameter is defined in this way. However, it is possible to compare with previous results if both the dispersion in abundance and the slope of the radial gradient are provided. The most recent exploration of the abundance gradient over a large sample of galaxies was published by Sánchez et al. (2012bSánchez et al. ( , 2014b. They found that the common abundance gradient has a slope of α = −0.1 dex/r e , with a dispersion of ∼0.6 dex, for which an average mixing scale-length of r mix ∼0.6 r e is derived. This value is slightly larger than the one we find for NGC 6754. However, we should note here that the estimation derived from Sánchez et al. (2012bSánchez et al. ( , 2014b results is purely statistical, based on the average abundance gradient derived once the individual gradients for each galaxy are considered all together and normalized to the abundance at the effective radius. Measurements on individual galaxies have not been provided.
Bars have been proposed as an effective mechanism for radial migration (e.g., Athanassoula 1992;Sellwood & Binney 2002b). Hydrodynamical simulations have shown that bars induce angular momentum transfer via gravitational torques, that result in radial flows and mixing of both stars and gas (e.g., Athanassoula 1992). These radial motions can produce a mixing and homogenization of the gas, that leads to a flattening of any abundance gradient (e.g., Friedli 1998). Resonances between the bar and the spiral pattern speeds can shift the orbits of stars, mostly towards the outer regions (Minchev & Famaey 2010), a mechanism that also affects the gas. Another process that produces a similar effect is the coupling between the pattern speed of the spiral arms and the bar, that induces angular momentum transfer at the corotation radius (e.g., Sellwood & Binney 2002b). In a recent study Di Matteo et al. (2013) analysed the signatures of radial migration in barred galaxies on the basis of simulations. They found that the slope of the abundance gradient does not change significantly up to ∼1.5-2 r e (when the scalelength of their simulated disks are transformed to an effective radius), but a flatenning is predicted beyond these galactocentric distances. This pattern is very similar to the one observed in our galaxy. However, as we discussed before, the flatenning in the outer region seems to be present in spiral galaxies irrespective of the presence or absence of bars (Sánchez et al. 2014b).
Di Matteo et al. (2013) defined a parameter to quantify the amount of spatial redistribution of stars in a disk as the ratio between the maximum absolute variation of the metallicity with respect to the radial gradient compared to the slope of this gradient (δ [Fe/H] /∆ [Fe/H] ). With units of distance, this parameter is equivalent to our mixing scale-length (r mix ). They found that this parameter evolves with time and presents a weak radial dependence. At the peak of the radial migration it ranges between 1-1.5 kpc (0.17-0.26 r e ) for a galactocentric distance between 3 and 12 kpc (0.5-2 r e ). As time evolves it decreases, being slightly larger in the outer regions. Our derived r mix is larger, but certainly of the same order, than that predicted by Di Matteo et al. (2013). It also presents a weak radial dependence, that may indicate that the peak of the migration has already past. This is a good agreement considering that we are not comparing with ad hoc simulations specifically done to reproduce our galaxy.
Another prediction by simulations is that the radial mixing should not be homogeneous. These inhomogeneities are related to the way radial migration occurs in galaxies, following the arms pattern (e.g., Minchev et al. 2012): metal-rich stars which move to the outer disk are mostly from the region outside corotation (Brunetti et al. 2011), and migrate through spiral patterns to the outer parts of the disk. In other words, migration is not axisymmetric, but associated to the distribution of arms and bars. In Section 4.3 we explored the possible azimuthal variations of the oxygen abundance, once subtracted the radial dependence, and we found evidence of an asymmetrical distribution. The strongest feature is associated with the spiral arms in the south-east of the galaxy. The amplitude seems to be smaller than that predicted by Di Matteo et al. (2013), for the epoch of the strongest migration: ∼0.2 dex at ∼7 kpc (1 r e ), at t=1.1 Gyr in their simulations (Fig. 8, top panel, in that article). However, this effect is expected to become weaker with time ( Figure 8, bottom panel of that article), and our previous result indicates that this galaxy has already passed the peak of the strongest migration.
The proposed scenario assumes that the deviation of the abundances with respect to the radial gradient, due to the radial migration associated with arms and bars, should be stronger in barred galaxies. This is a consequence of the stronger radial movements expected to be induced by these morphological features. In this context it is interesting to note that recent results indicate that the stellar and gas kinematics of barred and unbarred galaxies seem to be very similar, without stronger distorsions induced by the presence of the bar, at least at large scales Barrera-Ballesteros et al. (2014).
Despite the fact of this significant advance in our understanding of the possible effects of radial mixing, it is important to highlight that all these results were derived for a single galaxy. Larger samples of galaxies are needed to explore whether the estimated mixing scale-lengths and azimuthal variations depend on other properties of the galaxies, such as the presence or absence of bars, the strength of the bars, the interaction stage, the morphological type, and the stellar mass or luminosity. In particular, if the picture outlined by Di Matteo et al. (2013) is valid, we should find different strengths in both r mix and the intensity of the azimuthal variations depending on the timing of the evolution of the bars and the coupling or not with the spiral arms. Another important caveat is that the results from the current simulations were focused on the effects of the migration on old stars, and in their metallicities. As clearly illustrated in recent results by González Delgado et al. (2014) the gas-phase abundance is better correlated with the metallicity of young stars (t <2 Gyr), with old stars being in general more metal poor. It is still unclear how these differences may affect the interpretation of our results on the basis of the simulations. However, there is a lack of similar simulations on the effects of radial migration for the gas-phase abundance.
The current results illustrate the capabilities of MUSE to accomplish this kind of studies in a very efficient way, and demonstrate that it is possible to derive reliable dispersions around the mean abundance gradient. In future articles we will apply the methodology outlined here to a sample of galaxies with similar characteristics of cosmological distances and projected sizes, observed with this instrument, in order to explore the dependence of the results on galaxy type, as outlined before.
Fig. 2 .
2Color coded map of the equivalent width of Hα in logarithmic scale. The areas with Hα density flux below 1.5 10 −18 erg s −1 cm −2 spaxel −1 (∼3σ detection limit) have been masked. The circles represent the detected H ii regions, width the radius proportional to the extraction aperture.
Fig. 3 .
3Detail of the spectrum of a typical H ii region extracted from the galaxy. The black line shows the input spectrum in logarithmic scale, together with the best fitted stellar model, in orange. The difference between the logarithms of the input spectrum and the model, shifted by 1, is shown as a blue line. The intensities of the emission lines are so high that the spectra are plotted in logarithmic scale to show them together with the detail of the underlying stellar population. The most prominent spectral features discussed along the article are marked. The spectral regions masked during the fitting of the underlying stellar population are not shown in the orange-solid line. The two boxes show an expanded view around Hβ and Hα respectively.
Fig. 5 .
5Comparison among the oxygen abundances derived using the different indicators described in the text: (i) S23 vs. O3N2 (top lefthand panel); (ii) S3O3 vs. O3N2 (top righthand panel); (iii) ArO3 vs. O3N2 (bottom lefthand panel); and (iv) ArO3 vs. S3O3 (bottom righthand panel). Each blue solid circle corresponds to an individual H ii region in the sample. The error bars represent half of the estimated error considering the propagation of the emission line errors and the uncertainty in the calibrator. The dashed-line corresponds in each panel to the one-to-one relation.
Fig. 6 .
6Radial distribution for the oxygen abundance derived using the O3N2 indicator for the 396 H ii regions detected in NGC 6754, color coded by the equivalent width of Hα. The blue colors and small symbols correspond to H ii regions with an emission EW(Hα) lower than 20Å, while the reddish-to-grey colors and larger symbols correspond to regions with higher values. The error-bars illustrate the individual errors of the abundance propagated from the estimated errors of the emission lines, not including the systematic error of the abundance calibrator. The dashed-dotted black line shows the result of the best linear regression to all the points. The blue and red solid lines correspond to the results of the best linear regressions to the H ii regions of lower and higher values of EW(Hα).
Fig. 7 .
7Azimuthal distribution of the residual of the oxygen abundance for the individual H ii regions once subtracted the average radial gradient. The colors and sizes indicate the galactocentric distance, with blue solid circles corresponding to H ii regions more nearer to the center of the galaxy, and red/orange ones corresponding to those farther away. The solid line corresponds to the average value at each azimuthal angle for H ii regions within 25 • of the considered angle.
shows the classical diagram using [O iii]/Hβ vs. [N ii]/Hα (Baldwin et al. 1981, BPT diagram hereafter), for the S. F. Sánchez et al.: Census of H ii regions in NGC 6754 derived with MUSE: Constraints on the metal mixing scale. log([NII]λ6583/Hα) log([OIII]λ5007/Hβ)Fig. 4. [O iii] λ5007/Hβ vs. [N ii] λ6583/Hα diagnostic diagram for the 396 H ii ionized regions detected in NGC 6754, color coded by the deprojected galactocentric distance (where bluer colors correspond to the central regions, and reddish-togrey colors correspond to the outer regions). Solid and dashed lines represent, respectively, the Kauffmann et al. (2003) and Kewley et al. (2001) demarcation curves. They are usually invoked to distinguish between classical star-forming objects (below the solid line), and AGN powered sources (above the dashed line). Regions between both lines are considered intermediate ones. The average error of the line ratios is represented by the error bar in the upper-right corner.-1
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http://www.caha.es/sanchez/HII_explorer/
http://www.caha.es/sanchez/HII_explorer/ fined bySánchez et al. (2014b). The effective radius derived at the distance of the galaxy was estimated as r e =10.7±0.9 kpc .
Acknowledgements. SFS thanks the director of CEFCA, M. Moles, for his sincere support.We thank the referee for his/her comments that have improved this manuscript.Based
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| []
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[
"TOWARDS HIGH PERFORMANCE COMPUTING (HPC) THROUGH PARALLEL PROGRAMMING PARADIGMS AND THEIR PRINCIPLES",
"TOWARDS HIGH PERFORMANCE COMPUTING (HPC) THROUGH PARALLEL PROGRAMMING PARADIGMS AND THEIR PRINCIPLES"
]
| [
"DrBrijender Kahanwal \nDepartment of Computer Science & Engineering\nGalaxy Global Group of Institutions\nDinarpur, AmbalaHaryanaIndia\n"
]
| [
"Department of Computer Science & Engineering\nGalaxy Global Group of Institutions\nDinarpur, AmbalaHaryanaIndia"
]
| [
"International Journal of Programming Languages and Applications ( IJPLA )"
]
| Nowadays, we are to find out solutions to huge computing problems very rapidly. It brings the idea of parallel computing in which several machines or processors work cooperatively for computational tasks. In the past decades, there are a lot of variations in perceiving the importance of parallelism in computing machines. And it is observed that the parallel computing is a superior solution to many of the computing limitations like speed and density; non-recurring and high cost; and power consumption and heat dissipation etc. The commercial multiprocessors have emerged with lower prices than the mainframe machines and supercomputers machines. In this article the high performance computing (HPC) through parallel programming paradigms (PPPs) are discussed with their constructs and design approaches. | 10.5121/ijpla.2014.4104 | [
"https://arxiv.org/pdf/1402.1287v1.pdf"
]
| 8,232,282 | 1402.1287 | e5a1496fa203fbb32e11ed19dff209cf2f61bb25 |
TOWARDS HIGH PERFORMANCE COMPUTING (HPC) THROUGH PARALLEL PROGRAMMING PARADIGMS AND THEIR PRINCIPLES
January 2014
DrBrijender Kahanwal
Department of Computer Science & Engineering
Galaxy Global Group of Institutions
Dinarpur, AmbalaHaryanaIndia
TOWARDS HIGH PERFORMANCE COMPUTING (HPC) THROUGH PARALLEL PROGRAMMING PARADIGMS AND THEIR PRINCIPLES
International Journal of Programming Languages and Applications ( IJPLA )
41January 201410.5121/ijpla.2014.4104Parallel programming languagesparallel programming constructsdistributed computinghigh performance computing
Nowadays, we are to find out solutions to huge computing problems very rapidly. It brings the idea of parallel computing in which several machines or processors work cooperatively for computational tasks. In the past decades, there are a lot of variations in perceiving the importance of parallelism in computing machines. And it is observed that the parallel computing is a superior solution to many of the computing limitations like speed and density; non-recurring and high cost; and power consumption and heat dissipation etc. The commercial multiprocessors have emerged with lower prices than the mainframe machines and supercomputers machines. In this article the high performance computing (HPC) through parallel programming paradigms (PPPs) are discussed with their constructs and design approaches.
INTRODUCTION
The numerous computational concentrated tasks of the computer science like weather forecast, climate research, the exploration of oil and gas, molecular modelling, quantum mechanics, and physical simulations are performed by the supercomputers as well as mainframe computer. But these days due to the advancements in the technology multiprocessors systems or multi-core processor systems are going to be resembled to perform such type of computations which are performed by the supercomputing machines.
Due to the recent advances in the hardware technologies, we are leaving the von Neumann computation model and adopting the distributed computing models which have peer-to-peer (P2P), cluster, cloud, grid, and jungle computing models in it [1]. All these models are used to achieve the parallelism and are high performance computing (HPC) models.
Concurrency and Parallelism:
The terms concurrency and parallelism must be clear in our minds first. It can be well explained with the help of the threads (light weight processes). When two or more threads are in the middle of execution process at the same time, actually, they may or may not be executing at the same time, but they are in the middle of it.
It is called concurrency [25]. The threads may or may not execute at the single processor or a multiprocessor machine in it. When two or more threads are actually running at the same time on different CPUs, it is known as parallelism [25]. To achieve parallelism, we always require at least two CPUs which may be on a single machine (multiprocessor machine) or more machines. The parallel events may also be called as concurrent events, but the reverse is not true always. It is well described with the help of set theory that parallelism ⊂ concurrency (Parallelism is contained in Concurrency) as shown in Fig.1.
Figure 1: Concurrency is the superset of Parallelism
On the Von Neumann computing machines, the programming is a single execution sequence. But there might be various subroutines that can be executed simultaneously within a single program. These are called sequential due to the execution of subroutines proceeds in predetermined sequence. In general cases, the programs are termed as concurrent or parallel in which the subroutines can be executed concurrently and these subroutines are known as tasks [2]. Now a day, it is a common practice now to execute various programs concurrently by the computing machines. It may have the architecture with multiprocessors (various CPUs) which share the common memory space as shown in the Fig.2 (a) or another architecture that may have multiprocessors with their independent memories or distributed memories as shown in the Fig.2 It is the big challenge for the scientists to utilize these hardware technologies efficiently, effectively, and these processors may work cooperatively. In the present scenario, software technologies (STs) are not having well compatibility with the hardware technology's growth that the STs can utilize them efficiently and effectively. Hence the parallel computing community is going to be aware that they can build software technologies which are efficient and effective. But till now the programmers as well as the scientists are incompetent to find the solution. So it is the need to get more awareness regarding the parallelism, so we can find the better solutions for high performance computing (HPC). The article contains more sections which are organized as follows: the related works will be available in the section 2.
(a) (b)
RELATED WORKS
The parallelism is not a novel concept of computing. The Law of Amdahl is the key principle to estimate maximum improvements in the components of the system [3] which brings the idea of parallel computing to find the optimum performance. The time 1960-70 was the boom time for the parallel computing and during this time, we have solved many problems in achieving the optimum performance, but they have encountered today. Multi-core chips are a new paradigm in parallelism. Parallel computation is the never-ending desire for much faster and much cheaper computation of level of supercomputers as well as mainframe computers [4]. Until now we have not remarkable progress in building the efficient and optimal softwares for utilizing the parallel computer architectures of today [5].
PARALLEL PROGRAMMING CONSTRUCTS OR PRINCIPLES
It is complicated to write the parallel programs as compared to write sequential programs. We design algorithms and express them in some programming languages to execute on the computing machines. In the case of parallel programming we have to develop the same functioning, but it also adds more challenges to it. Such types of challenges are as follows: structured constructs [6]: structured region; thread based constructs [7]: synchronization, critical sections, and deadlock; and object-oriented constructs [8]: object replication, latency hiding, termination detection, and user-level scheduling; concurrency; data distribution; inter-process communication; computational load balancing; variable definitions [2]; parallel compositions [2]; program structures [2]; and easy implementation and debugging;. All of these are explored in the following sub sections.
Structured Construct -Structured Region
The structured parallel programming construct is introduced as a structured region. It has a region name and a region body which is enclosed with two barriers namely entry barrier as well as exit barrier. A par (or parfor) block has the starting instruction as the region name. A region name has a region keyword then an arbitrary name given to the region by the programmer and a list of the participants (processes) [6]. If the participants list is declared explicitly then the specified names becomes the participants and if the list is not mentioned then all the processes are the participants.
This structured region semantics is very easy and clear. The entry in the region is done only if all the participants reach at their specific entry points. There are the unique effects of the execution of the region body. The region is exited by all the participants after the complete execution of all the operations of the region body. A structured region has a single entry and exit point [6]. It wraps up inter-process communication as well as synchronization operations in it and makes the parallel programming easier to understand and less error-prone. It is opposite to the concept of mutual exclusion in which only a single process can enter in the critical region. But in the case hare all the processes can enter in the structured region.
Thread-Based Constructs
The process and the thread are much related terms. A process is program in execution and there may be some independent units within a process which are known as the threads. A thread is dispatch-able work unit. It is also known as the light-weight process. So it is concluded that the threads makes a process or it is a subset of a process [10]. Both the process as well as the thread is an active entity and a simple program before execution is a passive entity. The threads of a single process share same address space, so the context-switching as well as communication between threads is inexpensive [10]. The sharing between the threads creates some difficulties which are explored in the following sub-sections.
1. Synchronization: It is the construct which enforces the mechanism for controlling the execution order of the threads and resolves the conflicts among the threads [7]. It is a way of coordinating the execution of the threads and managing the shared address space. In synchronization, mutual exclusion and condition synchronization operations are used widely. In mutual exclusion, one of the threads block the critical section (shared data area by the threads) and other threads will wait for getting their turns one by one. The scheduler controls for the turns. But in the case of conditional synchronization, threads are blocked until some particular condition is satisfied. Here the thread has to wait until a particular condition is achieved. So the synchronization is well managed by the programmer or by the programming system, it is a critical construct for multi-threaded programming. 2. Critical sections: These sections have shared dependency variables and many threads are dependent on them [7]. It is the great programming construct for thread-based programming, so the threads can use these sections mutually exclusively and prevent to use these sections simultaneously. These sections should be minimized in size. 3. Deadlock: It is the situation when a thread holds a lock and waiting for another lock which is held by another thread and this thread is waiting for the lock first to be released. Such as the code: T1: lock (1); lock (2); and T2: lock (2); lock (1); in this code, the deadlock may or may not occur. The four basic conditions need to be hold which are mutual exclusion; hold-and-wait; no pre-emption; and circular wait.
Object-Oriented Constructs
The object-oriented parallel programming has complex computational as well as communicational structures to achieve the efficiency or optimization. For improving the performance in the object-oriented programming languages some of the constructs are discussed in the following sub-sections [8].
1. Object Replication: This construct highly improves the performance in the distributed memory architectures. When a program is frequently accessing an object then it is better to create a local replica of it for the processor and then there is a big fall in the number of remote messages [8]. 2. Latency Hiding: It is an optimization technique which reduces the waiting time for the remote messages. Here local computations and remote communications are overlapped. In it we break up a single thread into multiple threads manually by modifying the program [8]. 3. Termination Detection: In few parallel applications like search problems, it is a typical task to detect the termination point because of the invoking of the many threads and finding their termination points in absence of the global control on them [8]. 4. User-level Scheduling: A proper scheduling at application level also improves the performance of the parallelism. User-level scheduling facility is not offered in most of the programming language systems, so it becomes necessary to provide it by the programmers explicitly to control the order of execution [8].
Concurrency
These days, the processors are inexpensive as compared to the previous time, so we are constructing the distributed systems. Due to such things in the development the concurrency has a little importance. The programmers are working on various types of applications like DBMS, L-S parallel technical computations, real time applications, and embedded systems etc [9]. When a concurrent program shares one or more processors during execution is known as multiprogramming; when its sub-processes are to be executed on independent processor then it is known as multiprocessing; when there is the addition of the communication network then it is known as distributed processing; and any combination of these is known as hybrid approach [9]. Still, it is the fundamental construct to utilize optimally the parallel computing resources. We can't achieve the parallelism without dividing the operations to execute concurrently. A problem has many sub-problems in it for concurrent execution; there is a need to differentiate the concurrent tasks within the main problem. It is the dexterity of the programmers.
Data Distribution
It is a big challenge to distribute the data which creates problem. In the parallelism there are so many processors which are working cooperatively. Now a day, the principle of locality is important for the better performance of the systems. But in the case of parallelism it becomes the problem or a decision making event which data to be localized for the particular processor. It is due to the concept of independent cache memories for each processor in the shared memory systems. For the parallel programmers, it becomes issue to manage it carefully. The performance of the system increases as we store more data in the caches because the processor can access it quickly as compared to the shared memory area.
Inter-process Communication
When we are going to execute a process on two or more processors, it becomes necessary to make communication among them for transferring data from one processor's cache memory to another processor's cache memory. So here is the need of maintaining caches of the processors with the mechanism called cache coherence that may be implemented via hardware or cache coherence protocols. Another case may be that the processors may have distributed memories and all the processors need to be communicated properly. There may be the need of explicit calls to a library which require transferring values among processors. There may be the communication overheads which must be minimized to get the advantage of the parallelism.
Computational Load Balancing
In the parallelism, there are two or more processors or separate machines which are connected through the network, to take the advantage of the parallelism all the processors or machines must be utilized properly and equally. The total computation must be equally distributed among the processors or machines for getting the benefits of high performance computing.
Variable Definitions
Two types of variables may be used in the programming languages namely mutable and definitional. The mutable variables are the normal variables which are used in the sequential programming languages. The assignment may be done to the variables and that may change during the program execution. The definitional variables are those variables in which we can assign values only once and they can be accessed by any number of tasks. In such variables there is no need to maintain synchronization.
Parallel Compositions
In the execution process, the statements are executed one after another and they also have additional sequential as well as conditional statements in the sequential programming language. To achieve the parallelism, the parallel statements must be added which becomes the additional threads of control to start the execution.
Program Structures
There may be two types of parallel program execution models. Firstly, transformational in which the main task is to transform the input data into the correct output value. Secondly, reactive or responsive in which the programs works regarding the events which are the external one.
Ease of Programming and Debugging
This is the issue for every type of programming language. The parallel programs must be easily implemented by the programmers. They do not require thinking more about the parallelism. The parallelism should be tackled by the programming language platforms. It is common to be bugs in the program implementation and there are so many side-effects of these bugs. So these may be removed easily with the help of good debugging tools.
PARALLEL PROGRAMMING APPROACHES
There are basically three approaches to program high performance computers (parallel computers). These are as follows:
Implicit Parallelism
It is also known as automatic parallelism. This approach is headache free for the programmer's point of view; here the complete working is done by the compilers to make parallel all of the executions [11]. All the parallel language constructs are inherently implemented by the language platform. Such type of job is always done in the pure functional programming languages. With the help of this approach the existing code is utilized on parallel systems. No changing is required in the existing code. It saves the development costs. And it is attractive for the vendors of the high performance computing.
Such type of parallelism has its own advantages as well as drawbacks. The advantages are as follows: Firstly, programmer's attention is completely on the algorithms. Secondly, we require very less code for programming. Thirdly, the productivity of the programmers increases as he/she does not care about the parallel programming constructs. Fourthly, the definitions of the algorithms are separated from the parallel executions. Fifthly, the legacy systems are utilized properly and which is the concept of re-usability.
The drawbacks are as follows: Firstly, the complete parallelism is not achieved because the programmers have much more information of the parallel potential (not efficient). Secondly, the programmers have not the exact control over the parallelism. Thirdly, there is no optimum parallel efficiency achieved. Fourthly, the algorithms which are already implemented may be executed with a low configuration system (architecture + memory) a few or more decade ago. But the recent configurations are very high with more storage capacity and processor speeds. Fifthly, it is a tough task for the scientists and researchers to design the parallel compilers.
Explicit Parallelism
In it the existing programming languages are utilized. The proper extensions are made to them to achieve all the parallel programming constructs [11]. Here the parallel programming principles are defined explicitly by the programmers. Explicit threading is a sub-approach of explicit parallelism in which the programmers creates parallel threads explicitly [22,23].The explicit parallelism also has its own advantages and disadvantages.
The advantages are as follows: Firstly, the programmers are already trained in the existing language. Secondly, it is totally under the understanding and control of the programmers. The disadvantages are as follows: Firstly, it is very hard to debug and difficult to program for the programmers because everything is dependent on the creativity and thinking the programmers. Secondly, there is no standardization because there are so many extensions have been made by the developers with same functionality with different look.
Hybrid Parallelism
It is the mixed up approach which combines the features of implicit as well as explicit parallelism. It will take the advantages of both the above mentioned technique.
It is summarized that the language designers may design completely the new programming language paradigms which have all the parallel programming principles or constructs in it.
PARALLEL PROGRAMMING PARADIGMS
There are too many paradigms available for utilizing the current machines with parallel architectures. Some of the parallel programming languages are as follows:
Message Passing Interface (MPI)
It is a specification for message passing. It is de facto standard for the development of high performance computing applications for the distributed systems (heterogeneous networks) as well as parallel computers and clusters [12]. It has bindings for C, C++ and FORTRAN programming languages. It is highly portable environment. The workload partitioning as well as the work mapping are done explicitly be the programmers like Pthread [25] and UPC. All the communications between the processes take place with the help of message passing paradigm.
In it one process sends the data to another process through message passing.
Fortress
It is also a thread-based specification programming language to design the HPC applications [12]. The work management, workload portioning, as well as work mapping may be done implicitly by the compiler as well as explicitly by the programmers. All for loops are parallel by default as implicit approach. The synchronization principles like reductions as well as atomic expression are specified by the programmers when there is a data competition in a program.
POSIX Threads (Pthreads) Programming
It is actually a set of C language types as well as procedure calls and all these are maintained or defined in a library named as pthread.h [12]. It is the duty of the programmers to maintain the shared data among the threads for avoiding the deadlocks and data races [25]. The pthread's create function has four parameters the task run thread, attribute, tasks to run in routine call, and routine argument. All has been closed with the help of pthread's exit function call. The workload partitioning and work mapping is done explicitly by the programmers.
OpenMP
It is also thread based open specification for shared memory architectures. It provides compiler directives, callable runtime library, and environment variables which extends the existing programming languages C, C++, and FORTRAN. It is portable platform [12]. The worker management is done impliedly and a little programmer's effort is required for the workload partitioning and task mappings, they are also performed implicitly. Programmers are required to tell the parallel region with the help of the compiler directives. The synchronization is also maintained implicitly by the OpenMP.
CILK (pronunciation as 'silk')
It is a multi-threaded programming language. It is appropriate for the recent multi-core CPU architectures. It is based on the traditional programming language C. Cilk a true parallel extension to C semantically with good performance [13]. In 1994, it was designed by the MIT scientists. In it the work-stealing scheduler is efficiently utilized. A Cilk program is a collection of Cilk procedures and every procedure has a sequence of threads. Every thread is non-blocking C language function which can run independently without waiting or suspension.
OpenMPI
It is the programming tool which is specially designed for the poor scientific programmers for achieving simple and routine parallelism. It is based on the existing programming tool OpenMP [14]. It provides the sufficient directives for achieving the parallelism. All the directives are followed by the notation directive pragma ompi. The few of the directives are distvar (dim=dimension, sleeve=s size) for the distributed array on parallel processes; global for declaring the variable as global variable; for (reduction (operator: variable)) to parallelize the for loop; syn sleeve (var=variable list) for exchanging the sleeve data of the distributed array for correctness ; sync var (var=variable list, master=node id) for synchronizing the global variable by coping the master data to others; and single (master=node id) for executing the next block by one process only as a delegate for other processes.
JAVA
It is the most popular programming language these days because we can create common applications on it and it also supports the parallelism through its multi-threading concept. It uses the Just in time (JIT) compiler and automatic garbage collection to perform the critical task [15].
For transparent communication between Java Virtual Machines, it has Remote Method Invocation (RMI). It is utilized to develop high performance computing applications.
High Performance FORTRAN (HPF)
Its name conveys that it is the extension of Fortran 90. It supports the parallel programming principles [16]. It supports the data parallel programming pattern in which one program has the complete control for the distribution of data among all the processors. It works in the distributed memory environment. It is a portable programming language.
Z-level Programming Language (ZPL)
It is a language with parallelized compiler. It is particularly for the high performance computations such as the scientific as well as engineering. It abstracts the Flynn's MIMD (Multiple Instructions and Multiple Data) parallel architecture [17]. The applications developed in this language are portable and the performance is independent of the compiler as well as the machine. It is a good programming language, but the scientists and the engineers have not shown much interest in it.
Erlang
It is a functional programming language. Firstly, it was introduced by the telecommunication giant Ericsson to build the telecommunication switches. Lately in 1998, it becomes open source software [18]. Concurrency is achieved through threads. The applications developed in this language are highly available as well as reliable. In this programming paradigm the explicit threading parallel mechanism is utilized in which the programmers create the explicit threads to achieve the parallelism [23].
Unified Parallel C (UPC)
It supports both types of architectures shared memory as well as distributed memory. It is based on the partitioned memory principle [12]. The complete memory is partitioned into many small memory areas for every thread. Every thread has a private memory as well as global memory which are shared among the same class of threads. A new principle is used to get high performance that is thread affinity in which memory access performance among the threads of same class is optimized [19]. Workload management in it is implied and the work partitioning as well as workers mapping may be implied or programmer controlled. The thread communication is maintained with the help of pointers. Three types of pointers are utilized here which are as follows: (i) the private pointers who works on their own address spaces, (ii) sharing pointer who works on the shared memory area, and (iii) sharing pointers to share, these are the sharing pointers who works on the other shared memory. So many synchronization mechanisms are utilized in this language like barrier, split phase barriers, fence, locks, and memory consistency control. It resemble with the MPI platform in workload partitioning and worker mapping.
Streams and Iteration in a Single Assignment Language (SISAL)
It is a functional programming language. It offers automatic parallelism through its functional semantics. In it user-defined names are identifiers rather than variables. These identifiers are known as values rather than the memory locations [20]. These values are dynamic entities. The identifiers are defined and bound to the values only for the time of execution. The Sisal Compiler is optimizing one which converts the source program into the object code with the execution time system parts needed to automatic managing of memory, tasks, and input/output. The parallelism can be controlled by the users also. In conclusion the programming language has an optimizing compiler with better runtime performance.
Laboratory Virtual Instrumentation Engineering Workbench (LabVIEW)
It is visual programming language from National Instruments. It is a platform as well as development environment. It is also a data-flow programming language in which the execution is decided with the help of structure of a graphical block diagram by drawing the wires for the function nodes [21].These wires propagate the variables and the nodes starts to execute as soon as the input data is available. This programming language is basically utilized for the acquiring data and processing signals, instrument control, automating test and validation systems, and monitoring and controlling embedded systems. It may be run on a number of platforms like MS Windows, UNIX, Linux, and Mac OS X. Multiprocessing as well as multi-threaded hardware are automatically utilized by its inbuilt schedulers. We can also create distributed applications on this platform. Hence it is a good high performance computing technology. The nonprogrammers can also develop the good applications by dragging and dropping the virtual representations of the laboratory equipments to whom they are well-known.
Manticore Programming language
It is a new functional parallel programming language. It is a heterogeneous programming language that provides the parallelism at multiple levels. It provides coarse-grained, explicit parallelism based on Concurrent ML platform. It supports the explicit concurrency with finegrain and implicit threads [22]. The synchronization is provided with first class synchronization message passing which well fits to the nature of the functional programming paradigms [23]. The locally-concurrent/globally-sequential garbage collector is implemented.
CONCLUSIONS
In this construct, a little survey of parallel programming languages, their design approaches, and their constructs are presented. The current scenario is totally towards the parallelism to achieve the high performance computing (HPC) and the developers must be aware about the new concepts of the technology. And this article is good food for the novice parallel programming lovers who want to do much more in this field.
Figure 2 (
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Figure 2 (
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| []
|
[
"Cascade LSTM Based Visual-Inertial Navigation for Magnetic Levitation Haptic Interaction",
"Cascade LSTM Based Visual-Inertial Navigation for Magnetic Levitation Haptic Interaction"
]
| [
"Qianqian Tong ",
"Xiaosa Li ",
"Kai Lin ",
"Caizi Li ",
"Weixin Si ",
"Zhiyong Yuan "
]
| []
| []
| Haptic feedback is crucial to immersive experience in virtual and augmented reality applications. The existing promising magnetic levitation (maglev) haptic devices have advantages of none mechanical friction and low inertia. However, their performance is limited by the navigation approach, which mainly results from the challenge that it is difficult to obtain high precision, high frequency and good stability with lightweight design at the same time. In this study, we reformulate visualinertial navigation as a regression problem, and adopt deep learning to perform fusion navigation for maglev haptic interaction. A cascade LSTM based θ-increment learning method is first proposed to progressively learn the increments of target variables. Two cascade LSTM networks are then constructed to respectively estimate the increments of position and orientation which are pipelined to accomplish visual-inertial fusion navigation. Additionally, we set up a maglev haptic platform as the system testbed. Experimental results show that our cascade LSTM based visual-inertial fusion navigation approach can reach 200Hz while maintaining high-precision (the mean absolute error of the position and orientation is less than 1mm and 0.02 • , respectively) navigation for a maglev haptic interactive deformation application.Index Terms-Visual-inertial navigation; Cascade LSTM network; θ-increment learning; Maglev haptic interaction. | 10.1109/mnet.2019.1800371 | [
"https://arxiv.org/pdf/1901.09224v1.pdf"
]
| 59,316,475 | 1901.09224 | 2be2626e8bd73b936b18a1930dc1adb75df808f0 |
Cascade LSTM Based Visual-Inertial Navigation for Magnetic Levitation Haptic Interaction
Qianqian Tong
Xiaosa Li
Kai Lin
Caizi Li
Weixin Si
Zhiyong Yuan
Cascade LSTM Based Visual-Inertial Navigation for Magnetic Levitation Haptic Interaction
1
Haptic feedback is crucial to immersive experience in virtual and augmented reality applications. The existing promising magnetic levitation (maglev) haptic devices have advantages of none mechanical friction and low inertia. However, their performance is limited by the navigation approach, which mainly results from the challenge that it is difficult to obtain high precision, high frequency and good stability with lightweight design at the same time. In this study, we reformulate visualinertial navigation as a regression problem, and adopt deep learning to perform fusion navigation for maglev haptic interaction. A cascade LSTM based θ-increment learning method is first proposed to progressively learn the increments of target variables. Two cascade LSTM networks are then constructed to respectively estimate the increments of position and orientation which are pipelined to accomplish visual-inertial fusion navigation. Additionally, we set up a maglev haptic platform as the system testbed. Experimental results show that our cascade LSTM based visual-inertial fusion navigation approach can reach 200Hz while maintaining high-precision (the mean absolute error of the position and orientation is less than 1mm and 0.02 • , respectively) navigation for a maglev haptic interactive deformation application.Index Terms-Visual-inertial navigation; Cascade LSTM network; θ-increment learning; Maglev haptic interaction.
I. INTRODUCTION
The recent development of virtual reality (VR) and augmented reality (AR) has facilitated the advancement of related applications, such as surgical procedures, teaching-learning system, marketing research, and interactive recreation [1], [2]. In these applications, haptic sensation is an essential component of users' immersive interaction experience. Berkelman et al. [3], [4] developed a maglev haptic interface which provided haptic feedback via a penhandle or fingertip probe. Besides, a novel maglev haptic device with an adjustable coil configuration was deployed, and it can provide haptic feedback in a natural manner [5], [6]. For these maglev haptic devices, the position and orientation of their magnetic stylus/probe are firstly obtained to navigate users' interaction actions. Highprecision and high-speed navigation helps to capture subtle changes in users' actions. Conversely, if users' actions cannot be acquired accurately and quickly, the haptic experience will be distorted. Therefore, the navigation performance is crucial to providing immersive haptic feedback.
In the study of Berkelman et al. [3], [4], an Optotrak Certus 6 degrees-of-freedom (DOF) optical motion tracker (Northern Q. Tong, X. Li, C. Li, and Z. Yuan are with Wuhan University, China. Email: [email protected] K. Lin is with Dalian University of Technology, China W. Si is with Guangdong Provincial Key Laboratory of Machine Vision and Virtual Reality Technology, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China. Email: [email protected] Digital Inc.) provided real-time position and orientation feedback for their maglev haptic platform. Infrared LEDs with no wired connection were mounted on the back end of their user's probe. However, the infrared LEDs will be sheltered by each other when the probe is tilted at a large angle, leading to the loss of location information. Besides, the design of their tracking module is somewhat cumbersome because of the additional mass and bulk of the battery and electronics required in wireless mode.
Tong et al. [5], [6] designed a magnetic stylus consisting of several small rods and red markers were embedded in connections between these small rods. A visual module with two RGB cameras was utilized to track red markers in the magnetic stylus for obtaining user's interaction actions. Although this visual module has advantages of high precision, light weight and low cost, the positioning frequency is limited by cameras' low acquisition frequency while maintaining high precision, which will affect the resolution of haptic perception. Besides, there also exists occlusion problem in this visual module.
From the above observations, three challenges should be addressed for existing navigation methods to accomplish high quality navigation for maglev haptic interaction. Firstly, how to maintain high positioning frequency while providing high precision. Secondly, how to improve the stability and robustness when the occlusion problem occurs, i.e., how to tackle the possible occlusion problem when the probe is tilted at a large angle or when users operate several probes in the operation workspace at the same time. Thirdly, how to design a lightweight and cost-effective navigation module while addressing the above two challenges.
In this work, we resort to the fusion navigation scheme which is capable of taking advantages of different navigation methods to overcome the aforementioned challenges. Considering that inertial navigation has advantages of high sampling frequency and good stability, and these characteristics are complementary to the high precision of visual navigation, we adopt inertial measurement units (IMUs) to aid the visual module thus to advance the navigation performance for maglev haptic interaction.
Several researchers have explored many kinds of visualinertial (VI) navigation methods [7], [8], [9], [10]. These methods can be tightly-coupled or loosely-coupled according to the condition whether image features are part of the state vector. Although tightly-coupled fusion methods can provide longterm, high-precision navigation, they usually involve filter update based on a certain constraint or an optimization problem, leading to low positioning frequency. Loosely-coupled methods maintain the integrity of visual module and IMUs, which is convenient for their independent optimization. Inspired by the recent success of deep learning techniques [11], [12], especially the great advancement of the long short-term memory (LSTM) architecture for recurrent neural networks (RNNs) [13], we regard visual-inertial navigation as a regression problem and employ a deep learning approach to perform visual-inertial navigation. In our work, visual and inertial modules are calibrated with reference to the fusion navigation coordinate system, respectively. To accomplish high-speed and robust navigation, we present a cascade LSTM based θ-increment learning method and construct two cascade LSTM networks to estimate increments of position and orientation, respectively. The estimated increments are then pipelined to calculate the position and orientation of the moving object. Finally, this location information is used for navigating maglev haptic interaction applications.
The main contributions of this paper are as follows:
• We reformulate visual-inertial navigation as a regression problem and propose a novel visual-inertial fusion navigation approach based on deep learning for maglev haptic interaction. This approach is excellent in real-time performance while maintaining high precision.
• We present cascade LSTM based θ-increment learning for accomplishing visual-inertial navigation, and two cascade LSTM networks are constructed to estimate increments of position and orientation. The accuracy of our cascade LSTM based navigation approach is verified by experimental results.
• Our cascade LSTM based visual-inertial navigation approach is lightweight, cost-effective, as well as robust to the occlusion problem. Furthermore, it can be extended to other applications other than being utilized in the maglev haptic interaction application mentioned in this work.
The remainder of this article is organized as follows: In Section II, the problem statement and system architecture are introduced. The proposed cascade LSTM based visual-inertial navigation scheme is given in Section III. In Section IV, the system testbed and experimental results are presented. Finally, conclusions are given in Section V.
II. PROBLEM STATEMENT AND SYSTEM OVERVIEW
In this section, we first introduce the problem statement about the navigation method in maglev haptic interaction applications. Then, we provide the system overview, as shown in Fig. 1.
A. Problem statement
Visual navigation is usually used to capture users' actions in maglev haptic interaction applications [3], [4], [6] because of its high-precision. However, its output frequency of the position and orientation is low and unstable due to its low sampling frequency and environmental conditions of cameras. Besides, the navigation performance will be affected if markers are out of cameras' field of view.
Inertial navigation system (INS) is all-weather, and it can work in various environments and export in-motion data at a high frequency. To obtain the position and orientation of the moving object by INS, inertial navigation coordinates should be selected firstly. IMU including an accelerometer and a gyroscope is connected to the moving object to gather a raw acceleration a and an angular rate ω in the inertial navigation coordinates. The orientation can be determined by transforming the quaternion updated by Runge-Kutta Act method particularly to orientation angles (pitch, roll, yaw). The raw acceleration a should be converted to the motion acceleration a n in the inertial navigation coordinates before the integration for the position. The position can be simply obtained by the integration of acceleration. Note that the integration operation is usually done in the frequency domain through the fast Fourier transformation to reduce the error caused by biases and high-frequency noises. Though some measures are token to decrease the influence of biases and noises, computational errors still can not be eliminated completely. What's worse, the error will accumulate over time.
Considering that characteristics of visual and inertial navigation are complementary, cameras and IMUs are usually fused to acquire state estimations. Weiss et al. [7] coupled the visual framework and IMU loosely. They treated the visual framework as a black box, and showed how to detect failures and estimated drifts in it. Mourikis et al. [8] put forward a multistate constraint Kalman Filter (MSCKF) algorithm, which performed an Extended Kalman Filter (EKF) update based on geometric constraints. Apart from filter-based methods, there are also optimization-based methods such as keyframe-based visual-inertial SLAM (OKVIS) using nonlinear optimization proposed by Leutenegger et al. [9]. Besides, VINS-Mono presented by Qin et al. [10] is a nonlinear-optimization-based sliding window estimator using pre-integrated IMU factors.
The aforementioned visual-inertial navigation methods have been applied to state estimation problems in a variety of fields, such as autonomous vehicles and flying robots [14]. However, these methods still have many drawbacks. Specifically, the inertial and visual processing frequency of the method in [7] are only 75Hz and 25Hz, respectively. Although MSCKF [8] is robust and memory-efficient, its per-frame processing time is also long and its accuracy is low. Moreover, the accuracy of OKVIS [9] and VINS-Mono [10] is relatively high, but this achievement greatly sacrifices computational resources, leading low processing frequency. What's worse, the above methods can only achieve the accuracy of decimeter. Therefore, these existing visual-inertial navigation methods are not suitable for maglev haptic interaction which needs high precision and high frequency for immersive interaction experience. In our work, we takes advantages of visual and inertial navigation by reformulating visual-inertial navigation as a regression problem using deep leaning, aiming at improving the navigation frequency while maintaining high precision.
Although VINet [15] similarly regarded the visual-inertial odometry as a sequence-to-sequence regression problem, its fusion navigation frequency is limited by the low-frequency data stream, such as the visual or ground truth data stream. In this work, we present a cascade LSTM based θ-increment learning method to progressively learn increments of position and orientation at a small time step. Suppose that the time step of the ground truth is T , and the time step of our θ-increment learning method for navigation estimation is t. Note that t can be small than T in our study. From this perspective, our θ-increment learning based visual-inertial fusion navigation method can reach higher frequency than that of the ground truth. The implementation details of the presented cascade LSTM θ-increment learning method and our visual-inertial fusion navigation approach will be introduced in Section III.
B. System overview
As shown in Fig. 1, the maglev haptic interaction system is composed of a visual acquisition unit (stereo kit), two IMUs, a visual controller, an inertial controller, a haptic feedback interface, a current controller, an AI service and a visualization module, etc. In this study, the East-North-Up coordinate system is chosen as the fusion navigation coordinate system, and the navigation task is to capture the position and orientation of a moving object relative to the selected coordinate system. IMUs are fixed on the back end of the magnetic stylus. The visual and inertial controller are connected to a router by the Ethernet connector and send data collected by sensors to the AI service under the same LAN.
When an operator uses the magnetic stylus to interact with virtual scenes, the stereo kit with two cameras acquires RGB images and IMUs obtain acceleration and angular rate. Visual controller is used to calculate the position of the magnetic stylus and one inertial controller is used to calculate its orientation. The calculated position and orientation, and the collected acceleration and angular rate are used to estimate the final position and orientation of the magnetic stylus with high frequency through the AI service.
After the AI service calculates out the position and orientation of the magnetic stylus by using the proposed cascade LSTM based visual-inertial navigation approach, it sends the navigation information to the visualization module. The visualization module performs collision detection between the virtual stylus and virtual objects, and meanwhile computes the feedback force to be exerted on the magnetic stylus. Then, the current to be loaded for each coil in the coil array of the maglev haptic interface is calculated according to the calculated feedback force. The current controller intelligently adjusts the current of each coil [6], making the coil array generate effective magnetic field corresponding to the interactive process. Finally, the magnetic stylus receives the same force as the virtual stylus and transmits it to the operator.
III. CASCADE LSTM BASED VI NAVIGATION
In this section, we first give an overview of the presented cascade LSTM based θ-increment learning method. Then, we construct two cascade LSTM networks using the θ-increment learning method to estimate increments of position and orientation. Finally, our visual-inertial navigation approach based on cascade LSTM is described.
A. Cascade LSTM based θ-Increment Learning
Due to different sampling frequencies of visual and inertial sensors, data streams used for navigation are multi-rate, and the frequency of visual data is lower than that of inertial data. It is challenging to realize visual-inertial navigation with high frequency using these multi-rate data for deep learning models. To tackle this issue, we present a θ-increment learning method by constructing a cascade LSTM network unit to progressively learn increments of target variables, as shown in Fig. 2(a).
Giving an input X = (x 1 , x 2 , · · · , x N ) and 1 : N are timesteps of the sequence for our cascade LSTM network. Suppose that the corresponding label of X is ∆Y . Note that ∆Y denotes the total increment of n timesteps. In a certain practical application, if the time step of the ground truth is T , the time step of fusion navigation could be t = T /n(n > 1). To achieve high frequency navigation, the prediction for each time step t should be produced. We cascade n LSTMs to simulate incremental changes of n timesteps, and each LSTM is used to estimate the increment for one time step. In this study, his method is called θ-increment learning which learns increments of variables using the constructed cascade LSTM network for obtaining high frequency estimation, and θ denotes the target variable to be estimated.
Given that n LSTMs aim at learning the same relationship between their inputs and outputs, we let all these n LSTMs share parameters to be learned. This shared mode facilitates the training of the entire cascade LSTM network, and each LSTM is called a shared LSTM (S LSTM) cell. We assume the input of the i th S LSTM is X i = (x i , x (i+1) , · · · , x (m+i−1) ) and its output is ∆Y i , where 1 : m represent timesteps of each S LSTM and N = m + n − 1. The final estimation of the Xn (xn. ..m+n-1) Xi (xi...m+i-1) X2 (x2...m+1) X1 (x1...m)
Y1 Y2 Yi Yn
Xn -1 (xn-1...m+n-2 Fig. 2. The architecture of cascade LSTM network for visual-inertial navigation.
I((k-1)*n+1)...(N+(k-1)*n) I((i-1)*n+1)...(N+(i-1)*n) I(n+1)...(N+n) I1...N I((k-2)*n+1)...(N+(k-2)*n)
P1 P2 Pi Pk-1 Pk q1 Γ q2 qi qn-1 qn ( ) ( ) ( ) Y Inertial Orientation Position X Γ
cascade LSTM network is ∆ Y = ∆ Y 1 + ∆ Y 2 + · · · + ∆ Y n , and ⊕ in Fig. 2 denotes the summation operation. The entire cascade LSTM network is trained using the Adam optimizer according to the mse loss between the label ∆Y and the predicted result ∆ Y . Thanks to the adaptive characteristics of LSTM, the cascade LSTM based θ-increment learning method does not need endto-end training data and the shared parameters are updated every n timesteps in our study. Moreover, benefiting from these adaptive characteristics, the shared LSTM cell is capable to accurately predict the increment of one small time step. Therefore, the trained shared LSTM cell can obtain predictions with high frequency and high precision which could be higher than that of the training data. Note that Γ in Fig. 2 denotes the initialization operation. The usage of Γ depends on the relationship between the target variable and time, which will be described in detail below.
B. Cascade LSTM based Orientation and Position Estimation
In the maglev haptic system, 6DOF navigation information (3DOF position and 3DOF orientation) of the magnetic stylus should be acquired for capturing users' interaction operation. In this work, visual-inertial navigation is implemented by using the presented cascade LSTM based θ-increment learning method. Specifically, two cascade LSTM networks are separately trained for estimating the position and orientation of moving objects. Considering that IMUs are capable to obtain high-frequency sampling and the visual module can acquire high-precision positioning information [6], the inertial data with high frequency is utilized as the input of our deep learning model and the ground truth of position is obtained by adopting the visual navigation method described in [6]. Note that the ground truth of orientation is calculated by using one inertial controller with a high-precision on board Digital Motion Processor (DMP).
1) Cascade LSTM based orientation estimation:
The cascade LSTM network used for orientation estimation is called OCasLSTM. For the orientation estimation, the acceleration and angular rate I acquired by IMUs are the input of OCasLSTM, and the total increment ∆Q of n shared LSTM cells for the orientation is the output of OCasLSTM, as shown in Fig. 2(b). Because the input is the first-order derivative of orientation, the output of each shared LSTM cell in OCasLSTM exactly corresponds to the increment of orientation. Therefore, the orientation estimation does not need the initialization operation.
During the training, the mse loss is utilized to update OCasLSTM, and ∆Q is the summation of n increments (∆q 1 , ∆q 2 , · · · , ∆q n ) obtained from n shared LSTM cells. In practical applications, only one shared LSTM cell is needed to predict the increment of orientation for one time step, and the current estimated orientation is the sum of the predicted increment and the orientation of the previous moment.
2) Cascade LSTM based position estimation: The cascade LSTM network used for position estimation is called PCasLSTM. Different from the orientation, the position is the double integration of the acceleration and angular rate. According to the kinematics theory, an initial velocity except for the acceleration and angular rate should be provided for calculating the increment of position. To tackle this issue, we introduce the initialization operation Γ into PCasLSTM, as shown in Fig. 2(c).
For the initialization operation Γ, the increment of position and its corresponding time are known. To obtain the initial velocity, the motion is assumed to be a certain state which can be uniform velocity, uniform acceleration, etc. In order to alle-viate the impact of such an initialization operation, PCasLSTM is constructed by cascading k CasLSTM units, and these units are trained simultaneously. For the first unit, the increment of each timestep obtained using the initialization operation Γ and the inertial data sequence I 1 , I 2 , ..., I N are contacted as its input. For each following unit, the estimated position from the previous CasLSTM unit and the corresponding inertial data sequence are contacted as its input.
The output of the i th CasLSTM unit ∆P i is the summation of n increments (i.e. ∆p 1 , ∆p 2 , · · · , ∆p n ) obtained from n shared LSTM cells. Note that n shared LSTM cells of each CasLSTM share parameters, while k CasLSTM units of PCasLSTM have separate parameter configurations. During the training, multiple losses are used to update the entire PCasLSTM. In practical applications, we firstly perform the initialization operation using the trained PCasLSTM, and then the shared LSTM cell of the last cascade LSTM unit in PCasLSTM is used to predict the increment of position for one time step. The current estimated position is the sum of the predicted increment and the position of the previous moment.
C. Cascade LSTM based Visual-Inertial Fusion Navigation
After finishing the offline training of cascade LSTM based orientation and position estimation models, we use the trained models to provide accurate and real-time navigation for the maglev haptic interaction system. The concrete steps for accomplishing visual-inertial fusion navigation are as follows:
• Preprocessing: After receiving the visual and inertial data of the magnetic stylus, the AI service firstly preprocesses these data, such as formatting and normalization.
• Initialization: As described in Section III-B, the shared LSTM cell of the last CasLSTM unit in PCasLSTM is used for position estimation. We firstly perform the entire PCasLSTM to obtain an accurate initial state for the shared LSTM cell.
• Estimation: This step can be divided into three cases: a) If only raw acceleration and angular rate data I are received, predict increments of position and orientation. b) If position data is received except from I, predict the increment of orientation. c) If orientation data is received except from I, predict the increment of position.
• Update: If position and orientation data are received except from I, update the location information (position and orientation) of the magnetic stylus. Otherwise, perform the "Estimation" step and update the location information of the magnetic stylus by adding estimated increments and the location information of the previous moment together.
IV. EXPERIMENTATION RESULTS
A. System Tested
In this section, to verify the cascade LSTM based visualinertial navigation approach proposed by this article, we set up a system testbed in view of a maglev haptic interaction application. The maglev haptic interaction system is composed of a visual-inertial navigation module, a maglev haptic interface [5], [6], an AI service and a visualization module. The visual-inertial navigation module is used to obtain visual and inertial data. The maglev haptic interface [5], [6] includes a magnetic stylus, a coil array and a coil driver module, and it provides haptic feedback in the haptic interaction application. The AI service performs visual-inertial navigation using deep learning models. Besides, the visualization module is utilized to show virtual scenes. In summary, the specific haptic interaction process is as follows. When users move the magnetic stylus in the operation workspace of the maglev haptic interface, visual and inertial data collected or calculated by the visual-inertial module are sent to the AI service which performs cascade LSTM based visual-inertial navigation for haptic interaction. The visualization module displays the virtual heart deformation model in real time according to the navigation information, and meanwhile, the maglev haptic interface provides the corresponding haptic feedback to users. As shown in Fig. 3, the visual-inertial module, AI service, maglev haptic interface and visualization module communicate under the same LAN in real time. Two IMUs (MPU6050) are fixed on the back end of the magnetic stylus. One is used to gather the raw acceleration and angular rate data, and the other one is used for providing orientation angles. Cameras capture the position information of the magnetic stylus by tracking red markers. Our cascade LSTM based visual-inertial navigation approach can export the position and orientation of the magnetic stylus at a frequency of 200 Hz in the heart deformation haptic interaction application.
The specific experimental process is as follows: First, we performed interactive heart deformation simulation on the system tested and collected 30,000 data. The collected data included acceleration and angular rate, and the corresponding position and orientation data, and their sampling frequency were 200Hz, 20Hz and 100Hz, respectively. Then, we divided the collected data into training set, validation set and testing set with the ratio of 8:1:1. Cascade LSTM based position and orientation estimation models were trained on the training set. After that, trained models were evaluated on the testing set.
B. Experimental Results and Analysis
In order to verify the performance of our cascade LSTM based θ-increment learning method, we trained five OCasLSTM models, and the increase ratio of the frequency were 2, 4, 6, 8, and 10, respectively. To make sure fairness, the testing data were not used to train these models. Table I demonstrates the mean absolute error (MAE) between the predicted and actual orientation angles (pitch, roll and yaw). From Table I, we can see although the MAE value between the predicted and actual orientation data becomes higher as the ratio increases, the max MAE value is less than 0.02 • . Besides, Fig. 4 shows the comparison between the predicted orientation data of the OCasLSTM with the increase ratio of 10 and the actual orientation data, and the predicted results are very close to the actual ones. From Table I and Fig. 4, it can be seen that the presented cascade LSTM based θ-increment learning method is promising.
Then, we trained three PCasLSTM models with ten CasLSTM units by using three different initialization methods (uniform velocity-u, uniform acceleration-ua and random-r). Fig. 5 shows the mean absolute error between the predicted and the actual position data of these ten CasLSTM units for three initialization methods. The mean absolute error of three initialization methods is very close, demonstrating the robust of PCasLSTM. Furthermore, the mean absolute error of ten cascade LSTM units for three initialization methods is less than 1mm, showing high accuracy of our approach.
V. CONCLUSION
In this work, we focus on the problem of how to improve the navigation frequency and stability while maintaining highprecision and the lightweight design for maglev haptic interaction. To achieve this goal, we present a cascade LSTM based θ-increment learning method which is utilized to construct two separate cascade LSTM networks for accomplishing position and orientation estimation. This proposed cascade LSTM based visual-inertial navigation approach can yield position and orientation estimation of moving objects for a small time step, thus it can achieve high frequency navigation. Furthermore, the accuracy of our approach was verified by building a testbed. In future studies, we will research high- precision haptic rendering methods based on the proposed visual-inertial navigation approach, and extend the navigation approach to other applications.
Fig. 3 .
3The system tested.
Fig. 4 .
4Comparison between the predicted and actual orientation angular.
Fig. 5 .
5The MAE between the predicted and actual data of ten CasLSTM units for three initialization methods.
arXiv:1901.09224v1 [cs.HC] 26 Jan 2019Inertial
Measurement
Units (IMUs)
Inertial Controller
Stereo Kit
Visual Controller
Inertial Data
Visual Data
Current Controller
AI Service
Challenges
1. Maintain high positioning frequency
while providing high precision;
2. Improve the stability and robustness
when the occlusion issue occurs;
3. Design a lightweight and cost-effective
navigation module.
Surgery
Teaching
Marketing
Recreation
Haptic Interaction Applications
Magnetic Stylus
Visualization
Module
Maglev Haptic Interface
Coil Array
Fig. 1. An overall design for our maglev haptic interaction system.
TABLE I THE
IMAE BETWEEN THE PREDICTED AND ACTUAL ORIENTATION DATA.Orientation
ratio=2
ratio=4
ratio=6
ratio=8
ratio=10
Pitch ( • )
0.0086
0.0115
0.0136
0.0166
0.0198
Roll ( • )
0.0092
0.0113
0.0143
0.0173
0.0200
Yaw ( • )
0.0050
0.0060
0.0072
0.0080
0.0105
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| []
|
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"Language Dependencies in Adversarial Attacks on Speech Recognition Systems",
"Language Dependencies in Adversarial Attacks on Speech Recognition Systems"
]
| [
"Karla Markert [email protected] \nFraunhofer AISEC\nGermany\n\nTechnical University Munich\nGermany\n",
"Donika Mirdita [email protected] \nTechnische Universität Darmstadt\nGermany\n",
"Konstantin Böttinger [email protected] \nFraunhofer AISEC\nGermany\n"
]
| [
"Fraunhofer AISEC\nGermany",
"Technical University Munich\nGermany",
"Technische Universität Darmstadt\nGermany",
"Fraunhofer AISEC\nGermany"
]
| []
| Automatic speech recognition (ASR) systems are ubiquitously present in our daily devices. They are vulnerable to adversarial attacks, where manipulated input samples fool the ASR system's recognition. While adversarial examples for various English ASR systems have already been analyzed, there exists no inter-language comparative vulnerability analysis.We compare the attackability of a German and an English ASR system, taking Deepspeech as an example. We investigate if one of the language models is more susceptible to manipulations than the other. The results of our experiments suggest statistically significant differences between English and German in terms of computational effort necessary for the successful generation of adversarial examples. This result encourages further research in language-dependent characteristics in the robustness analysis of ASR. | 10.21437/spsc.2021-6 | [
"https://arxiv.org/pdf/2202.00399v2.pdf"
]
| 243,477,145 | 2202.00399 | 313028769b97ddbd0fd1813bb039124a180859f3 |
Language Dependencies in Adversarial Attacks on Speech Recognition Systems
2 Feb 2022
Karla Markert [email protected]
Fraunhofer AISEC
Germany
Technical University Munich
Germany
Donika Mirdita [email protected]
Technische Universität Darmstadt
Germany
Konstantin Böttinger [email protected]
Fraunhofer AISEC
Germany
Language Dependencies in Adversarial Attacks on Speech Recognition Systems
2 Feb 2022* authors contributed equallyIndex Terms: speech recognitionadversarial exampleslan- guage comparison
Automatic speech recognition (ASR) systems are ubiquitously present in our daily devices. They are vulnerable to adversarial attacks, where manipulated input samples fool the ASR system's recognition. While adversarial examples for various English ASR systems have already been analyzed, there exists no inter-language comparative vulnerability analysis.We compare the attackability of a German and an English ASR system, taking Deepspeech as an example. We investigate if one of the language models is more susceptible to manipulations than the other. The results of our experiments suggest statistically significant differences between English and German in terms of computational effort necessary for the successful generation of adversarial examples. This result encourages further research in language-dependent characteristics in the robustness analysis of ASR.
Introduction
Speech-controlled assistants perform a myriad of tasks based on voice commands, e.g., book appointments, manage contacts, make calls, send e-mails, or control IoT devices. The likelihood of these systems to misunderstand our commands is nonnegligible. Recently, research showed that voice commands can be successfully and efficiently manipulated so that audio transcription frameworks are fooled [1,2]. These manipulated inputs are called audio adversarial examples. They attack the system's security: For example, an adversarial attacker can play a phrase which to the human ear sounds innocuous like "Hello James, let's go for lunch" while the system interprets it as "Unlock the front door". Depending on the sensitivity of the command, these potential misunderstandings or manipulations can expose and endanger people's privacy [3,4].
Current methods for adversarial attacks on ASR models have mainly focused on the English language [4,5,6]. English ASR systems are based on very well curated and diverse audio databases. Since virtual assistants are deployed worldwide and often addressed in the user's mother tongue, other languages need to be considered equally, as they may differ in linguistic properties or in the amount of training data.
We therefore ask: Can we observe different vulnerable behavior in these local ASR systems when confronting them with language-specific adversarial attacks? We address this research question by comparing the generation process for adversarial samples attacking Deepspeech [7] trained on an English and a German dataset, respectively. Our evaluation shows that the German language model is more prone to a gradient-based attack [4], whereas both language models are equally robust if one also includes psychoacoustic hiding [5].
This paper is organized as follows. Section 2 provides an overview of the most relevant publications in the field of whitebox targeted adversarial attacks on ASR. The ASR framework, the datasets, and the language models are presented in Section 3, along with the parameters of interest for the adversarial methods. We introduce our experiments in Section 4, followed by a conclusion and an outlook on future work in Section 5.
Related Work
The first paper to document a successful targeted adversarial attack against an ASR systems in a white box attack environment was published by Carlini and Wagner (CW attack) [4]. Via gradient descent, they calculated noise to be added to the Mel-Frequency Cepstral Coefficients (MFCC), a representation of audio waves based on the power spectrum. As reported by the authors, this experiment was 100% successful on all adversarial attempts when directly fed to the speech recognition system. However, in many cases the noise was audible for a human listener and, further, does not allow for an over-the-air attack, where the audio is played by a speaker and recorded by a microphone before being passed to an ASR system 1 .
Another attack of interest was developed by Qin et al. (Qin attack) [5]. They built a more refined attackthat aims to make the noise imperceptible for human ears on top of the CW attack. The authors apply a psychoacoustic model to mimic the physiological characteristics of the human ear in order to generate imperceptible noise and, further, a room model to simulate reverberations. They showed experimentally that the results remain resistant in different room simulations.
With a similar approach, [6] apply room simulation and a psychoacoustic model to generate over-the-air resistant adversarial examples. They tested their adversarial attack in a real over-the-air setting. The researchers were able to fool the ASR system and concluded that the algorithm was room-agnostic, i.e., works in a variety of room settings with no specific room requirements.
There are many different ASR systems and for most of these architectures different attack techniques have been developed. In order to improve the comparability, we have used only one ASR system, Deepspeech v0.4.1. Since [6] and [5] follow very similar ideas, we have limited ourselves to the CW attack and the Qin attack and reimplemented the latter one for Deepspeech.
Preliminaries
In this section, we provide a detailed description of the ASR system used for this work and the reason why this system was chosen over other available ones. The system and algorithms presented in this section are the universal setup for both languages considered. Every experiment from Section 4 is run under the same parameters for the English and the German language model.
ASR Framework
There are currently several open source code ASR systems available for research, including DeepSpeech [7], Kaldi [8], and Lingvo [9]. In this work, we use DeepSpeech v0.4.1 as the ASR system of choice. This specific ASR was chosen due to its compatibility with existing attacks and easy access to its underlying fundamental audio wave processing methods in order to set up the attacks.
DeepSpeech 2 is a character-level ASR framework first developed back in 2014 [7]. It is provided as an open source repository and it has been under active development for years. This ASR system is end-to-end using recurrent neural networks (RNNs). The RNN-based architecture enables the model to develop robustness towards noise and speaker variations without the need for specialized components to engineer that robustness independently of the language model itself. The system then measures the prediction error by using the connectionist temporal classification loss (CTC loss) [10].
Like any machine learning model, DeepSpeech does not generalize well on words it has rarely or never seen before in its training dataset. In practise, it is hard to get a publicly available audio dataset that is rich enough in variety and frequency of words that it can model the entire language as a whole correctly and in a balanced way. Alongside the language model trained on the audio data, DeepSpeech also uses an additional N -gram language model to better recognize a certain word or sequence of characters. The purpose is to help the system to provide the most meaningful transcriptions when the word or phrase is something the audio model cannot easily recognize. For the English language, the N -gram model is generated from a corpus of 220 million sentences with a vocabulary of 495,000 words [11]. Here, we use the pre-trained model v0.4.1 that is available on the DeepSpeech platform. The German language N -gram model is generated from a corpus of 8 million sentences. This model was trained by us, see Section 3.3.2.
Dataset
The English training dataset, called LibriSpeech [12], was created using sound snippets from thousands of audiobooks, most common classical ones. It is comprised of 1000h of English audio data sampled at 16kHz. The readers are both male and female and predominantly speak American English.
The German training dataset is comprised of two different audio databases: voxforge and tuda-de [13]. Voxforge has about 35h of audio data and is one of the oldest comprehensive German language audio data. This data was generated using Ger-man Wikipedia articles and European parliament protocols. The tuda-de dataset is based on the same written sources as voxforge while contaning 127h of audio data. The tuda-de data is generated under more controlled and elaborate conditions than voxforge, with the assumption that the resulting models would be more accurate. The fact that there is more language data available for English than for German also holds true for most proprietary ASR models.
Language Models
We have picked English and German as the two languages to test and compare the resilience and susceptibility to attacks. The choice of languages was narrowed down to these two, since these are two languages with very good curated open source audio datasets.
English Model
We use the default English language model for that particular DeepSpeech version v.0.4.1, provided by the DeepSpeech platform. The model is reported to have a word error rate (WER) of 8.26%.
German Model
The German language model in this project was adapted from [14]. They describe the optimal creation of a language model for German based on DeepSpeech with respect to different speech datasets. There are three major datasets for the German language online: voxforge, tuda-de and Mozilla Common Voice, see Table 3. In [14], German ASR models were trained using different combinations of the datasets. The authors obtained the most meaningful results by using only tuda-de and voxforge.
For the purpose of this research, we retrained the German model using these two datasets with the original hyperparameters for which [14] achieved an WER of 15.1%. It is noticeable that this WER value is almost twice as high as that of the English language. This is potentially due to the system being provided less audio data to train on, compared to the English language model.
Adversarial Attacks
Carlini Wagner Attack
The CW attack is a targeted attack on DeepSpeech [4]. This attack is a white-box attack where the attacker needs access to the model. The adversary takes a benign audio sample and designates a target expression. The algorithm then iteratively calculates noise based on gradient descent which manipulates the original sample for the purpose of fooling the ASR system to transcribe it as the target label. The attacker needs to be able to extract gradients as well as manipulate the MFCC audio waves in order to generate an adversarial sample.
Parameters of Interest. For the CW attack, we log the following parameters for the evaluation of our experiments:
• First Hit (FH) → the first epoch the adversarial attack creates a successful adversarial sample that can fool the network.
• Best Hit (BH) → the epoch in which the best optimized successful adversarial sample is generated.
• Noise Loudness (NL) → calculates the loudness difference in Decibels between noise and original audio according to [4]: the larger the absolute value, the quieter is the noise compared to the original audio.
• Perturbation Bounds (PB) → the numerical higher and lower bounds of the perturbation for a successful attack.
Qin Attack
The second attack that we use to evaluate our language models is presented in [5]. This attack builds on top of the CW attack methodology. The first step of this attack utilizes the CW attack to generate a successful adversarial sample. Then, the researchers improve this attack by applying psychoacoustic auditory masking to have only impercetible noise. They also provide a model to simulate room reverberations for resistant adversarial examples, which we do not include in our analysis. The attack is also white-box as it needs access to the gradient and underlying MFCC transformations of the system in order to be successful. We have adjusted the original code for Lingvo such that it runs on DeepSpeech.
Parameters of Interest. We log the following parameters for the evaluation of our experiments that are unique to the Qin attack:
• Alpha → balance between adversarial attack accuracy and imperceptibility to human hearing: the higher the value, the higher the focus on imperceptibility compared to accuracy. A high value suggests that the sample is a strong adversarial attack and the psychoacoustic fine tuning does not cause instability on the sample effectiveness.
• Psychoacoustic Loss (PL) → imperceptibility loss value, measures the loss of the subroutine that ensures the psychoacoustic hiding of the adversarial sample: the lower the value, the higher the accuracy.
T-Test
The Welch's t-test is a statistical method to test the hypothesis that two populations have equal means. Following [15], we have applied this test directly without any pre-testing for normality.
In accordance with the implementation in [16], the null hypothesis H0 of our computation can be formulated as both sets of samples (German and English) have identical means. We evaluate whether the null hypothesis can be neglected or not by observing the two outputs of the computation: t-statistic and pvalue. We set the level of significance per test to α * = 0.05, which leads to a Bonferroni-corrected level of significance α = α * 4 = 0.0125 for every experiment [17].
For the purpose of providing a comprehensive picture of our data, we have also applied the non-parametric Kruskal-Wallis test and provide the results in Appendix A.3.
Experiments
In order to discover and analyse potential language dependencies in the process of audio adversarial samples' generation, we run five different experiments. Each experiment measures the generation process in terms of speed and efficiency. We use the parameters of interest described in Section 3.4 in order to measure the characteristics of the generation process.
We apply two different attacks for our experiments to measure the language-specific variability in adversarial attack efficiency. During stage one, we run the CW attack for 1000 epochs. We apply the attack with the hyperparameters that were used in the original paper [4]. Stage two is the psychoacoustic hiding attack [5], that runs for 4000 epochs (default value). Table 1 provides an overview of the experiments we carry out. We run five experiments per stage and all these experiments differ from one another in terms of the type of manipulation. We apply each experiment to both, the CW and the Qin attack. We created 40 different attack scenarios for experiments 1, 2, 3 and 4. Experiment 5 is a special experiment as the target phrases have to be constructed manually. As a result, the sample dataset for this experiment is considerably smaller. For all experiments, we use common words one might find in the original training vocabulary.
# Experiment Description 1 Randomized Manipulations
Target is different from the original transcription.
Phrase Expansion
Target phrase is longer.
Phrase Abbreviation
Target phrase is shorter.
Phrase Negation
Target phrase is negated (by one insertion). 5 Targeted Manipulations Target has specific letters changed. Table 1: Experiments overview.
In order to quantify the results in a more rigorous manner, we run the student t-test on the results for each language. By using the t-test, we are able to decide if the differences between English and German are statistically relevant.
In the following, each attack is introduced in more detail.
Randomized Lexical-based Manipulations
We take the original audio data and create an adversarial target that has the same length as the original ±10% difference but is a completely different transcription compared to the original. This experiment has a 100% hit rate on both stage one and stage two for both languages, i.e., the experiment was successful for every sample and every stage.
Phrase Expanding Manipulations
We take the original audio data and create an adversarial target that is 50% larger than the original audio transcription and different from the original label. Here, we test further expanding the sentence to include additional new information.
For the English language model, this experiment has a 97.5% hit rate on Stage 1 and 95% for Stage 2. This means one sentence could not be successfully manipulated on both stages (for further details, see Appendix A.2). We can see that the successfulness of the Qin attack depends on the similarity between the original and target audio, as stated in [5].
For the German language model, we have a 97.5% hit rate on both stages. This means that only one sentence (see Appendix A.2) could not be manipulated to be transcribed to its target through the first stage, and, consequently, the second stage failed as well because the psychoacoustic hiding cannot be performed successfully if the adversarial attack does not work.
Phrase Abbreviating Manipulation
We take the original audio and create an adversarial example that is only 50% the length of the original audio. In this case, we encounter again a 100% hit rate for both stages and both languages.
Phrase Negations
This experiment is designed around the semantic concept of negation. In each sample, we insert one negation word. The negation semantically nullifies the original command, so when the ASR receives the modified command, it will not act upon it. This experiment was chosen as a special case scenario when an attacker tries to nullify the command a user sends. The success rate of this attack is 100% for both stages and both languages.
Targeted Lexical-Based Manipulations
In this final experiment, we run a targeted attack based on phonetic features of each language. We try to measure the effort it takes for an attacker to replace a certain phonemes (consonant or vowel, including diphthongs and monophthongs) with another one from the same type. One phoneme replacement can alter the meaning of the word altogether i.e., Frau (woman) becomes frei (free), can becomes con.
When we run the generation process for adversarial samples, the algorithm analyzes the entire phrase and also unavoidably optimizes even those frames that already match between original and target transcript, which is due to the CTC loss. We attempt to lower the computational effort by making only one a targeted change of a phoneme. We further make sure that any letter flip results in a meaningful word that is represented in the training dataset.
We created seven different adversarial samples for each of the three phonemes of interest, for the German and English language, respectively: monophtong, diphtong, and consonant changes. We do not run a t-test here, as the dataset in this experiment is too small to be a meaningful sample set to run statistical tests on. Instead, we only observe how the parameters change and act for each phoneme type. In future work, this experiment should be expanded to a broader dataset covering more cases to deliver statistically relevant results.
Results
The results of the first four experiments are aggregated in Table 2. Each row represents one experiment and the column is set to ✓, if the null hypothesis can be neglected, hence, if there is a statistically significant difference between the two languages, and ✗ otherwise. For more detailed information on the pand t-values, see Table 4a and 4b in Appendix A.3.
For all four experiments in stage one, the German system is more susceptible to attacks (most of the times, statistically significantly). This might indicate that there are some languagespecific characteristics that are important for the generation of adversarial examples. Here, one might consider the different correlation between the written and spoken language for English and German: Whereas the spelling for an unknown German word is rather clear from its sound, this does not hold for the English language. This might also be a reason why in experiment 5, there is less work needed to fool the English ASR system.
There are not many experiments in stage two that reveal a statistically significant difference. Even if so, the values do not allow for one language model to be marked more robust than the other, since even if we encounter statistically significant differences, there is no single better language model. Further, the value of α, a variable that is adjusted according to the weight the algorithm puts on the psychoacoustic hiding loss function, is often considerably higher and more varied for the German model than for the English model. The algorithm does not spend much time optimizing the psychoacoustic loss function weight for the English model and more time optimizing that parameter for German instead. This suggests that doing psychoacoustic hiding for German is harder and needs more fine tuning compared to English. The German dataset is smaller than the English one but it is also cleaner with respect to the sound quality. We can assume that the hardship to replace German phonemes is a language feature and not a model vulnerability. However, further research should evaluate the relation between model accuracy and vulnerability.
Experim. FH BH NL PB 1 ✓ (G) ✓ (G) ✓ (G) ✗ 2 ✓ (G) ✓ (G) ✓ (G) ✓ (G) 3 ✓ (G) ✓ (G) ✓ (G) ✓ (G) 4 ✓ (G) ✓ (G) ✗ ✓ (E) (a) StageExperim. FH BH Alpha PL Experiment 1 ✗ ✓ (E) ✗ ✗ Experiment 2 ✗ ✗ ✗ ✗ Experiment 3 ✗ ✗ ✓ (E) ✓ (G) Experiment 4 ✗ ✓ (E) ✗ ✗ (b) Stage
Conclusion and Future Work
We ran five different targeted adversarial attack experiments for the purpose of discovering potential language-based differences in the generation process. Overall, we observed that meaningful differences between the two languages in terms of attack efficiency. The German model was both, the quicker to be fooled using the CW attack, needing on average quieter noises to successfully bypass the platform, and often the hardest one to psychoacoustically hide the noise. This cannot be explained given simply the WER of the models when trained. In a next project, it would be interesting to evaluate the correlation between the model's accuracy and its vulnerability to adversarial examples. Experiment 5 was observation-based, since we did not have enough datapoints to run proper statistical tests. The observations from experiment 5 indicate that the English ASR system is easier fooled by specific phoneme-based attacks, which might be due to the specific correlation between spoken and written language. As a result, we can say that while the German model exhibited some of the expected weaknesses due to its limited audio sources and model accuracy, we also discovered that there are phonetic aspects to the language that require more work to adequately hide the attack compared to English.
While there are some statistically relevant differences, it is important to note that both, German and English, have the same root. It would be of further interest to explore how the measured parameters behave for other languages that do not belong in the same family tree. Further, including more linguistic findings on the correlation between spoken and written language could provide useful insights for improving the robustness of ASR systems.
one experiments: CW attack.
two experiments: Qin attack.
Table 2 :
2Summary for the Welch's test: ✓, if null hypothesis can be neglected for the Bonferroni-corrected level of significance α = 0.0125; ✗, otherwise. In parentheses, the language with the lower mean value (the more vulnerable one) is noted.
In the original paper, the ASR system Deepspeech v0.1.0 was used.Since it was adapted to a newer version, we use the attack for Deepspeech v0.4.1, see https://github.com/carlini/audio_adversarial_examples, retrieved on September 21, 2020.
The ASR framework DeepSpeech v0.4.1 can be found at "https://github.com/mozilla/DeepSpeech/releases/tag/v0.4.1".
AcknowledgementsThis research was supported by the Bavarian Ministry of Economic Affairs, Regional Development and Energy.ReferencesA. Appendix A.1. German Language ModelsInTable 3, we list the three major datasets for German ASR models according to[14].A.2. Unsuccessful Phrases Experiment TwoSome manipulations in experiment 2 did not work successfully. In the following, we list these examples. If the target audio includes a lot of letters and sounds that were unavailable in the original, the process of manipulating becomes more complex and likely to fail. Due to the way the CTC loss function operates, a straightforward grammatical compartmentalization of the letters is not useful. The system looks for frames of sounds and uses grammatical knowledge as supplemental data to decide the spelling when the system is at crossroads.A.3. T -Statistics and P -ValuesIn 3.7s >1000 noisy readTable 3: German datasets for ASR training according to[14]. In this work, we use voxforge and tuda-de.ExperimentFirst
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| [
"https://github.com/carlini/audio_adversarial_examples,",
"https://github.com/mozilla/DeepSpeech/releases/tag/v0.4.1\"."
]
|
[
"Imbalanced magnetohydrodynamic turbulence modified by velocity shear in the solar wind",
"Imbalanced magnetohydrodynamic turbulence modified by velocity shear in the solar wind"
]
| [
"G Gogoberidze ",
"Y M Voitenko "
]
| []
| []
| We study incompressible imbalanced magnetohydrodynamic turbulence in the presence of background velocity shears. Using scaling arguments, we show that the turbulent cascade is significantly accelerated when the background velocity shear is stronger than the velocity shears in the subdominant Alfvén waves at the injection scale. The spectral transport is then controlled by the background shear rather than the turbulent shears and the Tchen spectrum with spectral index −1 is formed. This spectrum extends from the injection scale to the scale of the spectral break where the subdominant wave shear becomes equal to the background shear. The estimated spectral breaks and power spectra are in good agreement with those observed in the fast solar wind. The proposed mechanism can contribute to enhanced turbulent cascades and modified −1 spectra observed in the fast solar wind with strong velocity shears. This mechanism can also operate in many other astrophysical environments where turbulence develops on top of non-uniform plasma flows. | 10.1007/s10509-016-2950-6 | [
"https://arxiv.org/pdf/1610.07073v1.pdf"
]
| 118,633,797 | 1610.07073 | fc290c63974fa03daed956081dee0676a4b70359 |
Imbalanced magnetohydrodynamic turbulence modified by velocity shear in the solar wind
22 Oct 2016
G Gogoberidze
Y M Voitenko
Imbalanced magnetohydrodynamic turbulence modified by velocity shear in the solar wind
22 Oct 2016sun: solar wind -turbulence
We study incompressible imbalanced magnetohydrodynamic turbulence in the presence of background velocity shears. Using scaling arguments, we show that the turbulent cascade is significantly accelerated when the background velocity shear is stronger than the velocity shears in the subdominant Alfvén waves at the injection scale. The spectral transport is then controlled by the background shear rather than the turbulent shears and the Tchen spectrum with spectral index −1 is formed. This spectrum extends from the injection scale to the scale of the spectral break where the subdominant wave shear becomes equal to the background shear. The estimated spectral breaks and power spectra are in good agreement with those observed in the fast solar wind. The proposed mechanism can contribute to enhanced turbulent cascades and modified −1 spectra observed in the fast solar wind with strong velocity shears. This mechanism can also operate in many other astrophysical environments where turbulence develops on top of non-uniform plasma flows.
of magnetic and velocity fluctuations observed in the solar wind. He also suggested that the dissipation of the turbulence at high wave numbers could account for the anomalously high proton temperature observed in the solar wind at 1 AU.
The observations of the fast solar wind fluctuations (Bruno & Carbone 2013) show that below the ioncyclotron frequency spectrum of the fluctuations consist of two intervals. Below the spacecraft-frame frequency f b ≈ 10 −3 Hz, which is usually referred as energy containing range, the spectral slope is close to −1, while for higher frequencies the Kolmogorov spectrum is observed (this range is called inertial range). It is widely agreed that the formation of the Kolmogorov spectrum in the inertial range is related to the active turbulent cascade, as originally proposed by Coleman (1968), whereas the origin of the spectrum observed in the energy containing range is not entirely clear yet [see, e.g., Bruno & Carbone (2013) for a recent review]. First explanation was proposed by Matthaeus & Goldstein (1986). These authors suggest that the observed spectrum results from the superposition of uncorrelated samples of solar surface turbulence. Alternative possibility suggests that the formation of the spectrum can be related to the coronal dynamics (Matthaeus et al. 2007).
The viewpoint that velocity-shear driven instabilities could produce power spectra of magnetic and velocity fluctuations observed in the solar wind has two major shortcomings. Firstly, Belcher & Davis (1971) noticed that Alfvénic fluctuations in the fast flows of the solar wind are strongly imbalanced -the power of the Alfvén waves traveling outward from the sun is significantly larger than the power of inward propagating Alfvén waves and it is difficult to explain how the shear-driven instabilities can produce this asymmetry. Secondly, as mentioned by Bavassano et al. (1978), the Kelvin-Helmholtz instability cannot pro-duce observed large-scale fluctuations. For these reasons it is widely accepted nowadays (Bruno & Carbone 2013) that the dominant, outward traveling Alfvén waves are mainly generated near the Sun below the Alfvénic critical point, as has been originally proposed by Belcher & Davis (1971).
On the other hand, even if generated, the inward waves can not propagate above the Alfvénic critical point and should have the local origin. Moreover, analysis of Helios and Voyager data (Roberts et al. 1987) showed that fluctuations in the solar wind become less imbalanced with increasing distance from the sun. Roberts et al. (1987) also found that the regions of strong shear are associated with a rapid evolution from the purely Alfvénic state to a more balanced state with accelerated turbulent cascade. These observations are puzzling in view of long known result of Parker (1964) that the Kelvin-Helmholtz instability is inefficient in the solar wind. The same result, strong enhancement of the turbulence cascade in strong shear flows, has been confirmed later in numerical simulations (Goldstein & Roberts 1995). Therefore, although enhancement of the turbulent cascade by the shear flows in seen both in the solar wind observations and numerical simulations, many aspects of its physics still remain unclear.
From the theory of neutral fluid turbulence it is long known (Tchen 1954;Hinze 1975) that the strong background shears can significantly affect turbulent dynamics. Namely, if the background velocity shear exceeds the velocity shears in turbulent fluctuations, then the distortion of fluctuations driven by the background shear dominates over nonlinear interactions. This leads to the enhancement of the turbulence cascade rate and formation of so-called Tchen spectrum E(k) ∼ k −1 (here k is a wave number and E(k) is one dimensional spectrum of the fluctuations).
In this paper we consider strong incompressible imbalanced Alfvénic turbulence in the presence of background shear flows. By means of scaling analysis we show that, similarly to the fluid turbulence, the strong shear flow can significantly increase the energy cascade rate, resulting in the formation of the Tchen-like spectrum. This is especially crucial for the dominant Alfvén waves, because their evolution driven by the subdominant component is naturally weak because of the weak subdominant waves. Our analysis shows that this mechanism can explain strong enhancements of turbulent dynamics observed in the solar-wind shear flows.
The paper is organized as follows. Existing models of imbalanced magnetohydrodynamical (MHD) turbulence are reviewed in Sec. 2. Phenomenology of the Tchen model of strong imbalanced MHD turbulence in the presence of strong velocity shear is developed in Sec. 3. Application of the obtained results to the solar wind turbulence is discussed is Sec. 4 and conclusions are given in Sec. 5.
Existing Models of Anisotropic Imbalanced MHD Turbulence
We consider incompressible MHD turbulence in the presence of the background magnetic field B 0 . The Elsässer variables
w ± = v ± b/ 4πρ,(1)
representing eigenfunctions of counter propagating Alfvén waves, are considered as the fundamental variables most useful to study MHD turbulence (Dobrowolny et al. 1980;Biskamp 2003). In equation (1) ρ is the mass density, v and b are velocity and magnetic field fluctuations respectively. The dynamics of the Elsässer variables is governed by the incompressible MHD equations
∂ ∂t ∓ V A · ∇ w ± + (w ∓ · ∇)w ± + ∇p = 0.(2)
Here p is the total (hydrodynamic plus magnetic) pressure and V A ≡ B 0 / √ 4πρ is the Alfvén velocity. In equations (2) we have neglected viscous and resistive dissipative terms, which become important on smaller scales.
Alfvén waves represent exact solutions of the ideal incompressible MHD equations. This means that if in equations (2), say, w − is zero initially, than w + = w + (x, y, z − V A t) is a nonlinear solution of arbitrary form. Iroshnikov (1963) and Kraichnan (1965) realized that due to this property, the MHD turbulence can be described as nonlinear interactions of oppositely propagating Alfvén wave packets. The first model of MHD turbulence developed by Iroshnikov (1963) and Kraichnan (1965) assumed that the turbulence is isotropic. However, the mean magnetic field has a strong effect on the turbulence, in contrast to the mean flow in the hydrodynamic turbulence, which can be eliminated by the Galilean transformation. The anisotropy of MHD turbulence had been already seen in very early numerical simulations (Shebalin et al. 1983).
A theory of anisotropic balanced (under balanced we mean turbulence with equal energy of counterpropagating Alfvén waves) MHD turbulence was proposed by Goldreich & Sridhar (1995). This model implies that the dynamics of turbulence is dominated by the perpendicular cascade with respect to the mean magnetic field whereas the parallel size of turbulent 'eddies' (wave packets) is determined by the critical balance condition. For wave packets with characteristic parallel length scales Λ ± = Λ ∼ 1/k and perpendicular length scale λ ± = λ ∼ 1/k ⊥ , this condition implies that the characteristic time scale of wave packet collision Λ/V A is equal to the characteristic time scale of the energy cascade t cas ∼ λ/w λ , where w λ is characteristic value of the Elsasser variables at scale λ. As a result one arrives at Kolmogorov-like phenomenology with
w λ ∼ λ 1/3 . Equivalently, for 1-dimensional perpendic- ular energy spectrum E(k ⊥ ) we have E(k ⊥ ) ∼ k −5/3 ⊥ .
In the case of imbalanced MHD turbulence situation becomes more complicated. Assuming local turbulence, and noting that for Alfvén waves Elsasser fields w ± λ are perpendicular to the mean magnetic field, it can be readily estimated that the nonlinear terms (w ∓ · ∇)w ± are of the order ∼ w + λ w − λ /λ. Therefore, the straining rates for w ± λ are (Lithwick et al. 2007;Chandran et al. 2009)
ω ± sh ∼ w ∓ λ λ .(3)
If typical parallel length scale of colliding wave packets is Λ, then characteristic timescale of their collision τ col can be estimated as
τ col ∼ Λ V A .(4)
Note that if to packets of size Λ are counter-propagating with speed V A , then collision time τ col = Λ/(2V A ), but because we perform scaling analysis, this factor of 2 is ignored similar to other studies (Lithwick et al. 2007;Chandran et al. 2009). We assume that w + λ is the dominant component
(w + λ ≥ w − λ )
. Dynamics of the turbulence depends on the dimensionless parameter
χ + = τ col ω + sh ∼ w + λ Λ V A λ .(5)
If χ + 1, then subdominant wave packet is cascaded to smaller scale during one collision and we have the strong turbulence. Then for the energy cascade rate of the subdominant component we have
ε − ∼ (w − λ ) 2 ω sh ∼ (w − λ ) 2 w + λ λ .(6)
This does not imply that the dominant wave packet is also cascaded during one collision.
Regarding the cascade of the dominant waves, various models give different predictions. Here we shortly consider main features and predictions of several recent models of anisotropic imbalanced MHD turbulence. According to the model developed by Lithwick et al. (2007) the straining rate imposed by the subdominant waves on dominant ones, w − λ /λ, is imposed coherently over a time λ/w − λ and therefore cascade time for the dominant waves is
τ + ∼ λ w − λ .(7)
For the energy cascade rate of the dominant waves this equation gives
ε + ∼ (w + λ ) 2 τ + ∼ (w + λ ) 2 w − λ λ .(8)
According to the model developed by Chandran (2008), the strainings of the dominant waves by the subdominant ones are summed up randomly. This assumption makes cascade of the dominant waves weaker:
ε + ∼ w + λ w − λ 2 λ .(9)
Yet another model of strong imbalanced MHD turbulence was developed by Beresnyak & Lazarian (2008).
The key feature of this model is that the turbulent fluctuations of the dominant component cascade nonlocally, from k 1⊥ to significantly larger k 2⊥ where k 2z(−) = k 1z(+) . As a result, the subdominant waves become more anisotropic than the dominant waves. This model predicts the cascade rate of dominant waves between the cascade rates predicted by two other models (equations 8 and 9), but there is no simple analytical expression for this cascade rate.
3 Tchen spectrum of MHD turbulence Tchen (1954) was the first who recognized that the strong background shear can significantly affect the energy cascade rate and statistical properties of the hydrodynamic turbulence. In literature there exist several ways to obtain the Tchen spectrum, including spectral energy budget analysis (Tchen 1954), Heisenberg's eddy viscosity model (Katul et al. 2012) and scaling analysis (Perry et al. 1986).
Consider turbulent fluctuations of neutral fluid with characteristic excitation scale λ f and amplitude u f imposed in the mean flow with strong velocity shear, S ≡ dV 0 /dx ≫ u f /λ f . Then the distortion of a turbulent eddy by the background flow is stronger than the distortion by the turbulent flows (nonlinear interaction with other eddies). The main effect of the sheared mean flow is stretching the eddies along the flow, which in the wave number space is equivalent to the increasing perpendicular (with respect to the mean flow) wave number. Consequently, the background shear flow transfers the energy to higher wave numbers faster than the nonlinear interactions.
If the fluctuations can be treated at outer scales as quasi-isotropic, then at some scale λ where the mean shear is greater that the inverse eddy turnover time v/λ, the effective cascade timescale shortens and becomes equal τ cas ∼ 1/S (although it has to be noted that nonlinear interactions are still necessary to ensure decorrelation of fluctuations and isotropic redistribution of fluctuation energy). If the energy cascade rate is denoted by ε, then from equation
ε ∼ v 2 λ /τ cas we have v λ ∼ ε S .(10)
For one dimensional energy spectrum E(k) ∼ v 2 λ /k this gives
E(k) ∼ ε Sk .(11)
Therefore, Tchen's model predicts that at relatively large scales, where the shear imposed by the turbulent fluctuations is still weaker then the mean flow shear, the energy spectrum should be inversely proportional to the wave number, E(k) ∼ k −1 . When k increases, the shear associated with the turbulent eddies s λ ∼ kv λ also increases and starting from the wave number where s λ = S the turbulence is expected to follow Kolmogorov's phenomenology. There is significant evidence supporting Tchen spectrum both in boundary layer experiments and the atmospheric boundary layer measurements [see, e.g., Calaf et al. (2013) and references therein].
Here we develop an analogue of the Tchen phenomenology for the MHD turbulence. Consider incompressible imbalanced MHD turbulence is the presence of the background magnetic field B 0 z and background shear flow V 0 = (0, 0, Sx). Linear dynamics of MHD waves in such a flow have been studied by Gogoberidze et al. (2004). Along with other phenomena (such as possibility of over-reflection and mutual transformation of different MHD modes), one of the main effects produced by the velocity shear is distortion of waves. In the wave number space it is equivalent to the linear variation in time of the perpendicular wave number, k x (t) = k x − Sk t. Similarly to the hydrodynamic case, this is equivalent to the spectral transfer of energy in the perpendicular wave number space. Therefore, with strong velocity shear one can expect an enhancement of the cascade rate and formation of the Tchen-type spectrum in the MHD turbulence.
Here we consider the strongly imbalanced turbulence, the reason for which is twofold. First, the turbulence in the fast solar wind is strongly imbalanced, and there is plenty of in-situ observations to compare with our theoretical predictions. Second, in the imbalanced turbulence the cascade rate of the dominant component is reduced significantly because of the low amplitudes of subdominant waves responsible for the spectral transport in the dominant component. Consequently, even relatively weak background shear can strongly accelerate cascade in the dominant component.
Let us assume that the turbulence is excited isotropically at the (injection) outer scale λ o with the characteristic amplitudes of dominant and subdominant components w + o and w − o , respectively. Suppose that the background velocity shear is moderately strong, exceeding velocity shears in the subdominant component, but still smaller than the shears in the dominant component:
w + o λ o > S > w − o λ o .(12)
In this case the cascade of subdominant waves is not significantly affected by the background shear and the spectral flux is still given by equation (6),
ε − ∼ (w − o ) 2 w + o λ o .(13)
On the contrary, the strainings of dominant waves by the background shear exceed the strainings imposed by the subdominant waves. Then, as in the Tchen fluid model, the cascade time for dominant waves is effectively shortened to τ + cas ∼ 1/S and the cascade rate is accelerated to γ + cas ∼ 1/τ + cas ∼ S. In terms of this new cascade rate, the spectral flux in the wave number space at k ⊥ ∼ 1/λ is given by
ε + ∼ (w + k ) 2 S.(14)
Because of energy conservation, ε + is constant and all terms in this expression are k-independent, which results in the following one-dimensional wave number spectrum of energy:
E + (k) ∼ (w + k ) 2 k ⊥ ∼ ε + S k −1 ⊥ .(15)
The relative strength of the cascades generated by the background and turbulent velocity shears can be conveniently described by the critical parameter
η λ ≡ Sλ w − λ .(16)
The cascade is dominated by the background shear and the −1 spectrum (15) is formed at scales where η λ > 1.
The turbulent shears dominate at η λ < 1 forming the −5/3 spectrum. Equations (12-16) represent our model of the imbalanced MHD turbulence modified by the velocity shear. If the imbalanced MHD turbulence follows phenomenology by Lithwick et al. (2007), then formation of Tchen's spectrum is expected if the background shear is strong enough in sense of equation (12), i.e. when the cascade rate due to background shear (γ + cas ∼ S) is larger than the cascade rate due to the turbulent shears at the injection scale (
γ o ∼ w − o /λ o ): η λo > 1.(17)
In the cases where the turbulence follows phenomenology by Chandran (2008) with a weaker cascade of dominant waves, the Tchen spectrum can be formed by the proportionally smaller background shear (then the critical parameter η λ should be modified correspondingly). As the cascade generated by the background shear proceeds to smaller scales, the Tchen cascade rate S remain the same. On the contrary, the strainings imposed by the turbulent eddies become progressively stronger because of the stronger velocity gradients in the smallscale eddies. Then η λ decreases below η λo and the Tchen-type cascade eventually arrives to the spectral break
λ b = w − b S ,(18)
where η λ b = 1, i.e. the background and turbulent shears become the same. The Tchen wave number spectrum ∼ k −1 ⊥ is formed at scales λ o > λ > λ b , whereas the strongly turbulent spectrum ∼ k −5/3 ⊥ is formed at smaller scales λ < λ b .
Application to the solar wind turbulence
Recent studies based on in-situ observations have revieled that the fast-slow solar wind interface has two parts: a smooth "boundary layer" surrounding the fast wind, and a sharper "discontinuity" between the slow and intermediate solar winds (Schwadron et al. 2005). A relatively strong velocity shear was observed by Ulysses over its first orbit in the transition area between the fast and slow solar winds at 13 • − 20 • latitudes. The data analysis (McComas et al. 1998) showed that the boundary layer separating two winds consists of two regions, the first one with the width l 1 ≈ 2 × 10 7 km and velocity difference ∆V 1 ≈ 200 km/s and the second one with l 2 ≈ 8 × 10 7 km and ∆V 2 ≈ 100 km/s. The spectral brake between the "energy containing range" and the "inertial range" occurs at the spacecraft-frame frequency f b ≈ ×10 −3 Hz (Telloni et al. 2015;Bruno & Carbone 2013). The corresponding break scale. It is well know that the power of inward propagating Alfvén waves in the fast solar wind streams is about one order of magnitude lower than the power of outward waves [see, e.g., Wicks et al. (2011) and references therein]. As the typical values in the fast solar wind we take w − b ∼ 7 km/s for the subdominant wave amplitude at the scale λ b (Wicks et al. 2011;Gogoberidze et al. 2012) and V sw ≈ 600 km/s for the solar wind speed (Bruno & Carbone 2013).
Noting that λ b = V sw /f b and S = ∆V 1 /∆l 1 , with observed numerical values our model predicts
f b = V sw ∆V 1 ∆l 1 w − b ≈ 1.2 × 10 −3 Hz.(19)
As we see performed rough estimate gives the value which is the same order of magnitude as the observed spectral brake frequency. Below f b our model predicts the Tchen spectrum ∼ k −1 ⊥ . Although we do not claim that all observed ∼ k −1 ⊥ spectra are generated by our mechanism, the correspondence between the model and observations is good enough to motivate further observational studies. In particular, as the break wave number k ⊥b between ∼ k −1 ⊥ and ∼ k −5/3 ⊥ spectra is proportional to the background shear S, the presence of positive correlation between k ⊥b and S in various data sets of fast solar wind streams would strongly support our mechanism.
As it is known (Bruno & Carbone 2013) the −1 spectrum is not observed in the slow solar wind. Therefore another interesting direction of further research is to study weather this phenomenon is related to the absence of strong velocity shear in the slow solar wind.
Conclusions
We developed a semi-phenomenological model of incompressible imbalanced MHD turbulence in the presence of sheared background flows. Our results can be summarized as follows:
1) The Tchen-type spectrum ∼ k −1 ⊥ can be generated by the background velocity shear exceeding the shears of the subdominant Alfvén waves at the injection scale λ o .
2) The k −1 ⊥ spectrum breaks down at the scale λ b given by (18), where the turbulent shears of the subdominant component become as strong as the background shear. The k −1 ⊥ spectrum extends from λ o to λ b .
3) At smaller scales, λ < λ b , the Kolmogorov k −5/3 ⊥ spectrum is formed by the turbulent velocity shears.
It is long known, but still unexplained, that in the fast solar wind streams the spectral index of turbulent fluctuations at large scales is close to −1 and the spectral break frequency is close to 10 −3 Hz (see e.g. Marsch (1991)). These observations are compatible with the mechanism we propose here, which motivates its future verification by observations.
Our model can be applied to other astrophysical environments with strong velocity shears, like astrophysical jets and supernova explosions.
This manuscript was prepared with the AAS L A T E X macros v5.2.
Acknowledgements This work has been supported by Shota Rustaveli National Science Foundation grant FR/51/6-300/14.
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| []
|
[
"Heavy flavor jet production and substructure in electron-nucleus collisions",
"Heavy flavor jet production and substructure in electron-nucleus collisions"
]
| [
"Hai Tao Li \nSchool of Physics\nShandong University\n250100JinanShandongChina\n\nHEP Division\nArgonne National Laboratory\n60439ArgonneIllinoisUSA\n\nDepartment of Physics and Astronomy\nNorthwestern University\n60208EvanstonIllinoisUSA\n",
"Ze Long Liu \nDepartment of Physics and Astronomy\nNorthwestern University\n60208EvanstonIllinoisUSA\n",
"Ivan Vitev \nDepartment of Physics and Astronomy\nNorthwestern University\n60208EvanstonIllinoisUSA\n",
"\nTheoretical Division\nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n"
]
| [
"School of Physics\nShandong University\n250100JinanShandongChina",
"HEP Division\nArgonne National Laboratory\n60439ArgonneIllinoisUSA",
"Department of Physics and Astronomy\nNorthwestern University\n60208EvanstonIllinoisUSA",
"Department of Physics and Astronomy\nNorthwestern University\n60208EvanstonIllinoisUSA",
"Department of Physics and Astronomy\nNorthwestern University\n60208EvanstonIllinoisUSA",
"Theoretical Division\nLos Alamos National Laboratory\n87545Los AlamosNMUSA"
]
| []
| Deep inelastic scattering on nuclei at the Electron-Ion Collider will open new opportunities to investigate the structure of matter. Heavy flavor-tagged jets are complementary probes of the partonic composition and transport coefficients of large nuclei, but introduce a new mass scale that modifies the structure of parton showers and must be carefully accounted for in perturbative calculations. In the framework of soft-collinear effective theory with Glauber gluon interactions, we present the first calculation of inclusive charm-jet and bottom-jet cross sections in electron-nucleus collisions at next-to-leading order and compare them to the reference electron-proton case. We also show predictions for the heavy flavor-tagged jet momentum sharing distributions to further clarify the correlated in-medium modification of jet substructure. (Hai Tao Li), [email protected] (Ze Long Liu), [email protected] (Ivan Vitev) diagrams, the evolution of heavy-flavor SiJFs obeys DGLAP-like equations similar to the ones for lightflavor SiJFs. The renormalization-group equation (RGE) is given bywhere s = Q +Q. Here, P i j are usual Altarelli-Parisi splitting functions. To solve the above RGE, we work in Mellin moment space following the method outlined in[47]J Q/g (N) = 1 0 | 10.1016/j.physletb.2022.137007 | [
"https://arxiv.org/pdf/2108.07809v2.pdf"
]
| 237,194,664 | 2108.07809 | cda12501fff168f6c60d71625d4f7c2830e41393 |
Heavy flavor jet production and substructure in electron-nucleus collisions
Hai Tao Li
School of Physics
Shandong University
250100JinanShandongChina
HEP Division
Argonne National Laboratory
60439ArgonneIllinoisUSA
Department of Physics and Astronomy
Northwestern University
60208EvanstonIllinoisUSA
Ze Long Liu
Department of Physics and Astronomy
Northwestern University
60208EvanstonIllinoisUSA
Ivan Vitev
Department of Physics and Astronomy
Northwestern University
60208EvanstonIllinoisUSA
Theoretical Division
Los Alamos National Laboratory
87545Los AlamosNMUSA
Heavy flavor jet production and substructure in electron-nucleus collisions
Deep inelastic scattering on nuclei at the Electron-Ion Collider will open new opportunities to investigate the structure of matter. Heavy flavor-tagged jets are complementary probes of the partonic composition and transport coefficients of large nuclei, but introduce a new mass scale that modifies the structure of parton showers and must be carefully accounted for in perturbative calculations. In the framework of soft-collinear effective theory with Glauber gluon interactions, we present the first calculation of inclusive charm-jet and bottom-jet cross sections in electron-nucleus collisions at next-to-leading order and compare them to the reference electron-proton case. We also show predictions for the heavy flavor-tagged jet momentum sharing distributions to further clarify the correlated in-medium modification of jet substructure. (Hai Tao Li), [email protected] (Ze Long Liu), [email protected] (Ivan Vitev) diagrams, the evolution of heavy-flavor SiJFs obeys DGLAP-like equations similar to the ones for lightflavor SiJFs. The renormalization-group equation (RGE) is given bywhere s = Q +Q. Here, P i j are usual Altarelli-Parisi splitting functions. To solve the above RGE, we work in Mellin moment space following the method outlined in[47]J Q/g (N) = 1 0
Introduction
In the past several years significant progress has been made toward defining the physics of the future Electron-Ion Collider (EIC) and establishing detector requirements that will enable the proposed measurements. The recently released Yellow Report [1] is an important first step in summarizing the current status of deep inelastic scattering (DIS) studies at the future facility, but is far from complete in terms of science reach. In the years that lead to EIC operation its physics program will continue to expand to reflect the pertinent new developments in theory and phenomenology. In this letter we report one such developmentthe first calculation of heavy flavor jet production and substructure in electron-nucleus (e+A) collisions.
Theoretical studies of charm-quark jets (c-jets) and bottom-quark jets (b-jets) in nucleus-nucleus (A+A) collisions [2][3][4][5][6][7][8] and related experimental measurements at the Large Hadron Collider [9][10][11][12][13] have become readily available. Heavy flavor-tagged jets will also soon be studied at the Relativistic Heavy Ion Collider (RHIC) [14] with the sPHENIX experiment. On one hand these results complement the physics of inclusive jets dominated by light partons in heavy ion collisions and provide alternative diagnostics of the transport properties of nuclear matter. On the other hand there are unique aspects of quantum chromodynamics (QCD) that can only be accessed with heavy flavor measurements. First and foremost is the effect of heavy quark mass on parton showers dubbed generically the "dead cone" effect [15]. It was found to play an important role in non-Abelian parton energy loss at small and moderate heavy quark energies [16][17][18] and in the full medium-induced splitting kernels [19][20][21] -the analogues of Altarelli-Parisi branching in nuclear matter. Second, in this kinematic regime, the heavy quark mass can produce quantitatively and even qualitatively different modification of jet observables in reactions with nuclei in comparison to the massless case [3]. For an overview or heavy flavor physics, albeit with more emphasis on hadron production, see e.g. Refs. [22][23][24].
At the same time, theoretical studies of heavy flavor jets at the EIC have been extremely limited. In electron-proton (e+p) collisions c-jets produced in charge current reactions have been proposed [25] as a probe of the strangeness content of the proton at intermediate and large values of Bjorken-x. The transverse spin asymmetry of back-to-back heavy flavor-tagged jet production has been shown to be sensitive to the gluon Sivers function [26]. Experimental feasibility studies for heavy flavor jet measurements at the EIC have also been performed [25,27]. In electron-nucleus collisions this physics is not yet developed. Ref. [28] explored the possibility of using the total charm production cross section to constrain the gluon nuclear parton distribution function (nPDF). Semi-inclusive D-meson and B-meson production in DIS on nuclei can shed light on the physics of hadronization and differentiate between competing paradigms of hadron attenuation in cold nuclear matter [29,30]. In this letter we take the studies of heavy flavor in cold nuclear matter to the next level and present predictions for c-jet and b-jet cross section and substructure modification in electron-nucleus collisions.
To address this problem we can take guidance from the recent calculation of light jet rates and the jet charge in DIS on nuclei [31]. Even in the absence of nuclear matter, the heavy quark mass m must be accounted for in perturbative calculations. It was introduced in Refs. [32,33] in the framework of soft-collinear effective theory (SCET) [34][35][36] with focus on the m/p = λ 1 regime, yielding a new formulation with finite mass corrections SCET M . The role of the heavy quark mass in heavy flavor-tagged jet cross sections evaluated with the help of semi-inclusive jet functions (SiJFs) was also understood [4,37]. In addition to the logarithms of the ratio of the hard scale µ to the jet scale p T R that should be resumed for small radii R, logarithms of the ratio of the jet scale to the heavy quark mass have also been accounted for. For reactions with nuclei, SCET M has been generalized to describe parton shower interactions in matter via the exchange of off-shell Glauber gluons [19]. This development, which resulted in the derivation of the in-medium splitting functions for heavy quarks, has been complemented by the re-analysis of parton branching using the formalism of lightcone wavefunctions [20], confirming earlier results and allowing to compute splittings in QCD media to any order in the opacity of matter. Last but not least, the contribution of medium-induced parton showers to the SiJFs for inclusive jets [38] and charm-quark jets / bottom-quark jets [4] has been derived, the latter being particularly relevant to this work. Here we show how this approach can be applied to deep inelastic scattering, and in particular to e+A reactions. We will complement the calculation of heavy flavor-tagged jet quenching in cold nuclear matter with the evaluation of the c-jet and b-jet soft-dropped momentum sharing distributions [39]. This observable is especially illuminating since its modification in QCD matter can be quite different at small and moderate transverse momenta relative to large ones, depending on the quark mass [3].
The rest of this letter is organized as follows: in Section 2 we present the theoretical formalism for calculating the b-jet and c-jet cross sections and the soft-dropped momentum sharing distributions in e+p and e+A reactions. Phenomenological results for DIS on a gold (Au) nucleus are shown in Section 3. We conclude in Section 4.
Theoretical Framework
Semi-inclusive jet cross sections at the EIC
The factorization formula for the cross section for semi-inclusive jet production in collinear leadingtwist perturbative QCD can be written as [40]:
E J d 3 σ d 3 P J = 1 S i, f 1 0 dx x 1 0 dz z 2 f i/N (x, µ) J J Q / f (z, p T R, m, µ) σ i→ f + f γ/ ren −t s + u , µ σ γi→ f .(1)
Here f i/N is the parton distribution function (PDF) of parton i in nucleon N and J J Q / f is the SiJF from parton f to jet J Q containing heavy flavor. z and m denote the momentum fraction taken by the jet and the mass of the heavy flavor parton, respectively. p T and R are the transverse momentum and radius of the semi-inclusive jet.σ i→ f denotes the cross section for lepton-parton scattering with initial-state parton i and final-state parton f . Lastly s, t, u are the partonic Mandelstam variables defined as s = (k + l) 2 , t = (k − p) 2 and u = (l − p) 2 , where l µ , k µ and p µ are the momenta of the incoming lepton, the incoming parton and the fragmenting parton, respectively. Because kinematic constraints on the scattered lepton are not employed in jet production at the EIC, events with forward lepton scattering can be selected. In this case, the hard process can be described by an incoming quasi-real photon scattering: γq → q(g), γq → g(q), γg → q(q), which contributes to the cross section starting at order α 2 EM α s . Quasi-real photons originate from the incoming lepton and can be accurately described by the well known Weizsäcker-Williams (WW) distribution with a perturbative distribution function f γ/ ren (y, µ) [41][42][43][44]. The analytical expressions forσ i→ f ,σ γi→ f and f γ/ ren (y, µ) up to O(α 2 EM α s ) can be found in [40]. We note that there is also a resolved photon contribution, related to the partonic content of the γ [45]. Formally, it starts at O(α 2 EM α 2 s ) which is homogeneous with NNLO QCD corrections [46] and contributes at relatively small transverse momenta. Furthermore, the overall cross section normalization of high transverse momentum jets will cancel in the nuclear modification ratio that we study in Section 3. For these reasons, we did not consider the resolved photon component here and defer its investigation to future studies.
The NLO SiJFs for heavy flavor jets can be found in Ref. [37]. The evolution of heavy flavor SiJFs is briefly reviewed below. Since the heavy quark mass does not affect the ultraviolet (UV) behavior of
dz z N−1 J Q/g (z) .(3)
The solution for the jet function in this space is given by
J J Q /s (N, µ) J J Q /g (N, µ) = e + (N) α s (µ) α s (µ J ) −r − (N) + e − (N) α s (µ) α s (µ J ) −r + (N) · J J Q /s (N, µ J ) J J Q /g (N, µ J ) ,(4)
where r + (N) and r − (N) are the larger and smaller eigenvalue of the leading-order singlet evolution matrix,
r ± (N) = 1 2β 0 P qq (N) + P gg (N) ± P qq (N) − P gg (N) 2 + 4P qg (N)P gq (N) .(5)
The projector matrices e ± (N) in (4) are defined as
e ± (N) = 1 r ± (N) − r ∓ (N) P qq (N) − r ∓ (N) 2N f P gq (N) P qg (N) P gg (N) − r ∓ (N) .(6)
Eventually, the evolved SiJFs in z-space can be obtained by performing an inverse Mellin transformation.
J J Q /g (z, µ) = 1 2πi C N dN z −N J J Q /g (N, µ) ,(7)
where we chose the contour in the complex N plane to the right of all the poles of J J Q /g (N, µ). The NLO medium corrections to the Q → J Q and g → J Q SiJFs are similar to the case of heavy-ion collisions [4] and can be written as
J med,(1) J Q /Q (z, p T R, m, µ) = µ z(1−z)p T R d 2 q ⊥ f med Q→Q+g (z, m, q ⊥ ) − δ(1 − z) 1 0 dx µ x(1−x)p T R d 2 q ⊥ f med Q→Q+g (x, m, q ⊥ ) = µ z(1−z)p T R d 2 q ⊥ f med Q→Q+g (z, m, q ⊥ ) + ,(8)
and
J med,(1) J Q /g (z, p T R, m, µ) = µ z(1−z)p T R d 2 q ⊥ f med g→Q+Q (z, m, q ⊥ ) + + µ z(1−z)p T R d 2 q ⊥ f med g→Q+Q (z, m, q ⊥ ) ,(9)
respectively. Here f med i→ j+k is the medium induced splitting kernel. The scale µ was introduced as a UV cut-off for the medium corrections [4,38] which is set to be the jet transverse momentum in this work. In the presence of a QCD medium, it was demonstrated that the vacuum splitting must be replaced by the full splitting kernels for each possible branching channel
dN full dzd 2 q ⊥ = dN vac dzd 2 q ⊥ + f med (z, m, q ⊥ ) .(10)
Hence, the full in-medium SiJFs are obtained as
J J Q /i = J vac J Q /i + J med J Q /i ,(11)
where the vacuum contributions are calculated at the LL accuracy, while only the fixed-order medium corrections are included consistently.
Soft-dropped jet momentum sharing distribution
Jet substructure is a promising to study the mass effects in parton shower evolution. In this work, we focus on the jet momentum sharing variable based on the "soft drop grooming" [48] in 1→2 QCD splitting processes. It is defined as the distribution of
z g = min(p T 1 , p T 2 ) p T 1 + p T 2 , z g > z cut ,(12)
where p T 1 and p T 2 are the transverse momenta of the subjets in a reconstructed jet. For the heavy-flavor jet, we are interested in the kinematic region where the jet energy is much larger than the heavy quark mass, 0 < m p T .
Consider an off-shell parton of momentum [p + , p − , 0 ⊥ ], with p + the large lightcone component, that splits into two daughter partons [zp
+ , q 2 ⊥ /zp + , q ⊥ ] and [(1 − z)p + , q 2 ⊥ /(1 − z)p + , −q ⊥ ].
The massive splitting kernel Q → Qg, for example, in the vacuum reads
dN vac dzd 2 q ⊥ Q→Qg = α s 2π 2 C F q 2 ⊥ + z 2 m 2 1 + (1 − z) 2 z − 2z(1 − z)m 2 q 2 ⊥ + z 2 m 2 .(13)
Here C F is the quadratic Casimir invariant of the fundamental representation of SU (3). For q ⊥ m Eq. (13) reduces to the massless splitting functions, however when q ⊥ ≤ zm the heavy quark mass will significantly modify the momentum sharing observable -an effect that is amplified in nuclear matter [3]. If r g is the angular separation between the subjets and p T is the transverse momentum, then q ⊥ = z(1 − z)r g p T when r g is not too large. The interesting regime discussed above can easily be reached with the moderate transverse momenta available at the EIC, especially for b-jets. Here, we start with the vacuum case to set up the stage for the jet splitting function calculation in heavy ion collisions. After soft-drop grooming in the parton branching i → jk, the θ g and z g distribution for parton i is
dN vac dz g dθ g i = α s π 1 θ g j P vac i→ jk (z g ) ,(14)
with r g = θ g R, R being the jet radius. Resummation is necessary in the kinematic region with large splitting probability. It was performed to modified leading-logarithmic (MLL) accuracy in Ref. [48]. The resummed distribution for a i-type jet, initiated by a quark or a gluon, is
dN vac,MLL i dz g dθ g = j dN vac dz g dθ g i→ jk exp − 1 θ g dθ 1/2 z cut dz i dN vac dzdθ i→ jk .(15)
The normalized joint probability distribution then reads
p(θ g , z g ) i = dN vac,MLL i dz g dθ g 1 0 dθ 1/2 z cut dz dN vac,MLL i dzdθ ,(16)
and in for e+A collisions the corresponding medium contribution must be included according to Eq. (10).
Numerical results
3.1. Semi-inclusive heavy flavor jet cross section modification in e+A relative to e+p at the EIC In this section we present the main result of this work -heavy flavor jet cross section modification at the EIC. Here we consider two benchmarks energy combinations for electron-proton collisions (for electronnucleus collisions, the beam energy is per nucleon): 10 GeV (e) × 100 GeV (A) and 18 GeV (e) × 275 GeV (A).
In the following numerical calculations we use CT10nlo PDF sets [49] for the proton and the nCTEQ15 PDF sets [50] for the nucleus, and the associated strong coupling provided by Lhapdf6 [51]. The mediuminduced splitting kernel up to first order of opacity [19,21,52] were used, consistent with earlier EIC studies [29,31]. As a default choice, the nominal transport coefficient of cold nuclear matter for quarks is set to be q 2 ⊥ /λ q = 0.05 GeV 2 /fm from the above references. The theoretical uncertainties in this section are evaluated by varying the transport parameter up and down by a factor of two, which represents the sensitivity of the observable to the transport coefficient. To investigate the nuclear medium effects, we study the ratio of the cross sections in electron-gold (e+Au) collisions normalized by the number of nucleons to the one in e+p collisions.
R eA (R) = 1 A dσ dηd p T e+A dη dσ dηd p T e+p dη .(17)
In Figs. 1 we present R eA for c-jets and b-jets as a function of the transverse momentum p T in the laboratory frame in two rapidity bins 0< η <2 and 2< η <4. The left column of panels is for 10 GeV × 100 GeV e+Au collision and the right column of panels is for 18 GeV × 275 GeV ones. The in-medium shower corrections induced by the interactions between the final-state partons and the nucleus vary with the parton energy in the nuclear rest frame, where the lower energy parton receives larger medium corrections. Therefore, in the forward (nucleus-going) rapidity region 2 < η < 4 we can see more significant jet quenching due to final-state interactions in the large nucleus. Furthermore, a clear separation of jet suppression is observed as a function of the radius R. Initial-state effects reflecting the difference between proton and nuclear PDFs also play an important role in R eA . In the kinematic domains that we consider the smaller beam energies combination is primarily sensitive to the so called EMC region. At the higher beam energies combination we see a clear transition from the anti-shadowing region near midrapidity to the EMC at forward rapidity. Initial-state effects are large for c-jets and b-jets as their production channels are dominated by gluon and sea quarks.
To understand the structure and evolution of showers containing heavy quarks in cold nuclear matter and to use them as tomographic probes at the EIC, it is essential to reduce the effects of nPDFs and enhance the effects of final-state interactions. A successful strategy was developed on the example of inclusive light parton jets [29] and it involves measuring the ratio of the modifications with different jet radii, e.g. R eA (R)/R eA (R = 0.8). In such double ratio initial-state effects in e+A reactions will cancel for jets with a similar kinematics. This double ratio is also an observable sensitive to the angular distribution of in-medium branching processes [31,53]. Furthermore, it provides an opportunity to explore smaller center-of-mass energies where the final-state effects are expected to be sizable even though the cross section is small. Such measurements will take advantage of the high-luminosity design of the future facility. Our predictions for the double ratio of jet cross section suppression in two rapidity bins at the EIC are presented in Fig. 2, where the left and right panels correspond to the results for 10 GeV (e) × 100 GeV (A) and 18 GeV (e) × 275 GeV (A) collisions. The blue and red bands correspond to jet radii R =0.3 and 0.5, respectively, with the large normalization radius R = 0.8 and with variation of the cold nuclear matter transport coefficient. Since medium-induced parton showers are broader than the ones in the vacuum, for smaller jet radii the suppression from final-state interactions is more significant. For both c-jets and b-jets, we can identify larger in-medium effects at 10 GeV × 100 GeV e+Au collision than at 18 GeV× 275 GeV collision with the same p T range fixed. In fact, these are as large as the ones observed for light jets. Additionally, jet production in the forward rapidity region 2 < η < 4 receives the largest in-medium corrections. We flesh out the rapidity dependence of R eA in Fig. 3. Instead of rapidity, we have integrated the p Tdependent cross sections above 8 GeV. Once again we show 10 GeV× 100 GeV (left) and 18 GeV× 275 GeV e+Au collisions (right). The upper and bottom two panels correspond to c-jets and b-jets, respectively. It is very clear that the medium-induced suppression is much enhanced in the forward rapidity region where the jet has smaller energy in the nuclear rest frame -a region that should be well-instrumented for these key measurements at the EIC.
Heavy flavor-tagged jet substructure
The jet splitting-function observable in e+A, i.e. the distribution of z g , only depends on the final-state interactions between the jet and cold nuclear matter. At the EIC we probe a regime very different from heavy ion collisions. To illustrate this we study the p T dependence of p eA (z g )/p ep (z g ) and fix the jet energy in the rest frame of the nucleus to be 94 GeV for a jet radius R = 0.4. With the design energies, the transverse momentum of jets at EIC for practical purposes will be limited to about 30 GeV. However, a much larger range of jet transverse momentum 10 < p T < 80 GeV is used here to demonstrate the p T dependence and compare the large p T region with the previous analyses in Ref. [3]. 1 The results for charmquark jets (left) and bottom-quark jets (right) are presented in Fig. 4 for the nominal value of the medium's transport parameter. Even though at large jet p T the modifications of c-jet and b-jet substructure are similar because m p T , we are beginning to see hints of the quark mass effect. When p T ∼ E, the results have a qualitatively similar behavior with the z g modifications in heavy-ion collisions [3]. For the small jet p T a much larger mass effect is observed by comparing the c-jet and b-jet modifications. Not only is the magnitude larger for the bottom quark-initiated ones, but the shape of the modification differs consistent with the fact that the scale z g m plays an important role in the branching Eq. (13). In Fig. 5, we present the modification of the jet splitting functions for charm jets and bottom jets with fixed jet rapidity at the EIC. Upper panels correspond to the results for 10 GeV × 100 GeV collisions and bottom panels correspond to the results 18 GeV × 275 GeV collisions, respectively. The jet radius is chosen to be R = 1, thus there is still available, albeit limited, phase space to ensure m p T R. The blue, red and green bands correspond to our calculations for light, c-quark and b-quark jets, respectively, and the bands reflect the uncertainties to the variation of the transport parameter by a factor of two. It is clear that, just as in the case of the cross sections, the in-medium corrections are larger in the forward rapidity region of the jet, because the corresponding jet energy in the rest frame of the nucleus is smaller. At the transverse momenta accessible at the EIC the modification is quite different when compared to heavy ion collisions. By changing rapidity we can also see a difference in the modification pattern of bottom-quark jets which we attribute to the interplay between mass and parton energy in the non-Abelian Landau-Pomeranchuk-Migdal (LPM) effect for heavy quarks. These findings are intriguing and, clearly, more detailed future studies of heavy flavor-tagged jet substructure at the EIC will be quite important.
Conclusions
In summary, we presented the first calculation of semi-inclusive charm-quark jet and bottom-quark jet production and substructure in e+A relative to e+p collisions at the EIC. Our formalism allowed to obtain NLO results by consistently combining the parton level cross sections and semi-inclusive jet functions up to NLO, and included resummation for small jet radii in electron-hadron reactions. We found that heavy flavor-tagged jet production is more sensitive to the gluon and sea quark distributions in nucleons and nuclei in comparison to light jets. Thus, in kinematic regions where R eA is dominated by initial-state nPDF effects the modification was even stronger when compared to inclusive jets. Similar to the case of light jets, by applying the strategy of studying ratios of the nuclear modification with two different jet radii we were successful in eliminating nPDF effects, primarily the anti-shadowing and the EMC effect in the regions of interest. The remaining quenching of the jet spectra can be as large as a factor of two for small jet radii, for example R = 0.3, and can clearly be attributed to final-state interactions and in-medium modification of parton showers containing heavy quarks. This suppression is comparable to the one predicted for light jets and expected to be observed in the proton/nucleus going direction. In contrast, near mid rapidity and at backward rapidity the deviation of R eA (R)/R eA (R = 0.8) from unity is small since the energy of the parton/jet in the rest frame of the nucleus is very large. This, in turn, strongly reduces the contribution of in-medium parton shower due to the non-Abelian LPM effect. In fact, even at forward rapidity and smaller center-of-mass energies the parton energies in nuclear rest frame are quite sizeable and, therefore, there isn't much difference in the suppression of c-jets and b-jets.
We complemented the calculation of semi-inclusive jet cross sections with a calculation of the groomed, soft-dropped momentum sharing distribution. Our results show that the substructure modification in e+A relative to e+p reactions is relatively small -on the order of 10% or smaller. Still, just like in the case of heavy ion collisions at relatively small transverse momenta the differences in the subjet distribution are most pronounced for b-jets, followed by c-jets. In the kinematic regime accessible at the EIC the modification of light jets was found to be the smallest. In contrast to the heavy ion case, however, there is significant difference between the energy of the parton in the rest frame of the nucleus and the jet scale which determines the available phase space for substructure even for large radii R ∼ 1. Thus, the jet momentum sharing distribution at the EIC probes a different interplay between the heavy quark mass and suppression of small-angle medium-induced radiation -a regime that can only be accessed at the EIC and merits further investigation in the future. We conclude by pointing out that with the theoretical tools that are becoming available one can also look at how subeikonal corrections to in-medium branching, such as the effects of varying matter density [54], propagate into the observables that we predicted in this work.
Figure 1 :
1In-medium corrections for charm-quark jets and bottom-quark jets as a function of p T at the EIC in two rapidity regions. Green bands (solid lines), red bands (dashed lines), and blue bands (dotted lines) correspond to R = 0.8, R = 0.5 and R = 0.3, respectively. Results for 10 GeV(e) × 100 GeV(A) collisions are shown on the left and results for 18 GeV(e) × 275 GeV(A) collisions are shown on the right.
Figure 2 :
2The ratio of R eA normalized by R eA (R = 0.8) for c-jets and b-jets as a function of p T at the EIC. Blue bands (dotted lines) and red bands (dashed lines) correspond to R = 0.3 and R = 0.5, respectively. The kinematic regions and beam energy combinations are as inFig. (1).
Figure 3 :
3The ratio of R eA normalized by R eA (R = 0.8) for heavy flavor jet production as a function of η at the EIC. Blue bands (solid lines) and red bands (dashed lines) correspond to R = 0.3 and R = 0.5, respectively.
Figure 4 :
4The modification of the jet splitting functions for c → cg (left) and b → bg (right). The energy of the jet is fixed at 94 GeV at the rest frame of the nucleus. The orange dot-dashed, red dotted, green dashed and blue solid lines denote the distribution with p T = 10 GeV, 20 GeV, 40 GeV and 80 GeV, respectively.
Figure 5 :
5The modification of the jet splitting functions for c-jets and b-jets vs z g at the EIC. The upper and bottom two panels correspond to the result for 10 GeV × 100 GeV and 18 GeV × 275 GeV e+Au to e+p collisions, respectively. The jet radius R = 1 and two different rapidities are presented for each energy combination.
It is completely clear that the higher transverse momenta shown cannot be reached at EIC kinematics. That would be possible in heavy ion collisions. We ignore kinematic constraints to make a physics point.
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| []
|
[
"Ultrafast shift and rectification photocurrents in GaAs quantum wells: Excitation intensity dependence and the importance of bandmixing",
"Ultrafast shift and rectification photocurrents in GaAs quantum wells: Excitation intensity dependence and the importance of bandmixing"
]
| [
"Thanh Huynh ",
"Duc \nDepartment of Physics and CeOPP\nUniversität Paderborn\nWarburger Str. 100D-33098PaderbornGermany\n\nHo Chi Minh City Institute of Physics\nVietnam\n\nAcademy of Science and Technology\n\n",
"Reinold Podzimski \nDepartment of Physics and CeOPP\nUniversität Paderborn\nWarburger Str. 100D-33098PaderbornGermany\n",
"Shekhar Priyadarshi \nPhysikalisch-Technische Bundesanstalt\nBundesallee 100, D38116BraunschweigGermany\n",
"Mark Bieler \nPhysikalisch-Technische Bundesanstalt\nBundesallee 100, D38116BraunschweigGermany\n",
"Torsten Meier \nDepartment of Physics and CeOPP\nUniversität Paderborn\nWarburger Str. 100D-33098PaderbornGermany\n",
"Mac Dinh ",
"Chi Str \nDepartment of Physics and CeOPP\nUniversität Paderborn\nWarburger Str. 100D-33098PaderbornGermany\n",
"Ho Chi \nDepartment of Physics and CeOPP\nUniversität Paderborn\nWarburger Str. 100D-33098PaderbornGermany\n",
"Minh City ",
"Vietnam "
]
| [
"Department of Physics and CeOPP\nUniversität Paderborn\nWarburger Str. 100D-33098PaderbornGermany",
"Ho Chi Minh City Institute of Physics\nVietnam",
"Academy of Science and Technology\n",
"Department of Physics and CeOPP\nUniversität Paderborn\nWarburger Str. 100D-33098PaderbornGermany",
"Physikalisch-Technische Bundesanstalt\nBundesallee 100, D38116BraunschweigGermany",
"Physikalisch-Technische Bundesanstalt\nBundesallee 100, D38116BraunschweigGermany",
"Department of Physics and CeOPP\nUniversität Paderborn\nWarburger Str. 100D-33098PaderbornGermany",
"Department of Physics and CeOPP\nUniversität Paderborn\nWarburger Str. 100D-33098PaderbornGermany",
"Department of Physics and CeOPP\nUniversität Paderborn\nWarburger Str. 100D-33098PaderbornGermany"
]
| []
| A microscopic approach that is based on the multisubband semiconductor Bloch equations formulated in the basis of a 14-band k · p model is employed to compute the temporal dynamics of photocurrents in GaAs quantum wells following the excitation with femtosecond laser pulses. This approach provides a transparent description of the interband, intersubband, and intraband excitations, fully includes all resonant as well as off-resonant excitations, and treats the light-matter interaction non-perturbatively. For linearly polarized excitations the photocurrents contain contributions from shift and rectification currents. We numerically compute and analyze these currents generated by the excitation with femtosecond laser pulses for [110]-and [111]-oriented GaAs quantum wells. It is shown that the often employed perturbative χ (2) -approach breaks down for peak fields larger than about 10 kV/cm and that non-perturbative effects lead to a reduction of the peak values of the shift and rectification currents and to temporal oscillations which originate from Rabi flopping. In particular, we find a complex oscillatory photon energy dependence of the magnitudes of the shift and rectification currents. Our simulations demonstrate that this dependence is the result of mixing between the heavy-and light-hole valence bands. This is a surprising finding since the bandmixing has an even larger influence on the strength of the photocurrents than the absorption coefficient. For [110]-oriented GaAs quantum wells the calculated photon energy dependence is compared to experimental results and a good agreement is obtained which validates our theoretical approach. | 10.1103/physrevb.94.085305 | [
"https://arxiv.org/pdf/1606.07002v1.pdf"
]
| 119,236,065 | 1606.07002 | 92bd4bb9acbff7d0086596e68e0e32f5fb8be345 |
Ultrafast shift and rectification photocurrents in GaAs quantum wells: Excitation intensity dependence and the importance of bandmixing
22 Jun 2016
Thanh Huynh
Duc
Department of Physics and CeOPP
Universität Paderborn
Warburger Str. 100D-33098PaderbornGermany
Ho Chi Minh City Institute of Physics
Vietnam
Academy of Science and Technology
Reinold Podzimski
Department of Physics and CeOPP
Universität Paderborn
Warburger Str. 100D-33098PaderbornGermany
Shekhar Priyadarshi
Physikalisch-Technische Bundesanstalt
Bundesallee 100, D38116BraunschweigGermany
Mark Bieler
Physikalisch-Technische Bundesanstalt
Bundesallee 100, D38116BraunschweigGermany
Torsten Meier
Department of Physics and CeOPP
Universität Paderborn
Warburger Str. 100D-33098PaderbornGermany
Mac Dinh
Chi Str
Department of Physics and CeOPP
Universität Paderborn
Warburger Str. 100D-33098PaderbornGermany
Ho Chi
Department of Physics and CeOPP
Universität Paderborn
Warburger Str. 100D-33098PaderbornGermany
Minh City
Vietnam
Ultrafast shift and rectification photocurrents in GaAs quantum wells: Excitation intensity dependence and the importance of bandmixing
22 Jun 2016
A microscopic approach that is based on the multisubband semiconductor Bloch equations formulated in the basis of a 14-band k · p model is employed to compute the temporal dynamics of photocurrents in GaAs quantum wells following the excitation with femtosecond laser pulses. This approach provides a transparent description of the interband, intersubband, and intraband excitations, fully includes all resonant as well as off-resonant excitations, and treats the light-matter interaction non-perturbatively. For linearly polarized excitations the photocurrents contain contributions from shift and rectification currents. We numerically compute and analyze these currents generated by the excitation with femtosecond laser pulses for [110]-and [111]-oriented GaAs quantum wells. It is shown that the often employed perturbative χ (2) -approach breaks down for peak fields larger than about 10 kV/cm and that non-perturbative effects lead to a reduction of the peak values of the shift and rectification currents and to temporal oscillations which originate from Rabi flopping. In particular, we find a complex oscillatory photon energy dependence of the magnitudes of the shift and rectification currents. Our simulations demonstrate that this dependence is the result of mixing between the heavy-and light-hole valence bands. This is a surprising finding since the bandmixing has an even larger influence on the strength of the photocurrents than the absorption coefficient. For [110]-oriented GaAs quantum wells the calculated photon energy dependence is compared to experimental results and a good agreement is obtained which validates our theoretical approach.
A microscopic approach that is based on the multisubband semiconductor Bloch equations formulated in the basis of a 14-band k · p model is employed to compute the temporal dynamics of photocurrents in GaAs quantum wells following the excitation with femtosecond laser pulses. This approach provides a transparent description of the interband, intersubband, and intraband excitations, fully includes all resonant as well as off-resonant excitations, and treats the light-matter interaction non-perturbatively. For linearly polarized excitations the photocurrents contain contributions from shift and rectification currents. We numerically compute and analyze these currents generated by the excitation with femtosecond laser pulses for [110]-and [111]-oriented GaAs quantum wells. It is shown that the often employed perturbative χ (2) -approach breaks down for peak fields larger than about 10 kV/cm and that non-perturbative effects lead to a reduction of the peak values of the shift and rectification currents and to temporal oscillations which originate from Rabi flopping. In particular, we find a complex oscillatory photon energy dependence of the magnitudes of the shift and rectification currents. Our simulations demonstrate that this dependence is the result of mixing between the heavy-and light-hole valence bands. This is a surprising finding since the bandmixing has an even larger influence on the strength of the photocurrents than the absorption coefficient. For [110]-oriented GaAs quantum wells the calculated photon energy dependence is compared to experimental results and a good agreement is obtained which validates our theoretical approach.
I. INTRODUCTION
In systems of sufficiently reduced symmetry it is possible to generate electrical currents on ultrafast time scales without any applied bias simply by the optical excitation with suitable laser pulses. In particular, in noncentrosymmetric semiconductor systems the non-vanishing second-order optical susceptibility χ (2) allows one to generate photocurrents by excitation with a single optical pulse. As was shown by Sipe et al. 6 , χ (2) contains three different contributions which correspond to three kinds of photocurrent named injection, shift, and rectification currents. Injection currents are caused by asymmetric populations of spin-polarized carriers in k-space which originate from Dresselhaus and/or Rashba spin splittings in systems of sufficiently reduced symmetry. To excite spin-polarized carriers circularly polarized light fields are required. Without any additional electric or magnetic fields, injection currents do not exist in bulk GaAs, they are, however, present in GaAs quantum wells (QW) with lower symmetry. 10,19,20 Under the action of an optical field that induces interband transitions the electronic charge density in the noncentrosymmetric GaAs crystal is shifted in real space from the As atoms towards the neighboring Ga atoms and this process leads to the so-called shift current. 15 Additionally the spatial shift of bound electrons leads to a static polarization, i.e., optical rectification. If the op-tical rectification is time varying, e.g., due to the time dependence of the optical pulse envelope, it induces another kind of current: the rectification current. Unlike injection currents which are created by resonantly excited carriers and shift currents which require a resonant excitation in first order and therefore exist for excitation frequencies above the band gap, rectification currents are present for all photon energies.
Previously we have developed a microscopic approach 20 in which we employed the commonly used 14-band k.p method to obtain the electron bandstructure and the Bloch functions for GaAs QWs and used the wave functions to formulate the multiband semiconductor Bloch equations (SBE). We solved the SBE to describe the optoelectronic response excited by ultrashort laser pulses. Using this approach we analyzed injection currents generated by circularly polarized pulses in GaAs QWs and obtained results in quantitative agreement with experiment. 19 When limiting our approach to a perturbative analysis we were able to reproduce the magnitude and the dynamics of the GaAs shift current obtained in Ref. 15, where on the basis of a full ab-initio band structure a second-order (χ (2) ) analysis of the optical response was performed.
Here, we further extend our approach to describe besides injection and shift currents also rectification currents. Thus we are able to describe all three photocurrents that exist in the second-order response of noncen-trosymmetric semiconductors 6 in an unified way directly in the time domain and non-perturbatively in the lightmatter interaction. We would like to point out that (unlike to the case of injection currents) due to the involved non-resonant excitations to higher bands the rotating wave approximation cannot be applied for simulations of shift and rectification currents in the framework of Bloch equations which significantly increases the numerical effort. We apply our microscopic approach to analyze shift and rectification currents in GaAs QW systems. It is shown that for large enough excitation intensities nonperturbative effects arising from Pauli blocking lead to a reduction of the peak values of the shift and rectification currents and to intensity-dependent temporal oscillations which originate from Rabi flopping. The magnitude and direction of the photocurrents depend on the polarization direction of the incident pulse and the strength of different inter-and intersubband coherences to the currents strongly depends on the excitation geometry. Furthermore, we find an oscillatory dependence of the currents on the photon energy, which is also confirmed experimentally. Our simulations demonstrate that this dependence is caused by valence bandmixing. This is an interesting finding since is emphasizes the tremendous influence of bandmixing on transport phenomena and might be at the origin of new applications. To keep the numerical requirements within reasonable limits, excitonic effects are neglected in our present calculations. This paper is organized as follows. In Sec. II, we present our theoretical approach. In Sec. III several numerical results for shift and rectification currents in [110]and [111]-oriented QWs are discussed. Experimental results and a comparison between experiment and theory are shown in Sec. IV. Finally, the main results are briefly summarized in Sec. V.
II. THEORETICAL APPROACH
The energies and wave functions of electrons in a semiconductor QW are described by the Schrödinger equation
h 2 2m 0 ∇ 2 + V 0 + H SO + V conf ψ = εψ,(1)
where V 0 is the periodic lattice potential, H SO is the spin-orbit interaction, and V conf is the confinement potential. To obtain a realistic electronic band structure and wave functions near the Γ-point we employ a 14band k · p method 25-27 within the envelope function approximation. By choosing the z-axis as the growth direc-tion of the QW and writing the electron wave function as ψ = e ik ·r 14 n=1 f n k (z)u n , where k = (k x , k y ) is the in-plane wave vector, u n are band-edge Bloch functions, and f n k are slowly varying envelope functions Eq. (1) becomes
14 m=1 H k.p nm (k) + V n (z)δ nm f m k (z) = ε k f n k (z),(2)
where H k.p nm are the matrix elements of the 14-band k · p Hamiltonian withk = (k x , k y , −i∂/∂z) and V n (z) is the band-offset potential of the well which is taken to be centered at z = 0. To solve Eq. (2) we expand the envelope function into plane waves 27
f n k (z) = N j=1 c j nk φ j (z),
where φ j (z) = 2 L sin πj L (z + L 2 ) for −L/2 ≤ z ≤ L/2 and φ j (z) = 0 for otherwise. This leads to a 14N × 14N eigenvalue equation. By numerical diagonalizations of the matrix for several values of k the electronic band structure is obtained. The number of plane wave functions N and the width L are chosen to ensure convergence of the results.
To analyze the dynamics of photoexcited semiconductor QWs we use a Hamiltonian that contains the band structure and the light-matter interaction in velocity gauge
H = λ,k ε λk a † λk a λk + eA(t) · λ,λ ′ ,k v λλ ′ k a † λk a λ ′ k ,(3)
where ε λk is the energy and a † λk (a λk ) is the creation (annihilation) operator of an electron in band λ with wave vector k , A(t) is the vector potential of the light field, and v λλ ′ k is the velocity matrix element between Bloch states
v λλ ′ k = 1 h 14 n,m=1 dz f n * λk (∇ k H k.p ) nm f m λ ′ k . (4)
From the Heisenberg equation of motion we obtain the well-known multiband SBE 20,28,29 which describe the time evolution of the microscopic interband and intersubband polarizations p λλ ′ k = a † λk a λ ′ k with λ = λ ′ and the electron occupations n λk = a † λk a λk
∂ ∂t p λλ ′ k (t) = i(ω λλ ′ k + i/τ 2 ) p λλ ′ k + ī h eA(t) · v λ ′ λk (n λ ′ k − n λk ) + ī h eA(t) · ν =λ ′ v νλk p νλ ′ k − ν =λ v λ ′ νk p λνk ,(5)∂ ∂t n λk (t) = − 2 h Im eA(t) · λ ′ =λ v λ ′ λk p λ ′ λk − 1 τ 1 n λk − n eq λk (n, T ) ,(6)
where ω λλ ′ k = (ε λk − ε λ ′ k )/h is the transition frequency. τ 1 and τ 2 are introduced phenomenologically in order to model the relaxation of occupations to quasiequilibrium and the dephasing of polarizations due to carrier-phonon and carrier-carrier scattering. In the numerical calculations, we use typical values of τ 1 = 120 fs and τ 2 = 82.3 fs. 29,30 n eq λk (n, T ) is a quasi-equilibrium thermal occupation at T = 300 K which has the same carrier density n(t) as the actual time-dependent occupation.
The exciting light field with a central frequency ω is given by
E(t) = E ω (t)e iωt + c.c.,(7)
where E ω is the slowly-varying envelope function. To describe Gaussian-shaped pulses which propagate along the z-direction we use E ω in the form
E ω (t) = E 0 e − t 2 2τ 2 L (cos θ, sin θe iφ , 0),(8)
where E 0 is the maximal amplitude, τ L is the duration of the Gaussian envelope, θ is the polarization angle with respect to the x-axis, and φ is the phase difference between the x-and the y-components of the field. If φ = ±π/2 and θ = π/4 the light field is circularly polarized, whereas φ = 0, which is used in all simulations presented below, corresponds to linearly polarized pulses. The vector potential used in the SBE (5) and (6) is given by
A(t) = − t −∞ E(t ′ )dt ′ .
By solving the SBE for an initially unexcited QW system we obtain the time-dependent occupations n λk and microscopic polarizations p λλ ′ k . From the occupations we can evaluate the intraband charge current known as injection photocurrent via
j inject (t) = e λ,k v λλk n λk = e c,k v cck n ck + e v,k v vvk n vk ,(9)
where c (v) is a band index of the conduction (valence) band.
From the microscopic polarizations it is possible to compute the charge current induced by interband and intersubband transitions
j inter (t) = e λ,λ ′ =λ,k v λλ ′ k p λλ ′ k = e c,c ′ =c,k v cc ′ k p cc ′ k + e v,v ′ =v,k v vv ′ k p vv ′ k+ e c,v,k v cvk p cvk + c.c.(10)
as well as the interband and intersubband polarization
P inter (t) = e λ,λ ′ =λ,k r λλ ′ k p λλ ′ k = e c,c ′ =c,k r cc ′ k p cc ′ k + e v,v ′ =v,k r vv ′ k p vv ′ k + e c,v,k r cvk p cvk + c.c. ,(11)
where er λλ ′ k is the transition dipole moment which can be obtained from the velocity operator via the relation
v λλ ′ k = iω λλ ′ k r λλ ′ k .
The shift current j shift is defined as the zero-frequency contribution to the interband photocurrent j inter . Similarly, the optical rectification P rect is given by the slowly varying part of the interband polarization P inter . The time-derivative of the optical rectification provides the rectification current
j rect (t) = ∂ ∂t P rect (t)(12)
As explained in Sec. III in more detail, to obtain j shift (t) and j rect (t) we Fourier transform j inter (t) and ∂ ∂t P inter (t), respectively, into the frequency domain, apply a filter around ω = 0, and transform back to the time domain.
In the limit of weak excitation intensities the lightmatter interaction can be considered as a perturbation. Using a second-order perturbation expansion of the SBE (5) and (6), by applying the rotating wave approximation for the first-order microscopic polarization p (1) λλ ′ , and neglecting 2ω terms in the second-order microscopic po-larization p (2) λλ ′ we analytically derive approximate expressions for the injection current, the shift current, and the optical rectification, respectively, that read 1,6 ∂ ∂t j
(2) inject = 2e 3 h 2 ω 2 c,v,k (v cck − v vvk ) E ω · v cvk 2 1/τ 2 (ω − ω cvk ) 2 + 1/τ 2 2 − j (2) inject /τ 1 ,(13)j (2) shift = − 2e 3 h 2 ω 2 Im c,v,k E ω · v cvk ω − ω cvk + i/τ 2 λ =v r vλk (E * ω · v λck ) − λ =c (E * ω · v vλk )r λck ,(14)
and
P (2) rect = 2e 3 h 2 ω 2 Re c,v,k E ω · v cvk ω − ω cvk + i/τ 2 λ =v r vλk ω vλk (E * ω · v λck ) − λ =c (E * ω · v vλk ) r λck ω λck .(15)
III. NUMERICAL RESULTS
In this section, we present and discuss results obtained from numerical solutions of the SBE for GaAs/Al 0.35 Ga 0.65 As QWs grown in the crystallographic [110]-and [111]-directions. The band parameters for GaAs and Al x Ga 1−x As are taken from Ref. 26 and the temperature dependence of the band gap is described by the Varshni relation. In our calculations we consider room temperature (T = 300 K). For the optical excitation we use linearly polarized laser pulses which propagate perpendicular to the QW plane, i.e., in z-direction, and have a Gaussian envelope with a duration of τ L = 150 fs. The geometry of the excitation is illustrated in Fig. 1.
The band structure of the QW system is obtained from a matrix diagonalization using 14 N bands, where we use N ≥ 20. Due to this large number of bands, the direct evaluation of the multisubband SBE (6) and (5) beyond the rotating wave approximation is numerically very intensive. In order to keep the calculation feasible we limit the number of bands included in the numerics to 40. In particular we take into account the eighteen energetically highest valence subbands, six energetically lowest s-like conduction subbands, and sixteen energetically lowest p-like conduction subbands. Though the optical excitation to p-like conduction bands is well nonresonant the presence of these bands in the SBE is necessary. In second-order processes, the p-like conduction band states play the role of intermediate states for the transition of electrons from the valence to the conduction band and thereby contributions to the shift and rectification currents.
Obtaining shift and rectification currents from numerical solutions of the SBE involves basically the following steps: (i) solve the SBE, (ii) compute the current j inter (t) and the interband polarization P inter (t), (iii) Fourier transform j inter (t) and P inter (t) to the frequency domain and apply a frequency filter to remove the high-frequency components, (iv) inversely Fourier transform back to time domain to obtain the shift current j shift (t) and the optical rectification P rect (t). We carefully checked the convergence versus the number of bands and also compared the complete numerical solutions with the analytical second-order approximation, i.e., Eqs. (14) and (15), where we included all 14 N bands to ensure that our approache works properly within excitation conditions considered in the paper. In the following we analyze shift and rectification currents generated by linearly polarized laser pulses. The laser field has a photon energy of 1.54 eV, i.e., 80 meV above the band gap (E g = 1.46 eV). For light polarization along the y-axis we obtain the current flowing in the x-direction, named xyy current. The computed time evolution of xyy shift and rectification current densities in the GaAs QW for different laser amplitudes is displayed in Fig. 2(a)-(c) and Fig. 2(d)-(e), respectively. Here, solid (dashed) lines correspond to a nonperturbative complete (perturbative second-order) solution of the SBE. When the amplitude of the light field is small the perturbative approximation agrees well with the non-perturbative complete solution, see Fig. 2(a) and Fig. 2(d) for E 0 = 10 4 V/cm. In this perturbative regime the shift current follows the envelope of optical pulse intensity while the rectification current has the shape of its time derivative. The small difference between the two approaches is due to contributions from heavy to light hole transitions which are enabled by the finite bandwidth of the incident pulse and can be only described in the nonperturbative solution. When the light field amplitude is large enough such that band filling effects become relevant there are strong deviations between the approximate perturbative and the full non-perturbative results, see Fig. 2(b) and (e) for E 0 = 2 · 10 5 V/cm and Fig. 2(c), and (f) for E 0 = 4 ·10 5 V/cm. In this regime, the secondorder approximation fails to describe the shift current dynamics properly and strongly overestimates the current magnitude. In particular, the non-perturbative solutions of the SBE result in significantly smaller currents since phase-space filling limits the strengths of the optical excitations and due to Rabi flopping in the strongly excited regions of k-space an oscillatory dynamics is obtained, see solid lines in Fig. 2(b), (c), (e), and (f). For bulk GaAs the relevance of phase-space filling effects in limiting the perturbative χ (3) -scaling of two-color injection currents has been confirmed recently 31 and it can therofore be expected that also for shift and rectification currents nonperturbtive signatures should be observable.
In the following, we focus on the weak excitation regime (E 0 = 10 4 V/cm) where the perturbative second- order solution works properly and analyze the dependence of shift and rectification currents on the other excitation conditions, in particular, the light polarization and the photon energy. In Fig. 3 we show the x-and y-components of the peak value of shift current density ( Fig. 3(a)) and rectification current density ( Fig. 3(b)) as function of θ, where θ is measured with respect to the x = [001] direction and determines the polarization direction of the linearly polarized incident pulse. The computed current components are well described by a formula which also arises from a macroscopic symmetry analysis: j x = A + B cos 2θ and j y = C sin 2θ. 16,17 We note that for light polarization parallel to the xaxis (θ = 0 • ) there is a finite current flowing in the x-direction, i.e., xxx current. Such kind of current is not present in bulk GaAs but does exist in [110]-oriented GaAs QWs because of the symmetry reduction.
The peak values of the xyy (xxx) shift current densities versus the photon energy are presented by the black solid lines in Fig. 4 for θ = 90 • (θ = 0 • ), i.e., the polarization of the incident field is in y-direction (x-direction). With increasing the photon energy nearby and above the bandgap both xyy and xxx shift currents show a non-monotonic and complex variation. The xxx current shows a very interesting sign change which corresponds to a reversal of the current direction at the photon energy of 1.504 eV, see the solid line in Fig. 4(b). In order to understand the origin of the complex dependence of shift currents we separately calculate three different contributions to the net current according to Eq. (10). Currents originating from inter-conduction band polarizations p cc ′ (j cc ′ ), inter-valence band polarizations p vv ′ (j vv ′ ), and inter band polarizations p cv (j cv ) are evaluated separately and plotted as dotted, dashed, and dash-dotted lines, respectively. Since j cc ′ and j cv shown in Fig. 4(a) follow the step-like increase of the two-dimensional interband density of states the strong variation of j vv ′ in both magnitude and direction is responsible for the non-monotonic photon-energy dependence of the net xyy shift current.
In the case of the xxx current, because of very small contributions of j cc ′ and j cv , the net shift current is mainly given by j vv ′ . To obtain non-vanishing j vv ′ it is necessary that velocity matrix elements v cv , v cv ′ , and v vv ′ are simultaneously non vanishing. At k = 0, heavy hole and light hole states are mixed which allows the velocity matrix elements between all pairs of c, v and v ′ subbands to have finite and k-dependent values. If we artificially remove the valence band mixing by setting the coupling matrix elements between heavy hole and light hole to zeros the contribution j vv ′ vanishes and we obtain a monotonically increasing xyy shift current and a negligibly small xxx shift current shown as red solid lines in Fig. 4(a) and Fig. 4(b), respectively. The photon energy dependence of xyy and xxx rectification currents is presented in Fig. 5 showing also a complex, oscillatory variation. Using the same analysis as for shift currents we find that the dominant contribution to both the xyy and xxx rectification currents comes from inter-valence band polarizations, i.e., j vv ′ . The photon-energy dependence of rectification current is therefore governed by valence band mixing. Furthermore, the sign change of the optical rectification at the resonance excitation (see Eq. (15) for ω = ω cvk ) may lead to a direction reversal of the rectification current with increasing photon energy. 3,15 B. , and z = [111] is chosen. We compute the photocurrents in a 10 nm wide GaAs QW using an incident pulse with a peak amplitude of E 0 = 10 4 V/cm. Figure 6 shows the x-and the y-components of shift and rectification current densities versus the polarization angle θ for a photon energy of 1.52 eV. The calculated data are fitted well by the formulas: j x = A cos 2θ and
IV. COMPARISON TO EXPERIMENTS
Since the shift current involves real carriers its amplitude is typically much larger than the rectification current amplitude under the same excitation conditions, see also Figs. 4, 5, and 7. Therefore for an analysis of the experiments we only consider shift currents and neglect rectification currents. A comparison between experiments and simulations on injection currents is given in Ref. 19.
For the shift current experiments we employed a standard free-space terahertz (THz) setup in transmission geometry. [32][33][34] The samples consist of [110]-oriented GaAs QWs with well widths of 12 nm, 15 nm, and 20 nm. They were excited at normal incidence with a 150 fs laser pulse originating from a Ti:sapphire oscillator with a repetition rate of 76 MHz. The optical peak intensity in front of the samples was 60 MW/cm 2 resulting in two-dimensional carrier densities of approximately 2 × 10 11 cm −2 . The center photon energy of the femtosecond laser could be varied allowing for different excitation conditions. The polarization of the initially linearly polarized pump beam was aligned to the x direction of the samples.
After generation, the shift currents decay on a femtosecond time-scale. Since the emitted electromagnetic radiation is proportional to the time derivative of the current transients, THz radiation is emitted. The THz radiation was collected from the samples back surface with an off-axis parabolic mirror, guided to a second offaxis parabolic mirror and focused down onto 1 mm thick [110]-oriented ZnTe crystal. The probe pulse was guided through a small hole in the second off-axis parabolic mirror, collinearly overlapped with the THz pulse and read out the electric field-induced refractive index change of the electro-optic crystal. A silicon wafer and a THz polarizer were placed between the two off-axis parabolic mirrors. The silicon wafer blocked any scattered pump light. The THz polarizer allowed us to detect currents flowing in certain directions in the sample since the polarization of the THz radiation in the far field is parallel to the direction of the current flow. This is a large advantage of THz experiments over experiments based on charge collection at electrodes, since the THz setup is only sensitive to currents flowing in the plane of the sample. In the experiments the THz polarizer was oriented such that only shift currents flowing along the x direction were detected. The bandwidth of the experimental setup was limited by the velocity mismatch between the group velocity of the optical pulse and the phase velocity of the THz pulse in the electro-optic crystal. These restrictions allowed for the detection of frequencies up to approximately 3 THz. All experiments were done at room temperature.
In Fig. 8 we plot the THz amplitude emitted from the xxx shift currents in the three QW samples versus excitation photon energy. Additionally, the calculated peak magnitude of the xxx shift current is shown. Starting with the 20 nm QW, we obtain an excellent agreement between theory and experiment in nearly the complete photon energy range. In particular the peaks at 1.5 eV and 1.57 eV and also a zero crossing at 1.45 eV are nicely reproduced. The deviations at small photon energy are most likely because of neglect of Coulomb interaction in the simulations. Moreover the deviations at large photon energies might be the result of improper rescaling of the measured signals. At these photon energies the optical pump and probe power decreased due to the end of the tuning range of our femtosecond laser.
The deviations between experiment and theory at small and large photon energies also appear for the 15 nm and 12 nm QWs; otherwise the overall agreement between experiment and theory is also good for these QW samples. In general the good agreement between experiment and theory is very important since it (i) validates the theoretical approach and (ii) confirms that the measured signal is mainly caused by the shift of the electron charge during excitation and not by scattering contributions. 2
V. CONCLUSIONS
We present an unified microscopic approach which is capable of describing fully dynamically all three kinds of single-frequency photocurrents in noncentrosymmetric semiconductor QWs, i.e., injection, shift, and rectification currents. Our approach has been applied to analyze shift and rectification currents generated by linearly polarized incident pulses in [110]-and [111]-oriented GaAs/AlGaAs QWs.
The dependence of the photocurrents on the polarization direction of the incident laser pulses has been studied. For shift and rectification currents we compared the non-perturbative solution of the SBE to a second-order approximation. It is demonstrated that intrinsic nonlinearities arising from phase-space filling effects and Rabi flopping which are not included in the second-order approximation lead to a significant difference between the two solutions in the limit of strong excitation intensities (E 0 > 10 5 V/cm).
Unlike the optical interband absorption which increases in steps with increasing photon energy, we find an unexpected complex and non-monotonic dependence of shift and rectification currents on the photon energy in both theory and experiments. In certain spectral regions both currents may even change their direction. It has been demonstrated that this dependence is the result of bandmixing of the heavy-and light-hole valence bands. This excitation frequency dependence of the photocurrents might be employed for new applications such as opto-electronic converters or modulators.
1. (color online) Schematic illustration of the shift and rectification current generation using linearly polarized light pulses. θ is the light polarization angle with respect to the x-axis.
FIG. 2 .
2(color online) Dynamics of shift and rectification currents in a 10 nm wide [110]-oriented GaAs QW for different peak amplitudes of the laser pulses. The dashed lines are obtained by a perturbative second-order analysis of the light-matter interaction whereas the solid lines are obtained by solving the full SBE.
A. [110]-oriented quantum wells We consider a GaAs QW of 10 nm width grown along the [110] crystallographic direction using a coordinate system in which x = [001], y = [110], and z = [110].
FIG. 3 .
3Components in x-and y-directions of the peak value of shift (a) and rectification (b) current densities as function of the light polarization angle θ in a 10 nm [110]-oriented GaAs QW.
FIG. 4 .
4(color online) Peak values of the xyy (a) and xxx (b) shift current densities versus photon energy for a 10 nm wide [110]-oriented GaAs QW. Black solid line: total shift current. Blue dotted line: inter-conduction band contribution j cc ′ . Green dashed line: inter-valence band contribution j vv ′ . Magenta dash-dotted line: inter band contribution jcv. Red solid line: without heavy hole-light hole coupling. The vertical arrows highlight the photon energies corresponding to interband transitions at k = 0.
FIG. 5 .
5(color online) Peak values of the xyy (a) and xxx (b) rectification current densities versus the photon energy for a 10 nm wide [110]-oriented GaAs QW. The description of the lines is the same as inFig. 4.
FIG. 6 .
6Peak value of shift (a) and rectification (b) current densities in x-and y-directions as function of the light polarization angle θ in a 10 nm wide [111]-oriented GaAs QW.
[111]-oriented quantum wells For [111]-oriented QWs a coordinate system of x = [112], y = [110]
FIG. 7 .
7Peak value of shift (a) and rectification (b) current densities as function of photon energy in a 10 nm wide [111]oriented GaAs QW. Vertical arrows show the photon energies corresponding to interband transitions at k = 0. j y = −A sin 2θ. The polarization-direction dependence of the shift and rectification currents in [111]-oriented QW is quite similar to [111]-oriented bulk GaAs. 3The photon-energy dependence of the shift and the rectification current densities is shown inFig. 7for the light polarization parallel to the y = [110] direction (θ = 90 • ). Similarly to the case of [110]-oriented QWs, also for [111]oriented QWs a complex and non-monotonic dependence of the shift and the rectification currents on photon energy is obtained.
FIG
. 8. (color online) Measured THz amplitudes emitted from shift currents (solid black) and calculated absolute peak values of shift currents (dashed red) in x-direction in [110]oriented GaAs QWs of different well width: (a) 20 nm, (b) 15 nm, and (c) 12 nm.
PACS numbers: 72.40.+w, 73.63.Hs, 78.47.J-, 78.67.De
For providing computing time we thank the PC 2. Paderborn Center for Parallel Computing262012.26. For providing computing time we thank the PC 2 (Paderborn Center for Parallel Computing).
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| []
|
[
"Learning Meta-Embeddings by Using Ensembles of Embedding Sets",
"Learning Meta-Embeddings by Using Ensembles of Embedding Sets"
]
| [
"Wenpeng Yin [email protected] \nCenter for Information and Language Processing\nUniversity of Munich\nGermany\n",
"Hinrich Schütze \nCenter for Information and Language Processing\nUniversity of Munich\nGermany\n"
]
| [
"Center for Information and Language Processing\nUniversity of Munich\nGermany",
"Center for Information and Language Processing\nUniversity of Munich\nGermany"
]
| []
| Word embeddings -distributed representations of words -in deep learning are beneficial for many tasks in natural language processing (NLP). However, different embedding sets vary greatly in quality and characteristics of the captured semantics. Instead of relying on a more advanced algorithm for embedding learning, this paper proposes an ensemble approach of combining different public embedding sets with the aim of learning meta-embeddings. Experiments on word similarity and analogy tasks and on part-of-speech tagging show better performance of metaembeddings compared to individual embedding sets. One advantage of meta-embeddings is the increased vocabulary coverage. We will release our meta-embeddings publicly. | null | [
"https://arxiv.org/pdf/1508.04257v2.pdf"
]
| 3,195,655 | 1508.04257 | c7cf38bc337760dee4fc6766060178a5c1904127 |
Learning Meta-Embeddings by Using Ensembles of Embedding Sets
Wenpeng Yin [email protected]
Center for Information and Language Processing
University of Munich
Germany
Hinrich Schütze
Center for Information and Language Processing
University of Munich
Germany
Learning Meta-Embeddings by Using Ensembles of Embedding Sets
Word embeddings -distributed representations of words -in deep learning are beneficial for many tasks in natural language processing (NLP). However, different embedding sets vary greatly in quality and characteristics of the captured semantics. Instead of relying on a more advanced algorithm for embedding learning, this paper proposes an ensemble approach of combining different public embedding sets with the aim of learning meta-embeddings. Experiments on word similarity and analogy tasks and on part-of-speech tagging show better performance of metaembeddings compared to individual embedding sets. One advantage of meta-embeddings is the increased vocabulary coverage. We will release our meta-embeddings publicly.
Introduction
Recently, deep neural network (NN) models have achieved remarkable results in NLP (Collobert and Weston, 2008;Sutskever et al., 2014;Rocktäschel et al., 2015). One reason for these results are word embeddings, compact distributed word representations learned in an unsupervised manner from large corpora (Bengio et al., 2003;Mnih and Hinton, 2009;Mikolov et al., 2010;Mikolov, 2012;Mikolov et al., 2013a).
Some prior work has studied differences in performance of different embedding sets. For example, Chen et al. (2013) showed that the embedding sets HLBL (Mnih and Hinton, 2009), SENNA (Collobert and Weston, 2008), Turian (Turian et al., 2010) and Huang (Huang et al., 2012) have great variance in quality and characteristics of the semantics captured. Hill et al. (2014;2015a) showed that embeddings learned by NN machine translation models can outperform three representative monolingual embedding sets: word2vec (Mikolov et al., 2013b), GloVe (Pennington et al., 2014) and CW (Collobert and Weston, 2008). Bansal et al. (2014) found that Brown clustering, SENNA, CW, Huang and word2vec yield significant gains for dependency parsing. Moreover, using these representations together achieved the best results, suggesting their complementarity. These prior studies motivate us to explore an ensemble approach. Since each embedding set is trained by a different NN on a different corpus and can be treated as a distinct description of words, our expectation is that the ensemble contains more information than each component embedding set. We want to leverage this diversity to learn betterperforming word embeddings.
The ensemble approach has two benefits. First, enhancement of the representations: metaembeddings perform better than the individual embedding sets. Second, coverage: meta-embeddings cover more words than the individual embedding sets. The first three ensemble methods we introduce are CONC, SVD and 1TON and they directly only have the benefit of enhancement. They learn meta-embeddings on the overlapping vocabulary of the embedding sets. CONC concatenates the vectors of a word from the different embedding sets. SVD performs dimension reduction on this concatenation. 1TON assumes that a meta-embedding for the word exists and uses this meta-embedding to predict representations of the word in the indi-vidual embedding sets -the resulting fine-tuned meta-embedding is expected to contain knowledge from all individual embedding sets.
To also address the objective of increased coverage of the vocabulary, we introduce 1TON + , a modification of 1TON that learns meta-embeddings for all words in the vocabulary union in one step. Let an out-of-vocabulary (OOV) word w of embedding set ES be a word that is not covered by ES (i.e., ES does not contain an embedding for w). 1 1TON + first randomly initializes the embeddings for OOVs and the meta-embeddings, then uses a prediction setup similar to 1TON to update meta-embeddings as well as OOV embeddings. Thus, 1TON + simultaneously achieves two goals: learning metaembeddings and extending the vocabulary (for both meta-embeddings and invidual embedding sets).
An alternative method that increases coverage is MUTUALLEARNING. MUTUALLEARNING learns the embedding for a word that is an OOV in embedding set from its embeddings in other embedding sets. We will use MUTUALLEARNING to increase coverage for CONC, SVD and 1TON, so that these three methods (when used together with MU-TUALLEARNING) have the advantages of both performance enhancement and increased coverage.
In summary, meta-embeddings have two benefits compared to individual embedding sets: enhancement of performance and improved coverage of the vocabulary. Below, we demonstrate this experimentally for three tasks: word similarity, word analogy and POS tagging.
If we simply view meta-embeddings as a way of coming up with better embeddings, then the alternative is to develop a single embedding learning algorithm that produces better embeddings. Some improvements proposed before have the disadvantage of increasing the training time of embedding learning substantially; e.g., the NNLM presented in (Bengio et al., 2003) is an order of magnitude less efficient than an algorithm like word2vec and, more generally, replacing a linear objective function with a nonlinear objective function increases training time. Similarly, fine-tuning the hyperparameters of the embedding learning algorithm is complex and 1 We do not consider words in this paper that are not covered by any of the individual embedding sets. OOV always refers to a word that is covered by at least one embedding set. time consuming. In many cases, it is not possible to retrain using a different algorithm because the corpus is not publicly available. But even if these obstacles could be overcome, it is unlikely that there ever will be a single "best" embedding learning algorithm. So the current situation of multiple embedding sets with different properties being available is likely to persist for the forseeable future. Metaembedding learning is a simple and efficient way of taking advantage of this diversity. As we will show below they combine several complementary embedding sets and the resulting meta-embeddings are stronger than each individual set.
Related Work
Related work has focused on improving performance on specific tasks by using several embedding sets simultaneously. To our knowledge, there is no work that aims to learn generally useful metaembeddings from individual embedding sets. Tsuboi (2014) incorporated word2vec and GloVe embeddings into a POS tagging system and found that using these two embedding sets together was better than using them individually. Similarly, Turian et al. (2010) found that using Brown clusters, CW embeddings and HLBL embeddings for NER and chunking tasks together gave better performance than using these representations individually. Luo et al. (2014) adapted CBOW (Mikolov et al., 2013a) to train word embeddings on different datasets -a Wikipedia corpus, search click-through data and user query data -for web search ranking and for word similarity. They showed that using these embeddings together gives stronger results than using them individually.
These papers show that using multiple embedding sets is beneficial. However, they either use embedding sets trained on the same corpus (Turian et al., 2010) or enhance embedding sets by more training data, not by innovative learning algorithms (Luo et al., 2014). In our work, we can leverage any publicly available embedding set learned by any learning algorithm. Our meta-embeddings are generically useful and are learned by supervised training of an explicit model of the dependencies between embedding sets and (except for CONC) not by simple concatenation.
In this work, we use five released embedding sets. (i) HLBL. Hierarchical log-bilinear (Mnih and Hinton, 2009) (Pennington et al., 2014). 1,193,514 word embeddings, 300 dimensions; training corpus: 42 billion tokens of web data, from Common Crawl. (iv) CW (Collobert and Weston, 2008). Released by Turian et al. (2010); 5 268,810 word embeddings, 200 dimensions; training corpus: same as HLBL. (v) word2vec (Mikolov et al., 2013b) CBOW; 6 929,022 word embeddings (we discard phrase embeddings), 300 dimensions; training corpus: Google News (about 100 billion words).
The intersection of the five vocabularies has size 35,965, the union has size 2,788,636.
Ensemble Methods
This section introduces the four ensemble methods: CONC, SVD, 1TON and 1TON + .
CONC: Concatenation
In CONC, the meta-embedding of w is the concatenation of five embeddings, one each from the five embedding sets. For GloVe, we perform L2 normalization for each dimension across the vocabulary as recommended by the GloVe authors. Then each embedding of each embedding set is L2-normalized. This ensures that each embedding set contributes equally (a value between -1 and 1) when we compute similarity via dot product.
We would like to make use of prior knowledge and give more weight to well performing embedding sets. In this work, we give GloVe and word2vec Figure 1: Performance vs. Weight scalar i weight i > 1 and weight 1 to the other three embedding sets. We use MC30 (Miller and Charles, 1991) as dev set, since all embedding sets fully cover it. We set i = 8, the value in Figure 1 where performance reaches a plateau. After L2 normalization, GloVe and word2vec embeddings are multiplied by i and remaining embedding sets are left unchanged. The dimensionality of CONC meta-embeddings is k = 100 + 50 + 300 + 200 + 300 = 950.
SVD: Singular Value Decomposition
We do SVD on above weighted concatenation vectors of dimension k = 950.
Given a set of CONC representations for n words, each of dimensionality k, we compute an SVD decomposition C = U SV T of the corresponding n×k matrix C. We then use U d , the first d dimensions of U , as the SVD meta-embeddings of the n words. We apply L2-normalization to embeddings; similarities of SVD vectors are computed as dot products.
d denotes the dimensionality of meta-embeddings in SVD, 1TON and 1TON + . We use d = 200 throughout and investigate the impact of d below.
1TON
Figure 2 depicts the simple neural network we employ to learn meta-embeddings in 1TON. White rectangles denote known embeddings. The target to learn is the meta-embedding (shown as shaded rectangle). Meta-embeddings are initialized randomly. Let c be the number of embedding sets under consideration, V 1 , V 2 , . . . , V i , . . . , V c their vocabularies and V ∩ = ∩ c i=1 V i the intersection, used as training set. Let V * denote the meta-embedding space.
We define a projection f * i from space V * to space V i (i = 1, 2, . . . , c) as follows:
w i = M * i w * (1) where M * i ∈ R d i ×d , w * ∈ R d is the meta- embedding of word w in space V * andŵ i ∈ R d i is the projected (or learned) representation of word w in space V i .
The training objective is minimizing the sum of (i) squared error:
E = i |ŵ i − w i | 2(2)
and (ii) L2 cost (sum of squares) of the projection weights M * i . As for CONC and SVD, we weight GloVe and word2vec by i = 8. For 1TON, we implement this by applying the factor i to the corresponding loss part of the squared error.
The principle of 1TON is that we treat each individual embedding as a projection of the metaembedding, similar to principal component analysis. An embedding is a description of the word based on the corpus and the model that were used to create it. The meta-embedding tries to recover a more comprehensive description of the word when it is trained to predict the individual descriptions.
1TON can also be understood as a sentence modeling process, similar to DBOW (Le and Mikolov, 2014). The embedding of each word in a sentence s is a partial description of s. DBOW combines all partial descriptions to form a comprehensive description of s. DBOW initializes the sentence representation randomly, then uses this representation to predict the representations of individual words. The sentence representation of s corresponds to the meta-embedding in 1TON; and the representations of the words in s correspond to the five embeddings for a word in 1TON.
1TON +
Recall that an OOV (with respect to embedding set ES) is defined as a word unknown in ES. 1TON + is an extension of 1TON that learns embeddings for OOVs; thus, it does not have the limitation that it can only be run on overlapping vocabulary. Figure 2, we assume that the current word is an OOV in embedding sets 3 and 5. Hence, in the new learning task, embeddings 1, 2, 4 are known, and embeddings 3 and 5 and the meta-embedding are targets to learn.
We initialize all OOV representations and metaembeddings randomly and use the same mapping formula as for 1TON to connect a metaembedding with the individual embeddings. Both meta-embedding and initialized OOV embeddings are updated during training.
Each embedding set contains information about only a part of the overall vocabulary. However, it can predict what the remaining part should look like by comparing words it knows with the information other embedding sets provide about these words. Thus, 1TON + learns a model of the dependencies between the individual embedding sets and can use these dependencies to infer what the embedding of an OOV should look like.
CONC, SVD and 1TON compute metaembeddings only for the intersection vocabulary. 1TON + computes meta-embeddings for the union of all individual vocabularies, thus greatly increasing the coverage of individual embedding sets.
MUTUALLEARNING
MUTUALLEARNING is a method that extends CONC, SVD and 1TON such that they have increased coverage of the vocabulary. With MUTU-ALLEARNING, all four ensemble methods -CONC, SVD, 1TON and 1TON + -have the benefits of both performance enhancement and increased coverage and we can use criteria like performance, compactness and efficiency of training to select the best ensemble method for a particular application. MUTUALLEARNING is applied to learn OOV embeddings for all c embedding sets; however, for ease of exposition, let us assume we want to compute embeddings for OOVs for embedding set j only, based on known embeddings in the other c − 1 embedding sets, with indexes i ∈ {1 . . . j − 1, j + 1 . . . c}. We do this by learning c − 1 mappings f ij , each a projection from embedding set E i to embedding set E j .
Similar to Section 4.3, we train mapping f ij on the intersection V i ∩ V j of the vocabularies covered by the two embedding sets. Formally,ŵ
j = f ij (w i ) = M ij w i where M ij ∈ R d j ×d i , w i ∈ R d i
denotes the representation of word w in space V i and w j is the projected meta-embedding of word w in space V j . Training loss has the same form as for 1TON. A total of c − 1 projections f ij are trained to learn OOV embeddings for embedding set j.
Let w be a word unknown in the vocabulary V j of embedding set j, but known in V 1 , V 2 , . . . , V k . To compute an embedding for w in V j , we first compute the k projections f 1j (w 1 ), f 2j (w 2 ), . . ., f kj (w k ) from the source spaces V 1 , V 2 , . . . , V k to the target space V j . Then, the element-wise average of f 1j (w 1 ), f 2j (w 2 ), . . ., f kj (w k ) is treated as the representation of w in V j . Our motivation is that -assuming there is a true representation of w in V j and assuming the projections were learned well -we would expect all the projected vectors to be close to the true representation. Also, each source space contributes potentially complementary information. Hence averaging them is a balance of knowledge from all source spaces.
Experiments
We train NNs by back-propagation with AdaGrad (Duchi et al., 2011) and mini-batches. Table 1 gives hyperparameters.
We report results on three tasks: word similarity, word analogy and POS tagging.
Word Similarity and Analogy Tasks
We evaluate on SimLex-999 (Hill et al., 2015b), WordSim353 (Finkelstein et al., 2001), RG (Rubenstein and Goodenough, 1965) and RW (Luong et al., 2013). For completeness, we also show results for MC30, the validation set.
The word analogy task proposed in (Mikolov et al., 2013b) consists of questions like, "a is to b as c is to ?". The dataset contains 19,544 such questions, divided into a semantic subset of size 8869 and a syntactic subset of size 10,675. Table 2 gives results on similarity and analogy. Numbers in parentheses are line numbers in what follows. Block "ind-full" (1-5) lists the performance of individual embedding sets on the full vocabulary. Results on lines 6-34 are for the intersection of the vocabularies of the five embedding sets: "ind-overlap" contains the performance of individual embedding sets, "ensemble" the performance of our four ensemble methods and "discard" the performance when one component set is removed.
The four ensemble approaches are very promising (31-34). For CONC, discarding HLBL, Huang or CW does not hurt performance: CONC (31), CONC(-HLBL) (11), CONC(-Huang) (12) and CONC(-CW) (14) beat each individual embedding set (6-10) in all tasks. GloVe contributes most in SimLex-999, WS353, MC30 and RG; word2vec contributes most in RW and word analogy tasks. SVD (32) reduces the dimensionality of CONC from 950 to 200, but still gains performance in SimLex-999 and RG. GloVe contributes most in SVD (larger losses on line 18 vs. lines [16][17][19][20]. Other embeddings contribute inconsistently.
1TON performs well only on word analogy, but it gains great improvement when discarding CW embeddings (24). 1TON + performs better than 1TON: it has stronger results when considering all embedding sets, and can still outperform individual embedding sets while discarding HLBL (26), Huang (27) or CW (29).
These results demonstrate that ensemble methods using multiple embedding sets produce stronger embeddings. However, it does not mean the more embedding sets the better. Whether an embedding set helps, depends on the complementarity among the sets as well as how we measure the ensemble results. (21) Table 2: Results on five word similarity tasks and analogical reasoning. The number of OOVs is given in parentheses for each result.
"ind-full/ind-overlap": individual embedding sets with respective full/overlapping vocabulary; "ensemble": ensemble results using all five embedding sets; "discard": one of the five embedding sets is removed. If a result is better than all methods in "ind-overlap", then it is bolded. CONC, the simplest ensemble, has robust performance. However, using embeddings of size 950 as input may mean too many parameters to tune for deep learning. The other three methods -SVD, 1TON, 1TON + -all have the advantage of smaller dimensionality. SVD reduces CONC's dimensionality dramatically and still keeps competitive performance, especially on word similarity. 1TON is competitive on analogy, but weak on word similarity. 1TON + performs consistently strongly on word similarity and analogy.
Not all state-of-the-art results are included in Table 2. One reason is that a fair comparison is only possible on the shared vocabulary, so methods without released embeddings cannot be included. However, GloVe and word2vec are widely recognized as state-of-the-art embeddings. In any case, our main contribution is to present ensemble frameworks which show that a combination of complementary embedding sets produces better-performing meta-embeddings.
System comparison of learning OOV embeddings. In Table 3, we extend the vocabularies of each individual embedding set ("ind" block) and our ensemble approaches ("ensemble" block) to the vocabulary union, reporting results on RW and analogy -these tasks contain the most OOVs. As both word2vec and GloVe have full coverage on analogy, we do not rereport them in this table. For each embedding set, we can compute the representation of an OOV (i) as a randomly initialized vector (RND); (ii) as the average of embeddings of all known words (AVG); (iii) by MUTU-ALLEARNING (ml) and (iv) by 1TON + . 1TON + learns OOV embeddings for individual embedding sets and meta-embeddings simultaneously, and it would not make sense to replace these OOV embeddings computed by 1TON + with embeddings computed by "RND/AVG/ml". Hence, we do not report "RND/AVG/ml" results for 1TON + . Table 3 shows four interesting aspects. (i) MUTU-ALLEARNING helps much if an embedding set has lots of OOVs in certain task; e.g., MUTUALLEARN-ING is much better than AVG and RND on RW, and outperforms RND considerably for CONC, SVD and 1TON on analogy. However, it cannot make big difference for HLBL/CW on analogy, probably because these two embedding sets have much fewer OOVs, in which case AVG and RND work well enough. (ii) AVG produces bad results for CONC, SVD and 1TON on analogy, especially in the syntactic subtask. We notice that those systems have large numbers of OOVs in word analogy task. If for analogy "a is to b as c is to d", all four of a, b, c, d are OOVs, then they are represented with the same average vector. Hence, similarity between b − a + c and each OOV is 1.0. In this case, it is almost impossible to predict the correct answer d. Unfortunately, methods CONC, SVD and 1TON have many OOVs, resulting in the low numbers in Table 3. (iii) MUTUALLEARNING learns very effective embeddings for OOVs. CONC-ml, 1TON-ml and SVDml all get better results than word2vec and GloVe on analogy (e.g., for semantic analogy: 88.1, 87.3, 88.2 vs. 81.4 for GloVe). Considering further their bigger vocabulary, these ensemble methods are very strong representation learning algorithms. (iv) The performance of 1TON + for learning embeddings for OOVs is competitive with MUTUALLEARNING. For HLBL/Huang/CW, 1TON + performs slightly better than MUTUALLEARNING in all four metrics. Comparing 1TON-ml with 1TON + , 1TON + is better than "ml" on RW and semantic task, while performing worse on syntactic task. Figure 4 shows the influence of dimensionality d for SVD, 1TON and 1TON + . Peak performance for different data sets and methods is reached for d ∈ [100, 500]. There are no big differences in the averages across data sets and methods for high enough d, roughly in the interval [150,500]. In summary, as long as d is chosen to be large enough (e.g., ≥ 150), performance is robust.
We will release the meta-embeddings produced by methods SVD, 1TON and 1TON + for d = 200 and also the meta-embeddings for method CONC.
Domain Adaptation for POS Tagging
This section evaluates individual embedding sets and meta-embeddings on POS tagging. FLORS (Schnabel and Schütze, 2014), the best performing POS tagger for unsupervised domain adaptation, acts as testbed, in which POS tagging is a window-based, multilabel classification problem using a linear SVM. A word's representation consists of four feature vectors based on suffix, shape, left and right distributional neighbors respectively. We Table 4: POS tagging results on six target domains. "baselines" lists representative systems for this task, including FLORS. "+indiv / +meta": FLORS with individual embedding set / meta-embeddings. Bold means higher than "baselines" and "+indiv". insert word's embedding as the fifth feature vector. All embedding sets (except for 1TON + ) are extended to the union vocabulary by MUTUALLEARN-ING. We follow Schnabel and Schütze (2014) for all feature learning and also train on sections 2-21 of Wall Street Journal (WSJ) and evaluate on the development sets of six different target domains: five SANCL (Petrov and McDonald, 2012) domainsnewsgroups, weblogs, reviews, answers, emailsand sections 22-23 of WSJ for in-domain testing. Table 4 gives results for some representative systems ("baselines"), FLORS with individual embedding sets ("+indiv") and FLORS with metaembeddings ("+meta"). Following conclusions can be drawn. (i) Not all individual embedding sets are beneficial in this task; e.g., HLBL embeddings make FLORS perform worse in 11 out of 12 cases. (ii) However, in most cases, embeddings improve system performance, which is consistent with prior work on using embeddings for this type of task (Xiao and Guo, 2013;Yang and Eisenstein, 2014;Tsuboi, 2014). (iii) Meta-embeddings generally help more than the individual embedding sets, except for SVD (which only performs better in 3 out of 12 cases).
Conclusion
This work presented four ensemble methods -CONC, SVD, 1TON and 1TON + -for learning meta-embeddings from multiple embedding sets. Experiments on word similarity, word analogy and POS tagging indicated the high quality of these meta-embeddings. The ensemble methods have the added advantage of increasing vocabulary coverage. We will release the meta-embeddings.
Figure 2 :
21toN
Figure 3 :
31toN + Figure 3 depicts 1TON + . In contrast to
(-HLBL) 46.0 (3) 76.5 (21) 86.3 (0) 82.5 (1) 63.0 (1211) 93.2 (8486) 74.0 (1859) 74.8 12 CONC (-Huang) 46.1 (3) 76.5 (21) 86.3 (0) 82.5 (1) 62.9 (1212) 93.2 (8486) 74.0 (1859) 74.8 13 CONC (-GloVe) 44.0 (3) 69.4 (21) 79.1 (0) 75.6 (1) 61.5 (1212) 89.3 (8486) 72.7 (1859) 73.4 14 CONC (-CW) 46.0 (3) 76.5 (21) 86.6 (0) 82.5 (1) 62.9 (1212) 93.2 (8486) 73.9 (1859) (-HLBL) 48.5 (3) 76.1 (21) 85.6 (0) 82.7 (1) 61.5 (1211) 90.6 (8486) 69.5 (1859) 70.4 17 SVD (-Huang) 48.8 (3) 76.5 (21) 85.4 (0) 83.5 (1) 61.7 (1212) 91.4 (8486) 69.(-HLBL) 46.3 (3) 75.5 (21) 82.4 (0) 81.0 (1) 60.1 (1211) 91.9 (8486) 75.9 (1859) 76.5 22 1TON (-Huang) 46.5 (3) 75.4 (21) 82.4 (0) 82.3 (1) 60.2 (1212) 93.5 (8486) 76.3 (1859) 77.0 23 1TON (-GloVe) 43.4 (3) 66.5 (21) 76.5 (0) 75.3 (1) 56.5 (1212) 89.0 (8486) 73.8 (1859) 74.5 24 1TON (-CW) 47.4 (3) 76.5 (21) 84.8 (0) 83.4 (1) 62.0 (1212) 91.4 (8486) 73.1 (1859) 73.8 25 1TON (-W2V) 46.3 (3) 75.9 (21) 80.1 (0) 78.6 (1) 56.8 (1212) 92.2 (8486) 72.2 (1859) 73.0 26 1TON + (-HLBL) 46.1 (3) 75.8 (21) 85.5 (0) 83.3 (1) 62.3 (1211) 92.2 (8486) 76.2 (1859) 76.9 27 1TON + (-Huang) 46.2 (3) 76.1 (21) 86.3 (0) 83.3 (1) 62.2 (1212) 93.8 (8486) 76.1 (1859) 76.8 28 1TON + (-GloVe) 45.3 (3) 71.2 (21) 80.0 (0) 75.7 (1) 62.5 (1212) 90.0 (8486) 73.3 (1859) 74.0 29 1TON + (-CW) 46.9 (3) 78.1 (21) 85.5 (0) 83.9 (1) 62.7 (1212) 91.8 (8486) 73.3 (1859) 74.1 30 1TON + (-W2V) 45.8 (3) 76.2 (21) 84.4 (0) 83.1 (1) 60.9 (1212) 92.4 (8486) 72.4 (1859) 73.2 ensemble 31 CONC 46.0 (3) 76.5 (21) 86.3 (0) 82.5 (1) 62.9 (1212) 93.2 (8486) 74.0 (1859) 74.8 32 SVD 48.5 (3) 76.0 (21) 85.7 (0) 82.7 (1) 61.5 (1212) 90.6 (8486) 69.5 (1859) 70.4 33 1TON 46.4 (3) 74.5 (21) 80.7 (0) 80.7 (1) 60.1 (1212) 91.9 (8486) 76.1 (1859) 76.8 34 1TON + 46.3 (3) 75.3 (21) 85.2 (0) 82.7 (1) 61.6 (1212) 92.5 (8486) 76.3 (1859) 77.0
vs. d of SVD
embeddings released byTurian et al. (2010); 2 246,122 word embeddings, 100 dimen-
sions; training corpus: RCV1 corpus (Reuters En-
glish newswire, August 1996 -August 1997). (ii)
Huang. 3 Huang et al. (2012) incorporated global
context to deal with challenges raised by words with
multiple meanings; 100,232 word embeddings, 50
dimensions; training corpus: April 2010 snapshot of
Wikipedia. (iii) GloVe 4
Table 1 :
1Training setup. bs: batch size; lr: learning rate.
AVG ml 1TON + RND AVG ml 1TON + RND AVG ml 1TON + RND AVG ml 1TON +RW(21)
semantic
syntactic
total
RND ind
HLBL
7.4 6.9 17.3
17.5 26.3 26.4 26.3
26.4 22.4 22.4 22.7
22.9 24.1 24.2 24.4
24.5
Huang
4.4 4.3 6.4
6.4 1.2 2.7 21.8
22.0 7.7 4.1 10.9
11.4 4.8 3.3 15.8
16.2
CW
7.1 10.6 17.3
17.7 17.2 17.2 16.7
18.4 4.9 5.0 5.0
5.5 10.5 10.5 10.3
11.4
ensemble
CONC 14.2 16.5 48.3
-4.6 18.0 88.1
-62.4 15.1 74.9
-36.2 16.3 81.0
-
SVD
12.4 15.7 47.9
-4.1 17.5 87.3
-54.3 13.6 70.1
-31.5 15.4 77.9
-
1TON
16.7 11.7 48.5
-4.2 17.6 88.2
-60.0 15.0 76.8
-34.7 16.1 82.0
-
1TON +
-
-
-
48.8
-
-
-
88.4
-
-
-
76.3
-
-
-
81.1
Table 3 :
3Comparison of effectiveness of four methods for learning OOV embeddings. RND: random initialization. AVG: averageof embeddings of known words. ml: MUTUALLEARNING. RW(21) means there are still 21 OOVs for the vocabulary union.
OOV ALL OOV ALL OOV ALL OOV ALL OOV ALL OOV FLORS+HLBL 90.01 62.64 92.54 74.19 94.19 79.55 90.25 62.06 89.33 62.32 96.53 91.03 FLORS+Huang 90.68 68.53 92.86 77.88 94.71 84.66 90.62 65.04 89.62 64.46 96.65 91.69 FLORS+GloVe 90.99 70.64 92.84 78.19 94.69 86.16 90.54 65.16 89.75 65.61 96.65 92.03 FLORS+CW 90.37 69.31 92.56 77.65 94.62 84.82 90.23 64.97 89.32 65.75 96.58 91.36 FLORS+W2V 90.72 72.74 92.50 77.65 94.75 86.69 90.26 64.91 89.19 63.75 96.40 91.03 +meta FLORS+CONC 91.87 72.64 92.92 78.34 95.37 86.69 90.69 65.77 89.94 66.90 97.31 92.69 FLORS+SVD 90.98 70.94 92.47 77.88 94.50 86.49 90.75 64.85 89.88 65.99 96.42 90.36 FLORS+1TON 91.53 72.84 93.58 78.19 95.65 87.62 91.36 65.36 90.31 66.48 97.66 92.86 FLORS+1TON + 91.86 73.36 93.14 78.77 95.65 87.29 91.73 66.28 90.53 66.72 97.75 92.55Dimension of O2M
50
100
150
200
250
300
350
400
450
500
Performance (%)
50
55
60
65
70
75
80
85
WC353
MC
RG
SCWS
RW
(b) Performance vs. d of 1TON
Dimension of O2M+
50
100
150
200
250
300
350
400
450
500
Performance (%)
50
55
60
65
70
75
80
85
90
WC353
MC
RG
SCWS
RW
(c) Performance vs. d of 1TON +
Figure 4: Influence of dimensionality
newsgroups
reviews
weblogs
answers
emails
wsj
ALL baselines
TnT
88.66 54.73 90.40 56.75 93.33 74.17 88.55 48.32 88.14 58.09 95.76 88.30
Stanford
89.11 56.02 91.43 58.66 94.15 77.13 88.92 49.30 88.68 58.42 96.83 90.25
SVMTool
89.14 53.82 91.30 54.20 94.21 76.44 88.96 47.25 88.64 56.37 96.63 87.96
C&P
89.51 57.23 91.58 59.67 94.41 78.46 89.08 48.46 88.74 58.62 96.78 88.65
FLORS
90.86 66.42 92.95 75.29 94.71 83.64 90.30 62.15 89.44 62.61 96.59 90.37
+indiv
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| []
|
[
"KeV scale new fermion from a hidden sector",
"KeV scale new fermion from a hidden sector"
]
| [
"We-Fu Chang \nDepartment of Physics\nNational Tsing Hua University\nNo. 101, Section 2, Kuang-Fu RoadR.O.C30013HsinchuTaiwan\n\nTRIUMF\n4004 Wesbrook MallV6T 2A3VancouverBCCanada\n",
"John N Ng "
]
| [
"Department of Physics\nNational Tsing Hua University\nNo. 101, Section 2, Kuang-Fu RoadR.O.C30013HsinchuTaiwan",
"TRIUMF\n4004 Wesbrook MallV6T 2A3VancouverBCCanada"
]
| []
| We studied a simple model of hidden sector that consists of a Dirac fermion χ and a spontaneously broken U (1) s symmetry. The dark sector is connected to the Standard Model(SM) via three righthanded SM singlet neutrinos, N R 's, and the kinetic mixing between U (1) s and U (1) Y . A mixing between the scalar φ that breaks U (1) s and the SM Higgs boson, H, is implemented via the term φ † φH † H and this provides a third connection to the SM. Integrating out the N R at a high scale not only gives the active neutrinos, ν, masses but generates effective Dirac-type couplings between ν and χ. This changes the usual Type-I seesaw results for active neutrino masses and makes χ behave like a sterile neutrino even though its origin is in the hidden sector. Note that χ is also split into a pair of Majorana fermions. The amount of splitting depends on the parameters. If the lighter of the pair has a mass around keV, its lifetime is longer than the age of the Universe and it can be a warm dark matter candidate. Signatures of χ in high precision Kurie plots of nuclei β decays and low energy neutrino nuclei coherent scatterings are discussed. The model also induces new invisible Z decay modes that can be searched for in future Z factories. * Electronic address: [email protected] † Electronic address: [email protected] | 10.1103/physrevd.101.035028 | [
"https://arxiv.org/pdf/1903.12545v3.pdf"
]
| 88,516,920 | 1903.12545 | 8d5fdc32864922f72ad5af554c337072a6365bd9 |
KeV scale new fermion from a hidden sector
20 Feb 2020
We-Fu Chang
Department of Physics
National Tsing Hua University
No. 101, Section 2, Kuang-Fu RoadR.O.C30013HsinchuTaiwan
TRIUMF
4004 Wesbrook MallV6T 2A3VancouverBCCanada
John N Ng
KeV scale new fermion from a hidden sector
20 Feb 2020(Dated: February 21, 2020)
We studied a simple model of hidden sector that consists of a Dirac fermion χ and a spontaneously broken U (1) s symmetry. The dark sector is connected to the Standard Model(SM) via three righthanded SM singlet neutrinos, N R 's, and the kinetic mixing between U (1) s and U (1) Y . A mixing between the scalar φ that breaks U (1) s and the SM Higgs boson, H, is implemented via the term φ † φH † H and this provides a third connection to the SM. Integrating out the N R at a high scale not only gives the active neutrinos, ν, masses but generates effective Dirac-type couplings between ν and χ. This changes the usual Type-I seesaw results for active neutrino masses and makes χ behave like a sterile neutrino even though its origin is in the hidden sector. Note that χ is also split into a pair of Majorana fermions. The amount of splitting depends on the parameters. If the lighter of the pair has a mass around keV, its lifetime is longer than the age of the Universe and it can be a warm dark matter candidate. Signatures of χ in high precision Kurie plots of nuclei β decays and low energy neutrino nuclei coherent scatterings are discussed. The model also induces new invisible Z decay modes that can be searched for in future Z factories. * Electronic address: [email protected] † Electronic address: [email protected]
I. INTRODUCTION
Some time ago we investigated a very simple shadow U(1) s sector which consists of a scalar φ that spontaneously breaks the gauged Abelian symmetry [1]. The resulting massive gauge boson X µ was allowed to kinetically mix with the hypercharge gauge boson B µ of the Standard Model (SM). The scalar φ also couples to the SM Higgs field H via the term φ † φH † H. In today's parlance, this will be the simplest two portal model respecting the SM gauge symmetry. The first portal is a vector one with the hypercharge as the mediator and the second is a scalar portal mediated by the Higgs boson. U(1) s symmetry breaking scale characterized by v s is taken to be above the electroweak breaking scale given by v = 247GeV.
A scale-invariant version for the scalar sector was also constructed [2]. Moreover, the models did not yield a dark matter candidate. In this paper, we extend the model by adding a massive Dirac fermion χ which is a SM singlet but is charged under the local U(1) s . We also included at least two heavy righthanded SM singlet neutrinos, N R , which are singlets under U(1) s . Doing so enables us to use the type-I seesaw mechanism for active neutrino masses. For clarity's sake, much of our discussion will be given for one N R and extending to the realistic case of 3 N R is straightforward. Here, N R will also play the dual role of a fermion portal to the hidden sector. This constitutes a very simple complete minimal model with all three portals present.
Since the physics of the U(1) s gauge boson and the scalar has been discussed thoroughly before, we concentrate here on the fermion χ. In particular, we explore the parameter space which allows χ to be a dark matter candidate. Using the minimal content of the hidden sector and the conventional breaking of the U(1) s does not leave us with a symmetry that can protect χ from decaying; thus, in general, it cannot be stable. Without imposing an ad hoc symmetry, the only open option is to arrange χ to be long-lived, and it plays the role of warm dark matter (WDM); similar to that of a sterile neutrino. Recently, WDM receives increasing attention due to its ability to address the small scale problem of the cold dark matter plus cosmological constant (ΛCDM) paradigm. Note that ΛCDMs have DM masses in the GeV to TeV range, and they predict too many satellite galaxies in the Milky Way and cusped DM profiles which contradicts current observations. On the other hand, DM with masses in the keV range are capable of accommodating the number of observed satellites as well as cored profiles of dwarf galaxies which are believed to be DM dominated.
The satellite problem arises because free relativistic particles do not cluster and they erase structures of scale smaller than the particle free-streaming length ℓ fs which is approximately the distance traveled before the particle becomes non-relativistic ∼ c/3. For the keV scale ℓ fs ∼ 100kpc. On the contrary, for cold dark matter ( CDM ) which is heavier and slower, has ℓ fs that are a million times smaller, resulting in too many small scale structures [3]. While CDM is very successful in accounting for large scale structures and many other cosmological observations (see [4] for a review), it gives a steep cusp at the center for the galaxy density profile ρ ∼ r −1 [5]. In contrast, WDM gives a finite constant density at the center ρ ∼ ρ 0 which is more in line with observations [6].
In addition, there are claims of the detection of a monochromatic line at 3.56 keV X-ray data towards the Andromeda and Perseus cluster [7]and [8]. This can be interpreted as the radiative decay of a fermion, usually taken to be a sterile neutrino, into an active neutrino plus a photon. While this is suggestive, a more mundane astrophysical explanation is also possible. Here we explore the possibility that monochromatic gamma can come from the radiative decay of χ → ν + γ for a range of masses of interest in explaining the small scale structure conundrum. This motivates us to focus the χ mass in the range of 2 − 10keV.
We shall see later that the astrophysical and cosmological properties of χ we arrived at is almost indistinguishable from those of a sterile neutrino. A lucid review of sterile neutrinos as warm dark matter can be found in [9]. As expected, if χ were a viable WDM candidate, the parameter space of the model will be restricted. We emphasize that χ is conceptually and physically different from a sterile neutrino since it is not connected to active neutrino mass generation. Moreover, the χ fermions in our model come in as a vector pair. If they acquire a sizable mass splitting, the lighter one still serves as the WDM while the heavier one is much less restricted than the keV sterile neutrino WDM, and could have low energy phenomena. Exploration of this is one of the purposes of this paper.
The paper is organized as follows. In Sec.II, we discuss in detail the model and the role of the high scale type-I seesaw. This leads to the lifetime of χ in Sec.III. The implications of χ for low energy precision neutrino physics are given in Sec.IV. Effects on β decays of nuclei, neutrinoless double beta decays of nuclei and recently observed coherent low energy neutrino scattering producing χ will be examined. Next, we give miscellaneous considerations of χ at higher energies in Sec.V. The main new result is the additional invisible decay of the SM Z. In Sec.VI, we discuss the cosmological requirements of χ as the viable WDM and the limits they set on the parameters of the model. Sec.VII contains our conclusions.
Field ℓ L H N R χ L χ R φ SU (2) L 2 2 1 1 1 1 U (1) Y -1 2 1 2 0 0 0 0 U (1) s 0 0 0 1 1 1
II. TYPE I SEESAW AND THE THREE PORTALS TO THE HIDDEN WORLD
In the usual notation, the gauge group of the model is SU(2) L × U(1) Y × U(1) s where the color sector is omitted. We begin by discussing only one generation. The fields beyond the SM ones we use are the SM singlet righthanded neutrino, N R , the hidden Dirac fermion χ which can be considered as a pair of different chirality Weyl fermions χ L,R , a hidden sector scalar, φ, and the gauge field X µ of U(1) s . The fields and their relevant quantum numbers are given in Table I.
The complete gauge invariant Lagrangian is given by
L = L SMI + L sh + L N χ , L SMI = L SM + N R i / ∂N R − yl L N RH + 1 2 M N N c R N R + h.c. , L sh = − 1 4 X µν X µν − ǫ 2 B µν X µν + ∂ µ − ig s X µ φ 2 +χ(i / ∂ − g s / X)χ − M χχ χ − V (φ, H) , L N χ = −f L χ L N R φ − f R χ c R N R φ * + h.c. , V (H, φ) = −µ 2 s φ * φ + λ s (φ * φ) 2 + κ(H † H)(φ * φ) − µ 2 H † H + λ(H † H) 2 ,(1)
where ℓ L and H are the SM lepton doublet and the Higgs field, respectively. Also,H = iσ 2 H * as in the standard notation. Note that B µν is the field strength tensor of U(1) Y , and g s is the gauge coupling of U(1) s . We have arbitrarily chosen χ and φ to have unit shadow charge, with the convention given in Eq. (1). It is interesting to note that a conventional lepton number of plus/minus one unit can be assigned to χ L/R and 0 for φ. But this is unnecessary since the Majorana mass term for N R breaks it explicitly by two units. The three portal terms are ǫ in L sh , L N χ , and the κ term in V (H, φ).
Next, we implement the type-I seesaw mechanism. Namely, we assume that M N is much heavier than any other masses, and we integrate out N R . The easiest way to do this is diagrammatically. As depicted in the Feynman diagrams given in Fig.1, some dimensionfive terms are generated below the scale M N . The effective theory below the seesaw scale then consists of L SM + L sh and a lepton-
ℓ L H N R N R ℓ L H y 2 2M N ℓ c L ℓ L HH (a) ℓ L H N R N R χ R φ yf R M N χ R ℓ L φH (c) χ R φ N R N R χ R φ f 2 R 2M N χ c R χ R φ * φ * (e) ℓ L H N R N R χ L φ yf L M N χ c L φ * ℓ L H (b) χ L φ N R N R χ L φ f 2 L 2M N χ c L χ L φ * φ * (d) χ L φ N R N R χ R φ f L f R M N χ L χ R φφ *number-violating L 5 −M N L 5 = 1 2 y 2 ℓ c L ℓ L HH + yf L χ c L φ * ℓ L H + yf R χ R ℓ L φH + 1 2 f 2 L χ c L χ L φ * φ * + 1 2 f 2 R χ c R χ R φ * φ * + f L f R χ L χ R φφ * + h.c.(2)M ν ∼ y 2 v 2 M N yf L vvs M N yf R vvs M N yf L vvs M N f 2 L v 2 s M N M χ + f L f R v 2 s M N yf R vvs M N M χ + f L f R v 2 s M N f 2 R v 2 s M N .(3)
We reiterate that M ν is a consequence of generalizing the high scale type-I seesaw mechanism to include the fermion portal Lagrangian with SSB taken afterward. Thus, the hierarchy of the scale we are interested in is M N ≫ v s v. It is easy to see that with this hierarchy, the entry of active neutrino masses, i.e., the uppermost left corner of Eq.
= f R = 0.1, we have M ν ≃ y 2 v 2 M N yf L vvs M N yf R vvs M N yf L vvs M N f 2 L v 2 s M N M χ yf R vvs M N M χ f 2 R v 2 s M N .(4)
It is easy to see that the splitting of χ L and χ R arises from the f 2 L,R terms. For f L/R ∼ 0.1 this is O(1keV). Thus, χ will remain essentially a Dirac fermion for this range of parameters. As a reminder, M χ is not the physical mass. Even smaller splitting can be obtained by taking f L/R ≪ 1 . In that case, the physical states are actually two Majorana neutrinos χ 1 , χ 2 with masses so close to each other that most experiments cannot resolve them.
For larger splitting we have to explore a different parameter region. If we take To make the physics more transparent for how the above considerations can alter the type-I seesaw mechanism for active neutrino mass generation and the value of the eventual physical mass of χ, we first consider the case where f L = f R = f , y = 1, M N = 10 14 GeV, and ignore the splitting discussed above. The mixing of active ν is with a Dirac shadow fermion χ. The simplified neutral fermion mass matrix M s ν becomes
f L = f R = f ∼ 1, then f 2 v 2 s M N ∼ 100keV. For M χ ∼ 80keV,M s ν ≃ v 2 M N f vvs M N f vvs M N M χ .(5)
The eigenvalues are
M ± 0 = M χ 2 (1 + a) ± (1 − a) 2 + 4b 2 ,(6)
where a = v 2 MχM N and b = f vvs MχM N and a, b ≪ 1 for M χ > 100eV. Thus,
M 2 ≡ M + 0 ≃ M χ 1 + f vv s M χ M N 2 ,(7)M 1 ≡ M − 0 ≃ v 2 M N 1 − f 2 v 2 s M χ M N .(8)
The physical mass of χ is pushed up, but it does not change by much. On the other hand, the physical mass of the active neutrino is pushed down compared to the type-I seesaw value.
It is instructive to look at some typical numbers. Note that M χ is a free parameter. If we take v s = 1TeV, M N = 10 14 GeV (here we set y = 1), the correction to M χ = 1keV is negligibly small. In contrast, the correction to the seesaw active neutrino mass is ∼ f 2 % (see. Eq.(8)), which can be substantial if f > 1. Furthermore, the correction is more significant for smaller values of M χ .
The mixing angle is given by
θ ≃ yf vv s M χ M N .(9)
As expected, the heavier χ is, the smaller the mixing with the SM active neutrino is. Furthermore, its effect on the active neutrino mass is also less. In many aspects, it behaves very much like a sterile neutrino, although it originates from a hidden sector.
It is important to note that L 5 also gives rise to dimension-4 operators when only one of the scalar fields picks up a VEV. An example is vs M N f R χ R ℓ L H, which is not present in the original Lagrangian. This can lead to invisible decay modes for the Higgs boson if χ is sufficiently light. However, the effective coupling is expected to be O(10 −10 ), and the decay cannot be detected in the near future. Similar terms can be read off from Eq.(2), and they all have seesaw suppressed couplings.
Generalizing to the three active neutrino case is straightforward. For simplicity, we set f L = f R = f , and the neutral fermion mass matrix is now a 5 × 5 matrix since χ is now split into two Majorana fermions denoted by χ 1,2 . In the weak interaction basis, {ν α (α = e, µ, τ ) , χ L , χ c R }, and ignore the seesaw suppressed Majorana masses to χ R,L , this is given by
M ν ≃ yeev 2 M N yeµv 2 M N yeτ v 2 M N y ef vvs M N y ef vvs M N yeµv 2 M N yµµv 2 M N yµτ v 2 M N y µf vvs M N y µf vvs M N yeτ v 2 M N yµτ v 2 M N yττ v 2 M N y τ f vvs M N y τ f vvs M N yef vvs M N yµf vvs M N yτ f vvs M N 0 M χ y ef vvs M N y µf vvs M N y τ f vvs M N M χ 0 ,(10)
where we have restored the various Yukawa couplings of active neutrino N R couplings and y αα ′ = y α y α ′ , y αf = y α f . The mass basis ν i , i = 1 · · · 5 is related to the weak basis by a unitary transformation :
ν α = i U αi ν i . This diagonalizes Eq.(10), i.e. M diag ν = U † M ν U.
The weak charged current in the mass basis can be obtained from
ig √ 2 α=e,µ,τ 5 i=1 U αi e α γ µL ν i W µ,− + h.c. .(11)
Without going into the details of the numerical analysis of neutrino oscillation data, one can expect that U α4,5 is approximately given by Eq. (9). Similarly, with the help of the unitarity of U, the neutral current involving neutrinos in the mass basis can be deduced as
ig 2 cos θ w 5 i=1ν i γ µL ν i − 5 i,j=1 (U † ) jχ L U χ L iνj γ µL ν i + 5 i,j=1 (U † ) jχ c R U χ c R iνj γ µL ν i Z µ .(12)
For i, j = 4, 5 there is a small off-diagonal coupling that can be neglected. For the diagonal term the coupling strength is essentially that of the SM since the second term is negligible.
It is clear that since χ can mix with the active neutrino with a small mixing angle, it behaves very much like a sterile neutrino. We emphasize that although χ and the commonly studied sterile neutrino have similar charged and neutral current interactions, their origins are very different. There are numerous models for sterile neutrinos. It is useful to compare our case with a complete model of the sterile neutrino to bring out the differences. A wellknown example is the Neutrino Minimal Standard Model (νMSM) [11], which is a low scale seesaw model, i.e., the lepton number breaking scale is below the Fermi scale. Three sterile neutrinos correspond to the three N R 's, two of which are in the GeV range, and the third is the keV range. The last one is identified as WDM. It is the only state in this mass range.
On the other hand, χ consists of two nearly degenerate states. If the splitting is > 100keV, optimistically, this can be seen in near future experiments(see Sec.IV) and thus offers a distinction from the sterile neutrino scenario. Otherwise, χ will be a pseudo-Dirac fermion.
In this case, it will be more difficult to tell the two scenarios apart. It may be necessary to examine other signals.
III. STABILITY OF THE SHADOW FERMION
It is clear from the previous discussions that none of the hidden sector fields can be stable after SSB. For χ to play the role of WDM, we assume that χ is the lightest of the hidden particles. Denoting the mass eigenstates by χ ± and requiring that they have masses in the O(10keV) range, the only available decays are χ ± → 3ν and χ ± → νγ. The width of the invisible decays is given by
Γ(χ ± → 3ν) = G 2 F M 5 χ ± 96π 3 3 i=1 |U χi | 2(13)
as they are Majorana states.
The radiative decay is a 1-loop effect. Unlike most loop effects, the model dependence can be reduced to only the mixing angle by calculating the width in the U-gauge. The width is given by 1
Γ(χ ± → γν) = 9αG 2 F 256π 4 M 5 χ ± α=e,µ,τ i=1,2,3 |U 4α | 2 |U αi | 2 .(14)
The invisible decays will be faster than the radiative mode. For M χ = 10 keV and a lifetime longer than that of the Universe, we get the constraint
3 i=1 |U χi | 2 < 1.8 × 10 −2 .(15)
This is to be compared with the expectation of ∼ 10 −4 given by Eq. (9) for v s = 1TeV. This validates χ − as a WDM candidate. This decay will provide a monochromatic X-ray line for each of the χ's if the splitting is larger than the experimental resolution but still small enough to be in the < 10keV range.
Next, we consider the scenario that χ splits into two Majorana neutrinos, one with mass 10 keV, and the other 100 keV. The lifetime of the heavier one is estimated to be ∼ 3.3 × 10 5
years with the same mixing as in Eq. (15). This case will not affect the cosmic microwave background measurements and thus can also be a viable cosmological scenario. It may have later time cosmological implications that are beyond the scope of this investigation.
IV. IMPLICATIONS FOR LOW ENERGY NEUTRINO PHYSICS
A. β decay spectrum
It is well known that a detailed study of the β decay spectrum of nuclei can reveal the existence of one or more heavy neutrinos. This has been studied in the context of Kaluza-Klein extra-dimensional models [13] where there can be many such neutrinos. More recently, a detailed study has been conducted for tritium decays [14]. The KATRIN experiment [15] can also be used to look for neutral leptons of mass lower than 18 keV.
For a nuclear β decay with mass ≫ Q, E e , m να , the differential decay rate as a function of electron energy E e is given by the leading approximation
dR dE e = K β E e (Q + m e − E e )(E 2 e − m 2 e ) 1 2 |U e5 | 2 (Q + m e − E e ) 2 − M 2 2 1 2 + |U e4 | 2 (Q + m e − E e ) 2 − M 2 1 1 2 + 3 i=1 |U ei | 2 (Q + m e − E e ) 2 − m 2 i 1 2 .(16)
where K β includes the nuclear matrix element, the Fermi function, and G F . Note that Q is the Q-value of 18.59 keV for tritium. We have also separated the heavy Majorana fermions χ 2 , χ 1 with masses M 2 , M 1 from the physical active neutrinos, described by the last term.
The spectrum consists of three branches if the energy resolution is smaller than M 2 − M 1 .
The first one will cut off at
E e = Q + m e − M 2 ,(17)
and give a kink at that point. The second kink appears at
E e = Q + m e − M 1 .(18)
To illustrate, we display the differential decay rates in Fig.2 The dotted blue line is for (M 1 , M 2 ) = (7, 5) keV, and the red one is for (M 1 , M 2 ) = (7, 100) keV.
E e − m e (keV) 1 − (dR/dEe)χ (dR/dEe) SM /(2|UThe mixings are U e4 = U e5 = 0.4. (b) 1 − (dR/dEe)χ (dR/dEe) SM in units of 2|U e4,5 | 2 with (M 1 , M 2 ) = (7, 5) and (7, 100) keV.
In Fig.2(b), we display the ratio of the same Kurie plot to that of the SM derivation from the unit for two representative sets of M 1 , M 2 .
If the energy resolution is not sufficient to resolve the two masses, then one smeared kink will appear in the spectrum before it cuts off at E e ≃ 18.59keV and gives the usual result of
m νe = 3 i=1 |U ei | 2 m 2 i .(19)
To be able to observe a kink-like structure in the spectrum, the energy resolution will have to be approximately 300 eV in this energy range, and a dedicate experiment has been proposed [16]. A limit on the mixing of |U e4,5 | 2 10 −7 can be set if no kinks are found.
B. 0νββ decays of nuclei
For Majorana neutral fermions lighter than 100 keV that mix with the SM active neutrinos, the usual neutrino exchange mechanism for 0νββ decays of nuclei is still applicable.
The effective Majorana mass for ν e denoted by m ee is given by
m ee = U 2 e1 m 1 + U 2 e2 e 2iα 2 m 2 + U 2 e3 e 2iα 3 m 3 + U 2 e4 e 2iα 4 M 1 + U 2 e5 e 2iα 5 M 2 ,(20)
where α j , j = 2 . . . 5 are the Majorana phases. There are now four such phases in addition to the Dirac phases in U ei which now total 3. It is instructive to compare the contributions of the SM active neutrinos versus the χ's. Without going into detail, one expects the active neutrinos to contribute 10 −2 − 10 −3 eV to m ee . Using Eq.(9) as a guide, they contribute 10 −4 − 10 −2 eV, depending on the Yukawa, for M 1,2 ∼ 10keV. Hence, the mixing of χ with the SM neutrinos will significantly change the expectations of 0νββ decays.
C. Coherent low energy neutrino production of χ
We have argued that χ can be a warm dark matter candidate if its mass is in the keV range. It is a prime candidate for production in low energy coherent neutrino-nucleus scattering (CEνNS), which has been observed recently in the COHERENT experiment [17] at a spallation neutron source(SNS). Such experiments can also be carried out at high powered reactors. The principal experimental challenge is to detect very low nuclear recoils.
Cryogenic bolometers harbor the promise to detect sub-100 eV recoils [18].
The crucial physics requirement for CEνNS is a small momentum transfer to the nucleus.
It has to be smaller than the inverse radius of the nucleus to maintain coherence. The scattering process must also not alter the quantum state of the nucleus. Nuclear excitations must not be triggered; otherwise, the process will break the coherence of nucleons that are scattered. We studied the process
ν + N → χ + N ,(21)
where N denotes the nucleus. We are interested in how Eq.(21) can be used to limit the parameter space of the model.
The two main fundamental processes are due to Z and H exchanges, as shown in Fig.3.
The one-photon exchange process is very small in our model and can be ignored. The two main processes have the same kinematics but give different differential cross-sections dσ dT where T is the recoil energy of the nucleus. The Z exchange process is similar to the SM coherent scattering and thus is suppressed by the ν − χ mixing. It is well known that it is On the other hand, Fig.3(b) has no SM equivalence since it requires the existence of a righthanded singlet fermion such as sterile neutrino. Here χ plays that role, although it is not a sterile neutrino per se. Furthermore, Fig.3(b) is sensitive to Higgs nucleon coupling, g Hnn = gM N 2M W η, where η ≃ 0.3 [19], and M N is the nucleon mass. This coupling is both experimentally and theoretically interesting. Roughly speaking, we can take M ≈ AM N (A = N + Z) by neglecting the small neutron-proton mass difference.
ν(p 1 ) χ(k 1 ) Z N (p 2 ) N (k 2 ) (a) ν(p 1 ) χ(k 1 ) H 0 , φ N (p 2 ) N (k 2 ) (b) ν(p 1 ) χ(k 1 ) N (k 2 ) θ(
The two important kinematic quantities are the scattering angle θ (see Fig.(3 c)), and the recoil energy T . In terms of T , the scattering angle is
cos θ = E ν T + MT + 1 2 m 2 χ E ν √ T 2 + 2MT ,(22)
where E ν is the incoming neutrino energy. From this, we obtain the maximum T + and minimum T − allowed recoil energies
T ± = ME 2 ν − 1 2 m 2 χ (M + E ν ) ± E ν M 2 E 2 ν − m 2 χ M(M + E ν ) + 1 4 m 4 χ 1 2 M(M + 2E ν ) .(23)
We recover the SM values by taking m χ = 0. Hence T SM + = 2E 2 ν (M +2Eν ) and T SM − = 0. The differential cross-section can be written as
dσ a dT = 1 32π 1 ME 2 ν 1 2 spins |M| 2 ,(24)
where M is the invariant amplitude. The cross sections are the same forν. For the Z exchange process, we have
dσ (Z) dT = G 2 F Q 2 W |U ℓ4 | 2 4π M 1 − MT 2E 2 ν − T E ν + T 2 2E 2 ν − m 2 χ 4E 2 ν 2E ν M − T M + 1 F 2 Z (q 2 ) ,(25)
where F is the nuclear form factor for the specific nucleus used in the detector, and q 2 = −2MT is the momentum transfer squared. For m χ = 0, it reduces to the well-known SM result [20]. Here ℓ is the flavor of the incoming neutrino. For reactor neutrinos ℓ = e, and ℓ = µ, e for a spallation neutron source . Similarly, the cross-section due to scalar exchanges
is dσ (H) dT = y 2 χ g 2 HN N 4π M cos 2 α 1 M 2 H + v v s tan α m 2 φ 2 1 + T 2M MT E 2 ν + m 2 χ 2E 2 ν F 2 H (q 2 ) ,(26)
where y χ parameterizes the new dimension-4 H − ν − χ coupling (see Fig.(1)), and α is the mixing angle of the Higgs and φ. We have included φ exchange, although it is suppressed by the ratio of VEV's and the mixing α, as seen above. In the range of m φ ∼ O(1)GeV
, it can be comparable to the Higgs exchange effect. The Higgs coupling to nucleus g HN N is an unknown quantity. However, we take it to be g Hnn with the substitution m n → M.
Admittedly this is a gross attempt to capture the nucleon coherent effect. Furthermore, in general, the form factor F H is different from F Z in Eq. (25).
For the signal, we have to integrate over the appropriate neutrino spectrum. In general, the differential number of events per unit time is given by
dN (a) dT = n i Eνmax E νmin dE ν φ(E ν ) dσ (a) dT (T, E ν ) ,(27)
where n i is the number of target nuclei in the detector, φ(E ν ) is the flux of the incoming neutrinos, and E νmax is the maximum source neutrino energy. The differential cross-sections for a = SM, Z, H can be read off from Eqs. (25) and (26). The minimum required neutrino energy for a specific recoil T is given by
E νmin = MT + m 2 χ /2 √ 2MT + T 2 − T ≃ MT /2 .(28)
For a target with atomic number A and recoil energy of T = 1keV, the minimal required neutrino energy is ∼ 7× A/100 MeV. For a neutrino source with energy E ν , and m χ , E ν ≪ M, the maximal recoil energy is about T + ∼ 20 × (E ν /MeV) 2 × (100/A) eV. Therefore, the maximal recoil energy in the ν + N → χ + N process is about O(1) and O(40) keV for neutrinos from a nuclear power plant and a spallation neutron source, respectively.
The COHERENT experiment [17] utilizes neutrinos from a spallation source. There are three flavors of incoming neutrinos from π + decays almost at rest into µ + + ν µ and the muon subsequently decays into e + ν µ ν e . The ν µ from the first decay gives a monochromatic flux.
The muon decays are usually taken to be almost at rest. However, it is easy to take into account the energy of the muon which is given by E µ = m 2 π +m 2 µ 2mπ = 109.78MeV. The fluxes are calculated to be
φ νµ (E ν ) = φ 0 δ E ν − m 2 π − m 2 µ 2m π , (29a) φ νe (E ν ) = φ 0 192 m µ E ν E µ m 2 µ 2 m µ 2E µ − E ν m µ 1 + p 2 µ 3E 2 µ , (29b) φν µ (E ν ) = φ 0 64 m µ E 2 ν m 2 µ 3 4 − E ν E µ m 2 µ 1 + p 2 µ 3E 2 µ ,(29c)
where p µ is the muon 3-momentum and numerically |p| = 29.79MeV, and also E νmax = 1 2 E µ . Note that φ 0 is a normalization factor which depends on factors such as number of protons on In order to get a setup independent prediction, we can focus on the ratios of the signal with χ to the expected SM one. We first focus on the Z-mediated CEνNS. For our model, the total differential rate consists of the ones from SM neutrinos and those from χ ± ,
dN χ dT = dN SM dT × (1 − |U 4 | 2 − |U 5 | 2 ) + |U 4 | 2 dN (Z) (M 1 ) dT + |U 5 | 2 dN (Z) (M 2 ) dT = dN SM dT + |U 4 | 2 dN (Z) (M 1 ) dT − dN SM dT + |U 5 | 2 dN (Z) (M 2 ) dT − dN SM dT .(30)
Note that here we use a notation that is slightly different from Eq. (27) dT . For notational simplicity, we have also dropped the flavor content of the incoming neutrinos in the mixing elements, the other notation is obvious. Therefore, the ratio of the total differential rate to the SM one deviates from 1 by an amount of
1 − dNχ dT dN SM dT = |U 4 | 2 1 − dN (Z) (M 1 ) dT dN (Z) (0) dT + |U 5 | 2 1 − dN (Z) (M 2 ) dT dN (Z) (0) dT .(31)
For simplicity, we assume |U 4 | 2 = |U 5 | 2 = |U χ | 2 . Then, at fixed T ,
1 − dN χ dT dN SM dT |U χ | −2 = 2 − dN (Z) (M 1 ) dT dN (Z) (0) dT − dN (Z) (M 2 ) dT dN (Z) (0) dT ,(32)
which is displayed in Fig.4(a) and (c) for SNS 2 and nuclear power plant neutrino sources, respectively. The reactor neutrino flux in [22] is adopted. However, note that there is no SM counterpart for the scalar-mediated coherent scattering, and the mixings are different from the Z-mediated ones. Therefore, the scalar-mediated part is separately compared to the Z-mediated SM one,
dN H dT dN SM dT |U H | −2 = dN (H) (M 1 ) dT dN (Z) (0) dT + dN (H) (M 2 ) dT dN (Z) (0) dT .(33)
Again, we assume the couplings of the two shadow fermions are the same, and the quantity |U H | 2 is defined as
|U H | 2 ≡ y 2 χ g 2 HN N M 4 H G 2 F Q 2 W P 2 χ ≃ 7 × 10 −5 A N 2 y 2 χ P 2 χ ,(34)
where
P χ = c α + s α v v s M 2 H M 2 φ(35)
is of order unit as long as s α 10 −3 and M φ ∼ O(GeV). If taking P 2 χ ∼ 1, then |U H | 2 ∼ 10 −4 × y 2 χ . On the other hand, if s α > 10 −3 , the process will be dominated by the light φ and P 2 χ ≫ 1. These contributions are displayed in Figs.4(b) and (d) for SNS and nuclear power plant neutrino sources, respectively.
Note that the scalar-mediated CEνNS is not very sensitive to M 1,2 . Also, observe the jumps at around T ∼ 13keV in the SNS CEνNS. They are due to a sharp T max cutoff from the monochromatic muon neutrino line, Eq.(29a). Assuming that the Z-mediated process dominates, from Fig.4 The second portal of X µ can be studied independently of the other two. First, we note that the kinetic terms, including the mixing, can be recast into canonical form through a GL(2) transformation. Explicitly, this is given by
X B = c ǫ 0 −s ǫ 1 X ′ B ′ ,(36)
where
s ǫ = ǫ √ 1 − ǫ 2 , c ǫ = 1 √ 1 − ǫ 2 .(37)
After SSB, X ′ and B ′ will mix and result in a shift in the SM Z mass. The physical neutral bosons consist of three states γ, Z, Z s . The transformation relating the weak and mass bases is given by
B ′ A 3 X ′ = c W −s W 0 s W c W 0 0 0 1 1 0 0 0 c η −s η 0 s η c η γ Z Z s ,(38)
where s W (c W ) denotes sin θ W (cos θ W ) and similarly for the rotation angle η. The first rotation is the standard one that gives rise to the SM Z, and the second one diagonalizes the mixing of the two Z bosons. The extra mixing angle is given by
tan 2η = 2s W s ǫ c 2 W (M 3 /M W ) 2 + s 2 W s 2 ǫ − 1 ,(39)
where M 3 ≡ g s v s and φ = vs √ 2 . We use the shorthand notation c η (s η ) = cos η(sin η) and t 2η = tan 2η. In general, ǫ is a free parameter; however, the success of the SM indicates that it has to be small. Clearly, the photon will remain massless, and the W bosons will be unchanged from the SM. Our notations follow that of [1] where details can be found.
The masses for the two massive neutral gauge bosons are readily found to be [1]:
M 2 Z = M 2SM Z c 2 η − s 2η s W s ǫ + s 2 η s 2 W s 2 ǫ + s 2 η M 2 3 , M 2 Zs = M 2SM Z s 2 η + s 2η s W s ǫ + c 2 η s 2 W s 2 W s 2 ǫ + c 2 η M 2 3 .(40)
We see that the Z-boson has its mass shifted from the SM value of M SM Z = M W c W , and receives a small contribution from the hidden sector. Similarly, the physical Z s mass comes mainly from the hidden sector with a small contribution from the visible sector. We see later that if we want χ to be warm dark matter, it is more natural to have M Zs ≫ M W . With that in mind, Eqs.(39) and (40) become
η ∼ s W s ǫ c 2 W M 2 W M 2 3 s ǫ ,(41)M 2 Z ∼ M W c W 2 1 − s 2 W s 2 ǫ c 2 W M 2 W M 2 3 .(42)
The very precise measurement of the ρ parameter gives a stringent limit on the mixing parameters. The shift δρ is
δρ = s 2 W s 2 ǫ c 2 W M 2 W M 2 3 < 3.7 × 10 −4 .(43)
In turn, we get
M 3 > 2.29 |s ǫ | TeV .(44)
There are many electroweak precision tests (EWPT) that can set limits on ǫ, η. In particular, the measurements at the Z-pole are independent of the mass of Z s but are sensitive to the modifications to the SM fermion-fermion-Z couplings. These couplings are flavor universal and explicitly given by
Z µf f : iγ µ g 2 c W c η g L f − s η s W s ǫ Y L f L + c η g R f − s η s W s ǫ Y R f R ,(45)Z µ sf f : iγ µ g 2 c W −s η g L f − c η s W s ǫ Y L f L + −s η g R f − c η s W s ǫ Y R f R ,(46)
where g f L,R = T 3 (f L,R ) − s 2 W Q f is the coupling of the SM Z to fermions andL = 1−γ 5 2 ,R = 1+γ 5 2 . A comprehensive list of couplings used to constrain ǫ can be found in [1].
B. Invisible Z decays
The two U(1)'s mixing will also modify the SM Z boson invisible decay width Γ inv . There are two changes : (i) the modification of the Z − ν − ν couplings as given in Eq.(45), and (ii) the opening of the new channel Z → χ L χ L , χ R χ R from the mixing with Z s . The new invisible width is
Γ(Z → χχ) = c 2 ǫ s 2 η g 2 s 12π M Z ,(47)
where we have neglected the masses of χ. The experimental value of Γ inv = 499 ± 1.5MeV
agrees well with three nearly massless active neutrinos. Thus
s 2 η c 2 ǫ g 2 s 3 − g 2 2 8c 2 W ≤ 2.1 × 10 −4 .(48)
Other precision measurement constraints such as muon g − 2, atomic parity violation, and
Møller scattering are given in [1] and will not be repeated here.
C. Z → ffφ 0 decays
As will be discussed in the next section, for χ to be WDM, the new physical scalar φ 0 , mainly stemming from the real part of φ, is expected to be light, m φ ∼ a few GeV, and long-lived, τ φ ∼ 1 sec. Through the Higgs portal, the SM Z boson can now have a tree-level 3-body decays Z → Z * φ 0 →f f φ 0 , where f is the SM fermions. The mixing between φ 0 and SM Higgs depends on κ (see Eq. (1)) as well as other parameters in the scalar potential.
The details are not relevant here, and we will denote this resulting mixing by κ s . Thus, φ 0 couples to the SM fields with strengths of the SM Higgs couplings times κ s . The decay branching ratio is calculated to be [19,24]:
Br(Z → φ 0 ff ) = κ 2 s × F (m φ /M Z ) × Br(Z → ff ) ,(49)
where
F (x) = G F M 2 Z 24 √ 2π 2 3x(20 − 8x 2 + x 4 ) √ 4 − x 2 cos −1 x 2 (3 − x 2 ) − 3(4 − 6x 2 + x 4 ) ln x − 1 2 (1 − x 2 )(47 − 13x 2 + 2x 4 ) .(50)
Due to its long lifetime, φ 0 will escape the detector, and the apparent signal will be Z → f f + E. The SM background will be Z →f fνν. And f = µ, b are ideal options to search for such processes and probe the scalar mixing squared, κ 2 s , down to 10 −7 − 10 −8 with 10 12 fiducial Z's [24]. The smallest κ s that can be probed by this process is still roughly one order too big for φ 0 to dilute the DM relic density, Eq.(66). However, this search provides an interesting experimental cross-check for whether this model can accommodate the WDM in the way described in the next section.
VI. χ AS WARM DARK MATTER
The previous discussions are independent of whether and how χ can become WDM. Here we examine the parameter space that allows χ to become WDM. We intend to give a broadbrush description of a possible scenario and will leave many interesting details for a future study. Most of the discussions given below do not depend on the fact that physical χ ± are Majorana fermions. If the splitting is small, then they will behave as one Dirac particle. If the splitting is large, then only the lighter one χ − will serve as DM.
For clarity, we take χ to be Dirac. The Majorana case can be obtained byχ → χ c , and we use chiral projections where needed. We specifically explore the hierarchy of scale
σv ∼ ηg s g 2 2 G 2 F T 2 ≡ A 2 χ G 2 F T 2 ,(51)
where g 2 is the SM SU(2) L gauge coupling. If this rate falls below the Hubble expansion rate, χ will freeze out. The freeze-out temperature T f can be estimated by setting n σv to be equal to the Hubble expansion rate, and n is the number density of the freeze-out particle. Thus,
3ζ(3) 2π 2 A 2 χ G 2 F T 5 f = 8π 3 90 1 2 T 2 f M pl √ g * ,(52)
with g * being the effective number of degrees of freedom. It is given by
g * = bosons g i + 7 8 fermions g i ,(53)
and g i is the number of spin states.
From Eq.(52), the freeze-out temperature of χ is controlled by the parameter A χ . From the electroweak constraints, Eq.(48), ηg s O(10 −2 ). Hence, it is natural to take A χ = 0.01, and we get
T f = 87.5 g * 100 1 6 0.01 A χ 2 3
MeV.
For a keV χ, which we are interested in, it is relativistic at the freeze-out. Such a situation will lead to overclosure of the Universe.
To see this, we note that the number density per entropy is
Y ≡ n χ s ≃ 135ζ(3) 4π 4 g * (T f ) .(55)
Note that Y is thermally conserved and that it gives the relic density of χ as
Ω χ = Y m χ s ρ c ≃ 2.5 × m χ keV 100 g * (T f ) ,(56)
where we have used the present-day entropy density s = 2891.2cm −3 , critical density ρ c = 1.05371 × 10 −5 h 2 GeVcm −3 and h = 0.678 [25]. Comparing with the observed dark matter relic abundance [25] of
Ω DM = 0.258 ± 0.011 ,(57)
the estimate of Eq.(56) clearly overcloses the Universe unless g * is of order 1000. In our model g * (T f ) ∼ 31 since T f is below 1 GeV. Hence, some mechanism is required to bring down the value of Ω χ . Moreover, Eq.(56) shows that the higher T f is, the less severe the overclosure problem is since g * will be larger.
One way to get around this obstacle is to dilute the density by producing more entropy.
The dilution can come from the decay of the scalar φ 0 if it has a long enough lifetime, and decays into SM particles during the era that χ is freezing out. Note that, due to parity, φ 0 φ 0 cannot annihilate into SM fermions via the Z s − Z mixing. However, it can do so via mixing with the Higgs boson. It must be relativistic when χ freezes out at T f so that it has no Boltzmann suppression at decay. It is easier to make such arrangement than adjusting T f given in Eq.(52). The ballpark estimation is as follows. The φ 0 φ 0 H coupling is ∼ κv,
Eq.(1), and gives
σ(φ 0 φ 0 →μµ) ≃ κv M 2 H m µ M W 2 .(58)
The freeze-out takes place when temperature yields nσ
T 2 M pl , or T ∼ κ −2 × 10 −9 GeV .(59)
Therefore, with κ < 10 −5 , it is ensured that φ 0 decouples relativistically and before χ freezes out. As φ 0 is unstable and can decay into SM fermions via mixing with Higgs, its decay will transfer the energy density into radiation while doing so. Using the sudden decay approximation that all the φ 0 's decay at t ≃ τ φ and reheat the Universe to a temperature of T r [26],
T r ≃ 0.78 (g * (T r )) − 1 4 Γ φ M pl = 1.11 MeV 1sec τ φ 1 2 .(60)
In the above, we have used g * = 10.75 and taken φ 0 lifetime to be ∼ 1 sec. The reason for taking this value is the constraint impose by Big Bang Nucleosynthesis(BBN). In order not to upset a successful BBN, T r should be larger than 1 MeV, and this sets τ φ O(1) second.
Next, we use energy conservation
m χ Y s(T f ) = ρ(T r ) = 3 4 s(T r )T r(61)
and obtain the dilution factor
D ∼ s(T r ) s(T f ) = 280.4 × g * (T r ) 1 4 g * (T f ) m φ 1 GeV 1 sec τ φ 1 2 .(62)
Inserting this dilution factor into Eq.(56), the relic density of χ is
Ω χ = 0.89 g * (T r ) 1 4 1 GeV m φ τ φ 1 sec. 1 2 .(63)
Interestingly, the dependence on g * (T f ) cancels out with this entropy dilution mechanism.
If we take T r to be slightly higher than 1 MeV, then Eq.(53) gives g * (T r ) = 15.25. And with m φ = 2 GeV, one obtains the right amount of entropy dilution.
Note that φ 0 can decay into SM particles via the mixing with the SM Higgs. As discussed in Sec.V C, we parameterize the unknown φ 0 -Higgs mixing as κ s . The width of φ 0 → ff where f is a SM fermion is given by
Γ = κ 2 s N c αm 2 f 8M 2 W m φ β 3 f ,(64)
where N c is the color of f , β f = 1 − 4m 2 f /m 2 φ , and α is the fine structure constant. However, for m φ ≃ 2 GeV, the main decay is into a gluon pair. The rate is estimated to be [19] Γ gg = α s 3π
2 m 2 φ m 2 µ [6 − 2β 3 π − β 3 K ] 2 β 3 µ Γ µ + µ − .(65)
Demanding that τ φ ≃ 1 sec. leads to
10 −5 κ s 1.3 × 10 −9 ,(66)
whereas the upper bound comes from previous considerations.
While we have identified the parameter space for χ as a WMD, we still have to ensure that there are no large processes that can generate significant numbers of χ during or after thermal freeze-out. One process in which χ's can be produced is φ 0 φ 0 → χχ. This is suppressed by large seesaw mass M N , see Eq.
(2), and can be neglected. Another source will be φ 0 → χχ. If assuming f L = f R = f , the effective coupling is given by y χ ef f ≡ f 2 vs M N . We will require this mode to be less than the SM, and this leads to a loose bound
y χ ef f < κ s m µ M W .(67)
A third process of producing χ is via active ν and χ oscillations [27,28]. The χ production rate via this mechanism peaks at temperatures of ∼ 0.1 − 1 GeV. This deserves a detailed study which is beyond the scope of this paper. We note that if this mechanism saturates the bound on dark matter relic density, it will give an upper bound on the mixing between the active neutrino and the lighter χ − , |U iχ − | 2 10 −9 (10 keV/M χ ) [9]. However, the entropy dilution mechanism operates at a lower temperature ∼ 1 MeV, Eq.(60), and will likely loosen the above constraint. Moreover, our model has one crucial difference from the νMSM.
Although we have so far only described phenomenology by treating χ as a Dirac fermion, it is, in fact, two Majorana fermions. If the splitting is small, they become pseudo-Dirac.
Otherwise, they are a pair of Majorana fermions with very different masses. This can be arranged as discussed in Sec.II. One can always arrange the model parameters such that the lighter one, χ − , fulfills all constraints and becomes the DM. On the other hand, if the mass splitting is sizable, the heavier one, χ + , quickly decays. Therefore, constraints for χ − do not apply to χ + . Our previous discussions of the low energy experiments will now be applied to χ + and can be used to set bounds on the mixing with active neutrinos.
To conclude this section, we note that φ 0 in the mass range of 1 − 10 GeV is notoriously challenging for experiments to discover. A direct detection will be impossible at the LHC [29]. However, the Higgs boson invisible decay can be searched for via H → φ 0 φ 0 , and the pair of φ 0 's will act as missing energy as noted before. The width is
Γ(H → 2φ 0 ) = κ 2 s v 2 32πM H 1 − 4m 2 φ M 2 H .(68)
ATLAS [30] and CMS [31] place a limit on invisible branching ratio B H→inv 24 − 30% at 95%C.L., which gives the constraint κ s 0.02. Hence, rare Z decays will still be the best probe of light and long-lived scalars.
VII. CONCLUSIONS
We have explored a simple model of the secluded sector that consists of only a Dirac fermion χ charged under a U(1) s . It has vector couplings to U(1) s and hence is not anomalous. Furthermore, it can acquire a bare mass term M χχ χ. Although M χ is a free parameter, it is of particular interest if it has a value 1MeV that will make it a candidate for warm dark matter.
The secluded sector can be connected to the SM via three kinds of portals. The first is the seesaw portal (SP). It consists of three SM singlet righthanded neutrinos N R 's which allows us to implement type-I seesaw for active neutrino masses. The second portal is due to the kinetic mixing of U(1) s and the SM hypercharge U(1) Y . This is the gauge portal (GP).
Note that U(1) s symmetry is broken spontaneously at a scale v s below the seesaw scale by a SM singlet scalar φ. We also take v s > v. The gauge invariant term φ † φH † H provides the third portal. This is commonly known as the Higgs portal (HP). Moreover, the gauge invariant χN R φ Yukawa couplings provide an indirect connection between χ's and the SM sector after N R is integrated out. All three portals have been discussed independently as simplified models for dark matter. Taking all three together reveals features that are not present in the simplified 1-portal models.
SP not only provides the seesaw mechanism for active neutrinos, but it also splits χ into two Majorana fermions. They can form Dirac mass terms with the active neutrinos, and act very much like sterile neutrinos, although they are not. These singlet fermions originate from the hidden sector. They add to the structure of the neutrino mass matrix and contribute to 0νββ decays. They make their presence felt in the Kurie plots of β decays of nuclei as well as coherent low energy scattering of nuclei. The latter has the added advantage that they can probe the new ν − χ − φ 0 vertex, while β decay experiments do not.
In addition to producing a new gauge boson Z s , the GP also allows χ to interact with the SM. This allows χ to be a thermally produced dark matter candidate. The phenomenology of the Z s , whose mass is expected to be in TeV, can be probed in electroweak precision measurements [1] and directly searched for at the LHC. The new invisible decay of the SM Z will also be an exciting channel for the Z-factory option of future lepton colliders such as the FCC-ee and CEPC [32,33].
In this study, we also found a new role for the portal scalar φ 0 . If it has a mass in the GeV range, it can act as an agent for entropy dilution and bring the relic density of the thermally produced χ to the range of the observed value. It can also be looked for in precision measurements of the Z-boson at the Z-factory [24].
Although the model we explored is self-contained and renormalizable, it suffers from the same hierarchy problem that plagues the SM because of the use of an elementary scalar.
The U(1) s symmetry also has a Landau pole problem as in QED. Hence, it is reasonable to assume that it should be embedded in a more elaborate dark sector. If that is the case, one need not demand that χ is a dark matter candidate. This opens up regions of parameter space that we have not discussed. We reiterate that the region of parameter space we have studied was motivated by looking at χ as a WDM candidate. The search of signatures for χ, Z s and φ in experiments that we have discussed should be conducted with this general setting in mind.
FIG. 1 :
1The effective Lagrangian L 5 terms, after integrating out the heavy N R , and the corresponding Feynman diagrams. The crosses represent the Majorana mass of N R .
( 3 )
3, is in the sub-eV range for y 2 M N ≃ 10 −14 (GeV ) −1 . We use the benchmark point of M N = 10 10 GeV, y = 10 −2 and v s = 1 TeV. Thus, v 2 s M N ≃ 100keV and vv S M N ≃ 10keV. We also work in the perturbative regime and hence take f L/R 1. Specifically if we take M χ ∼ 10keV and f L
the original Dirac χ will now split into two Majorana fermions one with mass ∼ 100keV and the other with mass ∼ 10keV. Another example is to take M N = 10 8 GeV with y = 10 −3 so that the active neutrino masses will be in the sub-eV range via Type-I seesaw. With f ≃ 0.3, M χ ∼ MeV, this yields a mass at around 10keV, with the other at about MeV. From this exercise, it is clear to see that the mass splitting becomes more substantial if the lepton number violation scale M N is lower and/or v s , M χ are raised. Moreover, only the lighter one can be the WDM to solve the small scale problem of CDM. Nevertheless, the more massive partner can have interesting phenomenology as we shall see later.
FIG. 2 :
2(a) dR/dE e , in arbitrary units, vs E e − m e (in keV). The black line denotes the SM curve.
FIG. 3 :
3(a) and (b): The Feynman diagrams for χ production in the coherent neutrino-nucleus scattering. The fixed target kinematics is depicted in (c). sensitive to the weak charge of the nucleus Q w = N − Z(1 − 4 sin 2 θ W ), and N(Z) represents the number of neutrons(protons)in the nucleus. Due to the accidental cancelation of the proton weak charge, this cross-section is expected to scale as N 2 . Moreover, we can view Fig.3(a) as an additional branch to the SM process much in the same way it adds to the β decay spectrum studied earlier.
target and the number of pions produced per incident proton. Specific to the COHERENT experiment, φ 0 = rN POT 4πL 2 . The number of protons on target N POT = 1.76 × 10 23 , r = 0.08 is the number of neutrinos per flavor produced for each proton on target, and L = 19.3 m is the distance between the source and the CsI detector. Also, the number of target nuclei in Eq.(27) is given by n CsI = N A M det M CsI , where N A is the Avogadro number, M det = 14.6 kg is the detector mass, and M CsI = 259.8 is the molar mass of CsI. For the total signal, one multiplies Eq.(27) by the lifetime of the experiment. An acceptance factor that depends onT is omitted but can be easily included.
with the χ − ν mixing squared factored out, namely, dN SM dT = dN (Z) (mχ=0)
FIG. 4 :
4(a) The deviation from the SM CEνNS at the SNS neutrino source from Z−mediated scattering on the CsI target. Here we consider the simplified scenario where |U 4,5 | 2 = |U χ | 2 . (b) Same as (a) but mediated by the scalar exchange. The deviations are in units of |U χ | 2 and |U H | 2 for sub-diagrams (a) and (b), respectively. In other words, we set |U χ | 2 = 1 and |U H | 2 = 1 for the plots. (c, d): Same as (a) and (b) but with a nuclear power plant neutrino source.
M
Zs M Z ≫ M φ > M χ .The small Z s − Z mixing will be denoted by η, see Eq.(39).After SSB of U(1) s , the physical degrees of freedom in the hidden sector are Z s , χ, φ 0 , where φ 0 is the physical scalar mainly constituted by ℜφ. We also use the benchmark values M χ = 10 keV and M φ = 2 GeV to focus our discussion.The secluded sector and the SM have feeble interactions through the portals. If the portal connections are switched off, the two sectors will not establish thermal equilibrium, and cosmologically they evolve separately. If the portal interactions are small but not negligible, then the SM and the hidden sector can interact. In particular, χ can interact with the SM fields via the Z s − Z mixing. The process χχ ↔ ff , where f is a SM fermion, can proceed via such a mixing. The cross-section at temperature T can easily be estimated to be
TABLE I :
IU (1) s and SM quantum numbers for relevant fields.
3×3 effective neutrino mass matrix M ν . In the weak basis {ν, χ L , χ c R },After spontaneous symmetry breaking (SSB) with H = v/
√
2, one gets the familiar
Weinberg operator [10] for active neutrinos with mass ∼ y 2 v 2
2M N , see Fig. (1a). Besides, we
also have SSB for U(1) s via φ = v s /
√
2. Figs.(1b) and (1c) induce mass mixings with the
hidden fermions given by ∼
yf L/R vvs
2M N . Repeating this for all the diagrams of Fig.(1) yields, for
one SM generation, a it is given by
(a) with an unrealistic sizable mixing |U e4,5 | = 0.4 for two representative sets of M 1 , M 2 .0
5
10
15
20
E e − m e (keV)
dR/dE
e
SM
(7,100)
(5,7)
0
5
10
15
20
, one sees that the SNS experiments are more sensitive to the M 1,2 splitting. For the small splitting cases, such as {M 1 , M 2 } = {7, 100}keV, this will be extremely challenging for the SNS experiments.On the other hand, the signal is about a thousand times bigger at the reactor experiments but requires a very low threshold for recoil energy, i.e., T < 1keV. However, this is beyond current capabilities for most proposed experiments. The successful development of cryogenic detectors such as the proposed[23] experiment may bring these measurements to reality. As compared to Kurie plot experiments, neutrino-nucleus coherent scatterings cannot probe χ splittings less than 100keV in the foreseeable future. On the positive side, they are sensitive to low energy scalars that can mix with the Higgs boson. These fields are popular in Higgs portal constructions but are notoriously difficult to get an experimental handle on.V. ELECTROWEAK PRECISION TESTS
A. Effects of kinetic mixing of U (1) Y and U (1) s
This is in agreement with the result of[12]. The calculation there was done in the Feynman gauge and is valid for the sterile neutrino in the Type-I seesaw model. Our U-gauge calculation shows that this result holds for any SM singlet fermion that mixes with the active neutrinos, independent of how the individual masses are obtained.
We have included the acceptance function of the COHERENT experiment[21] for the estimation.
Note Added After the paper was completed, we were informed that similar considerations of new results where they overlap with ours. the National Research Council of Canada and the Natural Science and Engineering Research Council of CanadaRef [35] considered a similar setup with radiative neutrino mass generationsthe National Research Council of Canada and the Natural Science and Engineering Research Council of Canada. Note Added After the paper was completed, we were informed that similar considerations of new results where they overlap with ours. Ref [35] considered a similar setup with radiative neutrino mass generations.
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| []
|
[
"Modus ponens and modus tollens for the compositional rule of inference with aggregation functions",
"Modus ponens and modus tollens for the compositional rule of inference with aggregation functions"
]
| [
"Dechao Li \nSchool of Information and Engineering\nZhejiang Ocean University\n316000ZhoushanChina\n",
"Qingxue Zeng \nSchool of Information and Engineering\nZhejiang Ocean University\n316000ZhoushanChina\n"
]
| [
"School of Information and Engineering\nZhejiang Ocean University\n316000ZhoushanChina",
"School of Information and Engineering\nZhejiang Ocean University\n316000ZhoushanChina"
]
| []
| The compositional rule of inference (CRI) proposed by Zadeh has been widely applied in artificial intelligence, control, data mining, image processing, decision making and so on. Recently, Li and Zeng [Li, D., Zeng, Q. Approximate reasoning with aggregation functions satisfying GMP rules, Artificial Intelligence Review (2022), https://doi.org/10.1007/s10462-022-10136-1] shown an A-compositional rule of inference (ACRI) method in which generalizes the t-norm to any aggregation function in CRI method and studied its validity using GMP rules. In this paper, we continue to investigate the validity of ACRI method from a logical view and an interpolative view. Specifically, to discuss the modus ponens (MP) and modus tollens (MT) properties of ACRI method based on well-known fuzzy implications with aggregation functions. | 10.48550/arxiv.2205.01269 | [
"https://arxiv.org/pdf/2205.01269v1.pdf"
]
| 248,506,108 | 2205.01269 | f89173ae51da75208866bc6230edc1d88de12747 |
Modus ponens and modus tollens for the compositional rule of inference with aggregation functions
Dechao Li
School of Information and Engineering
Zhejiang Ocean University
316000ZhoushanChina
Qingxue Zeng
School of Information and Engineering
Zhejiang Ocean University
316000ZhoushanChina
Modus ponens and modus tollens for the compositional rule of inference with aggregation functions
10.1007/s10462-arXiv:2205.01269v1 [math.LO] 3 May 2022Fuzzy implicationAggregation functionMP and MT propertiesACRI method
The compositional rule of inference (CRI) proposed by Zadeh has been widely applied in artificial intelligence, control, data mining, image processing, decision making and so on. Recently, Li and Zeng [Li, D., Zeng, Q. Approximate reasoning with aggregation functions satisfying GMP rules, Artificial Intelligence Review (2022), https://doi.org/10.1007/s10462-022-10136-1] shown an A-compositional rule of inference (ACRI) method in which generalizes the t-norm to any aggregation function in CRI method and studied its validity using GMP rules. In this paper, we continue to investigate the validity of ACRI method from a logical view and an interpolative view. Specifically, to discuss the modus ponens (MP) and modus tollens (MT) properties of ACRI method based on well-known fuzzy implications with aggregation functions.
Introduction
Motivation of this paper
In order to deduce some meaningful conclusions from some imprecise or vague premises, fuzzy modus ponens (FMP), as a generalized from modus ponens (MP) in the classical logic, is modelled as follows [45]:
Premise 1: IF x is D THEN y is B Premise 2: x is D ′ Conclusion: y is B ′ ,
where D and D ′ are fuzzy sets on U while B and B ′ are fuzzy sets on V . And then the compositional rule of inference (CRI) method is presented by Zadeh in 1973 to calculate the conclusion B ′ from FMP model [45]. After, the generalized CRI method for FMP problem is developed using commonly t-norms and fuzzy implications. Substituted aggregation functions for t-norms, Li and Zeng also extended the generalized CRI method to the A-compositional rule of inference (ACRI) as follows [26]:
B ′ ACRI (y) = x∈U A(D ′ (x), I(D(x), B(y))),
where A is an aggregation function and I a fuzzy implication.
Although there are still some deficiencies [6,36,42,43,46], the CRI method has been widely utilized in practice. In order to measure the validity of inference methods to solve the FMP problem, some commonly accepted axioms are proposed by many researchers [6,15,28]. The following are the most well-known axioms:
• The axioms presented by Baldwin and Pilsworth [6]. • The axioms presented by Fukami et al. [15]. It is necessary to mention that the axioms (A4) and (A5) have been studied for some fuzzy implications in generalized CRI method [2,10,16,25,26,[29][30][31]41].
It is well known that the fuzzy logical connectives mainly include negation, conjunction, disjunction, conditional and biconditional [4,22]. Many families of fuzzy implications, such as well-known R-, S-and QL-implications, f -and g-implications, probabilistic implication, probabilistic S-implication, have been utilized to interpret the conditional and biconditional.
The t-norms and t-conorms are usually employed to interpret the conjunction and disjunction [4,22]. However, as what pointed out by de Soto et al., a mathematical fuzzy model should be not always symmetrical [9]. Moreover, it is not necessary that the associativity or commutativity of the conjunction and disjunction in image processing, classification problems and decision making [8,11,12,14]. Aggregation functions, as a better substitute for the t-norms and t-conorms (They indeed are some spacial aggregation functions, See Definition 2.6) have been applied extensively in the fuzzy logic, actual classification problems and decision making [8,10,14,19,29,30,[37][38][39][40].
The axioms presented by Baldwin and Pilsworth give a logical view of reasoning while the axioms presented by Fukami et al. give an interpolative view of reasoning [16]. These inspire us to investigate the validity of ACRI method from not only a logical view but also an interpolative view. Considering the ACRI method does not satisfy the axiom (A8) (See Theorem 4.2 in [26]), our motivation of this paper is mainly to study the validity of ACRI method for well-known fuzzy implications using the axioms (A4) and (A5).
Contribution of this paper
As what mentioned above, it is worth to study the MP and MT properties of the fuzzy implications in fuzzy inference methods. Therefore, in this paper, we first investigate some properties of fuzzy implications and aggregation functions when they satisfy (A4) and (A5).
And then seek the aggregation functions for the well-known fuzzy implications such that they satisfy (A5). Finally, we give the conditions for the well-known fuzzy implications satisfying (A4). In a word, the contributions of this paper include:
(1) To investigate the properties of aggregation functions and fuzzy implications which satisfy (A4) and (A5).
(2) To construct the aggregation functions for the well-known fuzzy implications such that they satisfy (A5).
(3) To show the conditions for the well-known fuzzy implications satisfying (A4) with a strong negation.
This paper is organized as follows. Section 2 recalls some basic concepts and definitions utilized in this paper. In Section 3, we study the properties of fuzzy implications and aggregation functions when they satisfy (A4) or (A5). Section 4 constructs some aggregation functions such that R-, (A, N )-, QL-, f -, g-, probabilistic-, probabilistic S-implications and T -power implications satisfy (A5) with them, respectively. In Section 5, we investigate the conditions for these fuzzy implications satisfying (A4) with a strong negation.
Preliminary
To make this paper complete, this section will recall some main concepts and properties utilized in the remainder of this paper. Let F (U ) denote the set of fuzzy sets on U .
(N1) N (0) = 1, N (1) = 0; (N2) N (x) ≥ N (y) if x ≤ y, ∀ x, y ∈ [0, 1].
Further, a fuzzy negation N is strict if it satisfies the following properties:
(N3) N is continuous;
(N4) N (x) > N (y) if x < y.
A fuzzy negation is strong if it is involutive, i.e.,
(N5) N (N (x)) = x, ∀ x ∈ [0, 1].
Examples 2.2 [27] • The fuzzy negation N c (x) = 1 − x is strong. It also is called the standard fuzzy negation.
• The smallest and the greatest fuzzy negations are respectively given by
N ⊥ (x) = 1 x = 0 0 otherwise and N ⊤ (x) = 0 x = 1 1 otherwise .
• The natural negation of a fuzzy implication I (See Definition 2.11) is defined by N I (x) = I(x, 0).
Let ϕ be an automorphism on [0,1] (that is, an increasing bijection on [0,1]) and f an n-ary
function on [0,1]. We can define a function f ϕ (x 1 , x 2 · · · , x n ) = ϕ −1 (f (ϕ(x 1 ), ϕ(x 2 ) · · · , ϕ(x n ))).
Then, the following statement holds. (A1) A satisfies the boundary conditions: A(0, 0, · · · , 0) = 0 and A(1, 1, · · · , 1) = 1;
(A2) A is non-decreasing in each variable.
Obviously, A ϕ is again an aggregation function and is called a ϕ-conjugate of A.
ii. associative if A(x, A(y, z)) = A(A(x, y), z) for any x, y, z ∈ [0, 1],
iii. a semi-copula if 1 is a neutral element, iv. a uninorm if it is associative, commutative and having a neutral element e ∈ (0, 1), v. a t-norm if it is an associative and commutative semi-copula, vi. a t-conorm if it is dual to a t-norm;
vii. a copula if it is a semi-copula which is two-increasing, i.e., A(x 1 , y 1 ) − A(x 1 , y 2 ) −
A(x 2 , y 1 ) + A(x 2 , y 2 ) ≥ 0 holds for all x 1 , y 1 , x 2 , y 2 ∈ [0, 1] such that x 1 ≤ x 2 and y 1 ≤ y 2 .
Definition 2.7 [22] The aggregation function A 1 is not greater than
A 2 if A 1 (x, y) ≤ A 2 (x, y)
holds for any x, y ∈ [0, 1], for simplicity, it is denoted by A 1 ≤ A 2 .
Definition 2.8 [39] Let A be an aggregation function.
i. A is a conjunctor if it satisfies A(1, 0) = A(0, 1) = 0, ii. A is a disjunctor if it satisfies A(1, 0) = A(0, 1) = 1,
iii. A has zero divisors if there exist x, y ∈ (0, 1] such that A(x, y) = 0, iv. A has one divisors if there exist x, y ∈ [0, 1) such that A(x, y) = 1.
Example 2.9 [17,40] The following are some distinguished disjunctors:
• The greatest disjunctor, D ⊤ (x, y) = 0 x = y = 0 1 otherwise ;
• The smallest disjunctor, D ⊥ (x, y) = 1 x = 1 or y = 1 0 otherwise ;
• t-conorms; • Continuous generated functions with a neutral element e ∈ (0, 1) is defined as A e,g (x, y) = g (−1) (g(x) + g(y)), where g : [0, 1] → [−∞, ∞] is a continuous and strictly monotone function such that g(e) = 0, g(1) = ±∞ and g (−1) its pseudo inverse of g defined as
g (−1) (x) = g −1 (x) x ≤ g(1) 1 otherwise .
Considering the non-commutativity of aggregation function, we extend the non-contradiction principle in Ref. [40] as follows.
Definition 2.10 [40] Let A be a binary aggregation function and N a fuzzy negation. We say that A satisfies the law of non-contradiction (LNC) with respect to N if
A(N (x), x) = A(x, N (x)) = 0, ∀x ∈ [0, 1].(LNC)
By duality, we say that A satisfies the law of excluded middle (LEM) with respect to N if
A(N (x), x) = A(x, N (x)) = 1, ∀x ∈ [0, 1].(LEM)
Obviously, A is a conjunctor (disjunctor) if it satisfies (LNC) ((LEM)).
I A,N (x, y) = A(N (x), y).
If N is the standard negation, I A,N is called an A-implication. Moreover, I S,N generated by a strong negation N and a t-conorm is called a strong implication or S-implication.
Theorem 2.16 [39] I is a fuzzy implication if and only if I is an
A-implication, i.e. there exists a disjunctor A such that I(x, y) = I A,N (x, y) = A(1 − x, y).I A (x, y) = sup{t ∈ [0, 1] | A(x, t) ≤I A1,A2 (x, y) = A 1 (N (x), A 2 (x, y)), x, y ∈ [0, 1].
Especially, a QL-operation is a QL-implication if it satisfies (I1).
Definition 2.19 [44] Let f : [0, 1] → [0, +∞] be a strict decreasing and continuous mapping
with f (1) = 0. An f -generated implication, which is a function I f : [0, 1] 2 → [0, 1] with an f -generator, is defined by I f (x, y) = f −1 (xf (y)) with the understanding 0 × ∞ = 0.
Definition 2.20 [44] Let g : [0, 1] → [0, +∞] be a strict increasing and continuous mapping
with g(0) = 0. A g-generated implication, which is a function I g : [0, 1] 2 → [0, 1] with a g- generator, is defined by I g (x, y) = g (−1) g(y)
x with the understanding 0 × ∞ = ∞. [34] Let T be a continuous t-norm and I T its power implication.
Definition 2.21 [18] Let C be a copula. A function I C : [0, 1] 2 → [0, 1] given by I C (x, y) = C(x,y) x x > 0 1 otherwise is called a probabilistic implication if it satisfies (I1). Definition 2.22 [18] Let C be a copula. A function I C : [0, 1] 2 → [0, 1] given byĨ C (x, y) = C(x, y) − x + 1 is called a probabilistic S-implication. Definition 2.23 [34] A function I T : [0, 1] 2 → [0, 1] is said to be a T -power implication if there exists a continuous t-norm T such that I T (x, y) = ∨{r ∈ [0, 1]|y (r) T ≥ x} for all x, y ∈ [0, 1]. Lemma 2.24i. If T = T M is the minimum t-norm, then I TM (x, y) = 1 x ≤ y 0 x > y ;
ii. If T is an Archimedean t-norm with additive generator t, then
I T (x, y) = 1 x ≤ y t(x) t(y)
x > y .
Satisfaction of the axioms with aggregation functions and fuzzy implications
This section will study some properties of aggregation functions and fuzzy implications when they satisfy (A4) and (A5) in the ACRI method. Considering the normal fuzzy sets play an important role in ACRI method, we always assume the fuzzy sets involved in ACRI method are normal in the rest of this paper.
Lemma 3.1 Let I satisfy (A4) with an aggregation function A and a fuzzy negation N . Then
A is a conjunctor.
Proof. Since I satisfies (A4) with A and N , we have N (D(x)) = Proof. (=⇒) Obviously.
(⇐=) y∈V A(N I (B(y)), I(D(x), B(y))) ≥ A(N I (0), I(D(x), 0)) = A(1, N I (D(x))) = N I (D(x)).
In this paper, we say that the fuzzy implication I satisfies the dual of A-conditionality with respect to N if the above inequality holds for any x ∈ U and y ∈ V . And then, this inequality is shorten as Proof. It is sufficient to verify that I does not satisfy (DAC). Let 1 > a 0 > b 0 > 0 and Proof. Since D is normal, there exists x 0 ∈ U such that D(x 0 ) = 1. By (A4), we have A(N (B(y)), I(1, B(y)) = 0 holds for any y ∈ V . This implies that A has zero divisors. Remark 1. i. Similar to Ref. [4,10,25,30], we say that the fuzzy implication I satisfies Aconditionality if the above inequality holds for any x ∈ U and y ∈ V . And then, this inequality is shorten as
A(N (b), I(a, b)) ≤ N (a), ∀ a, b ∈ [0, 1]. (DAC)I(a 0 , b 0 ) = 1. This implies that A(N (b 0 ), I(a 0 , b 0 )) = A(N (b 0 ), 1) = N (b 0 ) > N (a 0 ) holds.A(a, I(a, b)) ≤ b, ∀ a, b ∈ [0, 1].(AC)
ii. If I fulfills (IP) and A has a right neutral element 1, then (I, A) satisfies (A5) if they satisfy (AC). Proof. By Lemma 3.8, it is sufficient to verify that J satisfies (AC) with A. Indeed, the monotonicity of A implies that A(a, J(a, b)) ≤ A(a, I(a, b)) ≤ b.
A ′ (a, J(a, b)) ≤ A(a, I(a, b)) ≤ b.
Lemma 3.18 Let A ϕ , I ϕ and N ϕ be the ϕ-conjugate of A, I and N , respectively. We have the following statements:
i. If I satisfies (A4) with A and N , then I ϕ satisfies (A4) with A ϕ and N ϕ ;
ii. If I satisfies (A5) with A, then I ϕ satisfies (A5) with A ϕ .
Proof. We only verify I ϕ satisfies (A5) with A ϕ . The other can be verified similarly. For
any y ∈ V , we have x∈U A ϕ (D(x), I ϕ (D(x), B(y))) = x∈U ϕ −1 (A(ϕ(D(x)), ϕ(ϕ −1 (I(ϕ(D(x)),
ϕ(B(y))))))) = x∈U ϕ −1 (A(ϕ(D(x)), I(ϕ(D(x)), ϕ(B(y))))) = ϕ −1 ( x∈U A(ϕ(D(x)), I(ϕ(D(x)), ϕ(B(y))))) = ϕ −1 (ϕ(B(y))) = B(y).
Lemma 3.19
Let I and J be two fuzzy implications. We have the following statements:
i. If I and J satisfy (A4) with A and N , then I ∨ J and I ∧ J satisfy (A4) with A and N , too.
ii. If I and J satisfy (A5) with A, then I ∨ J and I ∧ J satisfy (A5) with A, too.
Proof. Obviously.
Modus ponens of ACRI method for well-known fuzzy implications
In this section, we shall discuss the modus tollens property of ACRI method with wellknown fuzzy implications. That is, to seek some aggregation functions for well-known fuzzy implications such that they satisfy (A5). Proof. By Lemmas 3.9 and 4.1, it is sufficient to verify that A I has a left neutral element 1.
Indeed, A I (1, b) = inf{c|I(1, c) ≥ b} = inf{c|c ≥ b} = b.
We further study the satisfaction of (A5) with the R-implications. Obviously, we can obtain the following statement. (=⇒) Suppose that (I A , A ′ ) satisfies (A5). We have A ′ (a, I A (a, b)) ≤ b for any a, b ∈ [0, 1]. A(a, b)) holds for any a, b ∈ [0, 1]. A(a, b))) ≤ A(a, b). Proof. This proof is similar to that of Proposition 2.5.9 in [4].
The residuation property of I
A with A implies that b ≤ I A (a,
We therefore have
A ′ (a, b) ≤ A ′ (a, I A (a,
We now focus on the the R-implications generated by non left continuous with respect to second variable aggregation functions. It is not difficult to see that the R-implication I
Inspired by the idea of [20], we shall extend a border continuous aggregation function to a left continuous one.
A * (x, y) = sup{A(u, v)|u < x, v < y} x, y ∈ (0, 1) A(x, y) otherwise .(1)
Obviously, A * is a left continuous with respect to second variable aggregation function and Proof. We firstly assert that I A = I A * . Obviously, I A ≤ I A * holds. We further have I A (a, b) = I A * (a, b) = 1 in cases where a ≤ b. Thus, it needs to consider the case 1 > a > b > 0. On the contrary, suppose that there exist 1 > a 0 > b 0 > 0 such that Proof. This can be proved similarly to Theorem 4.8. Proof. In this case, I A,N becomes the greatest fuzzy implication, that is, I A,N (x, y) = 0 x = 1, y = 0 1 otherwise . Let A ′ be an aggregation function. For any b ∈ (0, 1), we have A ′ (1, x ∨ y otherwise .
A * ≤ A.c = I A (a 0 , b 0 ) < I A * (a 0 , b 0 ) = c ′ . By intermediate value theorem, there exists c 0 such that c < c 0 < c ′ . Since A * is left continuous with respect to second variable, c 0 ≤ I A * (a 0 , b 0 ) = c ′ implies that A * (a 0 , c ′ ) ≤ b 0 holds. This means b 0 ≥ A * (a 0 , c ′ ) = sup{A(u, v)|u < a 0 , v < c ′ } ≥ sup{A(u, c 0 )|u < a 0 } = A(a − 0 , c 0 ). Therefore, c 0 ≤ I A (a − 0 , b 0 ) holds. We then have 1 = I A (c 0 , I A (a − 0 , b 0 )) = I A (a − 0 , I A (c 0 , b 0 ))if A ≤ A I f , where A I f (x, y) = f −1 ( f (y) x ) x = 0 0 x =I A,N (1, b)) = 1 > b.A IS,N (x, y) = 0 N (x) ≥ y a α + (e α − a α )f −1 α f α N (x)−aα eα−aα − f α y−aα eα−aα N (x) ∈ [a α ,
With some tedious calculations, we can obtain A IS,N as follows For a QL-operation I A1,A2 , it is easily seen that I A1,A2 satisfies (I3) and (I5) when A 1 is a disjunctor and A 2 is a conjunctor. Further, let A 2 have a right neutral element 1. We then have the fact that A 1 satisfies (LEM) (that is, A 1 (N (x), x) = 1 holds for any x ∈ [0, 1]) if I A1,A2 is a QL-implication. We therefore only consider the case where I A1,A2 is obtained from a disjunctor having a left neutral element 0, a conjunctor having a neutral element 1 and a fuzzy negation in the rest of this section.
A IS,N (x, y) = 0 N (x) ≥ y a α + (e α − a α )f −1 α f α N (x)−aα eα−aα − f α y−aα eα−aα N (x) ∈ [a α ,
Lemma 4.14 Let I A1,A2 be a QL-implication generated by a disjunctor without one divisor A 1 , A 2 and a fuzzy negation N . Then I A1,A2 satisfies (A5) with any conjunctor A having a left neutral element 1.
Proof. Since A 1 has not one divisor, A 1 satisfies (LEM) with respect to N if and only N = N ⊤ .
Therefore, I A1,A2 is an (A, N )-implication generated by A 1 and N ⊤ . It is not difficult to verify that I A1,A2 satisfies (A5) with conjunctor A having a left neutral element 1. ii. The probabilistic S-implication
Further, we have I A1,A2 (x, y) = A 1 (N (x), A 2 (x, y)) ≤ A 1 (N (x), A 2 (1, y)) = A 1 (N (x), y) = I A1,N (x,I C satisfies (A5) with A if and only if A ≤ A IC , where A IC (x, y) = 0 x + y ≤ 1 c −1 (c(x + y − 1) − c(x)) otherwise
for any x, y ∈ [0, 1].
Proof. The proof is similar to that of Theorem 4.13.
Modus tollens property of ACRI method for well-known fuzzy implications
This section will investigate the modus tollens property of ACRI method with well-known fuzzy implications. Obviously, we have firstly the following statements. Proof. According to Lemma 3.5, it is sufficient to verify that the equals sign holds in (DAC).
Indeed, y∈V A(N ⊥ (B(y)), I(D(x), B(y))) = N I (D(x)).
Lemma 5.2 Let the fuzzy implication I fulfill I(1, b) > 0 for some b ∈ (0, 1). Then there does not any conjunctor such that I satisfies (A4) with the greatest fuzzy negation.
Proof. This proof is similar to that of Lemma 5.1.
However, it is not easy to investigate (A4) with ordinary non-continuous fuzzy negation.
We therefore only consider the case where N is strong in the rest of this section. As pointed out in Remark 1, the (CP(N)) acts as a bridge between modus ponens property and modus tollens property of ACRI method. Thus, we assume that I fulfills (NP) and A ′ has a left neutral element 1. And then begin to study the law of contraposition with a strong negation for well-known fuzzy implications. (⇐=) Since A is commutative and N I is strong, it is easy to verify that I A (x, y) = N I (A(x, N I (y)) satisfies (CP(N)).
Remark 5. i. In this case, I A can be rewritten as
I A (x, y) = A N (N I (x), y) which is an (A, N )-implication, where A N (x, y) = N (A(N (x), N (y))) is the dual of A with N .
ii. Obviously, A has zero-divisors. And then A(x, y) = 0 iff x ≤ N I (y).
For a QL-implication I A1,A2 , the following equation holds if it fulfills (CP(N)).
A 1 (N (x), A 2 (x, y)) = A 1 (y, A 2 (N (x), N (y))).
It is not difficult to see that A 1 satisfies (LEM) if A 2 has a right element 1. However, it still is not easy to solve this equation. So, we only consider the case when A 1 is a continuous t-conorm and A 2 a t-norm. And the corresponding result can be found in [14]. holds.
Lemma 5.9 [33] i. If T is the Minimum t-norm, then I T satisfies (CP(N)) if and only if N is strictly decreasing;
ii. If T is strict with additive generator t, then I T satisfies (CP(N)) if and only if N is a strong negation given by N (x) = t −1 ( k t(x) ) for some positive constant k; iii. If T is a non-strict Archimedean t-norm, then I T does not satisfies (CP(N)) with any fuzzy negation. Next, let us consider the case when the fuzzy implications do not satisfy (CP(N)). We firstly modify the fuzzy implication by the method in [1]. Let N be a strong negation and I a fuzzy implication. The N -lower (upper)-contrapositivisation of I, denoted as I lc I,N (I uc I,N ) is defined as
I lc I,N (x, y) = I(x, y) y ≥ N (x) I(N (y), N (x)) otherwise , I uc I,N (x, y) = I(x, y) y ≤ N (x) I(N (y), N (x)) otherwise .
It can be obtained the fact that I lc I,N and I uc I,N are two fuzzy implications and satisfy (CP(N)) (See Theorems 1 and 4 in [1]). Further, the aggregation functions A I lc I,N and A I uc I,N can be constructed according to Lemma 4.1. We have immediately the following result. Lemma 5.11 Let N be a strong negation and I a fuzzy implication. I lc I,N and I uc I,N satisfy (DAC) with A I lc I,N and A I uc I,N , respectively. However, I lc I,N and I uc I,N are not equal to I unless I fulfills (CP(N)) itself. So, we begin to consider the satisfaction of (A4) for some special conjunctors. Lemma 3.7 implies that I does not satisfy (A4) with any conjunctor A without zero divisor. Thus, we move on the next case when A has zero divisors. Inspired by Ref. [13], we further demand that A meets the following conditions:
(C1) A has a left neutral element 1;
(C2) A(x, 1) < 1 and φ(x) = A(x, 1) is strictly continuous increasing;
(C3) A(x, A(y, z)) = A(y, A(x, z)).
According to Theorem 5.1 in [13], there exists a t-norm T such that A(x, y) = T (φ(x), y). In this case, we can easily see that A has zero divisors if and only if T has zero divisors. Therefore, we have the following results. (N (b), N (a)) holds for any a, b ∈ [0, 1], where I A is the R-implication generated by A. Then, we have I A (N (b), N (a) N (a)) + 1 − ϕ(φ (N (b)))) ∧ 1). Therefore, N (a)) + 1 − ϕ(φ (N (b)))) ∧ 1).
) = {z|A(N (b), z) ≤ N (a)} = {z|T (φ(N (b)), z) ≤ N (a)} = {ϕ −1 ((ϕ(φ(N (b))) + ϕ(z) − 1) ∨ 0) ≤ N (a)} = ϕ −1 (ϕ(I(a, b) ≤ ϕ −1 (ϕ(
Especially, (I, A) satisfies (A4) with respect to N I if A has a left neutral element 1 by (=⇒) Assume that I A,N satisfies (A4) with N and A ′ . Since A has a right neutral element 0, we have N IA,N = N ≤ N A ′ = N ϕ . Further, there exists an automorphism ϕ on [0,1] such that (ϕ(φ(N (b))) + ϕ(A(a, b)) − 1) ∨ 0 ≤ ϕ(a). This implies that (ϕ(A(a, b)) ≤ (1 − ϕ(φ(N (b)))+ ϕ(a))∧1. We then have A(a, b) ≤ ϕ −1 (1 − ϕ(φ(N (b)))+ ϕ(a))∧1). Therefore, N ϕ (φ(N (b)))), where (S LK ) ϕ is the ϕ-conjugate of Lukasiewicz t-conorm S LK .
A(a, b) ≤ (S LK ) ϕ (a,
Theorem 5.14 Let I T is an R-implication generated by continuous t-norm T . If the continuous conjunctor A fulfills (C1)-(C3), then (I T , A) satisfies (DAC) if and only if there exist an automorphism ϕ on [0,1] and some continuous additive generators t α such that the following inequality holds for any a, b ∈ [a α , e α ]: N (a)) + 1 − ϕ(φ (N (b)))) ∧ 1) − a α e α − a α .
t α b − a α e α − a α − t α a − a α e α − a α ≤ t α ϕ −1 (ϕ(
Especially, (I T , A) satisfies (A4) with respect to N I if and only if there exist an automorphism ϕ on [0,1] and some continuous additive generators t α such that the following inequality holds for any a, b ∈ [a α , e α ]:
t α b − a α e α − a α − t α a − a α e α − a α ≤ t α ϕ −1 (2 − ϕ(a) − ϕ(φ(ϕ −1 (1 − ϕ(b)))) ∧ 1) − a α e α − a α .
Proof. (⇐=) This can be verified directly.
(=⇒) Let (I T , A) satisfy (A4). According to Theorem 5.12, we have the fact that I T (a, b) ≤ ϕ −1 (ϕ(N (a)) + 1 − ϕ(φ (N (b)))) ∧ 1). Since T is a continuous t-norm, I T can be rewritten as follows by Theorem 2.5.24 in [4]: a ≤ g(ϕ −1 (ϕ(a) + 1 − ϕ(φ (N (b))) ∧ 1)) holds for any a, b ∈ [0, 1]. Especially, (I g , A) satisfies (A4) with respect to N I if and only if there exists an automorphism ϕ on [0,1] such that g(b) a ≤ g(ϕ −1 (1 + ϕ(a) − ϕ(φ(ϕ −1 (1 − ϕ(b)))) ∧ 1)) holds for any a, b ∈ [0, 1]. Theorem 5.18 Let I C be an probabilistic implication. If the continuous conjunctor A fulfills
I
(
A1) B ⊆ B ′ ; (A2) If D ′ ⊆ D ′′ , then B ′ ⊆ B ′′ ; (A3) If D ′ = D C , then B ′ = V , where D C is the complement of D; (A4) If D ′ = B C , then B ′ = D C ;
( A5 )
A5If D ′ = D, then B ′ = B; (A6) If D ′ = very D, then B ′ = very B; (A7) If D ′ = more or less D, then B ′ = more or less B; (A8) If D ′ = D C , then B ′ = B C ;
•
The dual of representable aggregation functions, A(x, y) = f −1 ((f (x∨y)−f (N (x∧y)))∨0), where f : [0, 1] → [0, +∞] is continuous strictly decreasing with f (1) = 0 and N is a strong negation; • Weighted quasi-arithmetic mean (WQAM), M λ,f (x, y) = f −1 ((1 − λ)f (x)+ λf (y)), where f : [0, 1] → [−∞, +∞] is continuous and strictly monotone with f (1) = ±∞ and λ ∈ (0, 1); • Group functions (that is are commutative and continuous functions such that A(x, y) = 0 if and only if x = y = 0 and A(x, y) = 1 if and only if xy = 1);
Definition 2 .
211 [4] A fuzzy implication is a function I : [0, 1] 2 → [0, 1] which satisfies:(I1) Non-increasing in the first variable, i.e. I(x, z) ≥ I(y, z) if x ≤ y;(I2) Non-decreasing in the second variable, i.e. I(x, y) ≤ I(x, z) if y ≤ z; 2.11, we directly obtain the fact that a fuzzy implication satisfies the followingproperties: (LB) Left boundary condition, I(0, y) = 1, ∀ y ∈ [0, 1]; (RB) Right boundary condition, I(x, 1) = 1, ∀ x ∈ [0, 1]. Definition 2.12 [4] A fuzzy implication I : [0, 1] 2 → [0, 1] satisfies: (NP) Left neutrality property, if I(1, y) = y, ∀ y ∈ [0, 1]; (IP) Identity principle, if I(x, x) = 1, ∀ x ∈ [0, 1]; (EP) Exchange principle, if I(x, I(y, z)) = I(y, I(x, z)), ∀ x, y, z ∈ [0, 1]; (CP(N)) Law of contraposition with a fuzzy negation N if I(x, y) = I(N (y), N (x)), ∀ x, y ∈ [0, 1]; (OP) Ordering property, if I(x, y) = 1 ⇐⇒ x ≤ y, ∀ x, y ∈ [0, 1]. Definition 2.13 [39] Let I be a fuzzy implication and A an aggregation function. I is said to satisfy the law of importation with an aggregation function A (LIA) if for all x, y, z ∈ [0, 1], I(A(x, y), z) = I(x, I(y, z)). (LIA) Definition 2.14 [4] Let I and J be two fuzzy implications. We can define (I ∨ J)(x, y) = max(I(x, y), J(x, y)), (I ∧ J)(x, y) = min(I(x, y), J(x, y)) and I ϕ (x, y) = ϕ −1 (I(ϕ(x), ϕ(y))). It is not difficult to see that I ∨ J, I ∧ J and I ϕ are fuzzy implications, too. Definition 2.15 [39] An (A, N )-implication is a function I A,N : [0, 1] 2 → [0, 1] associated with a disjunctor A and a fuzzy negation N defined by
A(N (B(y)), I(D(x), B(y)))for any x ∈ U . Especially, for x 0 and y 0 such that D(x 0 ) = 1 and B(y 0 ) = 1, we have A(0, I(1, 1)) = A(0, 1) = 0. A(1, 0) = 0 can be similarly obtained. Thus, A is a conjunctor.Lemma 3.2 Let I satisfy (NP) and (A4) with a conjunctor A being a left neutral element 1 and a fuzzy negation N . Then N I ≤ N ≤ N A holds, where N A (x) = sup{y ∈ [0, 1]|A(x, y) = 0} is the natural negation of A. Proof. Since I satisfies (A4) with A and N , we have N (D(x)) ≥ A(N (0), I(D(x), 0)) = N I (D(x)). Similarly, we obtain N (1) = A(N (B(y)), I(1, B(y))) = A(N (B(y), B(y)). This implies that N ≤ N A . Lemma 3.3 Let I satisfy (IP) and A have a right neutral element 1. Then I satisfies (A4) with A and N if and only if A(N (B(y)), I(D(x), B(y))) ≤ N (D(x)) holds for any x ∈ U and y ∈ V .Proof. (=⇒) Obviously.(⇐=) Suppose that A(N (B(y)), I(D(x), B(y))) ≤ N (D(x)) holds for any x ∈ U and y ∈ V . We immediately havey∈V A(N (B(y)), I(D(x), B(y))) ≤ N (D(x)). On the other hand, y∈V A(N (B(y)), I(D(x), B(y))) ≥ A(N (D(x)), I(D(x), D(x))) = A(N (D(x)), 1) = N (D(x)).Lemma 3.4 Let A have a left neutral element 1. Then I satisfies (A4) with A and N I if and only if A(N I (B(y)), I(D(x), B(y))) ≤ N I (D(x)) holds for any x ∈ U and y ∈ V .
Lemma 3. 5
5The fuzzy implication I satisfies (DAC) with respect to N ⊥ and any conjunctorA.Proof. It is sufficient to A(N ⊥ (b), I(a, b)) = 0 holds for any a = 0. We consider two cases. i.if b = 0. This implies that A(N ⊥ (b), I(a, b)) = A(0, I(a, b)) = 0. ii. if b = 0. In this case, we have A(1, I(a, 0)) = A(1, 0) = 0.Lemma 3.6 Let N be an one-by-one fuzzy negation. If there exist 1 > a > b > 0 such that I(a, b) = 1, then I does not satisfy (A4) with any aggregation function A having a right neutral element 1.
Lemma 3. 7
7Let I satisfy (A4) with A and N . If N = N ⊥ and I(1, b) > 0 holds for any b > 0, then A has zero divisors. Especially, if I fulfills (NP), then A satisfies (NC).
Lemma 3. 8
8Let I satisfy (A5) with A being a left neutral element 1. Then I(1, b) ≤ b holds for any b ∈ [0, 1]. Proof. We have B(y) = x∈U A(D(x), I(D(x), B(y))) ≥ A(1, I(1, B(y)) = I(1, B(y)) for any y ∈ V . Lemma 3.9 Let I satisfy (NP) and A have a left neutral element 1. Then I satisfies (A5) with A if and only if A(D(x), I(D(x), B(y))) ≤ B(y) holds for any x ∈ U and y ∈ V . Proof. (=⇒) Obviously. (⇐=) Suppose that A(D(x), I(D(x), B(y))) ≤ B(y) holds for any x ∈ U and y ∈ V . We immediately have x∈U A(D(x), I(D(x), B(y))) ≤ B(y). On the other hand, y))) ≥ A(D(x 0 ), I(D(x 0 ), B(y))) = A(1, I(1, B(y)) = I(1, B(y)) = B(y).
iii. Obviously, (AC) implies (DAC) if I fulfills (CP(N)). Further, (DAC) implies (AC) if I satisfies (CP(N)) and N is continuous. Owing to the relationship between (AC) and (DAC), we can parallelly obtain the following results.
Lemma 3 .
310 If I satisfies (A5) with A, then A is a conjunctor.
Lemma 3 .
311 Let I satisfies (A5) with A. If N I > N ⊥ , then A has zero divisors.Lemma 3.12 If there exist 1 > a > b > 0 such that I(a, b) = 1, then I does not satisfy (A5) with any aggregation function A having a right neutral element 1.Remark 2. According to Lemmas 3.6 and 3.12, we can directly obtain the fact that I(a, b) = 1 implies a ≤ b if I satisfies (A4) or (A5) with A having a right neutral element 1. However, we can not ensure that I satisfies (OP) in this case as shown in the following examples.
Example 3. 13
13Let I f be an f -implication. We know that I f does not satisfy (OP). However, we can define an aggregation function A as A(x, y)= f −1 ( f (y) x ) x = 0 0 x = 0 . Then, (I f , A)satisfies (A5).
Example 3. 14
14Let I be an R-implication generated by a nilpotent t-norm. Obviously, I fulfills (OP). However, I does not satisfies (A5) with A and N ⊤ unless A is the smallest conjunctor.
Lemma 3. 15
15Let I satisfy (OP) and (LIA) with A. If A is commutative, then I satisfies (AC) with A. Proof. By (LIA), we have I((A(a, I(a, b)), b) = I(a, I(I(a, b), b)) = I(I(a, b), I(a, b)) = 1 by Lemma 3.1 in Ref. [24]. Since I satisfies (OP), A(a, I(a, b)) ≤ b holds.Lemma 3.16 Let I and J fulfill (NP) and J ≤ I. If I satisfies (A5) with A having a left neutral element 1, then J satisfies (A5) with A, too.
Lemma 4. 1
1Let I be a fuzzy implication fulfilled the condition I(1, b) < 1 for any b ∈ [0, 1). Then there exists an aggregation function defined as A I (a, b) = inf{c|I(a, c) ≥ b} such that (I, A I ) satisfies (AC). Proof. According to Lemma 3.3 in [26], A I is an aggregation function. Further, we have A I (a, I(a, b)) = inf{c|I(a, c) ≥ I(a, b)} ≤ b.Corollary 4.2 Let I fulfill (NP). Then (I, A I ) satisfies (A5).
Theorem 4. 3
3Let A and A ′ be two conjunctors being a left neutral element 1. If A is left continuous with respect to second variable, then (I A , A ′ ) satisfies (A5) if and only if A ′ ≤ A. Proof. (⇐=) Since A has a left neutral element 1, I A satisfies (NP). According to Lemma 3.9, it is sufficient to verify that (I A , A ′ ) satisfies (AC). Since A is left continuous with respect to second variable, A and I A satisfy the residuation property (RP), i.e. A(a, c) ≤ b ⇐⇒ c ≤ I A (a, b) holds for any a, b, c ∈ [0, 1]. This means that A ′ (a, I A (a, b)) ≤ A(a, I A (a, b)) ≤ b.
A generated by an aggregation function A having a right neutral element 1 satisfies I A (a, b) = 1 if a ≤ b. This means that the R-implication I A fulfills (OP) when (I A , A) satisfies (A5) according to Lemma 3.12. In order to investigate (A5), it is therefore sufficient to consider the case where the R-implication I A fulfills (OP) and A has a right neutral element 1. We then have the following statement.
Lemma 4. 4
4Let I A be an R-implication generated by an aggregation function A having a right neutral element 1. Then, I A satisfies (OP) if and only if A is border continuous (that is, A is continuous at the border of [0, 1] 2 ).
Definition 4. 5
5Let A be a border continuous aggregation function. A function A * on [0, 1] 2 is defined as follows:
Theorem 4. 6
6Let I A be an R-implication generated by an aggregation function A having a neutral element 1 and A ′ a conjunctor being a left neutral element 1. If I A fulfills (OP) and (EP), then (I A , A ′ ) satisfies (A5) if and only if A ′ ≤ A * , where A * is defined as Eq.(1).
by (OP) and (EP). Again, a − 0 ≤ I A (c 0 , b 0 ) by (OP). On the other hand, 1 > I A (c 0 , I A (a 0 , b 0 )) = I A (a 0 , I A (c 0 , b 0 )) by (OP) and (EP). We then obtain a 0 > I A (c 0 , b 0 ). This implies that I A (c 0 , b 0 ) = a 0 . However, I A (c 0 , c) = I A (c 0 , I A (a 0 , b 0 )) = I A (a 0 , I A (c 0 , b 0 )) = I A (a 0 , a 0 ) = 1 by (OP) and (EP). Again, we obtain c 0 ≤ c. This is a contradiction. For the f -, g-and T -power implications, it is easy to obtain the aggregation functions generated by them according to Lemma 4.1. By Corollary 4.2, we immediately have the following results.
Theorem 4. 8
8Let I f be an f -implication. Then (I f , A) satisfies (A5) if and only
0 for any x, y ∈ [0, 1]. Proof. (⇐=) Since I f fulfills (NP), (I f , A) satisfies (A5) if A ≤ A I f by Corollary 4.2. (=⇒) A I f (a, I f (a, b)) = b holds for any a ∈ (0, 1] and b ∈ [0, 1]. Since (I f , A) satisfies (A5), we have A((a, I f (a, b)) ≤ b = A I f (a, I f (a, b)) for any a ∈ (0, 1] and b ∈ [0, 1]. The continuity of I f implies that A(a, b) ≤ A I f (a, b) holds for any a ∈ (0, 1] and b ∈ [0, 1]. For a = 0, we have A(0, b) = 0 = A I f (0, b). Thus, A ≤ A I f .Theorem 4.9 Let I g be a g-implication. Then (I g , A) satisfies (A5) if and only if A ≤ A Ig , where A Ig (x, y) = g −1 (xg(y)) x = 0 0 x = 0 for any x, y ∈ [0, 1].
Theorem 4. 10
10Let T be a continuous t-norm and I T its power implication. i. If T = T M is the minimum t-norm, then (I TM , A) satisfies (A5) if and only if A has a right neutral element 1;ii. If T is an Archimedean t-norm with additive generator t, then (I T , A) satisfies (A5) if and only if A ≤ A I T , where A I T (x, y) = t −1 ( t(x)y ) for any x, y ∈ [0, 1]. Proof. This proof is similar to that of Theorem 4.8.However, it is difficult to obtain the aggregation functions generated by(A, N )-, QL-, probabilistic and probabilistic S-implications by Lemma 4.1. So, we only consider some spacial (A, N )-, QL-, probabilistic and probabilistic S-implications in the last of this section.
Lemma 4.11 Let I A,N be an (A, N )-implication generated by the smallest disjunctor D ⊥ and a fuzzy negation N . There does not exist any conjunctor such that they satisfy (A5). Proof. The (A, N )-implication generated by the smallest disjunctor D ⊥ is the smallest fuzzy implication I A,N (x, y) = 1 x = 0 or y = 1 0 otherwise . Let A ′ be a conjunctor. For any fuzzy set B on V such that B(y) < 1, we have x∈U A ′ (D(x), I A,N (D(x), B(y))) = 0 < B(y).
Lemma 4 .
412 Let I A,N be an (A, N )-implication generated by the greatest disjunctor D ⊤ and a fuzzy negation N . There does not exist any aggregation function such that they satisfy (A5).
Theorem 4 .
413 Let I S,N be an (A, N )-implication generated by a continuous t-conorm S with the ordinal sum structure and a continuous fuzzy negation N . Then, I S,N satisfies (A5) if and only if A ≤ A IS,N , which is defined as
e α ] and N (x) < y y otherwise , where f α is the continuous additive generator of Archimedean t-conorm S α . Proof. By Corollary 5.12 in [22], there exists a uniquely determined (finite or countably infinite) index set A, a family of uniquely determined pairwise disjoint open subintervals {(a α , e α )} α∈A of [0, 1] and a family of uniquely determined continuous Archimedean t-conorms (S α ) α∈A such that the continuous t-conorm S can be rewritten as S(x, y) = a α + (e α − a α )S α x−aα eα−aα , y−aα eα−aα x, y ∈ [a α , e α ]
e α ] and N (x) < y y otherwise , where f α is the continuous additive generator of Archimedean t-conorm S α . (⇐=) Since I S,N fulfills (NP), (I S,N , A) satisfies (A5) if A ≤ A IS,N by Corollary 4.2. (=⇒) Suppose that (I S,N , A) satisfies (A5). Let us consider the following three options: i. If N (a) ≥ b. In this case, we have A(a, b) ≤ A(a, N (a)) = 0 = A IS,N (a, b).
ii. If N (a) ∈ [a α , e α ] and N (a) < b. This case implies that A IS,N (a, I S,N (a, b)) = b ≥ A(a, I S,N (a, b)) holds. By the continuity of I S,N , we have A(a, b) ≤ A IS,N (a, b).iii. If N (a) / ∈ [a α , e α ] and N (a) < b. In this case, we have I S,N (a, b) = b. Therefore, A(a, b) = A(a, I S,N (a, b)) ≤ b = A IS,N (a, b).
Remark 4 .
4We can similarly obtain the results about the (A, N )-implications generated by dual representable aggregation functions, weighted quasi-arithmetic mean, group functions generated by generators and continuous generated functions with a neutral element e, respectively. Here, the repetitious details are shown no longer.
y). This means that I A1,A2 satisfies (A5) with the same conjunctor A if I A1,N satisfies (A5) with A. In a general way, there exists a disjunctor A such that I A1,A2 (x, y) = I A,N (x, y) = A(1 − x, y) according to Theorem 2.16. We then have the following result.
Theorem 4 . 15 I
415A1,A2 satisfies (A5) with the same conjunctor A ′ if and only if I A,N satisfies (A5) with A ′ . Proof. Obviously.Theorem 4.16 Let C be an Archimedean copula with additive generator c. Then, i. The probabilistic implication I C satisfies (A5) with A if and only if A ≤ A IC , where A IC (x, y) = 0 x = 0 or y = 0 c −1 (c(xy) − c(x)) otherwise for any x, y ∈ [0, 1];
Lemma 5. 1
1Let I be a fuzzy implication and A a conjunctor. If N I is the small fuzzy negation, then (I, A) satisfies (A4) with the smallest fuzzy negation.
Lemma 5. 3
3Let I A,N be an (A, N )-implication with a strong negation. Then, I A,N satisfies (CP(N)) if and only if A is commutative. Proof. Obviously. Lemma 5.4 Let I A be an R-implication generated by the left continuous with respect to second variable aggregation function A. If A is commutative, then I A satisfies (CP(N)) if and only if I A (x, y) = N I (A(x, N I (y)). Proof. (=⇒) Suppose that I A satisfies (CP(N)) with a strong negation N . We have N = N I according to Corollary 1.5.7 in [4]. Since A is left continuous with respect to second variable, A and I A satisfy the residuation property by Lemma 3.1 in [26]. Therefore, A(x, y) ≤ z ⇐⇒ y ≤ I(x, z) ⇐⇒ y ≤ I(N I (z), N I (x)) ⇐⇒ A(N I (x), y) ≤ N I (z) ⇐⇒ z ≤ N I (A(N I (x), y)). This implies that I(x, y) = max{z ∈ [0, 1]|A(x, z) ≤ y} = max{z ∈ [0, 1]|z ≤ N I (A(N I (x), y))} = N I (A(N I (x), y)).
Lemma 5.5 [4] I f fulfills (CP(N)) with a strong negation if and only if f (0) < ∞.Lemma 5.6 [4] I g does not satisfy (CP(N)) with any fuzzy negation. Lemma 5.7 [3] I C does not satisfy (CP(N)) with any fuzzy negation. Lemma 5.8 [3]Ĩ C satisfies (CP(N)) if and only if the equation C(x, y) = x+y−1+C(1−y, 1−x)
Theorem 5.10 Let I be R-, (A, N )-, QL-, f -, probabilistic S-and T -power implications fulfilling (CP(N)), respectively. If I satisfies (A5) with A, then I satisfies (A4) with the same A.
Theorem 5.12 Let A be a continuous conjunctor fulfilling (C1)-(C3). Then, (I, A) satisfies (DAC) with respect to N if and only if there exists an automorphism ϕ on [0,1] such thatI(a, b) ≤ ϕ −1 (ϕ(N (a)) + 1 − ϕ(φ(N (b)))) ∧ 1) holdsfor any a, b ∈ [0, 1]. Especially, (I, A) satisfies (A4) with respect to N I if and only if there exists an automorphism ϕ on [0,1] such that I(a, b) ≤ ϕ −1 (2 − ϕ(a) − ϕ(φ(ϕ −1 (1 − ϕ(b)))) ∧ 1) holds for any a, b ∈ [0, 1]. Proof. (⇐=) Obviously. (=⇒) Let (I, A) satisfy (DAC) with respect to N . The continuity of A implies that I(a, b) ≤ I A
Lemma 3 . 4 .
34This means that (I, A) satisfies (A4) with respect to N I if and only if there existsan automorphism ϕ on [0,1] such that I(a, b) ≤ ϕ −1 (2 − ϕ(a) − ϕ(φ(ϕ −1 (1 − ϕ(b)))) ∧ 1) holds for any a, b ∈ [0, 1].Theorem 5.13 Let I A,N be an (A, N )-implication generated by a strong negation N and a disjunctor A having a right neutral element 0. If the continuous conjunctor A fulfills (C1)-(C3), then (I A,N , A ′ ) satisfies (A4) with respect to N if and only if there exists an automorphism ϕ on [0,1] such that A(a, b) ≤ ϕ −1 (ϕ((N (a)) + 1 − ϕ(φ(N (b)))) ∧ 1) holds for any a, b ∈ [0, 1]. Especially, (I, A) satisfies (A4) with respect to N I if and only if there exists an automorphism ϕ on [0,1] such that A(a, b) ≤ (S LK ) ϕ (a, N ϕ (φ(N (b)))) holds for any a, b ∈ [0, 1], where (S LK ) ϕ is the ϕ-conjugate of Lukasiewicz t-conorm S LK . Proof. (⇐=) This can be verified directly.
y ∈ [a α , e α ] y otherwise ,where I Tα is an R-implication generated by the Archimedean t-norm T α . Obviously, it is sufficient to study the case when a, b ∈ [a α , e α ]. Considering that T α is a continuous Archimedean t-norm, there exists a continuous additive generator t α such that T α (x, y) = t −1 ((t(x) + t(y)) ∧ t(0)). We therefore have t we have the following statements for QL-, f -, g-, probabilistic, probabilistic Sand T -power implications.Theorem 5.15 Let I A1,A2 be a QL-implication. If the continuous conjunctor A fulfills (C1)-(C3), then (I A1,A2 , A) satisfies (DAC) if and only if there exists an automorphism ϕ on [0,1] such that A 1 (a, A 2 (N (a), b)) ≤ ϕ −1 (ϕ(a) + 1 − ϕ(φ(N (b)))) ∧ 1) holds for any a, b ∈ [0, 1]. Especially, (I A1,A2 , A) satisfies (A4) with respect to N I if and only if there exists an automorphism ϕ on [0,1] such that A 1 (a, A 2 (N (a), b)) ≤ ϕ −1 (1 + ϕ(a) − ϕ(φ(ϕ −1 (1 − ϕ(b)))) ∧ 1) holds for any a, b ∈ [0, 1]. Theorem 5.16 Let I f be an f -implication. If the continuous conjunctor A fulfills (C1)-(C3), then (I f , A) satisfies (DAC) if and only if there exists an automorphism ϕ on [0,1] such that af (b) ≤ f (ϕ −1 (ϕ(a) + 1 − ϕ(φ(N (b))) ∧ 1)) holds for any a, b ∈ [0, 1]. Especially, (I f , A) satisfies (A4) with respect to N I if and only if there exists an automorphism ϕ on [0,1] such that af (b) ≤ f (ϕ −1 (1 + ϕ(a) − ϕ(φ(ϕ −1 (1 − ϕ(b)))) ∧ 1)) holds for any a, b ∈ [0, 1]. Theorem 5.17 Let I g be an f -implication. If the continuous conjunctor A fulfills (C1)-(C3), then (I f , A) satisfies (DAC) if and only if there exists an automorphism ϕ on [0,1] such that g(b)
(
C1)-(C3), then (I C , A) satisfies (DAC) if and only if there exist an automorphism ϕ on [0,1] such that C(a, b) ≤ aϕ −1 (ϕ(a) + 1 − ϕ(φ(N (b))) ∧ 1) holds for any a, b ∈ [0, 1]. Especially, (I C , A) satisfies (A4) with respect to N I if and only if there exists an automorphism ϕ on [0,1]
Theorem 2.3 [23] N is a strong negation if and only if there exists an automorphism ϕ on [0,1] such that N = (N c ) ϕ . Definition 2.4 [17] A function A : [0, 1] n → [0, 1] is said to be an n-ary aggregation function if the following conditions meet:
Definition 2.5 [17] Let A be a binary aggregation function. e ∈ [0, 1] is called a left (right)neutral element if A(e, x) = x (A(x, e) = x) for any x ∈ [0, 1]; e ∈ [0, 1] is a neutral element if
A(e, x) = A(x, e) = x for any x ∈ [0, 1].
Definition 2.6 [17] A binary aggregation function A is
i. symmetric or commutative if A(x, y) = A(y, x) for any x, y ∈ [0, 1],
Remark 3 .
3We can similarly obtain the fact that J satisfies (DAC) with A and N if I satisfies (DAC) with A and N . However, we cannot ensure whether J satisfies (A4) with A and N when J satisfies (A4) with A and N .Proof. We only verify I satisfies (A4) with A ′ and N . The other can be verified similarly.Lemma 3.17 Let I satisfy (NP) and A have a left neutral element 1. We have
i. if A ′ is a conjunctor being a left neutral element 1 such that A ′ ≤ A and I satisfies (A4)
with A and N I , then I satisfies (A4) with A ′ and N I , too;
ii. if A ′ is a conjunctor being a left neutral element 1 such that A ′ ≤ A and I satisfies (A4)
with A, then I satisfies (A5) with A ′ , too.
By Lemma 3.3, it is sufficient to verify that I satisfies (DAC) with A ′ . Indeed, we have
Conflict of interest Author declares that he has no conflict of interest.Human and animal rights This article does not contain any studies with human participants or animals performed by the authors.
that C(a, b) ≤ aϕ −1 (1 + ϕ(a) − ϕ(φ(ϕ −1 (1 − ϕ(b)))) ∧ 1) holds for any a. b ∈ [0, 1that C(a, b) ≤ aϕ −1 (1 + ϕ(a) − ϕ(φ(ϕ −1 (1 − ϕ(b)))) ∧ 1) holds for any a, b ∈ [0, 1].
then ( I C , A) satisfies (DAC) if and only if there exists an automorphism ϕ on [0,1] such that C(a, b) ≤ ϕ −1 (ϕ(a) + 1 − ϕ(φ(N (b))) ∧ 1) + a − 1 holds for any a, b ∈ [0, 1]. Especially, ( I C , A) satisfies (A4) with respect to N I if and only if there exists an automorphism ϕ on. Theorem 5.19 Let I C be a probabilistic S-implication. If the continuous conjunctor A fulfills (C1)-(C3). 0,1] such that C(a, b) ≤ ϕ −1 (1 + ϕ(a) − ϕ(φ(ϕ −1 (1 − ϕ. ∧ 1) + a − 1 holds for any a, b ∈ [0, 1Theorem 5.19 Let I C be a probabilistic S-implication. If the continuous conjunctor A fulfills (C1)-(C3), then ( I C , A) satisfies (DAC) if and only if there exists an automorphism ϕ on [0,1] such that C(a, b) ≤ ϕ −1 (ϕ(a) + 1 − ϕ(φ(N (b))) ∧ 1) + a − 1 holds for any a, b ∈ [0, 1]. Especially, ( I C , A) satisfies (A4) with respect to N I if and only if there exists an automorphism ϕ on [0,1] such that C(a, b) ≤ ϕ −1 (1 + ϕ(a) − ϕ(φ(ϕ −1 (1 − ϕ(b)))) ∧ 1) + a − 1 holds for any a, b ∈ [0, 1].
Theorem 5.20 Let T be a continuous t-norm and I T its power implication. i. If T = T M is the minimum t-norm, then (I TM , A) satisfies (DAC) if and only if A has a right neutral. element 1Theorem 5.20 Let T be a continuous t-norm and I T its power implication. i. If T = T M is the minimum t-norm, then (I TM , A) satisfies (DAC) if and only if A has a right neutral element 1;
then (I T , A) satisfies (DAC) if and only if there exists an automorphism ϕ on [0,1] such that t(a) t(b) ≤ ϕ −1 (ϕ(a) + 1 − ϕ(φ(N (b))) ∧ 1) holds for any a, b ∈ [0, 1]. Especially, (I T , A) satisfies (A4) with respect to N I if and only if there exists an automorphism ϕ on. T is an Archimedean t-norm with additive generator t and the continuous conjunctor A fulfills (C1)-(C3). 0,1] such that t(a) t(b) ≤ ϕ −1 (1 + ϕ(a) − ϕ(φ(ϕ −1 (1 − ϕ(b)))) ∧ 1) holds for any a, b ∈ [0, 1ii. If T is an Archimedean t-norm with additive generator t and the continuous conjunctor A fulfills (C1)-(C3), then (I T , A) satisfies (DAC) if and only if there exists an automorphism ϕ on [0,1] such that t(a) t(b) ≤ ϕ −1 (ϕ(a) + 1 − ϕ(φ(N (b))) ∧ 1) holds for any a, b ∈ [0, 1]. Especially, (I T , A) satisfies (A4) with respect to N I if and only if there exists an automorphism ϕ on [0,1] such that t(a) t(b) ≤ ϕ −1 (1 + ϕ(a) − ϕ(φ(ϕ −1 (1 − ϕ(b)))) ∧ 1) holds for any a, b ∈ [0, 1].
Conclusions Fuzzy implications and aggregation functions play a vital role in fuzzy inference and decision making. Therefore, we have studied the modus ponens and modus tollens properties of ACRI method with well-known fuzzy implications in detail. Concretely, we have (1) Analyzed the properties of fuzzy implications and aggregation functions satisfying. A4) or (A5)Conclusions Fuzzy implications and aggregation functions play a vital role in fuzzy inference and de- cision making. Therefore, we have studied the modus ponens and modus tollens properties of ACRI method with well-known fuzzy implications in detail. Concretely, we have (1) Analyzed the properties of fuzzy implications and aggregation functions satisfying (A4) or (A5);
Constructed the aggregation functions for well-known fuzzy implications such that they satisfy (A5). (2) Constructed the aggregation functions for well-known fuzzy implications such that they satisfy (A5);
) given the conditions for well-known fuzzy implications when they satisfy (A4) with a strong negation. (3) given the conditions for well-known fuzzy implications when they satisfy (A4) with a strong negation.
Considering that some linguistic modifiers are involved in (A6) and (A7), we will extend them as follows. A6 ′ ) If D ′ = m(D), then B ′ = m(B). In the future, we wish to study the validity of ACRI method using (A6) and (A7)These results contribute to improve the effectiveness of ACRI method. In the future, we wish to study the validity of ACRI method using (A6) and (A7). Considering that some linguistic modifiers are involved in (A6) and (A7), we will extend them as follows: (A6 ′ ) If D ′ = m(D), then B ′ = m(B);
(A7 ′ ) If D ′ = m(D), then B ′ = B, where m is a fuzzy modifier. (A7 ′ ) If D ′ = m(D), then B ′ = B, where m is a fuzzy modifier.
Acknowledgement This work was supported by the National Natural Science Foundation of China. Grant NoAcknowledgement This work was supported by the National Natural Science Foundation of China (Grant No.
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"Field-induced low-temperature electronic specific heat of boron nitride nanotubes",
"Field-induced low-temperature electronic specific heat of boron nitride nanotubes"
]
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"Feng-Lin Shyu [email protected]@fax:88677194170 \nDepartment of Physics, R.O.C. Military Academy\n830KaohsiungTaiwan, R.O.C\n"
]
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"Department of Physics, R.O.C. Military Academy\n830KaohsiungTaiwan, R.O.C"
]
| []
| We use the tight-binding model to study the effect of transverse electric field on the low-temperature electronic specific heat (C v ) for armchair and zigzag boron nitride nanotubes (ABNNTs and ZBNNTs). For wide-band-gap BNNTs, electric field could significantly modulate their energy dispersions and shift many electronic states close to the Fermi energy. Under a critical electric field (F c ), the density of states show special peak structures and the vanishing specific heat at zero field jumps to a giant one. C v , at F c 's, has a value comparable to that of the phonon specific heat and reveals strongly non-linear dependence on temperature. The critical field strength and the value of giant specific heat are closely related to nanotube's geometry. In the presence of F c 's, the extra longitudinal magnetic flux could enhance the value of C v again at low temperature for ZBNNTs, whereas it is not always true for ABNNTs. | null | [
"https://arxiv.org/pdf/1511.08442v1.pdf"
]
| 119,239,085 | 1511.08442 | 18e8bf82854691177ad102a5c4a8495653f23aa5 |
Field-induced low-temperature electronic specific heat of boron nitride nanotubes
November 26, 2015
Feng-Lin Shyu [email protected]@fax:88677194170
Department of Physics, R.O.C. Military Academy
830KaohsiungTaiwan, R.O.C
Field-induced low-temperature electronic specific heat of boron nitride nanotubes
November 26, 2015boron nitride nanotubeelectronic specific heattight-binding modelelec- tric fieldmagnetic field
We use the tight-binding model to study the effect of transverse electric field on the low-temperature electronic specific heat (C v ) for armchair and zigzag boron nitride nanotubes (ABNNTs and ZBNNTs). For wide-band-gap BNNTs, electric field could significantly modulate their energy dispersions and shift many electronic states close to the Fermi energy. Under a critical electric field (F c ), the density of states show special peak structures and the vanishing specific heat at zero field jumps to a giant one. C v , at F c 's, has a value comparable to that of the phonon specific heat and reveals strongly non-linear dependence on temperature. The critical field strength and the value of giant specific heat are closely related to nanotube's geometry. In the presence of F c 's, the extra longitudinal magnetic flux could enhance the value of C v again at low temperature for ZBNNTs, whereas it is not always true for ABNNTs.
Introduction
Carbon nanotubes (CNTs) described as a rolled-up graphene sheet have stirred many intensive studies owing to their unique physical properties and widely potential application in nanodevices. Due to similar hexagonal symmetry in layered boron nitride, the existence of boron nitride nanotubes (BNNTs) was first theoretically predicted by Rubio et al. [1].
Following the theoretical prediction, synthesis in experiment was performed by Chopra et al. using an arc-discharge method [2]. Unlike the strongly geometry-dependent electronic structures of CNTs, semiconducting BNNTs have a wide-band-gap ranging from 4 eV to 5 eV independent of their geometry [3]. Based on band-gap engineering, these results anticipate more practical applications in nanoscaled electronic and photonic devices for BNNTs than CNTs.
Some previous studies showed that applying transverse electric field could modulate electronic structures of CNTs and BNNTs [4][5][6], except for mechanical deformation and impurity doping. As a result, band gap was reduced and even semiconductor-metal transition occurred. The giant Stark effect was further validated experimentally by Ishigami et al. through the bias-dependent scanning tunneling microscopy [7]. From the point of view of perturbation, the Stark effect could cause a strong coupling for neighboring subbands with nearly equivalent state energy. That would lead to a modulation of energy dispersions and further change optical and magnetic properties [8][9][10][11]. Since BNNTs are III-V compounds, their boron-nitride bonding with strong ionicity is more sensitive to the giant Stark effect than the C-C bonding in CNTs. Therefore, a significant change in electronic and physical properties induced by electric field is expected for BNNTs.
With the rapid advance of nano-devices, heat removal has become a crucial issue due to increased levels of dissipated power. Searching for novel materials that conduct heat well has become essential for designing the new integrated circuits and photoelectronic devices. Therefore, thermal properties such as specific heat and thermal conductivity have been extensively investigated in the past decades. The ability, for materials, to conduct heat is related to their atomic structures. Especially, rich changes in thermal properties of materials are revealed when they are structured on nanometer scale. The specific heat of nanomaterials coming from phonons (C ph ) or electrons (C v ) is determined by the details of phonon or electron spectra. The low-temperature behavior of the specific heat includes informations involving thermal excitations and the low-dimensional quantum confinement.
A two-dimensional graphene, for instance, the ratio of C ph to C v at the low-temperature is about 10 4 , whereas it is down to 10 2 for an one-dimensional CNT [12]. The specific heat, in these two carbon-related system, is primarily dominated by phonon's contribution even for T → 0. It could be roughly explained by that the Fermi velocity in electron spectra is two orders of magnitudes greater than the sound velocity in phonon spectra. In order to increase the low-temperature electronic specific heat, reducing the Fermi velocity of electron spectra, i.e., increasing band-edge states near the Fermi energy, is required. Our previous study showed that applied transverse electric field, for BNNTs, could effectively change electronic energy dispersions around the Fermi energy. Critical electric field could enhance the magnitude of persistent current and change the magnetism [11]. It is predicted that electric field could also effectively modulate the specific heat of wide-gap BNNTs.
For BN-related systems, thermal properties have been studied for hexagonal boron nitride [13] and multi-walled BNNTs [14][15] in experiments. Specific heat of hexagonal BN at low temperature (T < 10 K) showed a T 3 -dependence while it revealed a T 2.4dependence for thermal conductivity. The different T -dependence between specific heat and thermal conductivity was due to electron-phonon scattering. As for multi-walled BN-NTs, the T -dependence of thermal conductivity in experimental measurements reflected low-dimensional quantum confinement and revealed that BNNTs might be better thermal conductors than CNTs. For single-walled BNNTs, thermal properties have been studied theoretically [16][17][18]. The calculated results predicted that a higher specific heat for BN-NTs than that for CNTs. Moreover, thermal conductance of BNNTs exhibited a universal quantization at low temperature, independent of nanotube's geometry. As mentioned in the above, the low-temperature thermal properties, due to the wide-band-gap of BNNTs, are all dominated by phonon effects, thoroughly independent of electronic states. Our study thus intends to raise the electronic contribution to specific heat by applying external fields.
It is expected that external fields could cause an increase of electronic states close to the Fermi energy that might largely enhance the contribution of electrons to low-temperature thermal properties.
In this work, we use 2p z orbital tight-binding model to study electronic structures and low-temperature electronic specific heat for ABNNTs and ZBNNTs in the transverse electric field. Our study shows electric field strongly modulates energy dispersions and largely reduces band gap to a value comparable to the low-temperature thermal energy. The DOS reveals special peak structures causing a giant electronic specific heat, while electric field is at a critical strength. Furthermore, the extra magnetic field would re-enhance the giant specific heat. Modulations of external fields on electronic specific heat such as the magnitude and temperature-dependence are strongly dependent on the geometric structure of BNNTs.
The tight-binding model for electronic structures in external fields
A boron nitride nanotube is made of a monolayer hexagonal boron nitride (hBN) rolled up into a hollow cylinder. Its geometric structure could be built from a hBN and characterized by two vectors, i.e., the first R x = ma 1 + na 2 in the circumferential direction and the second R y = pa 1 + qa 2 along the longitudinal direction. Where (m, n, p, q) are integers, a 1 and a 2 are primitive lattice vectors of a hBN. Due to the orthogonality R x · R y = 0, the parameters (m,n) uniquely define the geometric structure of a BNNT. The radius and chiral angle are r = R x /2π = b √ m 2 + mn + n 2 /2π and θ = tan −1 [− √ 3n/(2m + n)], respectively. ZBNNT. The number of atoms in a unit cell is N u = 4 (p 2 + pq + q 2 )(m 2 + mn + n 2 )/3 , and both the ABNNT and ZBNNT have the same N u (=4m).
The hermitian Hamiltonian matrix, in the tight-binding model, is built from the subspace spanned by the N u wave functions of 2p z orbitals. In the presence of electric and magnetic fields, the Hamiltonian including nearest-neighbor interactions is given by BNNTs, unlike CNTs, are III-V compound nanotubes with strong ionicity, their electronic structures are expected to be much more sensitive to applied electric field than magnetic field. While the transverse electric field is applied, the coordinate-dependent electric potential breaks the rotational symmetry such that the transverse momenta are no longer good quantum numbers. The neighboring subbands with nearly equivalent energy would be coupled and the coupling is getting stronger with increasing field strength that leads to a strongly k y -dependent energy dispersion. The magnitude of electric field leading to band gap with a value smaller than 10 −3 eV is considered, since the low-temperature (T < 5 K) thermal properties are our concerns. Electric field not only reduces band gaps but also significantly changes energy dispersions (or density of states). For a (12,12) BNNT, Fig. 1(a) shows the variation of energy dispersion with various F 's which could reduce band gap to 10 −3 eV . Energy dispersion is changed from a parabolic into a two-local-minima structure except the reduction of band gap while electric field increases from F = 0.3078 to F = 0.3081. There is a tiny energy difference between the two local minima that is reflected in the DOS ( Fig. 1(b)). Moreover, the symmetry of conduction and valence subbands about the Fermi energy is unaltered.
H = i i c + i c i + t 0 i,j e i(2π/φ 0 ) j i A·dr c + i c j ,(1)where i = E i + F rcosα i is
The characteristics of field-induced energy dispersions are unveiled in the DOS which could further give a detailed elucidation to thermal properties. The special (divergence) structures of the DOS imply a drastic change in thermal properties; the magnitude and Tdependence of the low-temperature electronic specific heat are determined by peak positions and heights of the DOS. It is defined as
D(ω) = 1 N u h,σ 1stBZ dk y 2π 1 π δ [ω − E h (F, k y ; σ, φ)] 2 + δ 2 ,(2)
where h = c, v represents conduction or valence bands and the phenomenological broadening parameter is δ = 10 −4 eV . For a (12,12) BNNT, solid black curves (F = 0.3078) in Fig. 1(b) show that the special structure for E > E F diverges in the form 1/ E c (k ye ) − ω from the concave upward edge-state with a wave vector k ye = 0.688. Oppositely, the
other (E < E F ) is in the form 1/ ω − E v (k ye ) from the concave downward edge-state.
While critical field F A c = 0.3079 is applied, the parabolic subbands are flattened leading to much sharper peak structures (solid red curves); that would make a big contribution to the low-temperature thermal properties. As electric field continuously increases from (13,13) and (14,14) BNNTs. Moreover, the critical field is no more determined by the peak height of the DOS; the peak position is also a key factor.
For example, critical field of a (13,13) BNNT is F A c = 0.2806 because its edge-states are closer to E F than those of F = 0.2805; that could provide much more low-temperature electronic states.
As for (m,0) ZBNNTs, (21,0), (22,0), and (23,0) BNNTs having the same radii as the mentioned ABNNTs are chosen as a comparison. The F -dependent electronic structures have strong geometric dependence described as follows. The doubly degenerate states closest E F are split into two singlet states by electric field, as shown in Fig. 2(a) for a (21,0) BNNT. As electric field gradually increases, energy dispersion oscillates and shows stronger k y -dependence. Moreover, edge-states are shift from k y = 0 to larger k y 's and band gap is reduced as shown in the DOS (Fig. 2(b)). Each pair of peaks in the DOS is from two split singlet states while it is from two local minima of a singlet state for ABNNTs.
Similarly, there also exists a critical electric field (F z c = 0.3040) inducing a sharp peak structure in the DOS and making a contribution to low-temperature thermal properties.
It is noticed that for ABNNTs, the reduction of band gap by electric field is enhanced with increasing their radii. However, the rule is not true for ZBNNTs shown in the DOS from
Field-modulated electronic specific heat
The electronic specific heat is defined as the derivative of the total energy with respect to temperature. In the presence of electric and magnetic fields, the total energy per mole carrier at temperature T is written as
U(T ) = 3 √ 3b 2 N A 8πr σ,h 1stBz dk y 2π [E h (F, k y ; σ, φ) − µ]f(E h (F, k y ; σ, φ), T ),(3)
where N A is the Avogadro's number, and
f = 1/{exp[β(E h (F, k y ; σ, φ) − µ)] + 1} is the
Fermi-Dirac distribution function with β = 1/k B T and k B denotes the Boltzmann constant.
The electronic specific heat at temperature T could be calculated by
C v (T ) = ∂U(T ) ∂T = 3 √ 3b 2 8πr N A k B T 2 σ,h 1stBz dk y 2π (E h (F, k y ; σ, φ) − µ) 2 e β(E h (F,ky;σ,φ)−µ) (1 + e β(E h (F,ky;σ,φ)−µ) ) 2 .(4)
In the above equation, the term related to the derivative of chemical potential with respect to temperature is omitted, since the chemical potential µ, due to the π-band symmetry, is fixed at E F for arbitrary field strength and temperature.
The electronic specific heat, due to the wide band gap, vanishes in the absence of electric field. Our main study thus is how electric field modulates the electronic specific heat at low temperature (T ≤ 5 K). For a (12,12) BNNT, the detailed F -dependent specific heat at T = 1 K is shown in the inset of Fig. 3(a). The specific heat keeps vanishing for insufficient field strength. While the first critical electric field is applied (F A 1c = 0.3079), energy dispersions are flattened and edge-states are closer to the Fermi energy. Band gap is comparable to the thermal energy (∼ 10 −4 eV ) at T = 1 K. There are sufficient electronic states to initiate low-temperature specific heat with a jump structure which has a value C v = 153 (unit µJ/mole K, here and henceforth). Here, C v , due to critical electric fields, jumps from a zero (or very tiny) value to a very large one which is called the giant electronic specific heat (GESH). With continuously increasing electric field, the edge-states become two-local-minima states with higher curvature that reduces the DOS and thus the value of C v . As temperature gradually increases, more and more thermally excited electrons occupy the electronic states above the chemical potential, and leaving holes at original states. These two kinds of excited carriers equivalently make contribution to the specific heat. Therefore, the contribution to the specific heat made by the Fermi function is extended to wider energy range. The jump structure in the F -dependent specific heat reveals higher peak and wider width with increasing T .
Experimentally, the field-enhanced specific heat is more easily performed for larger BNNTs owing to the smaller applied electric field. Thus, the F -dependent specific heat with varying nanotube's radius is further studied. We first choose (m,m) ABNNTs with BNNT, respectively. Besides, C v -F curves of a larger BNNT also reveal richer special structures within strong field region, e.g., 0.38 < F < 0.4 for a (14,14) BNNT. Because the larger BNNT under the same applied field is subjected to the larger electric potential than the small one that causes a significant oscillation in F -dependent band gaps. Except F A 1c , there exists the second critical field inducing a special jump structure in C v -F curves, e.g., F A 2c = 0.3615 for a (13,13) BNNT in Fig. 3(b). At the same time the third critical field F A 3c = 0.3908 inducing a giant specific heat is also clearly observed for a (14,14) BNNT ( Fig. 3(c)). Also notice that the peak height of special jump structures in C v -F curves increases with increasing critical electric field. The main reason is the stronger critical field could flatten edge-states and then induce the larger DOS. That fully reflects the stronger dependence of electronic structures on electric field for larger armchair BNNTs. Thus, larger armchair BNNTs are suitable to create the GESH by applying electric field. Its value is comparable to that induced by the phonon [16][17] and could not be ignored for thermal properties.
As for zigzag BNNTs, the F -dependent specific heat of a (21,0) BNNT is first studied. Fig. 3(d) shows that a very small specific heat C v = 9.1 at T = 1 K, due to the larger band gap (Fig, 2(b)), is induced by the first critical field F Z 1c = 0.3040. The features of C v -F curves with increasing temperature are similar to those of a (12,12) BNNT, might be different from the delta-function-like structures for zigzag CNTs induced by linear energy dispersions. The F -dependent electronic properties of ZBNNTs, unlike ABNNTs, do not follow a regular rule as the radius increases that is also reflected in C v -F curves. For example, a (22,0) BNNT with m = 3I + 1, as mentioned in the above, has a larger band gap leading to a vanishing specific heat at T < 3 K and F < 0.32, as shown in the left inset of Fig. 3(d). The specific heat, even at higher temperature T = 5 K, increasingly reaches to C v = 800 which is one order of magnitude less than those of (21,0) and (23,0)
BNNTs. As for a (23,0) BNNT, field-reduced band gap is smaller than that of a (22,0)
BNNT such that at T = 2 K the critical field F Z 1c = 0.2736 could induce a specific heat with a value C v = 102. However, except the first critical field, there is no critical field to induce giant specific heat as electric field increases to F = 0.4. The second critical field less than F = 0.4 appears just for larger BNNTs, e.g., F Z 2c = 0.3308 for a (24,0) BNNT shown in the right inset of Fig. 3(d). Therefore, it is much more easily to induce many special structures in specific heat for ABNNTs than ZBNNTs with large radii. The result, for ABNNTs, is beneficial to the modulation of thermal properties by applying electric field.
From the above, the differences of F -modulated energy dispersions between a (m,m)
ABNNT and a (m,0) ZBNNT are pronouncedly reflected in the F -dependent specific heat.
They include (I) for the former, the threshold electric field acquired to induce the specific heat is smaller at low temperature (T < 2 K); (II) the magnitude of giant specific heat, for ABNNTs, induced by critical field is reduced with increasing nanotube's radius, but it depends on the modulus of m with respect to three for the later; (III) it is much more easily to induce many special structures in C v -F curves with increasing radius for the former.
On the other hand, the distortion of energy dispersions induced by electric field could also be validated according to the dependence of C v on temperature. In Fig. 4(a) While applied field is at the critical value F A 1c = 0.3079, much more electronic states get together close to the Fermi energy such that C v grows quickly at T > 1 K and shows nonlinear T -dependence. As for the (21,0) BNNT, Fig. 4(b) shows that the non-vanishing C v at F Z 1c = 0.3040, due to the larger band gap, occurs at higher temperature and has similar dependence on T as compared with that in the (12,12) BNNT. However, at non-critical fields F = 0.32− 0.36 the energy dispersions keep parabolic form and have larger band gaps so that C v -T curves show non-linear behaviors and small values (in the inset). Therefore, the enhancement of critical fields on electronic specific heat is more significant and not to be negligible with increasing temperature.
As nanotube's radius increases, except the first critical field, the second one with larger field strength also enhances specific heat. For (13,13) and (14,14) BNNTs, C v 's, at critical fields F A 1c and F A 2c , have similar behaviors with increasing T shown in Fig. 4(c). The two C v -T curves created by F A 1c and F A 2c respectively cross at certain temperature T c . At T < T c (T > T c ), the value of C v for F A 1c is larger (smaller) than that for F A 2c . It is because a modulation of F A 2c on energy dispersion extends to higher energy subbands leading to a more significant enhancement of C v at high temperature. As for larger ZBNNTs, BNNTs, e.g., a (21,0) BNNT in Fig. 5(b). The value of specific heat, at T = 1 K, increases from C v = 0 at φ = 0 to a maximum C v = 2048 at φ c = 0.0127 φ 0 .
Since the effect of magnetic flux on specific heat is more significant at low temperature, the φ-dependent specific heats at T = 1 K are further studied for larger BNNTs at different critical electric fields. For a (13,13) BNNT, magnetic flux could re-enhance the GESHs at the first and second critical fields, as shown in Fig. 5(c). Especially, at F A 2c = 0.3615, C v = 112 (φ = 0) reaches to a very large value C v = 2888 at φ c = 0.0142 φ 0 . However,
C v -φ curves of a (14,14) BNNT show different behaviors between critical fields F A 1c and F A 2c . At F A 2c = 0.3277 the GESH is enhanced with increasing φ while it is reduced at F A 1c = 0.2576. From the above, it is found that an increases of magnetic flux, at the first electric field, could easily enhance the low-temperature GESH for smaller BNNTS, e.g., (12,12) and (13,13) BNNTs. The main reason is Zeeman energy is inversely proportional to the radius squared. The smaller BNNTs through larger Zeeman energy could have much more electronic states close to the Fermi energy. Moreover, Fig. 5(c) also shows that the value of the maximum C v and the φ c is larger at F A 2c than F A 1c . The smaller band gap at In contrast to a (13,13) BNNT (Fig. 5(c)), the φ c of a (24,0) BNNT has a larger value at F Z 2c but the maximum of C v is smaller as compared with that at F Z 1c . It is related to the feature of field-modulated energy dispersion, except band gaps. From the above, the modulation of external fields on thermal properties has strong geometry-dependence that may be used to identify the different geometric structures of BNNTs.
Conclusion
In this work, the tight-binding model is used to study electronic structures and the lowtemperature electronic specific heat of ABNNTs and ZBNNTs in the presence of electric field. Electric field significantly affects energy dispersions, edge-states, and band gaps.
When BNNTs are under a critical electric field, the more electronic states emerge close to the the Fermi energy leading to the large density of states and the giant specific heat.
The specific heat has a value comparable to that induced by phonon and reveals strongly non-linear dependence on temperature. The critical field strength, and the value and temperature-dependence of the giant specific heat are profoundly dependent on nanotube's geometry. Additionally, under F c 's, the additional magnetic flux further modulates lowenergy dispersions of BNNTs. The interplay between the AB-oscillation and the Zeeman splitting strongly depend on nanotube's geometry. Consequently, the low-temperature specific heats, at F c 's and φ = 0, are always enhanced by certain magnetic flux for ZBNNTs but they are enhanced or diminished for ABNNTs. The modulation of electric and magnetic fields on electronic and low-temperature thermal properties is strongly dependent on the field strength and nanotube's geometry that is expected to be verified by experiments. temperatures. The φ-dependent electronic specific heats at the first and second critical electric fields and T = 1 K for (13,13) and (14,14) BNNTs are shown in (c). (d)
Figure Captions
is the same plot as (c) but for (22,0), (23,0), and (24,0) BNNTs.
r = 3mb/2π and θ = −30 • . They are r = √ 3mb/2π and θ = 0 • for a (m,0)
the on-site energy due to the 2p z atomic orbital and the external electric field. α i is the angle between the position vector of the ith atom and the transverse electric field F (unit eV/Å). t 0 = −2.92 eV is the nearest-neighbor hopping integral. The 2p z on-site energies (E i ) of boron atom and nitrogen atom are E B = 4.78 eV and E N = 0.48eV , respectively. c + i (c i ) is the creation (annihilation) operator. exp[i(2π/φ 0 ) j i A · dr] isthe magnetic phase, where A is the vector potential. When the transverse electric field and the longitudinal magnetic field (B) are applied, the period along the nanotube's axis is not destroyed. The longitudinal wave vector (k y ) is still a good quantum number and specify energy dispersions of BNNTs. The first Brillouin zone has the range −π/R y ≤ k y ≤ π/R y .For ABNNTs and ZBNNTs, the band structures could be calculated by diagonalizing the 4m × 4m hermitian Hamiltonian matrix.All BNNTs without external field are wide-gap semiconductors but the energy dispersions are strongly dependent on geometric structures. Here we give a brief review for band structures of BNNTs[11].For the ABNNTs, conduction and valence bands are symmetric about the Fermi energy (E F = 2.63 eV ). Most of subbands are doubly degenerate except the first (closest to E F ) and the last subbands. The band-edge states are located at k y = 2/3 (unit π/R y ). Energy spacing decreases with increasing wave vector k y and all subbands are merged together at k y = 1. As for the ZBNNTs, energy dispersions show similar band-symmetry and degeneracy, but the two singlet states correspond to the fifth and last subbands. In contrast to an ABNNT, the different features in band structures of a ZBNNT include (I) the band-edge states are at k y = 0 and (II) all subbands, at k y = 1, are merged into one doubly and four four-fold degenerate states. While a BNNT (or CNT) is threaded by uniform magnetic field along the nanotube axis, the induced magnetic phase could change the band degeneracy, energy dispersion, energy spacing, and band gap. With increasing magnetic flux, electronic structures at φ = φ 0 are restored to those at φ = 0, i.e., Ahanorov-Bohm (AB) oscillation. Compared with the sharp sensitivity of energy dispersions in CNTs to magnetic field, magnetic field just slightly changes band-width and band gap for BNNTs. According to our previous studies [12], the maximum change in the magnitude of band gap is about 0.08 eV which is much smaller than band gap (∼ 4.5 eV ) at φ = 0. Moreover, magneto-energy gap weakly depends on the nanotube's radius. Thus, effective modulations of energy dispersions should be considered by other methods.
F
= 0.3080 to F = 0.3081, the two-local-minima states are created showing two lower split peak structures in the DOS. The number of low-temperature thermal electronic states is also reduced. While the radius of ABNNTs increases, the features of energy dispersions are almost unchanged. But the stronger electric potential makes electronic states closer to the Fermi energy, and then the band gap and the critical electric field are thus effectively reduced, as shown in Figs. 1(c) and 1(e) for
Figs. 2
2(b) and 2(d). For a (m,0) ZBNNT, a nanotube with m = 3I (I an integer), e.g., a (21,0) BNNT, has the smallest band gap while one with m = 3I + 1 (a (22,0) BNNT) shows the largest band gap. The above rule also holds even for larger m, e.g., (24,0), (25,0), and (26,0) BNNTs. Therefore, F -dependent band gaps, for ZBNNTs, are profoundly related to geometric structures that would be fully reflected in the low-temperature electronic specific heat.
m=12, 13, and 14 as a model study. The comparison among Figs. 3(a), 3(b) and 3(c) showsthat the critical field initiating the first jump structure in C v decreases with increasing nanotube's radius, e.g., F A 1c = 0.2806 for a(13,13) BNNT and F A 1c = 0.2576 for a(14,14)
for a(12,12) BNNT, the specific heats, at non-critical fields, have small values and show nearly linear (at F = 0.31) and linear (at F = 0.32 and F = 0.34) T -dependences. Because electric field, increasing from F = 0.31 to F = 0.34, gradually exhibits linear energy dispersions close to the Fermi energy and reduces the band gap (∼ 10 −4 eV ), as shown in the inset.
C v -T curves of (22,0) and (23,0) BNNTs inFig. 4(d) are non-linear and C v 's have smaller values, since the first critical field exhibits larger band gap and parabolic subbands. It is noticeable that there are big differences in C v -T curves of a (24,0) BNNT created by F Z 1c and F Z 2c respectively. The main reason is that F Z 2c does not exhibit more electronic states close to the Fermi energy. The above clearly shows that C v -T curves have the strong geometry-dependence as well as the field-modulated energy dispersions.For carbon nanotubes, the magnetic phase, due to their cylindrical symmetry, couldinduce the quantum interference which strongly affects electronic structures near the Fermi energy; it causes an oscillatory behavior in energy dispersions and band gaps, i.e., the Ahanorov-Bohm oscillations. However, BNNTs have large band gaps and strong ionicity so that effects of magnetic field on electronic structures are less significant. Modulations of magnetic field on electronic structures and thermal properties will be possible if a BNNT is just under critical electric fields. For magnetic flux φ = 1 φ 0 , the corresponding magnetic field strength is about 938 T for the (12,12) (or (21,0)) BNNT (r = 8.3Å) that is difficultly achieved in experiments. The smaller magnetic flux φ ≤ 0.05 φ 0 (B ≤ 47 T ) is considered and the spin-B interaction (Zeeman energy) is included which is defined as E z = (gσ/m * r 2 )(φ/φ 0 ). The g factor is taken to the same that (≈ 2) of the pure graphite, σ = ± 1 2 is the electron spin, and m * is the bare electron mass. It causes a energy shift with the value E z ≈ ±5.5 × 10 −3 eV at φ = 0.05 φ 0 . First we study the change in specific heat of a (12,12) BNNT with increasing magnetic flux at F A 1c = 0.3079 and different temperatures. Figure 5(a) shows that the AB-oscillation of electronic properties induced by magnetic phase is also reflected in oscillatory behaviors of C v -φ curves. With increasing magnetic flux, the value of C v at T > 2 K is smaller than that at φ = 0, whereas it could be enhanced as T ≤ 2 K. Especially, the value of specific heat at T = 1 K increases from C v = 153 at φ = 0 to a maximum C v = 2256 at a critical magnetic flux φ c = 0.0086 φ 0 . For magnetic flux φ = 0.0086 φ 0 , the corresponding magnetic field strength is B ≈ 8 T which could be easily performed in experiments. The result reveals that magnetic flux significantly affects electronic states close to the Fermi energy and enhances the low-temperature GESH induced by critical electric field again. The similar result is also found in the zigzag
F
A 2c is the main reason. Different from ABNNTs, magnetic flux always enhances the low-temperature GESH of ZBNNTs while they are under the first electric fields, shown inFig. 5(d). For a (22,0)BNNT with the widest band gap, the required critical magnetic flux φ c inducing a maximum of C v is the largest while it is the smallest for a (24,0) BNNT with the narrowest band gap.
FIG. 1 FIG. 5
15(a) and (b) are low-energy band structures and the density of states at various electric fields for a(12,12) BNNT. (c)-(d) and (e)-(f) are for(13,13) and(14,14) BNNTs, respectively.FIG. 2 The same plots as Fig. 1, but for (21,0), (22,0), and (23,0) BNNTs. FIG. 3 The electric-field-dependent electronic specific heats at different temperatures are shown for (a) (12,12), (b) (13,13), (c) (14,14), and (d) (21,0) BNNTs. The inset in (a) is the detailed C v -F curve of a (12,12) BNNT at T = 1 K. The C v -F curves at different temperatures of (22,0) and (24,0) BNNTs are shown in the two insets of (d). FIG. 4 The T -dependent electronic specific heats at various electric fields are shown for (a) (12,12) and (b) (21,0) BNNTs. The inset in (a) is the detailed energy dispersion close to the Fermi energy for F = 0.31, F = 0.32, and F = 0.34. The inset in (b) is the detailed C v -T curves at larger electric fields for a (21,0) BNNT. The T -dependent electronic specific heats at the first and second critical electric fields for(13,13) and(14,14) BNNTs are shown in (c). (d) is the same plot as (c) but for (22,0), The φ-dependent electronic specific heats including the spin-B interaction are shown for (a)(12,12) and (b) (21,0) BNNTs at the first critical electric field and various
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"\nSayandip Dutta is with Indian Statistical Institute\n203 B.T Road, Kolkata 108, 203 B.T Road, Kolkata 108, Kolkata -108India., India., India\n"
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"Sayandip Dutta is with Indian Statistical Institute\n203 B.T Road, Kolkata 108, 203 B.T Road, Kolkata 108, Kolkata -108India., India., India"
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| In this paper, we address the basic problem of recognizing moving objects in video images using SP Theory of Intelligence. The concept of SP Theory of Intelligence which is a framework of artificial intelligence, was first introduced by Gerard J Wolff, where S stands for Simplicity and P stands for Power. Using the concept of multiple alignment, we detect and recognize object of our interest in video frames with multilevel hierarchical parts and subparts, based on polythetic categories. We track the recognized objects using the species based Particle Swarm Optimization (PSO). First, we extract the multiple alignment of our object of interest from training images. In order to recognize accurately and handle occlusion, we use the polythetic concepts on raw data line to omit the redundant noise via searching for best alignment representing the features from the extracted alignments. We recognize the domain of interest from the video scenes in form of wide variety of multiple alignments to handle scene variability. Unsupervised learning is done in the SP model following the 'DONSVIC' principle and 'natural' structures are discovered via information compression and pattern analysis. After successful recognition of objects, we use species based PSO algorithm as the alignments of our object of interest is analogues to observation likelihood and fitness ability of species. Subsequently, we analyze the competition and repulsion among species with annealed Gaussian based PSO. We've tested our algorithms on David, Walking2, FaceOcc1, Jogging and Dudek, obtaining very satisfactory and competitive results. | null | [
"https://arxiv.org/pdf/1704.07312v1.pdf"
]
| 7,073,565 | 1704.07312 | 3a8868fa5ff4ba8d2a0fe51657937b423ab6fcff |
Sayandip Dutta is with Indian Statistical Institute
203 B.T Road, Kolkata 108, 203 B.T Road, Kolkata 108, Kolkata -108India., India., India
Index Terms-Artificial IntelligenceCognitionComputer VisionInformation CompressionNatural Language ProcessingPattern RecognitionPerceptionPSOReasoningRepresentation of KnowledgeSP Theory of IntelligenceUnsupervised Learning
In this paper, we address the basic problem of recognizing moving objects in video images using SP Theory of Intelligence. The concept of SP Theory of Intelligence which is a framework of artificial intelligence, was first introduced by Gerard J Wolff, where S stands for Simplicity and P stands for Power. Using the concept of multiple alignment, we detect and recognize object of our interest in video frames with multilevel hierarchical parts and subparts, based on polythetic categories. We track the recognized objects using the species based Particle Swarm Optimization (PSO). First, we extract the multiple alignment of our object of interest from training images. In order to recognize accurately and handle occlusion, we use the polythetic concepts on raw data line to omit the redundant noise via searching for best alignment representing the features from the extracted alignments. We recognize the domain of interest from the video scenes in form of wide variety of multiple alignments to handle scene variability. Unsupervised learning is done in the SP model following the 'DONSVIC' principle and 'natural' structures are discovered via information compression and pattern analysis. After successful recognition of objects, we use species based PSO algorithm as the alignments of our object of interest is analogues to observation likelihood and fitness ability of species. Subsequently, we analyze the competition and repulsion among species with annealed Gaussian based PSO. We've tested our algorithms on David, Walking2, FaceOcc1, Jogging and Dudek, obtaining very satisfactory and competitive results.
and obtaining optimum solution via family resemblance or polythetic concept, analyzing a scene with high-level feature alignments recorded with more logical detailing about its existence in the raw data. Finally, we track the recognized object of interest using species inspired Particle Swarm Optimization (PSO).
In the past, there have been many attempts to achieve human like perception or to handle Computer Vision related problems strictly in a logical manner, i.e. using atomic symbols instead of using actual numerical data. Default logic based reasoning [13], [14] and bilattice based non-monotonic reasoning [25] have been applied in the field of visual surveillance. But it sometimes generates unexpected extensions. Conclusions drawn from default logic vindicates common sense, which in turn jeopardizes its soundness. Other logic based reasoning systems include Neutrosophic logic [4], which aims to improve on the basic prospects of fuzzy logic. However, it can often get paradoxical and sometimes unintuitive results. In these cases, SP systems provide a more simple and robust framework to interpret logic, which is ideal for problem solving in the domain of computer vision. In order to track the recognized objects of interest we have used species inspired Particle Swarm Optimization technique [19]. The multiple alignments and the class and subclass hierarchies that are derived from SP systems are analogous to the species based framework we have used in our tracking approach. Thus, unlike conventional state-of-the-art PSO algorithms [1], [2], [3], [8], [9], [12], [17], [22] species based PSO technique is more well equipped to process the multiple alignments generated from the SP framework.
In our experiment, we use natural language texts of multiple alignments for SP systems to extract necessary information from raw data line in form of Old pattern, and comprehend the knowledge base in test domain to derive and encode New information. The system, in turns, learns from its knowledge base exploring wide variety of alignments to create and compress information to form New relevant patterns, without any supervision via DONSVIC principle. Through its thorough experiences from the knowledge base, the system is capable of exploring objects, class of objects from images and form necessary patterns to update the Old information with New. Using DONSVIC principle and polythetic concept, SP systems are well equipped to handle partial occlusion by searching for relevant pattern from its Old knowledge base.
Detection, Recognition and Tracking of Moving Objects from Real-time Video via SP Theory of
Intelligence and Species Inspired PSO Kumar S. Ray, Sayandip Dutta, Anit Chakraborty E Furthermore, the optimum compressed pattern representing the knowledge base of our object of interest is tracked via species inspired PSO in successive frames. The contributions of this paper are:
• Development of the concept of simplicity with descriptive or explanatory power in the field of Computer Vision. • Development of brain-like visual inference system and its application in tracking objects from real-time video. • Versatility and flexibility of artificial systems by means of unsupervised learning, planning, pattern analysis to attain seamless integration of artificial vision with other sensory modalities. • Occlusion and redundant noise handling from the raw data line for accurate detection and recognition for tracking. • Accurate detection of the dominant object between two overlapping species for more persistent tracking of the object of interest. The organization of the paper constitutes: Review and overview of SP theory in Section II. Section III explains our proposed method for detection, recognition and tracking via SP systems and spices based PSO. Experimental observations are presented in Section IV. Section V concludes the paper and discuss future possibilities for further improvements in the field of Computer Vision and Machine Intelligence.
II. OUTLINE OF SP THEORY
Simplification and integration of concepts in cognition and computation can be achieved with the help of SP Theory of Intelligence with information compression as the overlying theme. This section briefly explains the basic concept of SP theory and some of its relevant application in the area of computer vision and natural vision, specifically, the basic problem of accurate detection, recognition of moving objects in video images and occlusion handling with significant advantages.
In broad terms, the SP theory has three principle elements:
• Knowledge is represented with patterns: One or two dimensional arrays of atomic symbols. • Information is compressed by pattern matching and unification (merging) concept via Multiple Alignment, as demonstrated in Fig. 1. • The learning of the system is achieved by compression of New patterns to create Old patterns and unification of alignments. The system learns by compressing New patterns to create Old patterns like those shown in columns 1 to 11 in Fig. 1. Because of the reliant connection in between information compression and concepts of probability and prediction [32], the SP system is intrinsically probabilistic. Frequency of occurrences of the best possible multiple alignment is intimately associated with its subsequent SP patterns. Probabilities can be formulated for each associate inferences of subsequent multiple alignments [26]. It is intended that the SP computer model will be the basis for the development of a high-parallel SP machine, an expression of the SP theory, a vehicle for research, and a means for the theory to be applied [15].
Although the main emphasis in the SP program has been on the development of abstract concepts in natural language processing, but its application in computer vision and video image processing is being currently explored by different researchers. Figure 2 demonstrates an example of multiple alignment in the SP system, where a sentence: `t w o k i t t e n s p l a y' is represented as a New pattern in row 0, whereas Old patterns are represented in rows 1 to 8 as a word with grammatical markers or a grammatical rule. This multiple alignment, parses the sentence in terms of grammatical structures or series of consecutive grammatical markers. This Multiple Alignment is the best of more than a few built by the SP62 model when it is provided with the New pattern and a series of consecutive grammatical markers of Old patterns that contains those shown in the figure and many others. In our example, `best' corresponds to the most economically encoded New pattern in terms of the Old patterns [26].
A. The Multiple Alignment Concept
The `Np' and `Vp' marks the inter grammatical dependency of the plural subject (`k i t t e n s') and the plural main verb (`p l a y') of the sentence, in terms of a Multiple Alignment Point of Interest (MAPI). There may exist a discontinuous dependency between one element and the another. The term `discontinuous' represents the presence of large amounts of arbitrary intervening structures. Discontinuous dependency marking is, conceivably, more well-designed and easier than other state-of-the-art grammatical systems.
B. The encoding of light intensities
Expressing the that light intensities in images as numbers is trivial while designing a Machine Learning and artificial systems for Computer Vision. But, the SP systems only recognizes atomic symbols of consecutive grammatical markers where every multiple alignment are matched with another in an all-or-nothing manner. In principle, it may interpret numerical values correctly if the machine is supplied with certain patterns that hold information that are similar to Peano's axioms [31]. Although, this has not yet been explored in the research areas of Computer Vision and Machine Intelligence, nevertheless, numerical values are not the best way to assess the principle of SP Theory of Intelligence.
Initially, for simplicity, we assume that all the images are in Binary, i.e. pixel values are either '0' or '1'. In such cases, the illumination variation of unit pixels at any given area of the image will be encoded as the distribution of pixel-intensities of Black and White in that area. This representation, somewhat, avoids the explicit numerical values of the corresponding pixels similar to dark and light monochrome photographs of old newspapers [27]. SP Systems welcomes the idea of atomic representation of unit pixel values as '0' or '1', without any numerical meanings.
C. Edge detection with the SP system
Recursion can simulate the outcome of run-length coding in the SP framework, as demonstrated in Fig. 3. Here, each instance of self-referential Old pattern, in the form of 'X 1 a b c X #X #X', in rows 1 to 4, is matched from row 0, which contains each appearance of 'a b c' in New pattern. The enclosing 'X #X' in the body of the pattern can be unified and matched with 'X ... #X' at the beginning and end. Due to this property, the structure of Old pattern is called self-referential.
Derivation of relatively shorter multiple alignment sequence 'X 1 1 1 1 #X' is encoded from the New pattern. The recording of the fact that pattern 'a b c' contains 4 instances, attains the lossless compression of the initial original sequence by unary arithmetic. Recording a sequence of instances of 'a b c', irrespective of the length of the initial sequence, may reduce the encoding to 'X #X' with lossy compression. As briefly mentioned earlier in this paper, two side-by-side consecutive encodings, would be a uniform economical boundary between one subsequent region to another.
At an abstract level, there may exist two set of similar productions as an outcome: Redundancy of uniform regions is extracted from the raw data, without any careful consideration of boundaries between subsequent regions as an economical depiction of the raw data, as mentioned by 'primal sketch' [36]. Moreover, SP concepts are generalized to two dimensions, as a tool to attain significant breakthroughs in the field of Computer Vision and Machine Learning Systems.
D. Orientations, Lengths, and Corners
In principle, the orientations of edges or their lengths may be mathematically encoded, very economically, with the help of vector graphics representation. Having said that, the aforementioned method may not be as useful for systems like: Molecular Biological Systems, Gene Technology etc.
Also, in real life, it is very difficult to attain human like capabilities in an artificial perception system following vector graphical method.
As briefly mentioned earlier, in natural vision, quite simply the edges may be directly encoded either by matching neuron type or by multiple alignment based artificial systems. [26] The orientation and length of a straight line may be obtained through sequence of codes containing significant amount of redundancy, as briefly mentioned earlier in Section II(C). Orientation of the sequence is repeated in succeeding parts of the raw line data. So, it is fair to assume, in natural vision and systems, redundancy is reduced with some runlength coding inside the body of the line. When repetition stops, the information is preserved at the beginning and ending points of the raw line data. [26] This method is susceptible to straight lines as well as uniform curvature. Such structures, either partially or its repeated instances are encoded to express the curvature of the entire line.
Relevant information regarding the presence of 'end stopped' hypercomplex cells that are selectively responsive to a corner or a bar of a definite length, can be extracted with regard to straight lines [20]. It is safe to assume that, in mammalian vision, the length of an edge and the orientation, line or slit, is majorly encoded via edge detection using neurons to record the end to end point associated corners. The input line for a 'higher' level of encoding is provided via orientation-sensitive neurons.
In terms of artificial systems, in principle, this kind of approach is adapted within the means of multiple alignment framework as mentioned in Section II (A).
E. Noisy data and low-level features
Visual data collected from raw images is hardly as clean as demonstrated above in Figure 4. Monochromic images likely to be carrying various kinds video frame impurities, such as: not purely black or purely white, shade of grey, and there are likely to be blots and smudges of various kinds. SP Systems are designed to search for optimal solutions and is not destructed by errors of commission, substitution and omission. There is more on this topic in Sections III (A, B).
III. PROPOSED METHOD
To successfully track objects across multiple frames, accurate detection of the objects of interest are of tremendous importance.
The primary objective of SP model is to figure out good full or partial match between different patterns with high efficiency, much like the standard models that are based on 'dynamic programming' for the sequence matching or alignment. However, the difference between the SP theory and the latter is that the former (SP model) delivers all the matching alternatives within the patterns; whereas the standard models (based on Dynamic Programming) are programmed only for the best solution. At every stage, multiple alignments are built by pairwise matching and unification of the patterns. The objective of this process is to encode New information in terms of Old information economically, so as to separate the subpar multiple alignments that are generated.
Despite the straightforwardness of the SP patterns, they are very much versatile in representing various kinds of knowledge, due to their processing within the multiple alignment frameworks.
This allows SP systems to process information such as, natural language grammar objects, part-whole hierarchies, class hierarchies, ontologies, if-then rules, relation tuples, decision trees, associations of medical symptoms with medical signs, causal relationships, and mathematical and logical concepts.
The SP system shows definite potential in the areas of natural pattern recognition, language processing, reasoning and inference frameworks, the efficient storage, compression and retrieval of information and unsupervised learning. As the leniency in multiple alignment process allows to filter out noisy and erroneous data, SP theory is quite robust in the face of errors. In this paper, we apply these traits to overcome the scenarios when an object is partially occluded from the camera viewpoint.
After successful detection and recognition of the object of interests we track them using species inspired Particle Swarm Optimization. This approach is very well equipped for processing multiple alignment and hierarchical data generated from the SP theory, which aids in a more persistent tracking of multiple moving objects of interests.
A. Object Detection and Recognition from Training Images
Object recognition, in some respect, is similar to parsing in natural language processing [16], [18], [28]. SP system is quite well equipped with parsing natural language, as outlined in Section II (A), thus it can be considered as a useful tool for the development of Computer Vision and Pattern Recognition areas. Logically, SP machine needs to be generalized for working with patterns with two dimensions. In our experiment, though, we would consider the system to be well equipped to detect and identify low level perceptual features, which are initially atomic in nature to balance the harmony with the SP theory.
To put in perspective, Fig. 4 demonstrates schematically how a person's face with all its atomic feature symbols [e.g. Ears, Nose, Eyes etc.] are parsed within the multiple alignment grammar. The New pattern, represented in row 0, contains the incoming information from the raw line data. Each instance of self-referential Old pattern, in the form of 'X 1 a b c X #X #X', in rows 1 to 4, is matched from row 0, which contains each appearance of 'a b c' in New pattern. [26] The stored knowledge of the structure of Ears, Nose, Eyes etc. is depicted in the Old patterns are aligned with every atomic feature of the object of interest. The updated multiple alignment is then matched with a pattern in row 2 as a relatively superior unit-feature of the object (i.e. Someone's head). Even though this method is schematic in nature, this approach has strong potential in our experiment, as explained in subsequent sections. Figure 5 is a pictorial representation of the set of human faces reduced to the extracted feature sets of atomic symbols [i.e. Ear-Eye-Nose-Eye-Ear]. In the class of Human, various unit elements bear different set of frameworks within the same alignment which helps in distinguishing the elements.
a. Noisy Data in Parsing and Recognition
Differing from the fundamental belief gathered from the earlier part of this paper, the SP system can also handle sequence of video images for detection and tracking.
In Fig. 4, we have shown that the SP System is quite adaptive to detect and omit errors, such as, partial occlusion, noisy data handling etc. As briefly illustrated in (Fig. 6), the newly formed pattern on the arrival of new raw data line in row 0 remains the same as in (Fig. 2) instead of the replacement of 'm' for 'n' in 'k i t t e n s', the absence of the 'w' in 't w o' and within the word 'p l a y', the erroneous addition of 'x'. In spite of these errors and noise addition, SP62 model derives the best possible multiple alignment, as shown in (Fig. 6), which in turn reflects the correct initial alignment of the feature set.
b. Family Resemblance
An alternative idea is, SP systems strongly accommodates 'Family resemblance', in terms of polythetic concepts: the method of parsing the raw data for visual detection and recognition is not dependent on the presence of any key feature or combination of features, as well as in the absence of it [15], [33].The system is well susceptible to errors in form of partial occlusion, noisy data etc., via searching for its optimal solutions [Sections III.A (a)], as it partly allows for the requirement of knowledge based alignments that may have various alternatives at any given point within the structure. Most of the SP system frameworks are polythetic. Although possession of a pair of legs seems to be a key feature to identify the concept of 'Human', yet the system should recognize Sam as a Human, even with partial occlusion which visually depicts a loss of one Leg. Similarly, this method is adapted for most of the concepts in any visual systems. In any logical system that aims to achieve human-like vision, the concept of 'Family resemblance' or polythetic is very essential. c. Hierarchies and their integration SP systems consists of various multiple alignments representing various objects of interest and domain of interests, which is simple yet effective for object detection and recognition in any visual system. Representation of classes of objects and processing of class hierarchies, part-whole hierarchies and their integration, as mentioned by [15], [26]. In Figure 7 (a, b, c), a multiple alignment of all the parts and sub parts of a human body is shown. This does not illustrate the visual appearance of a human body but it is sufficient to represent and process all the relative information to form a human body out of it.
It is safe to assume that, this system being efficient to work with two dimesons, has the capability to process all the parts and sub parts and relating to a hierarchy based on information extracted from raw data in form of multiple alignment. The integrated form is shown in [Fig 8].
d. Scene Analysis
Scene analysis is broadly a primary subsection of knowledge parsing, for example: In the process of analyzing a sea beach, high-level feature alignments are recorded of things that may typically be seen in a sea beach (i.e., rocks, boats, sea, beach, sky and so on), with more logical detailing about its existence in the raw data. The complications we face, as suggested by [35] in the process of a scene analysis are:
• Partial occlusion is one of the primary anomalies in the process of scene analysis. In a typical sea beach, various feature points of the data can be partially obscured by other mutually exclusive features, creates an ambiguity about the domain of the scene, i.e. a boat is partially occluded by other features, such as, waves, sea birds etc.
• Variability of the locations in the scene of all the feature points creates ambiguity about the scene, i.e. A boat may be on the beach or in the sea.
Although, people relate to the aforementioned anomalies quite easily, but in complicated scenarios, 'naïve' kind of parsing systems fail to address such issues. The SP systems retains these aspects, carefully, with respect to scene analysis in following ways:
• We have already established in Section II [A (a)], SP systems are well equipped with handling errors, noise, omissions, commissions and substitutions. Thus, it is safe to assume that, the SP models that are comprehensive to work with patterns in two dimensions, can handle partial visibility of the objects and recognize them successfully in the subsequent frames. • Inconsistency of scenes captured from any realtime video is similar to parsed sentences in natural language. SP systems, among other artificial systems, is capable of supporting the system with relevant information about the scene in form of wide variety of multiple alignments and phrases containing recursive forms. This principle is well applicable to Vision related domains. For example: "Politics is the art of looking for problem, finding it everywhere diagnosing it incorrectly and applying the wrong remedies." • Existing knowledge is not always palpable to varying domains and raw data line. In such scenarios, the system may learn from its experiences, as briefly mentioned in Section III (B).
B. Unsupervised learning and 'DONSVIC' principle
Learning is an essential part of computer vision since gaining new information and to monitor the changes happening around the world are primarily done by vision. In general, it is quite evident that, learning through vision is mostly unguided in nature that is 'unsupervised'. Intervention of a 'teacher' is not required when it learns through vision. The classification of samples from simpler to complex ones and provision of 'negative' samples are not required. We try and get information through our vision and try to comprehend that in our knowledge to make sense of it in the best possible way.
Unsupervised learning has been developed in the SP framework and rightly so, it works better than most of the well-developed knowledge based frameworks. In this section, we would like to demonstrate how unsupervised learning is developed in SP framework and applied in the vision via the 'DONSVIC' principle of unsupervised learning.
While dealing with our surroundings, there are certain kind of structures or objects or class of objects, that appear more useful and prominent than the others: for better understanding of visual appearances of 'discreet' objects (i.e. 'person', 'tree', 'house' etc.). These 'natural' kind of structures or class of objects are substantial in our information processing and compression of sensory information, which in turn, provides the key to learn and discover new objects. Even though, popular LZW algorithms based on information compression from JPEG images are more reliant to recognize words or objects in form of information and interpreting the knowledge in the application domain, but they are mostly designed to work on low-powered machines. In SP systems, programs are slower yet thorough and reveals natural structures in detail, as briefly explained below:
• Parsing of a corpus of natural language text, unsegmented, created by the MK10 program (Wolff, 1977), using only the information provided by the corpus of natural language text without any supervised knowledge provided dictionary or knowledge base about the structure of the language (Fig. 9). Even without all of its punctuations and spaces separating words are removed from the corpus, the system works exceedingly well in revealing the word structure of the text. • Similarly, the SP system works perfectly well, significantly better than chance, in detecting phrase structures from a corpus of natural language texts without reasonable punctuations or spaces, but with a symbol replacing words for its grammatical category. The process of replacing is done by a trained linguistic analysis, but the discovery of the structures of new phrases is done by the system, without supervision. • Derivation of a plausible grammar, from an unsegmented corpus of artificial language without any assistance is done by The SNPR program. The SNPR program for grammar discovery can learn new words from the text corpus, grammatical categories and the structure of phrases and words.
MK10 [26] and SNPR [26] programs are designed and equipped to search through the variety of alternatives among patterns which may be unified and matched to retain the set of patterns that yield a higher level of compression. This principle is not only applicable to discovery of words, grammars and pattern of words from artificial languages, but also in the area of vision: discovery of objects in images, class of entity in various kind of data. Principle is broadly termed as 'the discovery of natural structures via information compression', or 'DONSVIC'
A radically new conceptual information compression framework is developed with the concept of multiple alignment. As mentioned earlier, the SP70 system works on multiple alignments, deriving Old patterns from corpus of natural language texts and comprehending them into the knowledge base to create New Old patterns with economical and exceptional low-scoring tests.
SP learning system is illustrated schematically in Fig. (10). The SP system, as an abstract system works like a human brain, receiving 'new' information via its senses and deriving Old patterns in form of information. Suppose, the system hears someone saying "t h a t b o y r u n s". If the system never heard anything similar, then it stores New information as a relatively straightforward copy, as shown in row 1 of the multiple alignment in Fig. 10.
C. Tracking with Particle Swarm Optimization (PSO)
The PSO framework provides an effective way to track multiple object that are detected and recognized from aforementioned method (SP artificial systems). First, for singular object tracking, following analogies need to be assumed:
• The groundtruth of an object and surrounding region can be considered as ecological properties.
• State space particles correspond to a particular species.
• Each particle's observation likelihood and fitness capability of a particular species is analogous.
For multiple object tracking, these postulates can be easily extended by creating a tracker for each object. These trackers are managed independently. In case of occlusion, support regions of concerning objects may overlap, which implies, the intersectional area between two species are elementary to both. Subsequently, the repulsion and competition among the species arise as both of them aspire to the same resource, the stronger one has higher probability of winning the competition.
During the course of video scene there may be overlap between two object areas due to occlusion, and the related features between them become ambiguous. To handle this complication, we design a multiple-species-based PSO algorithm as suggested by [19]. The principle idea behind this approach is to divide the groundtruth particles of the object into various species according to the species object numbers and successfully model the relations and the partial visibility among varied species. Detailed description of the species inspired PSO algorithm is briefly described in the following sections.
a. Problem Construction:
Let Here t is the 2-D translation parameter. Formula of multiple object tracking is as follows:
* = arg max ( | )(1)
By independently maximization of the individual observation likelihood, the above optimization may be simplified, in case of no occlusion. In case of no occlusion, the above optimization may be simplified by maximizing the individual observation likelihood independently (here, we drop the superscript i, n for simplicity):
, * = arg max , ( , | , ), = 1, … ,
b. Competition Model:
When different object obscure one another, there is an overlap between corresponding support regions. In these Newly formed Old pattern. [26] circumstances, the competition between two objects elevates to subjugate the overlapping part (Fig. 11). In order to effectively design the competition phenomenon, the visual problem needs to be merged with the competition process.
To evaluate the fitness value on the overlapping part as the competition ability, the overlapping part is viewed as a whole and projected onto the learned subspace corresponding to each object. We define the power of each object or species in following manner:
= (̂, | , ) = (−‖̂, −̂̂̂, ‖ 2 ),(3)
where k and ̂ are the overlapping part of the object and its corresponding subspace respectively. In a similar way, the interactive likelihood of object 1 over the overlapping regions can be calculated:
(̂, 1 | , 1 , , 2 ) ⏟ = 1 ∑ =1,2 .(4)
ℎ
The mutual likelihood of each species describes the competition ability. Higher the competition ability of a species more like it is to win the competition. It means that the species which won the competition is more likely to be of the object that was occluding the other object species involved.
c. Annealed Gaussian Based PSO (AGPSO)
An annealed Gaussian based PSO algorithm [21] is considered in this paper, as in conventional PSO requires careful and fine tuning of various parameters. In this algorithm, the particles and corresponding velocities are updated as stated below:
, +1 = | 1 |( − , ) + | 2 |( − , ) +
, + = , + , +1
where | 1 | and | 2 | being the absolute values of the samples from Gaussian probability distribution N(0, 1). This is zeromean Gaussian disturbance that stops the algorithm from getting trapped in local optima. With the help of adaptive simulated annealing, the covariance matrix of is changed [34]: ∑ = ∑ − (7) Here, a transition distribution is predefined, and Σ is its covariance matrix, annealing constant c, and iteration number n. The components in Σ decrease in proportion to the iteration number which results in a fast rate of convergence. When 1 and 2 occlude each other at time t, a repulsion force is added to the evolution process of particles, and subsequently the iteration step for 1 becomes as follows:
where the parameter 3 is Gaussian random number sampling from N (0, 1). The third term on the right-hand side of the above equation depicts the shared effect between object 2 and 1 . In other words, the competition phenomenon on the observation level has been modelled in this paper. Also, the competition model of state space has been modelled to drive the evolution process of the species in the right direction.
d. Updating of the Appearance Model Selectively
In most of the tracking algorithms [24], [29], appearance models are not updated during occlusion. However, the appearance of the object under occlusion may change, and that can cause the tracker to fail to recapture the object appearance if it is not occluded anymore. A selective updating algorithm is implemented to cope with the appearance changes during occlusion: 1) pixels belonging to the visual part of the objects are cumulatively updated in the normal way, 2) pixels that are part of the overlapping region (Fig. 11) are projected onto the subsequent subspace of each object. Then the errors due to the reconstruction are calculated. If this error is smaller than a predefined threshold for pixels inside the overlapping area, then it is again updated in the subsequent subspace.
Due to this careful modelling of the updating strategy, the appearance changes can be easily accommodated, allowing more persistent tracking throughout the video stream.
IV. EXPERIMENTAL DATA AND ANALYSIS
Proposed method is observed on benchmark datasets to demonstrate the robustness and adaptivity of SP systems to cope with the varied challenges offered by them. Brief description of the benchmark datasets is shown followed by the experimental settings and experimental observations. The SP framework is implemented in SP70 Virtual Machine, provided by Wolf. J. G for research purposes (http://www.cognitionresearch.org/sp.htm#SOURCE-CODE) on VC++ with an Intel core 5th Gen i7, 2.10 GHz processor with 6 Gigabytes of RAM and 2 Gigabytes NVDIA GeForce GPU. Figure 11. Overlap between two object areas.
A. Observational Datasets
We observed the efficiency and robustness of the proposed algorithm in natural vision on benchmark datasets on some of the TB100 sequences, namely, Walking2 [29], Jogging (1,2) [22] and David [26]. In the following section, we discuss the aforementioned datasets and the challenges the datasets possesses in natural vision; especially object detection, recognition and tracking.
Walking2 [29] possess challenges like: Scale Variation (SV), Occlusion (OCC) and Low Resolution (LR). The video contains 500 frames, each of dimension 384x288 pixels. The video is of some people walking down the corridor of an office interior.
The David dataset [26] is much more challenging compared to Walking2 [29], as one has to consider in-frame challenges, like: Illumination Variation, Scale Variation, Occlusion, Deformation (DEF), Motion Blur (MB), In Plane Rotation (IPR) and Out Plane Rotation (OPR). The dataset contains 770 video frames of 320x240 resolution. The video is of a person walking down the corridor of an office interior.
Jogging (1,2) [22] contains complications like Deformation, Occlusion and Out Plane Rotation. The video contains 307 frames, each of dimension 352x288 pixels. The video is of two pedestrian jogging down the road.
In the following section, we demonstrate the adaptability and efficiency of SP theory to cope with the challenges provided by these datasets and how accurately problems can be dealt with.
B. Experimental Observation and Analysis
We observe our method on the aforementioned datasets with the SP70 Virtual Framework and others, as mentioned above. Initially, we have observed the original videoframes and the challenges it possesses. Firstly, we train the system with the multiple alignments of our object of interest, i.e. Human, as described in Fig. (8). The system stores the information in form of Old pattern, that is subjected to detect and to be tracked in the test domain. A pictorial demonstration is shown in Fig. (12).
As mentioned earlier, SP systems do not illustrate the visual appearance of human body, but it processes the visual inputs provided from other systems with the help of multiple alignment concept and can successfully interpret different body parts to gather the information associated with the human body (Fig. 13). Thus, in turns, helps to string together the part-whole hierarchies and class hierarchies of different parts of human body.
In a video scene, our object of interest may be occluded by some other object native to that domain. In these cases, SP system matches the multiple alignment stored as Old information with the visible parts from the target objects and derives the remaining part of object despite its invisibility due to occlusion. As mentioned earlier, in the Jogging dataset two pedestrians are being tracked. The primary challenge we have faced in this particular video sequence, is that the pedestrians are partially occluded by a post in some of the video frames (Fig. 14). In such scenarios, the SP system processes the visible parts of the object of interest (i.e. partial head, torso, leg), and matches them with the previously stored information in the Old pattern to derive the supposedly New pattern in order to make sense of the current knowledge base.
The multiple alignment structure for partial occlusion, as mentioned above, is derived as Fig. 15. Consequently, the extracted alignment of the object of interest is tracked via species inspired Particle Swarm Optimization (PSO) in successive frames. We have compared the performance of the present algorithm our vision system with the existing state-of-the-art algorithms and the result of comparison is briefly shown in Table 1. Specific parameter construction and technical details are incorporated while implementing various algorithms, hence, we have chosen only those algorithms whose source code or binary code was available publicly.
For comparisons, various parameters of considered trackers are fixed for all the experiments. The evaluation result of our experiment represents the lower bound of the average tracking performances. Despite of having varied approaches followed by any individual tracker, the primary features are somewhat similar and shared by all of state-of-the-art methods. Table 7 represents the average number of failures and average number of success of our algorithms with respect to six algorithms ( [12], [13], [35], [20], [7], [9]). For handling occlusion, the overlap threshold of 0.65 with respect to all it attributes is considered. Primarily, the initial runs from the beginning are generated at 60 frames, whereas, the virtual runs are raised for our threshold and experiment window. The trackers' performances are steady over large variations of experiment window by keeping a window size of 90.
As the overlap threshold increases, the number of failures increases and the success rate increases at first and then decreases rapidly over the couple of successive frames. To When the overlap threshold is low, a tracker is not restarted even when it actually loses track of a target object and the model of a tracker is likely to be incorrect.
The average success rate increases when the threshold values are low, since re-initialization significantly helps in keeping track of our object of interest even after failures of a frame or two.
Our algorithms perform better over other state-of the-art algorithms in various benchmark datasets. To support our claim, we have tested our tracking algorithm against the existing ones and formulated the quantification in Table 2 (David), 3 (Walking2), 4 (FaceOcc1), 5 (Jogging). In all the cases, our algorithm outperforms the existing state-of-the-art trackers.
Our method can steadily deal with partial occlusions and deduce the partial relationship of that object with its domain and surrounding based on its Old information and alignments from its knowledge base and interactive likelihood of species. The results, as demonstrated in Figure 11 and evaluated in Table 7, support our claim. The singularity of the object with relatively greater fitness value on the overlapping part will occlude the other object, more likely.
A brief illustration of other advantages presented in this paper, a quantitative analysis with other state-of-the-art methods is presented in Table 8: average number of failures in 1000 frames under occlusion, RMSE (root mean square error) of Position i.e. the distance between the estimated position and the groundtruth. It is quite evident from the results that our algorithm achieves optimum accuracy in localizing the object of interest, performs better under occlusion and shows significant breakthrough against the existing state-of-the-art tracking algorithms. Table 3. Tracking Accuracy on Walking2 Dataset Table 4. Tracking Accuracy on FaceOcc1 Dataset Table 5. Tracking Accuracy on Jogging Dataset Table 6. Detection Accuracy on Dudek Dataset Figure 17. Tracking performance on David Dataset and comparison with respect to RMSE. Figure 16 demonstrates the tracking accuracy of aforementioned algorithms in various benchmark datasets. The tracking performances of traditional PSO and species inspired PSO with SP theory on a fast-moving face is graphically represented in Figure 17. Here we used RMSE (root mean square error) as the performance metric, from which we can see that the latter can achieve a remarkably higher tracking accuracy due to the amalgamation of SP theory of intelligence and species inspired PSO. This paper aims to simplify the detection, recognition and tracking of moving objects in real-time video via SP Theory of Intelligence and species inspired Particle Swarm Optimization (PSO). Wide variety of multiple alignments extracted from the video scenes of our object of interests are stored as Old information. Subsequently, on arrival of New information, the knowledge base is updated in order to derive optimum compressed pattern to follow in the successive frames. The unsupervised learning potentiality is explored in section III (C), where the system derives objects and classes of objects from its existing knowledge base in form of multiple alignment. As in case of human perception, the SP system is quite adaptive in the face of substitution, omission or commission. SP theory possesses a brain-like knowledge interface which is more robust and thorough over other intelligent systems. As briefly explained in this paper, with the help of multilevel abstraction, part-whole hierarchy, class hierarchy and polythetic concepts, we are getting breakthrough results in the field of vision and pattern recognition. For tracking purposes, species inspired PSO is more persistent in processing natural language based multiple alignments, which in turn produces more satisfactory results than other state-of-the-art artificial systems. This has great potentials in the field of problem solving integrating vision and pattern recognition with more robustness and variability, with exciting opportunities to explore in near future.
Fig. 1 .
1Multiple alignment and information compression by pattern matching.[26]
Figure 2 .
2Demonstration of Grammatical Markers in SP system. New pattern is represented in row 0 and Old patterns are represented in rows 1 to 8.[26]
Figure 3 .
3Edge detection with SP systems.
Figure 4 .Figure 5 .
45Schematic description of a person's face. Its atomic symbols are parsed within the multiple alignment grammar.[26] Pictorial representation of the set of human faces reduced to the extracted feature sets of atomic symbols.
Figure 6 .
6Noisy Data in Parsing and Recognition. Best possible multiple alignment is extracted despite of presence of noise.[26]
Figure 7 .
7(a): Multiple alignment of human head. (b): Multiple alignment of human torso. (c): Multiple alignment of human leg.
Figure 8 .
8Integrated form of Fig. (7) Human body along with its multiple alignment structure.
us consider, M number of objects, surrounded with N number of particles,
Figure 9 .Figure 10 .
910Part of a parsing created by program MK10. [26] (a): Simple Old pattern is derived from SP70. (b):
Figure 13 :
13Full body symbolic representation.
Figure 14 :
14Occlusion handling.
Figure 12 :
12Atomic symbol representation of different body parts.
Figure 15 :
15Best possible multiple alignment of a Human body is extracted despite of presence of occlusion.
Table 1 .
1Evaluated Tracking Algorithms Trackers Representation CodeLocal
Template
Color
Histogram
Subspace
Sparse
Binary
or
Haar
Discriminative
Generative
Model
Update
C/C++
MATLAB
FPS
Published
ASLA
✓ ✓
✓ ✓
✓ ✓ ✓ ✓ 8.5
'12
BSBT
H ✓
✓ ✓
7.0
'09
CXT
B ✓
✓ ✓
15.3 '11
DFT
✓ ✓
✓ ✓
✓ 13.2 '12
IVT
✓
✓
✓ ✓ ✓ ✓ 33.4 '08
LIAPG
✓
✓ ✓
✓ ✓ ✓ ✓ 2.0
'12
LOT
✓
✓
✓ ✓
✓ 0.7
'12
LSHT
✓
✓ ✓
H
✓ ✓ ✓
20
'13
LSK
✓ ✓
✓ ✓
✓
✓ 5.5
'11
LSS
✓ ✓
✓ ✓
✓ ✓
✓ 15
'13
MIL
H ✓
✓ ✓
38.1 '09
MTT
✓
✓ ✓
✓ ✓ ✓
1.0
'12
ORIA
✓
✓
H ✓
✓ ✓
20.2 '11
PCOM
✓
✓ ✓
✓ ✓
✓ 20
'14
SMS
✓ ✓
✓
✓
19.2 '03
Proposed
Method
✓ ✓
✓
✓
✓ 34.3
_
Table 2. Tracking Accuracy on
David Dataset
Table 7 .
7Performance of different algorithms on different attributes Trackers FeaturesTable 8. Quantitative Results of Tracking under OcclusionsApproaches
ASLA
DFT
IVT
MIL
PCOM
LSS
Proposed
Average number of
failures in 1000
frames under
occlusion
Person A (red window)
3.5
5.3
5.2
6.1
5.7
4.7
0.5
Person B (blue window)
4.1
4.6
5.3
6.3
5.5
5.0
0.4
Person C (green window)
3.8
5.4
4.8
5.0
5.3
5.6
0.6
Average
3.8
5.1
5.1
5.8
5.5
5.1
0.5
RMSE of Position
(by pixels)
Person A (red window)
12.734
11.245
9.363
8.342
8.632
13.297
2.642
Person B (blue window)
5.246
7.147
5.754
5.329
5.387
10.298
2.167
Person C (green window)
13.154
5.267
7.839
4.329
11.423
7.684
1.924
Figure 16. Accuracy analysis against various datasets.
ApproachYear Accuracy ASLA[12]2012 82.24% DFT[13]2012 81.31% IVT[35]2008 83.86% MIL[20]2009 86.19% PCOM[7]2014 78.37% LSS[9]2013 79.34% Proposed method 91.3%Approach Year AccuracyASLA[12]2012 88.2% DFT[13]2012 87.9% IVT[35]2008 89.8% MIL[20]2009 90.2% PCOM[7]2014 88.3% LSS[9]2013 90.7% Proposed method 94.7%Approach Year AccuracyASLA[12]2012 91.3% DFT[13]2012 88.3% IVT[35]2008 93.9% MIL[20]2009 95.1% PCOM[7]2014 89.4% LSS[9]2013 87.2% Proposed method 97.7%Approach Year AccuracyASLA[12]2012 85.3% DFT[13]2012 86.8% IVT[35]2008 90.7% MIL[20]2009 92.4% PCOM[7]2014 86.7% LSS[9]2013[13]2012 86.3% IVT[35]2008 88.9% MIL[20]2009 91.2% PCOM[7]2014 86.7% LSS[9]2013 87.2% Proposed method 92.8%
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| []
|
[
"Daily Generation Scheduling : Decomposition Methods to Solve the Hydraulic Problems",
"Daily Generation Scheduling : Decomposition Methods to Solve the Hydraulic Problems"
]
| [
"J-P Chancelier \nENPC\nCERGRENE\nNoisy le GrandFrance\n",
"A Renaud \nElectricité de France\nClamartFrance\n"
]
| [
"ENPC\nCERGRENE\nNoisy le GrandFrance",
"Electricité de France\nClamartFrance"
]
| []
| Short-term hydro-generation management poses a non-convex or even non-continuous optimization problem. For this reason, the problem of systematically obtaining feasible and economically satisfying solutions has not yet been completely solved.Two decomposition methods, which, as far as we know, have not been applied in this field, are here proposed :• the first is based on a decomposition by prediction method and the coordination is a primal-dual relaxation algorithm,• handling the dynamic constraints by duality, the second achieves a price decomposition by an Augmented Lagrangian technique.Numerical tests show the efficiency of these algorithms. They will enable the process in use at Electricité de France to be improved. | 10.1016/0142-0615(94)90007-8 | [
"https://arxiv.org/pdf/2204.04978v1.pdf"
]
| 108,590,736 | 2204.04978 | bbe499113e847461613e496975cc24c9073029e7 |
Daily Generation Scheduling : Decomposition Methods to Solve the Hydraulic Problems
1994
J-P Chancelier
ENPC
CERGRENE
Noisy le GrandFrance
A Renaud
Electricité de France
ClamartFrance
Daily Generation Scheduling : Decomposition Methods to Solve the Hydraulic Problems
1994Hydro-thermal scheduling, Decomposition methods
Short-term hydro-generation management poses a non-convex or even non-continuous optimization problem. For this reason, the problem of systematically obtaining feasible and economically satisfying solutions has not yet been completely solved.Two decomposition methods, which, as far as we know, have not been applied in this field, are here proposed :• the first is based on a decomposition by prediction method and the coordination is a primal-dual relaxation algorithm,• handling the dynamic constraints by duality, the second achieves a price decomposition by an Augmented Lagrangian technique.Numerical tests show the efficiency of these algorithms. They will enable the process in use at Electricité de France to be improved.
Introduction
The optimization of a hydro-valley's daily generation schedules poses a problem of an appreciable size. For instance, a problem related to a valley of five or six reserves, with a half-hour step has no less than 250 or 300 constraints.
Since the 1970's, very efficient linear programming methods and softwares have been designed to handle such optimization problems. The most commonly used algorithms take advantage of the network flow structure of the dynamic constraints [1] and succeed in being about a hundred times faster than standard linear programming methods [2].
Over the last ten years, extensions to the nonlinear (but convex) case have been made. Using Frank-Wolfe or projected reduced gradient techniques, efficient softwares have been developed. Thanks to them, nonlinear efficiency curves, for example, can be coped with.
Nevertheless, the modelling of the hydro-problems which can be solved by these methods is not completely satisfactory. Downstream flow requirements or "spillage constraints" cannot be handled. Moreover, the efficiency curves, which are generally non-convex, have to be roughly approximated and no discontinuity in the generating domain can be dealt with.
The Electricité de France generation mix has over 150 thermal groups and 15 valleys. The global optimization of the daily generation schedules is achieved by using a price decomposition method. In this optimization, some reservoirs are aggregated and the generating domain of the hydro-plants is assumed to be convex. Considering the number of local hydroproblems which have to be solved during this optimization, such a simplifying hypothesis can easily be understood.
Nevertheless, at regional level, a second optimization stage is necessary [3]. For each valley, a schedule is computed independently :
• firstly, a linear problem is solved. It takes into account the dual variables that the global optimization yields and handles a detailed but convex modelling of the constraints,
• secondly, in the neighborhood of this schedule, a heuristic "smoothing" software processes a feasible solution with respect to the non-convex or even non-continuous constraints.
The study, whose results are here outlined, aims at improving this regional level twosteps process. Two decomposition strategies have been tested in order to solve this problem directly (in one stage). These algorithms only require small-sized nonconvex sub-problems to be solved at each iteration. Therefore, an exact method, such as dynamic programming, can be used to deal with this local problems.
The paper is organized as follows. In section 2, we formulate the considered optimization problem. Then, in section 3, we outline the theoretical background of the prediction and price decomposition strategy along with their application to the hydro-problem. Finally, in section 4, we present the results of the numerical tests which have been realized.
The Hydro Problem
As already emphasized, the "regional" problem we are considering consists in the schedule optimization of one valley. It may be written as follows :
min T i (.) i∈I t∈[0,H[ −c i (T i (t), t) − C end i (V i (H), H)(1)
subject to :
• ∀i ∈ I , ∀t ∈ [0, H[ : V i (t + 1) = V i (t) − D i (t) − T i (t) + A * i (t) + j∈Γ T i (D j (t i←j ) + T j (t i←j )) (2) V i (t) − V max i D i (t) = 0 (3) V i (t) ∈ [V min i , V max i ] , T i (t) ∈ [T min i , T max i ] , D i (t) ∈ [0, D max i ] ,(4)
• ∀i ∈ I :
T i ([0, H[) ∈ T ad i(5)
where :
• I is the set of water reservoirs of the valley under consideration -each reservoir is related to a plant having the same index -, • Γ T i is the set of plants located upstream i -whose discharges are inflows of i -, • av(i) is the reservoir located just downstream i,
• [0, H] is the studied period, • V i (t)
• t i←j def = t−δ i←j with δ i←j denoting the delay for discharge of plant j to reach reservoir i,
• A * i (t)
is the natural inflows of the reservoir i during the period [t, t + 1[. These inflows are supposed to be known,
• C end i (V i (H), H)
is the water value of the reservoir i at the end of the studied period,
• c i (T i (t), t) is the "value" of the generation related to the discharge T i (t) at time t.
It has already been pointed out that the generating constraints we consider define a noncontinuous domain T ad i . We illustrate this in Figure 1. Moreover, as is shown in Figure 2 the constraints (3) also introduce nonconvexities.
Ti(t)V i (t) − V max i ) D i (t) = 0 is nonconvex
The numerical tests, hereafter outlined, use the current EDF regional modelling. Concerning the generating constraints T ad i :
• only a finite number of values is allowed for the discharge T i (t) (See Figure 4),
• there is a minimum delay between two variations of the discharge. Moreover, these variations have to be smooth.
The cost function c i (., t) is a piecewiselinear function of the discharge and C end i (V i (H), H) is a linear function of the final content :
C end i (V i (H), H) = c wat i (V i (H) − V i (0)) .
This modelling allows us to compare decomposition methods to the current EDF process. Nevertheless, the decomposition methods framework hereafter presented, could indeed be applied to more general modelling
Resolution Methods
In order to present the coordination-decomposition methods we use, we will consider the following problem: min u∈U ad J(u) , subject to: Θ(u) = 0 ,
where U ad is a closed set of a vector space U, J : U → R and Θ : U → C an affine function from U to the vectorial space C. To allow decomposition, we also assume :
U = i∈I U i , U ad = i∈I U ad i , ∀u ∈ U ad : J(u) = i∈I J i (u i ) ,
where I is a finite set, U ad i ⊂ U i and J i : U i → R.
Today's resolution method
First of all, the convexified problem (problem (1), without constraints (3) and (5)) is solved by linear programming. Secondly, following the course of the river, for every plant, a feasible schedule is processed. In this aim, taking into account all the generating constraints 1 , discharge is computed by minimizing a mean-square distance to the solution of the convexified problem. These subproblems are solved by dynamic programming. It may be noticed that this heuristic has a major drawback : it does not ensure that a feasible solution will be found. The inflows being given, a subproblem may have no solution.
Price decomposition 3.2.1 Theoretical background
Provided that the cost function is separable, the Uzawa algorithm [4] is certainly the most commonly used to achieve this type of decomposition. If we suppose Θ(u) = i∈I Θ i (u i ),then, applied to the problem (1), this algorithm may be outlined as follows (k iteration):
∀i ∈ I : min u i ∈U ad i J i (u i ) + p k , Θ i (u i ) ⇒ u k+1 i p k+1 = p k + ε k Θ(u k+1 )
Therefore, at each iteration the i-subsystem minimizes a balance (the Lagrangian) which takes into account its own cost function J i and a "revenue" p k , Θ i (u i ) , from its contribution to the satisfaction of the constraint Θ(u) = 0.
More formally, the Lagrangian related to (6) is defined over U × C as follows :
∀(u, p) : L(u, p) def = J(u) + p, Θ(u) .(7)
The above mentioned algorithm may be understood as maximizing the dual function:
Ψ : p → min u∈U ad L(u, p)
by a gradient type algorithm.
If the function J is not strictly convex, as in the hydro-problem we are dealing with, this dual function is not differentiable. Consequently, to ensure the convergence, a sub-gradient algorithm must be used to maximize Ψ. The sequence (ε k ) k∈N must then be chosen as a sequence of type σ (i.e. k=+∞ k=0 ε k = +∞ and k=+∞ k=0 ε k 2 < +∞). The convergence is necessarily slow.
Nevertheless, the non-differentiability of the dual function is not the main difficulty. In this case, to find a price p * which maximizes the dual function Ψ is not enough : for p = p * , the primal minimization of the Lagrangian will not necessarily give a solution of (6) [5]. In practise, a "small" variation of the "prices" leads to a large variation of the primal variables. The primal variables "switch" from one value to another and never satisfy the coupling constraints. To our mind, these theoretical difficulties explain to a large extent the bad reputation that these dual methods have in terms of convergence.
However, Augmented Lagrangian can be used to reduce these difficulties. The Augmented Lagrangian L c related to the problem (6) is defined over U × C as follows :
∀(u, p) : L c (u, p) def = L(u, p) + c 2 Θ(u) 2(8)
In the convex case, the saddle-points of this Lagrangian are the same as those of L [6]. Then, the dual function Ψ c related to L c is differentiable [7]. Furthermore, solving min u∈U ad L c (u, p * ) where p * is a maximum of the dual function necessarily yields a solution of (6).
At first sight, this Augmented Lagrangian technique has a major drawback with regards to decomposition : it introduces non-separable terms c 2 Θ(u) 2 . But, this difficulty can be overcome by linearizing the non-separable terms at each iteration [6]. This strategy leads to considering the following algorithm (algorithm A) (iteration k + 1):
• for all i ∈ I, u k+1 i is computed by solving :
min u i ∈U ad i J i (u i ) + π k , Θ i (u i ) + b 2 u i − u k i 2 , • p k+1 = p k + εΘ(u k+1 ), with b ∈ R + * and π k = p k + cΘ(u k ).
In the convex case, even if the cost function J is not strictly convex, the convergence of this algorithm towards a saddle point of L has been proven provided that 0 < ε < 2c and
cτ 2 < b, where τ is the Lipschitz constant of Θ [6].
This algorithm has been shown to be particularly efficient in dealing with classical hydrothermal generation scheduling problems [8], [5].
Application to the hydro-scheduling
problem In the problem (1), two types of constraints have to be handled:
• state constraints -the volumes bounds:
V i (t) ∈ [V min i , V max i ] -,
• logical constraints concerning the controls which can be rather complex -T ∈ T ad -.
Suppose that dynamic constraints (2) do not have to be dealt with. It would not be necessary to handle these two types of constraints simultaneously and the decomposition of (1) in simple subproblems would be allowed.
This remark led us to dualize dynamic constraints (2). Then, the application of the Algorithm A to (1) results, at the k + 1 iteration, in the following steps 2 :
• for all i ∈ I, resolution of:
min T i ([0,H[) t∈[0,H[ −c i (T i (t), t) + b 2 (T i (t) − T k i (t)) 2 + π k,k i (t) − π k,k av(i) (t + δ av(i)←i ) T i (t)(9)
s.t. :
T i ([0, H[) ∈ T ad i yields T k+1 i ([0, H[),
• for all i ∈ I and t ∈ [0, H[, resolution of:
min D i (t),V i (t) −C end i (V i (t), t)(10)+ π k,k+1 i (t) − π k,k+1 av(i) (t + δ av(i)←i ) D i (t) + b 2 (D i (t) − D k i (t)) 2 + b 2 (V i (t) − V k i (t)) 2 + π k,k+1 i (t − 1) − π k,k+1 i (t) V i (t)
subject to:
V i (t) ∈ [V min i , V max i ] , D i (t) ∈ [0, D max i ] , V i (t) − V max i D i (t) = 0 , yields (V k+1 i (t), D k+1 i (t)),
• for all i ∈ I and for all t ∈ [0, H[ the dual variables p i (t) are updated as follows:
p k+1 i (t) = p k i (t) + cH k+1,k+1 i (t)
where:
• c > 0 and b > 0,
• ∀i ∈ I ∀t ∈ [0, H[ :
H k 1 ,k 2 i (t) def = V k 1 i (t + 1) − V k 1 i (t) + D k 1 i (t) + T k 2 i (t) − A * i (t) − j∈Γ T i [D k 1 j + T k 2 j ](t i←j ) , π k 1 ,k 2 i (t) def = p k 1 i (t) + cH k 1 ,k 2 i (t) , • ∀t ∈ [0, H[ : C end i (V i (t), t) = 0.
At each iteration, a dynamic subproblem related to each plant (9) is solved. This subproblem handles mixed integer constraints concerning the plant discharge but no state constraints. It is solved by dynamic programming. The problems (10) are very small and do not present any difficulties: only two real variables are optimized. Therefore, the subproblem resolutions that this algorithm requires, turn out to be quite simple.
Nevertheless, this is explained by the dualization of the most important constraints of the problem (1): the dynamic constraints. It may seem dubious, considering the mixed-integer constraints which have to be handled, that this dual method should achieve a feasible solution of (1).
To explain the numerical result which will be outlined further, it may be emphasized that this algorithm has been implemented in the following way:
• First of all, the "convexified" problem is solved using the price decomposition algorithm we have already described. In this case, the convexity assumptions being met, convergence is theoretically ensured and is obtained in practise. This first step yields a very good initial value of the dual variables p i (t).
• Then, every hundred iterations, until a feasible solution is reached, parameters are modified as follows:
the minimal bounds on the volumes V min i are slightly increased,
the value of parameter c of the Augmented Lagrangian is multiplied (by 3).
Even if it has not been explained theoretically, this progressive increasing of parameter c turns out to be a very efficient method for obtaining feasible solutions.
Interaction Prediction Principle
The second decomposition technique we consider lies on a simultaneous partitioning of variables and constraints. Every subproblem updates a set of variables handling a part of the constraints. Prices remunerate the sharing in the satisfaction of the constraints which are not coped with. Hence, this approach may be considered as mixing the price and resources decomposition techniques.
Theoretical Background
Takahara algorithm: Consider problem (6). Suppose that Θ = n i=1 Θ i where: for all i ∈ I : Θ i : U → C i and C = n i=1 C i . At the iteration k, Takahara algorithm [9], [10] substitutes to (1) a sequence of subproblems (11): min
u i ∈U ad i J i (u i ) + j =i p k j , Θ j (u i , u k −i ) Θ i (u i , u k −i ) = 0(11)
where (u i , u k −i ) denotes the vector whose components are equal to those of u k except u i . Resolution of each (11) yields a primal solution u k+1 i and dual variables (p k+1 i ) related to the local constraint Θ i (u i , u k −i ) = 0. (p k j ) j =i denote the dual variables that have been "predicted" by the other subproblems at step k. They are used to remunerate the participation of problem i to the other constraints.
A primal-dual relaxation algorithm: To explain the nature of this algorithm, let us assume that J is differentiable and U ad = U. Then, Kuhn and Tucker necessary optimality conditions related to (1) may be written as follows:
∀i ∈ I : J i (u i ) + Θ u i * p = 0 , Θ(u) = 0 .(12)
Furthermore, if (u k+1 i , p k+1 i ) is a (primal-dual) solution of (11) then:
J i (u k+1 i ) + j =i Θ j,u i * p k j + Θ i,u i * p k+1 i = 0 , Θ i (u k+1 i , u k −i ) = 0 .
Consequently, the Takahara algorithm appears to be a primal-dual relaxation algorithm applied to the resolution of (12).
Find a saddle-point of the Augmented Lagrangian: The algorithm we apply to the hydro-problem is built up in this way. However, this primal-dual relaxation framework is not used to find a saddle-point of L but a saddle-point of the Augmented Lagrangian L c (8).
With the notation introduced above, at iteration k, it leads to solving the following subproblems:
min u i ∈U ad i J i (u i ) + j =i p k j , Θ j (u i , u k −i ) + c 2 Θ(u i , u k −i ) 2 Θ i (u i , u k −i ) = 0 (13)
Although we will not present hereafter a comparative test on this purpose, it may be pointed out that, applied to the hydro-problem, (13) turns out to be more efficient than (11).
Application to the hydro-problem
In order to apply this algorithm, we first reformulate the hydro-problem (6). Variables A + i (t) representing the global inflows of each reservoir are introduced:
A + i (t) − A * i (t) − j∈Γ T i [D j + T j ](t i←j ) = 0 .(14)
With this definition, the constraint
V i (t + 1) = V i (t) + A + i (t) − D i (t) − T i (t) ,(15)
appears to be equivalent to the dynamic constraints (2). To split (1) into a sequence of subproblems, each related to a plant, algorithm (13) is then applied in the following way:
• vector A + i (t), T i (t), D i (t) t∈[0,H[ is u i ,
• constraints (14) are considered as being the coupling constraints Θ,
• constraints (15), (3), (4) and (5) define the domain U ad of (6),
• J i (u i ) = t∈[0,H[ −c i (T i (t), t) − C end i (V i (H), H).
With these choices, the algorithm (13) leads to solving, at iteration k + 1, the following subproblems (Algorithm B): (4) and (5) .
min T i ∈T ad i t∈[0,H[ −c i (T i (t), t) − C end i (V i (H), H) − p k av(i) (t) [T i + D i ](t av(i)←i ) + c 2 DT k i (t) − [D i + T i ](t av(i)←i ) 2 s.t.: V i (t + 1) = V i (t) − D i (t) − T i (t) + A * i (t) + j∈Γ T i D k j + T k j (t i←j ) (3),
with:
DT k i (t) = A + av(i) (t) k − A * av(i) (t) k − j∈Γ T av(i) −{i} [D k j + T k j ](t av(i)←j ) .
A sequential version of this algorithm has been implemented. Following the course of the river, each subproblem is solved taking into account the results of the current iteration (for the upstream informations) and of the preceding iteration (for the downstream). In this context, DT k i (t) is necessarily equal to [D k i + T k i ](t av(i)←i ) and at iteration k + 1, the subproblem related to the plant i may be written as follows:
min T i ∈T ad i t∈[0,H[ −c i (T i (t), t) − C end i (V i (H), H) (16) − p k av(i) (t)[T i + D i ](t av(i)←i ) + c 2 [D i + T i ] k (t av(i)←i ) − [D i + T i ](t av(i)←i ) 2 s.t.: V i (t + 1) = V i (t) − D i (t) − T i (t) + A * i (t) + j∈Γ T i [D k+1 j + T k+1 j ](t i←j )
(3), (4) and (5) .
Therefore, handling its generating constraints and taking into account the discharge the upstream plants have computed, each plant optimizes its schedule. The dual variable p k av(i) may be understood as being the price the downstream reservoir "would pay" its water inflows. Quadratic terms introduced by the Augmented Lagrangian appear to be a type of "brake" avoiding the oscillations of the algorithm.
It may also be noticed the subproblems (16) turn out to have exactly the same structure as the local problems of the heuristic process currently in use (See Today's methods). Therefore, from a practical point of view, Algorithm B appears to be an extension of this heuristic method. Moreover, providing there is no pumping unit, this sequential version generally 3 yields a feasible solution at the first iteration.
Numerical Tests
A hydraulic valleys sample
A sample of hydraulic valleys has been chosen so as to point out, as well as possible, the main resolution difficulties.
Size and topology: The hydraulic valley is illustrated in Figure 3. It contains six reservoirs and to each reservoir R i is related a plant U i . There are no natural inflows.
Cost function: The generation "revenue" c i (T i (t), t) is assumed to be a linear function of the discharge (i.e. c i (T i (t), t) = p gen (t)T i (t)). Three sequences (p gen (t)) t∈[0,H[ are considered (in Francs per MWh): P1 The first ranges from 99 to 101. It "switches" from one value to another every 4 hours.
This price vector enables the numerical accuracy of the algorithms to be tested.
P2
The second remains at 100 over the whole period except four hours during which it rises up to 500. Such a choice enables the spinning reserve over a four hour period to be computed. The ability to optimize feasible controls in real-time and in case of emergency is also measured in this way. From a numerical point of view, it is in this case that constraints (3) and (4) are actually active.
P3 Every four hours, the third switches between 80 and 120.
Discharge Constraints: For each plant U i , the discharge belongs to a discrete set of values (See Figure 4). The minimum delay between two discharge variations may be 0 (D1), 1 (D2) or 4 (D3) hours.
The (D4), (D5) and (D6) are deduced from (D1), (D2) or (D3) by supposing the plants 3 and 4 are out-of-order (have no discharge capacity).
Crossing these factors, it is a 56 valleys sample which is built up. It may be noticed the non-continuities in the discharge domain are actually sizeable. For the V2 storage/discharge ratio, the choice of a discharge level rather than another modify the hourly discharge by about half the storage capacity !
Numerical results
The two decomposition algorithms and the process currently in use at EDF have been compared over this sample of valleys.
The "scores" are computed in the following way. For each tested algorithm and each sample of valley, the maximum gain
t∈[0,H[ c i (T i (t), t) + C end i (V i (H)
, H) obtained by a feasible solution along the iterations is recorded. If no feasible solution is found, this gain is considered to be zero: because there are no natural inflows, a zero discharge solution is possible and its gain is zero.
Then, relative gains (or "scores") are computed by dividing these gains by those of the best solutions achieved by one of the three processes. Table 1 gives mean scores (m.s.) and feasibility average rates (f.a.) for the three types of "generation prices" which have been considered (P1, P2, P3) and for the whole sample (M). In every case, the two decomposition methods yield feasible solutions. Considering how significant the non-continuities are it is a remarkable achievement.
In spite of the efficiency of the heuristic process which is currently in use at EDF (Algorithm C) -in more than 90 percent of the (difficult) cases we have selected, feasible solutions are reached -these methods thus represent a real improvement. (5) and (3) are not handled, problem (1) is convex. It can be solved by linear programming. The mean "score" of the non feasible solutions obtained in this way are also indicated in Table 1 (Algorithm L). One may consider that about twenty percents of the current cost of the nonconvexities in the (1) modelling are saved thanks to these decomposition methods.
What CPU time ? For the valley of six reservoirs we consider, the heuristic resolution today in use at EDF takes about 10 seconds on a SUN 4/40 4 . One half of this time is dedicated to the linear optimization the other is used by the six dynamic programming resolution which are necessary to find a solution.
For the prediction strategy, the results presented above correspond to 25 iterations. Each of these having the same complexity as the heuristic research of a feasible solution, the CPU time required is more or less 2 minutes.
Our implementation of the price decomposition method uses about 1200 iterations. This number may seem important. Nevertheless, the subproblems are particularly simple and, for the six reservoirs valleys, CPU time does not exceed 2 minutes.
If these times are not huge, they multiply by ten CPU times of the current process. Therefore, work is currently undertaken to reduce these CPU times. To our mind, on average, they should be divided by about 5 in the final implementation by:
• improving the software design,
• avoiding to solve each subproblem at each iteration.
What is the best method ? We have already noticed the average "score" (M) of the price decomposition is 93.2%, the prediction one being 91.4%. Should the prediction method be rejected ? We have not made such a choice for two reasons:
• contrary to the price decomposition strategy, the relaxation algorithm generally yields, from the first iterations, a feasible solution,
• if, on average, price decomposition method reaches the best solutions, it does not in every case. Furthermore, if one choose the best of the two solutions these methods yield, it would not be 93.2% or 91.4% but a score of 100% which would be reached. In fact, the tools we are currently developing, on the basis of these first tests, will try to take advantage of each of these methods. By experimentations, we aim at establishing rules which, after considering the characteristics of the valley, choose the best of the two algorithms.
Conclusion
Over the 56 numerical tests which have been carried out, in spite of the mixed-integer constraints which are handled, the decomposition methods considered allow a feasible solution to be systematically found. Moreover, compared to the two-step process currently in use at EDF, those methods yield sizeable savings. For these reasons, these methods will be used to design new regional level software at Electricité de France.
Furthermore, by proving the robustness of these decomposition approaches, these tests open up new fields of research. A global optimization of generation schedules of several hydraulic valleys, handling coupling constraints (demand constraints), could be achieved in this way.
is the water content of the reservoir i at time t and V min i , V max i are respectively the upper and the lower bounds of the reservoir i, • T i (t) is the discharge of plant i over [t, t + 1[ and T min i et T max i are respectively the lower and the upper bounds of the discharge over the period, • T ad i is defined by the generating constraints of the plant i, • D i (t) is the spillage of plant i over [t, t + 1[ and D max i its upper bound over the period,
Figure 1 :Figure 2 :
12The generating domain of the plant i is non-continuous The domain defined by the constraints (
Figure 3 :
3Hydraulic test valleyVolume bounds: The upstream reservoirs 1 and 2 are supposed to have a large storage capacity: on a daily scale, no volume constraints have to be coped with. The other reservoirs are characterized by the ratio of their storage and hourly-discharge capacities. Three sets of storage/discharge ratios are considered:
Figure 4 :
4The generation is a piecewise linear function of the discharge Furthermore, if constraints
The inflows are known : feasible discharges of the upstream plants have already been processed.
For the sake of completeness, it must be emphasized that this algorithm is not exactly the algorithm (A). The discharges which solve (9) are used to define the cost function of the "volume" problems(10) which are solved at the same iteration. In practise, this sequential version turns out to be more efficient.
All the CPU time here mentioned have been measured on this computer
Optimization of the Flow through Networks with Gains. J F Maurras, Mathematical Programming. 32Maurras J.F., "Optimization of the Flow through Networks with Gains," Mathematical Programming, Vol. 3, n 0 2, pp. 135-144, 1972.
Optimization of Shorttrem Scheduling of EDF Hydraulic Valleys with Coupling Constraints: the OVIDE Model. A Merlin, B Lauzanne, J.-F Maurras, J Auge, M Ziglioli, Proc. PSCC. PSCCMerlin A., Lauzanne B., Maurras J.-F., Auge J. and Ziglioli M., "Optimization of Short- trem Scheduling of EDF Hydraulic Valleys with Coupling Constraints: the OVIDE Model", Proc. PSCC, pp. 345-354, 1981.
Daily Operational Planning of the EDF Plant Mix: The Octave Model optimizes Lake Plant Discharges. K Ea, M Monti, M Jouve, A Kiener, Proc. PSCC pp. PSCC ppEa K., Monti M., Jouve M., Kiener A. "Daily Operational Planning of the EDF Plant Mix: The Octave Model optimizes Lake Plant Discharges", Proc. PSCC pp. 175-181, 1987.
K Arrow, L Hurwicz, H Uzawa, Studies in Linear and Nonlinear Programming. Stanford, USAStandford University PressArrow K., Hurwicz L., Uzawa H., "Studies in Linear and Nonlinear Programming," Standford University Press, Stanford, USA.
Daily Generation Scheduling with Transmission Constraints : A New Class of Algorithms. J Batut, A Renaud, IEEE Transactions on Power Systems. 73Batut J., Renaud A., "Daily Generation Scheduling with Transmission Constraints : A New Class of Algorithms," IEEE Transactions on Power Systems, Vol.7, 3, pp. 982-989, August 1992.
Decomposition Coordination Methods in Large Scale Optimization Problems. The Nondifferentiable Case and the Use of Augmented Lagrangian. G Cohen, D L Zhu, Advances in Large Scale Systems Theory and Applications. J.B. CruzGreenwich, ConnecticutJAI PressICohen G., Zhu D.L., "Decomposition Coordination Methods in Large Scale Optimiza- tion Problems. The Nondifferentiable Case and the Use of Augmented Lagrangian", In: J.B. Cruz, Ed. Advances in Large Scale Systems Theory and Applications, Vol. I., JAI Press, Greenwich, Connecticut, 1984.
A Dual Approach to Solving Nonlinear Programming Problems by Unconstrained Optimization. R T Rockafellar, Mathematical Programming. 5Rockafellar R.T., "A Dual Approach to Solving Nonlinear Programming Problems by Unconstrained Optimization", Mathematical Programming, 5 , pp. 354-373, 1973.
A New Software for Generation Rescheduling in the future EDF national control center. J Batut, A Renaud, P Sandrin, Proc. PSCC Graz. PSCC GrazBatut J., Renaud A., Sandrin P., "A New Software for Generation Rescheduling in the future EDF national control center," Proc. PSCC Graz,1990.
Multilevel Approach to Dynamic Optimization. Y Takahara, SRC-50-C-64-18Cleveland, OhioCase Western Reserve UniversityReportTakahara Y., Multilevel Approach to Dynamic Optimization, Report SRC-50-C-64-18, Case Western Reserve University, Cleveland, Ohio, 1964.
Optimization with an Auxiliary Constraint and Decomposition. G Cohen, B Miara, SIAM J. of Control and Optimization. 281Cohen G., Miara B., "Optimization with an Auxiliary Constraint and Decomposition", SIAM J. of Control and Optimization, Vol. 28, No. 1, pp. 137-157, January 1990.
| []
|
[
"Super-Planckian thermal emission from a hyperlens",
"Super-Planckian thermal emission from a hyperlens"
]
| [
"C Simovski \nDepartment of Radio Science and Engineering\nAalto University\nP.O. Box 13000FI-00076AaltoFinland\n\nMechanics and Optics (ITMO)\nLaboratory of Metamaterials\nUniversity for Information Technology\n197101St. PetersburgRussia\n",
"S Maslovski \nMechanics and Optics (ITMO)\nLaboratory of Metamaterials\nUniversity for Information Technology\n197101St. PetersburgRussia\n\nDepartamento de Engenharia Electrotécnica\nInstituto de Telecomunicações\nUniversidade de Coimbra\nPólo II3030-290CoimbraPortugal\n",
"S Tretyakov \nDepartment of Radio Science and Engineering\nAalto University\nP.O. Box 13000FI-00076AaltoFinland\n",
"I Nefedov \nDepartment of Radio Science and Engineering\nAalto University\nP.O. Box 13000FI-00076AaltoFinland\n",
"S Kosulnikov \nMechanics and Optics (ITMO)\nLaboratory of Metamaterials\nUniversity for Information Technology\n197101St. PetersburgRussia\n",
"P Belov \nMechanics and Optics (ITMO)\nLaboratory of Metamaterials\nUniversity for Information Technology\n197101St. PetersburgRussia\n"
]
| [
"Department of Radio Science and Engineering\nAalto University\nP.O. Box 13000FI-00076AaltoFinland",
"Mechanics and Optics (ITMO)\nLaboratory of Metamaterials\nUniversity for Information Technology\n197101St. PetersburgRussia",
"Mechanics and Optics (ITMO)\nLaboratory of Metamaterials\nUniversity for Information Technology\n197101St. PetersburgRussia",
"Departamento de Engenharia Electrotécnica\nInstituto de Telecomunicações\nUniversidade de Coimbra\nPólo II3030-290CoimbraPortugal",
"Department of Radio Science and Engineering\nAalto University\nP.O. Box 13000FI-00076AaltoFinland",
"Department of Radio Science and Engineering\nAalto University\nP.O. Box 13000FI-00076AaltoFinland",
"Mechanics and Optics (ITMO)\nLaboratory of Metamaterials\nUniversity for Information Technology\n197101St. PetersburgRussia",
"Mechanics and Optics (ITMO)\nLaboratory of Metamaterials\nUniversity for Information Technology\n197101St. PetersburgRussia"
]
| []
| We suggest and theoretically explore a possibility to strongly enhance the steady thermal radiation of a small thermal emitter using an infrared hyperlens. The hyperbolic metamaterial of the hyperlens converts emitter's near fields into the propagating waves which are efficiently irradiated from the hyperlens surface. Thus, with the hyperlens, emitter's spectral radiance goes well beyond the black-body limit for the same emitter in free space. Although the hyperlens can be kept at a much lower temperature than the emitter, the whole structure may radiate, in principle, as efficiently as a black body with the same size as that of the hyperlens and the same temperature as that of the emitter. We believe that this study can lead to a breakthrough in radiative cooling at microscale, which is crucial for microlasers and microthermophotovoltaic systems. PACS numbers: 44.40.+a, 42.25.Bs, 78.20.nd In the classical theory of thermal radiation the power radiated from a unit surface of an optically large body in free space per unit interval of frequencies is given by Planck's formula:where k B and are Boltzmann's and Planck's constants, respectively, T is the temperature of the body surface and e s (ω) < 1 is the spectral emissivity of body's material. For a black body (BB) emitting, in accordance to the Planckian theory, maximal thermal radiation to free space, e s (ω) ≡ 1. For a material having no optical losses at frequency ω, i.e. for transparent media, e s (ω) = 0 and the emission is absent. It is commonly accepted that the thermal radiation is non-coherent and its spatial distribution is isotropic. Both these factors result in Lambertian pattern for a radiating halfspace. However, recent investigations have shown that thermal radiation can be partially coherent [1], directive [2], and combining coherence and directionality[3,4]. These deviations originate from intrinsic properties of metamaterials[5]. Especially, the so-called hyperbolic metamaterial (HMM) shows interesting responses to thermal radiation (see e.g. in[6][7][8][9]). In this paper we theoretically reveal a possibility for a sample of HMM to strongly enhance the far-field radiation from small (several micrometers) emitters exceeding the BB limit defined for the emitters of the same size in free space. To our knowledge, in previous works related with applications of HMMs for radiative heat transfer these materials were used only to control near-field thermal flows. Here, we use these media to enhance far-field radiation. Since classical works by Kirchhoff and Planck, the BB has been considered as a perfect thermal emitter whose spectral radiance cannot be exceeded in far-field zone (so-called Planckian limit). Despite that the photonic density of states (PDOS) and, consequently, the rate of spontaneous emission responsible for the thermal radiation may be enhanced considerably (e.g. in [10] by one order of magnitude) there is a belief that the photons that occupy the extra available states cannot be emitted out of a medium with high PDOS [11] due to the total internal reflection (TIR). However, it is not generally true.Long ago, in work [12], a possibility to exceed the BB limit for a hot particle having resonant sizes at infrared was pointed out. Recently, the authors of [13] have demonstrated super-Planckian radiation from a macroscopic emitter achieved due to a transparent dielectric dome. For a hemispherical emitter this idea is illustrated inFig. 1 (left half). As it follows from Eq. (1), filling free space with an isotropic transparent medium with refractive index n = √ ε h [respectively, c is replaced by c/n in Eq. (1)] increases the radiated power by n 2 , provided that e s (ω) stays unchanged. If such transparent medium forms a lens in a shape of a hemispherical dome as is shown inFig. 1, the emitted waves impinge on the lens surface and, after being partially reflected, pass onto free space. When the dome radius R is much larger than the emitter radius r, the power transmittance to the free space for these waves can be found in geometric optics (GO) approximation as for normally incident rays: t GO = 4n/(n + 1) 2 . Thus, the total gain in the power irradiated to the far zone due to the presence of the dome equals G = n 2 × t GO = 4n 3 /(n + 1) 2 . From here it may seem that one can achieve arbitrary high gain when n → ∞. However, this is not true, because when n R/r some of the incident rays start experiencing TIR at the output interface of the dome. This effect also ensures that the apparent diameter of the emitter as is seen from outside of the dome never exceeds the diameter of the dome, which sets an obvious upper bound for the total gain in this structure when R ≫ λ: G < R 2 /r 2 , i.e., the whole structure may not radiate more than a BB with radius equal to the outer radius of the dome.Because realistic thermal sources have e s < 1, and because the known transparent materials in the infrared range have rather small refractive indices n 3, the thermal lens[13]can hardly offer gain G BB which would exceed 7. Here, G BB is the ratio of power emitted by a realistic thermal source covered with a transparent dome to the power emitted | null | [
"https://arxiv.org/pdf/1406.1010v1.pdf"
]
| 118,566,633 | 1406.1010 | 94665655c031213add07b563f5e4062036c8a533 |
Super-Planckian thermal emission from a hyperlens
4 Jun 2014
C Simovski
Department of Radio Science and Engineering
Aalto University
P.O. Box 13000FI-00076AaltoFinland
Mechanics and Optics (ITMO)
Laboratory of Metamaterials
University for Information Technology
197101St. PetersburgRussia
S Maslovski
Mechanics and Optics (ITMO)
Laboratory of Metamaterials
University for Information Technology
197101St. PetersburgRussia
Departamento de Engenharia Electrotécnica
Instituto de Telecomunicações
Universidade de Coimbra
Pólo II3030-290CoimbraPortugal
S Tretyakov
Department of Radio Science and Engineering
Aalto University
P.O. Box 13000FI-00076AaltoFinland
I Nefedov
Department of Radio Science and Engineering
Aalto University
P.O. Box 13000FI-00076AaltoFinland
S Kosulnikov
Mechanics and Optics (ITMO)
Laboratory of Metamaterials
University for Information Technology
197101St. PetersburgRussia
P Belov
Mechanics and Optics (ITMO)
Laboratory of Metamaterials
University for Information Technology
197101St. PetersburgRussia
Super-Planckian thermal emission from a hyperlens
4 Jun 2014
We suggest and theoretically explore a possibility to strongly enhance the steady thermal radiation of a small thermal emitter using an infrared hyperlens. The hyperbolic metamaterial of the hyperlens converts emitter's near fields into the propagating waves which are efficiently irradiated from the hyperlens surface. Thus, with the hyperlens, emitter's spectral radiance goes well beyond the black-body limit for the same emitter in free space. Although the hyperlens can be kept at a much lower temperature than the emitter, the whole structure may radiate, in principle, as efficiently as a black body with the same size as that of the hyperlens and the same temperature as that of the emitter. We believe that this study can lead to a breakthrough in radiative cooling at microscale, which is crucial for microlasers and microthermophotovoltaic systems. PACS numbers: 44.40.+a, 42.25.Bs, 78.20.nd In the classical theory of thermal radiation the power radiated from a unit surface of an optically large body in free space per unit interval of frequencies is given by Planck's formula:where k B and are Boltzmann's and Planck's constants, respectively, T is the temperature of the body surface and e s (ω) < 1 is the spectral emissivity of body's material. For a black body (BB) emitting, in accordance to the Planckian theory, maximal thermal radiation to free space, e s (ω) ≡ 1. For a material having no optical losses at frequency ω, i.e. for transparent media, e s (ω) = 0 and the emission is absent. It is commonly accepted that the thermal radiation is non-coherent and its spatial distribution is isotropic. Both these factors result in Lambertian pattern for a radiating halfspace. However, recent investigations have shown that thermal radiation can be partially coherent [1], directive [2], and combining coherence and directionality[3,4]. These deviations originate from intrinsic properties of metamaterials[5]. Especially, the so-called hyperbolic metamaterial (HMM) shows interesting responses to thermal radiation (see e.g. in[6][7][8][9]). In this paper we theoretically reveal a possibility for a sample of HMM to strongly enhance the far-field radiation from small (several micrometers) emitters exceeding the BB limit defined for the emitters of the same size in free space. To our knowledge, in previous works related with applications of HMMs for radiative heat transfer these materials were used only to control near-field thermal flows. Here, we use these media to enhance far-field radiation. Since classical works by Kirchhoff and Planck, the BB has been considered as a perfect thermal emitter whose spectral radiance cannot be exceeded in far-field zone (so-called Planckian limit). Despite that the photonic density of states (PDOS) and, consequently, the rate of spontaneous emission responsible for the thermal radiation may be enhanced considerably (e.g. in [10] by one order of magnitude) there is a belief that the photons that occupy the extra available states cannot be emitted out of a medium with high PDOS [11] due to the total internal reflection (TIR). However, it is not generally true.Long ago, in work [12], a possibility to exceed the BB limit for a hot particle having resonant sizes at infrared was pointed out. Recently, the authors of [13] have demonstrated super-Planckian radiation from a macroscopic emitter achieved due to a transparent dielectric dome. For a hemispherical emitter this idea is illustrated inFig. 1 (left half). As it follows from Eq. (1), filling free space with an isotropic transparent medium with refractive index n = √ ε h [respectively, c is replaced by c/n in Eq. (1)] increases the radiated power by n 2 , provided that e s (ω) stays unchanged. If such transparent medium forms a lens in a shape of a hemispherical dome as is shown inFig. 1, the emitted waves impinge on the lens surface and, after being partially reflected, pass onto free space. When the dome radius R is much larger than the emitter radius r, the power transmittance to the free space for these waves can be found in geometric optics (GO) approximation as for normally incident rays: t GO = 4n/(n + 1) 2 . Thus, the total gain in the power irradiated to the far zone due to the presence of the dome equals G = n 2 × t GO = 4n 3 /(n + 1) 2 . From here it may seem that one can achieve arbitrary high gain when n → ∞. However, this is not true, because when n R/r some of the incident rays start experiencing TIR at the output interface of the dome. This effect also ensures that the apparent diameter of the emitter as is seen from outside of the dome never exceeds the diameter of the dome, which sets an obvious upper bound for the total gain in this structure when R ≫ λ: G < R 2 /r 2 , i.e., the whole structure may not radiate more than a BB with radius equal to the outer radius of the dome.Because realistic thermal sources have e s < 1, and because the known transparent materials in the infrared range have rather small refractive indices n 3, the thermal lens[13]can hardly offer gain G BB which would exceed 7. Here, G BB is the ratio of power emitted by a realistic thermal source covered with a transparent dome to the power emitted
We suggest and theoretically explore a possibility to strongly enhance the steady thermal radiation of a small thermal emitter using an infrared hyperlens. The hyperbolic metamaterial of the hyperlens converts emitter's near fields into the propagating waves which are efficiently irradiated from the hyperlens surface. Thus, with the hyperlens, emitter's spectral radiance goes well beyond the black-body limit for the same emitter in free space. Although the hyperlens can be kept at a much lower temperature than the emitter, the whole structure may radiate, in principle, as efficiently as a black body with the same size as that of the hyperlens and the same temperature as that of the emitter. We believe that this study can lead to a breakthrough in radiative cooling at microscale, which is crucial for microlasers and microthermophotovoltaic systems.
PACS numbers: 44.40.+a, 42. 25.Bs,78.20.nd In the classical theory of thermal radiation the power radiated from a unit surface of an optically large body in free space per unit interval of frequencies is given by Planck's formula:
P FS ω = πe s (ω)B ω = ω 3 e s (ω) 4π 2 c 2 e ω/(kB T ) − 1 −1 ,(1)
where k B and are Boltzmann's and Planck's constants, respectively, T is the temperature of the body surface and e s (ω) < 1 is the spectral emissivity of body's material. For a black body (BB) emitting, in accordance to the Planckian theory, maximal thermal radiation to free space, e s (ω) ≡ 1. For a material having no optical losses at frequency ω, i.e. for transparent media, e s (ω) = 0 and the emission is absent. It is commonly accepted that the thermal radiation is non-coherent and its spatial distribution is isotropic. Both these factors result in Lambertian pattern for a radiating halfspace. However, recent investigations have shown that thermal radiation can be partially coherent [1], directive [2], and combining coherence and directionality [3,4]. These deviations originate from intrinsic properties of metamaterials [5]. Especially, the so-called hyperbolic metamaterial (HMM) shows interesting responses to thermal radiation (see e.g. in [6][7][8][9]). In this paper we theoretically reveal a possibility for a sample of HMM to strongly enhance the far-field radiation from small (several micrometers) emitters exceeding the BB limit defined for the emitters of the same size in free space. To our knowledge, in previous works related with applications of HMMs for radiative heat transfer these materials were used only to control near-field thermal flows. Here, we use these media to enhance far-field radiation. Since classical works by Kirchhoff and Planck, the BB has been considered as a perfect thermal emitter whose spectral radiance cannot be exceeded in far-field zone (so-called Planckian limit). Despite that the photonic density of states (PDOS) and, consequently, the rate of spontaneous emission responsible for the thermal radiation may be enhanced considerably (e.g. in [10] by one order of magnitude) there is a belief that the photons that occupy the extra available states cannot be emitted out of a medium with high PDOS [11] due to the total internal reflection (TIR). However, it is not generally true.
Long ago, in work [12], a possibility to exceed the BB limit for a hot particle having resonant sizes at infrared was pointed out. Recently, the authors of [13] have demonstrated super-Planckian radiation from a macroscopic emitter achieved due to a transparent dielectric dome. For a hemispherical emitter this idea is illustrated in Fig. 1 (left half). As it follows from Eq. (1), filling free space with an isotropic transparent medium with refractive index n = √ ε h [respectively, c is replaced by c/n in Eq. (1)] increases the radiated power by n 2 , provided that e s (ω) stays unchanged. If such transparent medium forms a lens in a shape of a hemispherical dome as is shown in Fig. 1, the emitted waves impinge on the lens surface and, after being partially reflected, pass onto free space. When the dome radius R is much larger than the emitter radius r, the power transmittance to the free space for these waves can be found in geometric optics (GO) approximation as for normally incident rays: t GO = 4n/(n + 1) 2 . Thus, the total gain in the power irradiated to the far zone due to the presence of the dome equals G = n 2 × t GO = 4n 3 /(n + 1) 2 . From here it may seem that one can achieve arbitrary high gain when n → ∞. However, this is not true, because when n R/r some of the incident rays start experiencing TIR at the output interface of the dome. This effect also ensures that the apparent diameter of the emitter as is seen from outside of the dome never exceeds the diameter of the dome, which sets an obvious upper bound for the total gain in this structure when R ≫ λ: G < R 2 /r 2 , i.e., the whole structure may not radiate more than a BB with radius equal to the outer radius of the dome.
Because realistic thermal sources have e s < 1, and because the known transparent materials in the infrared range have rather small refractive indices n 3, the thermal lens [13] can hardly offer gain G BB which would exceed 7. Here, G BB is the ratio of power emitted by a realistic thermal source covered with a transparent dome to the power emitted by an uncovered BB with the same size and temperature as the original source. In [13] the gain G BB ≈ 3.1 has been experimentally demonstrated for a thermal lens of centimeter size with n ≈ 2.4. The above estimations predict G BB ≈ 4.8 for this case, when the emitter is an ideal BB.
The study that we are going to present next has been motivated by the following question: Since it is possible to enhance the thermal radiation of an emitter by 2-5 times by using a hemisphere of a transparent isotropic dielectric, can we go further using more advanced materials? Namely, can we approach the GO bound: G max = R 2 /r 2 with these materials? Note that here we are interested in the case when R ≈ 3λ or greater, because bodies with R λ can outperform this bound [12]. We show that a dome made of a hyperbolic metamaterial theoretically allows one to increase the spectral radiance of small emitters by up to two orders of magnitude, as compared to the limit dictated by Planck's law for BB emitters of the same size in vacuum. Hyperbolic metamaterials (HMM) which we propose for this purpose are uniaxial dielectric composites with the permittivity tensor defined by two components: transverse ε ⊥ and axial ε , such that Re(ε )Re(ε ⊥ ) < 0, Re(ε ,⊥ ) ≫ Im(ε ,⊥ ). The isofrequency surfaces (also called wave-dispersion surfaces) for HMM represent hyperboloids. A hot unit volume inside a HMM sample emits much more electromagnetic energy than a unit volume of a conventional lossy medium at the same temperature. This effect results from high Purcell's factor of a dipole located inside HMM. The concept of Purcell's factor (the gain in the spontaneous emission rate) historically referred to the case when the dipole radiation was enhanced by a closely located resonator (see e.g. in [14]). However, in work [15] the notion of Purcell's factor was extended to any environment of the dipole source different from free space. Purcell's factor of HMM dramatically exceeds the Purcell's factor F P,diel = n of a usual dielectric. In the lossless HMM without internal granularity the radiation resistance of a point dipole oriented orthogonally to the optical axis tends to infinity [16,17] because all the power is irradiated in the form of propagating waves. Therefore the thermal radiation of a unit hot volume in such an ideal HMM should be infinite. For realistic (lossy and internally granular) HMM the thermal radiation of a unit hot volume is finite but strongly super-Planckian as compared to vacuum [6,7].
The excessive super-Planckian radiative heat in HMM is contained in the modes with high transverse wavenumbers q = 2π/Λ tr ≥ ω/c. In flat uniaxial HMM slabs with the optical axis oriented orthogonally to the interface plane these modes experience TIR at the interface with free space and, thus, are confined inside the HMM (note that the coupling of such modes with free space can be carried out in asymmetric HMM [18][19][20], where the optical axis is tilted to the slab interface). As a result, the thermal radiation from such slabs into free space does not exceed the BB limit. However, it can be very close to it, and it is known that a half-space of HMM mimics the BB [5][6][7].
On the other hand, in locally uniaxial radially symmetric HMM samples the eigenmodes which are characterized with high local wavenumbers q(r) ≫ ω/c close to the center of the sample, may attain q(R) < ω/c at enough large radial distance R ≫ r, because in these modes, roughly, q(r) ∝ 1/r. Therefore, in radially symmetric HMM these modes can couple to the free space propagating waves if the radius R is large enough. This effect is known as hyperlensing [21][22][23]. Hyperlenses (HLs) were previously designed for obtaining magnified images of subwavelength objects.
In fact, HL is also a matching device for the radiation propagating from its central part to free space [24]. Here, we suggest to use a dome of radially symmetric HMM which operates as an infrared HL to extract the excessive super-Planckian heat otherwise confined within emitter's near field in the modes with high transverse wavenumbers. An implementation of such HMM in the infrared range is, for example, an optically dense array of aligned metal nanowires called wire medium (WM). WM is a spatially dispersive implementation of HMM [25]. The spatial dispersion of HMM for our purpose is not a harmful factor. On the contrary, in accordance to our estimations the spatial dispersion helps to match the hyperlens to free space.
Performing our emitter as a lossy dielectric body placed inside a transparent dielectric dome both comprising radially divergent nanowires, we arrive at the structure sketched in Fig. 1 (right half). For better matching of the HL to free space FIG. 1: Two hemispherical structures which offer super-Planckian thermal radiation from a finite emitter (just a half of each structure is shown). Left: Thermal lens analogous to the one considered in Ref. [13]. Right: Thermal hyperlens (HL) comprising radially diverging nanowires. The lens and HL's host are made of transparent glass with permittivity ε2 ≡ ε h . The emitter is formed by a lossy medium with complex permittivity ε1 and is partially filled with nanowires. The emitter is set under high temperature T1 ≫ T2, where T2 is the ambient temperature. the ends of the nanowires can be made free-standing as it is shown in the figure. To prevent direct thermal contact, the nanowires in the emitter may be separated from the HL by a sufficiently narrow nanogap. It is critical that the nanowires are radially oriented in the whole structure and that their density decreases with radial coordinate. The divergence angle φ between adjacent nanowires should be small enough so that the properties of HMM in the emitter volume are preserved, however, large enough so that the best possible matching to free space is achieved.
Formulas for the effective permittivity of WM operating at infrared can be found in [25,26]. We use highly radially anisotropic HMM, in which Re(ε ⊥ ) > 0, Re(ε ) < 0, |ε | ≫ |ε ⊥ |, and the energy propagates roughly in the radial direction, independently on the value of q. Thus, we notice that in this regard the situation is similar to the case of the simple dielectric dome considered previously, with a difference that when estimating the power transmittance through the outer interface of the HL we must use the effective complex index of refraction in the WM in the vicinity of the outer interface: n ⊥,out = √ ε ⊥,out . Hence, t HL = 4Re(n ⊥,out )/|n ⊥,out + 1| 2 . Note that still the modes with q(R) ≥ ω/c experience TIR at the dome-air interface and for these modes t HL = 0. From recent studies of the dipole radiation in WM [27,28] it is known that the irradiated wave beam is nearly as narrow as the WM period a. Therefore, in order to obtain high transmittance to free space (nearly as high as t HL ) for the dominant part of the spatial spectrum exited within HL, the divergence angle φ should be such that the separation between the nanowires at the outer surface of the HL is about λ/2 or larger. Hence, φ λ/(2R).
Let us now estimate how large can be the gain G HL in the HL configuration of Fig. 1. First, we note that inclusion of nanowires into a dielectric host increases the power radiated by an elementary dipole placed inside this medium by F P times, where F P is the Purcell factor for uniaxial WM. This factor was calculated in Ref. [17]. Being averaged over all possible locations of a transversely oriented electric dipole, F P equals
F tr P ≈ 3k 2 p 8k 2 h log 1 + K 2 m k 2 p ,(2)
where k h = √ ε h ω/c and k p = 2π/ log[a 2 /4r 0 (a − r 0 )]/a are the wavenumber in the host medium and the plasma wavenumber in WM [29], respectively, where a is the WM period and r 0 is the wire radius. In Eq. (2), K m is the spatial spectrum cut-off parameter, which equals 2 √ π/a in unbounded uniaxial WM [17]. Because only the modes with q < ω/c are irradiated from HL's outer surface, here we must limit this parameter by K m = min[2 √ π/a, (ω/c)(R/r)]. Strictly speaking, Eq. (2) refers to the case when the wires are perfectly conducting, however in [17] the estimations were done also for lossy wires and it was shown that (2) was applicable to realistic metal nanowires if they were thick enough (practically, their radius r 0 should be larger than the skin depth). For dipoles parallel to the wires the Purcell factor is much smaller than F tr P and can be neglected. Since a hot elementary volume of a lossy medium surrounded by nanowires can be treated as a set of three identical mutually orthogonal dipoles emitting thermal radiation, in thermal emission calculations we must use the average F HL P = 2F tr P /3, where F tr P is given by (2). Under these conditions, the total gain due to the effect of the HL dome can be estimated as follows:
G HL ≈ F HL P × Re(n ⊥,in ) 2 × t HL × e −2α(R−r) ≈ 8F tr P × Re(n ⊥,in ) 2 Re(n ⊥,out ) 3|n ⊥,out + 1| 2 e −2α(R−r) ,(3)
where n ⊥,in is the effective refractive index of the HL in the vicinity of the emitter (at the inner interface of the HL), and α ≈ (ω/c)Im( √ ε ⊥ ) is the decay factor due to the loss in the nanowires. Note that the structure suggested in this paper cannot radiate more than a BB with the same size as that of the dome and the same temperature as that of the emitter when R ≫ λ. Moreover, because the WM-based HL interacts mostly with P -polarized waves, the actual upper bound for the gain in this case is G max HL ∼ 0.5R 2 /r 2 . Due to optical losses in the metal the decay of thermal radiation over the path R ≫ λ is not negligible. This factor restricts the radius R by dozens of λ. However, Eq. (3) does not take into account the thermal radiation of heated wires inside the hyperlens. This additional emission can significantly increase G HL compared to (3), so that it may approach the GO bound: G max = R 2 /r 2 . In the same time, (2) slightly overestimates the Purcell factor for realistic nanowires. So, the implementation of our thermal HL with macroscopic dimensions (e.g. with R = 1 cm like in [13]) is disputable. In the present study we deal with a microscopic hyperlens with radius R = 10 µm and an emitter of radius r = 0.75 µm.
Estimations of the factor G HL were done using Eq. (3) and the effective-medium model of infrared WM [26]. The material parameters of gold were taken from Ref. [30]. We considered a hemispheric structure with concentric hemispheric emitter located on a perfect mirror as in Fig. 1. The HL is performed of non-tapered gold nanowires with the thickness 2r 0 = 50 nm located in a matrix with ε 2 = 3.16 (chalcogenide glass transparent in the range 50-150 THz). We calculate the gain G HL in the range 100-140 THz, where r 0 > δ (δ is the skin-depth of gold), and formula (2) for Purcell's factor is applicable. Internal ends of nanowires are located at r 1 = 0.5 µm from the geometrical center of the structure. An emitter comprises the hemisphere r = 0.75 µm and is partially filled with nanowires. The emitter is assumed to be a lossy dielectric which is well impedance-matched with the HL. The distance between the centers of nanowires at the surface r 1 = 0.5 µm equals 100 nm and within the emitter the averaged period of the WM equals a = 125 nm. Nanowires diverge with the angle φ ≈ 10 • . This angle is small enough to neglect the divergence of nanowires when calculating the effective permittivity of HMM in the domain of the emitter and its Purcell factor F tr P . However, it is large enough to offer good matching of the HL to free space, because for φ 7.5 • the distance A between the axes of nanowires at the outer surface of the HL exceeds λ/2 at frequencies 100-140 THz. Following to (2) Purcell's factor of the medium of parallel nanowires with the period a = 125 nm for a transverse electric dipole decreases from F tr P ≈ 18 to F tr P ≈ 7.5 over the range 100-140 THz. Then the relative enhancement G HL of the power spectrum radiated by an arbitrary dipole p located in between the wires near the internal surface of the HL in accordance to (3) is within approximately 40 . . . 20 over this frequency range. The range 100-140 THz is around the maximum of emission for the emitter temperatures T 1 of the order 700-800 • C. Higher temperatures are hardly actual for our HL since nanowires can melt. For lower temperatures thermal radiation is concentrated in lower frequencies where Purcell's factor is higher. For example, at 50 THz in unbounded WM F tr P ≈ 70. The same divergence angle at this frequency implies larger R needed for matching the HL to free space. The condition A > λ/2 holds at 50 THz for R = 20 µm. Then, taking into account the decay we obtain using (3) the gain G HL ≈ 170 at 50 THz. So, for emitters with temperatures T 1 <500-600 • C the thermal radiation of the emitter within HL may exceed the BB limit for the same emitter in vacuum by two orders of magnitude.
To check our estimations of the gain G HL we performed extensive numerical simulations. We studied a HL excited by a transverse dipole located in the middle between the ends of adjacent nanowires either at the surface of the central nanocavity or displaced from this surface -either embedded into the WM up (to 250 nm from the central cavity) or located inside it. The parameters of the HL in these simulations are as above besides one replacement -we substituted gold nanowires by perfectly conducting ones. This replacement dramatically reduced the computation time needed for the structure comprising many hundreds of metal nanowires and made simulations realistic. Simulations were performed using the CST Studio Suit software.
Although replacing gold by perfect conductor we removed the decay factor exp(−2αR), this is still a reasonable model of a HL. The decay factor is not the most relevant parameter and can be easily taken into account analytically. The absence of absorption makes the relative enhancement of radiation into free space equivalent to Purcell's factor. This equivalence allows us to concentrate on the hyperlensing effect, i.e., on emission enhancement and matching of our structure to free space. Our model source is a very short dipole antenna of perfectly conducting wire with bulbs mimicking the Hertzian dipole at the simulation frequency. First, we calculate the field distributions to inspect if the wave beam divergence is sufficient to prevent strong reflections from the effective surface of the HL. For divergence angles within the range φ = 6-9 • the concept of HL turned out to be fully adequate, and the result weakly depends on φ over this interval of values. The result weakly depends on the exact location of the transverse dipole embedded into the WM up to 250 nm from the internal cavity r 1 . However, if the dipole is moved to the central cavity to the distance more than 250 nm, Purcell's factor drops to unity. Also, the radiation decreases if φ < 6 • i.e. when the HL approaches to a block of parallel nanowires. In Fig. 2 a color map illustrates the hyperlensing of the dipole radiation for optimal divergence angle φ = 10 • . The horizontal dipole is located on top of the internal cavity r 1 in between two central nanowires and radiates at the frequency 120 THz which is between the bands of Fabry-Perot resonances. In both Eand H-planes we observed a sufficient width of the main radiation beam. The reflection from the effective surface of the HL in our simulations fits the estimation t ≈ t HL .
For a lossless HL G HL can be calculated in two ways: via the input resistance of the antenna and via the far-zone radiated power. Both these values were calculated and normalized to the corresponding values simulated for the same dipole when the HL is absent. In the first case we keep the same input voltage of the antenna in the absence or presence of the HL. In the second case we fix the antenna current. The coincidence of two results is expected at low frequencies where the short wire antenna is close to the Hertzian dipole. This equivalence is seen in Fig. 3 at 100-130 THz where the red and green curves nearly coincide (besides the small ripples of the red curve which are numeric errors). In this plot we observe several Fabry-Perot resonances at which the HL gain reaches very high values. These values are, however, hardly relevant for the thermal radiation because the emitter mimicking the BB will absorb all incoming waves. Therefore, the Fabry-Perot resonances in the HL in a more realistic configuration will be greatly suppressed. The blue curve shows the theoretical estimation for the HL gain calculated in accordance to Eq. (3) with the factor (2/3) exp(−2αR) excluded because only single orientation of the dipole in a lossless HL is considered in the simulations. In the range 100-130 THz our theoretical estimation agrees with the simulated gain when averaged between the Fabry-Perot resonances.
To conclude, we have suggested a structure that greatly enhances the radiative heat power produced by a small thermal emitter, which may go far beyond the limit enforced by Planck's law for the same radiator in free space. This is achieved by centering the emitter at the focal point of a hyperlens, which transforms emitter's near field into propagating waves which are matched well to free space and efficiently irradiated. However, the structure suggested in this paper still radiates less than a BB with the same size as that of the hyperlens and the same temperature as that of the emitter. A theoretical possibility to overcome this restriction for bodies of constrained radius is reserved for a future work (see [31]).
FIG. 2 :
2Electric field amplitude distribution in the H-plane (vertical cross section orthogonal to the dipole) produced by a dipole located in between the internal ends of perfect nanowires of radius r0 = 25 nm forming our HL. Its host material (between r1 = 0.5 µm and R = 10 µm) is glass.
FIG. 3 :
3Relative enhancement of radiation by a transverse dipole due to the presence of a HL of perfect wires calculated 1) directly via the radiated power spectrum (green curve) and 2) through the input resistance of a short wire dipole (red curve). The structure is the same as inFig. 2. The theoretical blue dashed curve and the green dotted line are explained in the main text.
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| []
|
[
"Closed and Open System Dynamics in a Fermionic Chain with a Microscopically Specified Bath: Relaxation and Thermalization",
"Closed and Open System Dynamics in a Fermionic Chain with a Microscopically Specified Bath: Relaxation and Thermalization"
]
| [
"Nicholas Sedlmayr \nDepartment of Physics and Research Center OPTIMAS\nTechnical University Kaiserslautern\nD-67663KaiserslauternGermany\n",
"Jie Ren \nDepartment of Physics and Research Center OPTIMAS\nTechnical University Kaiserslautern\nD-67663KaiserslauternGermany\n\nDepartment of Physics and Jiangsu Laboratory of Advanced Functional Material\nChangshu Institute of Technology\n215500ChangshuChina\n",
"Florian Gebhard \nDepartment of Physics\nPhilipps-Universität Marburg\n35032MarburgGermany\n",
"Jesko Sirker \nDepartment of Physics and Research Center OPTIMAS\nTechnical University Kaiserslautern\nD-67663KaiserslauternGermany\n"
]
| [
"Department of Physics and Research Center OPTIMAS\nTechnical University Kaiserslautern\nD-67663KaiserslauternGermany",
"Department of Physics and Research Center OPTIMAS\nTechnical University Kaiserslautern\nD-67663KaiserslauternGermany",
"Department of Physics and Jiangsu Laboratory of Advanced Functional Material\nChangshu Institute of Technology\n215500ChangshuChina",
"Department of Physics\nPhilipps-Universität Marburg\n35032MarburgGermany",
"Department of Physics and Research Center OPTIMAS\nTechnical University Kaiserslautern\nD-67663KaiserslauternGermany"
]
| []
| We study thermalization in a one-dimensional quantum system consisting of a noninteracting fermionic chain with each site of the chain coupled to an additional bath site. Using a density matrix renormalization group algorithm we investigate the time evolution of observables in the chain after a quantum quench. For low densities we show that the intermediate time dynamics can be quantitatively described by a system of coupled equations of motion. For higher densities our numerical results show a prethermalization for local observables at intermediate times and a full thermalization to the grand canonical ensemble at long times. For the case of a weak bath-chain coupling we find, in particular, a Fermi momentum distribution in the chain in equilibrium in spite of the seemingly oversimplified bath in our model. arXiv:1212.0223v2 [cond-mat.stat-mech] | 10.1103/physrevlett.110.100406 | [
"https://arxiv.org/pdf/1212.0223v2.pdf"
]
| 10,588,405 | 1212.0223 | 1fab0ceadb7e5a56e4d9c9eb191905727be88e7b |
Closed and Open System Dynamics in a Fermionic Chain with a Microscopically Specified Bath: Relaxation and Thermalization
Nicholas Sedlmayr
Department of Physics and Research Center OPTIMAS
Technical University Kaiserslautern
D-67663KaiserslauternGermany
Jie Ren
Department of Physics and Research Center OPTIMAS
Technical University Kaiserslautern
D-67663KaiserslauternGermany
Department of Physics and Jiangsu Laboratory of Advanced Functional Material
Changshu Institute of Technology
215500ChangshuChina
Florian Gebhard
Department of Physics
Philipps-Universität Marburg
35032MarburgGermany
Jesko Sirker
Department of Physics and Research Center OPTIMAS
Technical University Kaiserslautern
D-67663KaiserslauternGermany
Closed and Open System Dynamics in a Fermionic Chain with a Microscopically Specified Bath: Relaxation and Thermalization
(Dated: May 5, 2014)
We study thermalization in a one-dimensional quantum system consisting of a noninteracting fermionic chain with each site of the chain coupled to an additional bath site. Using a density matrix renormalization group algorithm we investigate the time evolution of observables in the chain after a quantum quench. For low densities we show that the intermediate time dynamics can be quantitatively described by a system of coupled equations of motion. For higher densities our numerical results show a prethermalization for local observables at intermediate times and a full thermalization to the grand canonical ensemble at long times. For the case of a weak bath-chain coupling we find, in particular, a Fermi momentum distribution in the chain in equilibrium in spite of the seemingly oversimplified bath in our model. arXiv:1212.0223v2 [cond-mat.stat-mech]
Introduction. The time evolution of classical and quantum systems is deterministic. If a system in the thermodynamic limit reaches thermal equilibrium at long times, we expect, however, that its physical properties will be determined by only a few parameters such as the temperature, chemical potential, and pressure. This thermalization process is often studied in two different settings: (a) The system is in contact with a thermal bath, i.e., a large reservoir of thermal energy. The key assumptions commonly used in this setting are a weak coupling between the bath and system and Markovian dynamics, i.e., a very short correlation time in the bath. In this case the microscopic details of the bath become unimportant [1][2][3]; for a simple example of classical thermalization, see Ref. [4]. (b) The system is closed, with particles being able to exchange energy and momentum among each other, so that the closed system can explore phase space, constrained only by the conservation laws such as total energy and particle number. An important difference between the two scenarios is that in the first case temperature, chemical potential, and pressure are parameters determined externally by the bath. In the latter case, on the other hand, these parameters are Lagrange multipliers fixing the values of the conserved quantities [5,6].
In this letter we want to study these two settings simultaneously using a model which can be either viewed as a closed quantum system or as a chain coupled to a simple bath. Thermalization, in both cases, requires: (I) Observables become time independent and all currents vanish (equilibration); (II) Time averages can be replaced by statistical averages over ensembles with a restricted number of intensive parameters [36], and are independent of initial conditions (ergodicity) [7]. The rather old but fundamental problem of nonequilibrium dynamics and thermalization in closed quantum systems has been put again into focus by experiments on cold quantum gases which are very well isolated from their surroundings [8][9][10][11], as well as by the development of new numerical techniques to study dynamics in many-body systems [12][13][14][15][16][17][18][19]. This has led to numerous simulations of nonequilibrium dynamics in closed quantum models where the question of whether or not thermalization occurs has not always been easy to answer due to the finite numerical simulation time [20][21][22][23].
Closed quantum systems. The time evolution of an initial state |Ψ 0 ≡ |Ψ(t = 0) is unitary and given by the Schrödinger equation. Therefore |Ψ(t) remains a pure state for all times t. Since ensemble averages describe mixed states such a description cannot apply to a finite closed quantum system as a whole. Only a subsystem can be in or close to a thermal state with the rest of the system acting as an effective bath. Furthermore, contrary to a classical system, every quantum system has exponentially many conserved quantities, e.g. the projection operators P n = |E n E n | onto the eigenstates of a system with a discrete spectrum, |E n [6]. However, it is usually assumed that only the local conserved quantities are of relevance for thermalization. A local conserved quantity can be represented for a lattice system as Q m = j q m j where q m j is a density operator acting on lattice sites j, j + 1, · · · , j + m with m finite. Here we want to concentrate on the case of generic one-dimensional quantum systems with a small number of local conservation laws, i.e. the total energy and particle number. Thermalization in closed integrable models, where the number of local conservation laws increases linearly with the system size [24,25], has been investigated with the help of numerical simulations in recent times as well [26][27][28][29].
We consider the nonequilibrium dynamics ensuing after preparing the system in a pure state |Ψ 0 which is not an eigenstate of the Hamiltonian. Using a Lehmann representation we can write |Ψ 0 = n c n |E n where |E n are the eigenstates of the Hamiltonian H the system evolves under. Furthermore, we restrict ourselves to typical states with a macroscopic number c n = 0 [37]. We can now easily calculate the long-time mean
O = lim τ →∞ n,m 1 τ τ 0 dt e i(Em−En)t c * n c m E m |O|E n = n |c n | 2 O nn(1)
of an observable O, where we have set = 1. The second line of Eq. (1) is often called the diagonal ensemble. Here we have assumed that the system is generic, i.e., that degeneracies play no role. If the observable becomes stationary at long times its value O ∞ = lim t→∞ ψ(t)|O|ψ(t) has to be equal to the longtime mean, O ∞ ≡Ō. Note that this is only possible in the thermodynamic limit. Otherwise observables show revivals on time scales of the order of the system size. Taking the thermodynamic limit is thus essential; a finite system can never thermalize.
If a subsystem of an infinite system containing the observable O equilibrates and the value O ∞ does not depend on details of the initial state, then the remaining open question is which ensemble describes the equilibrated system. If we have two statistically independent subsystems A and B, the density matrix ρ of the whole system is given by ρ = ρ A ⊗ ρ B , and thus ln ρ = ln ρ A ⊕ln ρ B . Second, the density matrix itself should become time independent once the system has equilibrated and the von Neumann equation impliesρ = −i[H, ρ] = 0. Thus the general density matrix under consideration has to be of the form ρ = exp(− n λ n Q n )/Z where Q n are the conserved quantities of the system [5]. The partition function Z is a normalization factor such that Tr ρ = 1. We stress again that the intensive parameters λ n are not given externally but rather are Lagrange multipliers determined by the set of equations Ψ 0 |Q n |Ψ 0 = Tr {Q n ρ} .
(
If we include all projection operators into our density matrix, Q n = P n , it follows immediately from Eq. (2) that O ρ ≡ Tr{ρO} is identical to the diagonal ensemble as given in Eq. (1) [6]. Having to use infinitely many Lagrange multipliers is expected because |Ψ(t) is always a pure state and the system as a whole therefore does not thermalize, because it does not fulfill condition (II).
In this Letter we focus on the generic situation where we split our system S = A ∪ B into a bath B and a subsystem A and consider observables acting only on subsystem A. We concentrate on the following questions: How does a subsystem A without intrinsic relaxation processes equilibrate when coupled to a strongly correlated but simple and possibly non-Markovian bath B? Which statistical ensemble gives the expectation values of observables in A in the equilibrated state?
Model Hamiltonian. To investigate some aspects of the questions raised above we consider a simple model system with Hamiltonian [30]
H = −J L−1 j=1 c † j c j+1 + h.c. + γ L j=1 s † j c j + H.c. +V s L−1 j=1 s † j s j − 1/2 s † j+1 s j+1 − 1/2 .(3)
The first term describes a chain of free spinless fermions with hopping amplitude J and is the subsystem A we study the thermalization of. The 'bath' B consists of extra sites, coupled to the chain sites via a hybridization γ (second term), and we also include a nearest-neighbor interaction V s between the bath sites (third term). As initial states for the time-evolution with the Hamiltonian (3) we will consider, on the one hand, the ground state |Ψ I 0 (J 0 , γ 0 ) ≡ |Ψ(J 0 , γ 0 , V s = 0) 0 of the noninteracting model with hopping parameters J 0 and γ 0 as well as the ground state
|Ψ II 0 (J 0 , J 0 , γ 0 ) ≡ |Ψ(J 0 , J 0 , γ 0 , V s = 0) 0 of Eq. (3)
with an additional hopping J 0 between the bath sites. In order to study the time evolution under the interacting Hamiltonian we use a time-dependent density-matrix renormalization group (DMRG) algorithm [31], a method which has already been applied to study other one-dimensional models [21,22,32]. We choose open boundary conditions with a chain length of L = 51. The number of states kept in the truncated adaptive Hilbert space varies between χ = 400 and χ = 800. For a global quench as considered here it is well known that the entanglement entropy between two subsystems usually increases linearly with time. Since the maximal entanglement which can be represented in a truncated Hilbert space is limited by ln χ, there is a maximum time t max up to which we can reliably simulate the time evolution. For the cases considered here this time scale is given by Jt max ≈ 15 − 25.
Results. First, we will concentrate on the relaxation dynamics at low particle densities. As an example, we show in Fig. 1 results for a quantum quench with N = 11 particles. Shown are results for the one-point correlation functions
C j (t) ≡ Ĉ j t = Ψ(t)|c † (L+1)/2 c (L+1)/2+j |Ψ(t) .
(4) In all correlation functions oscillations with a characteristic frequency are visible. These oscillations can be understood from an equation of motion approach. We define the three time-dependent expectation values f q (t) =
Ψ(t)|c † q c q |Ψ(t) , g q (t) = Ψ(t)|s † q s q |Ψ(t) , and ρ q (t) = Ψ(t)|c † q s q |Ψ(t)
where c q = 2/(L + 1) j sin(qj)c j with allowed momenta q = nπ/(L + 1), and n = 1, . . . , L. Then, using Heisenberg's equation of motion, a Hartree-Fock decoupling of the quartic terms, and the additional assumption of an instantaneous dephasing [33], we find the following system of coupled equations
f q (t) = −ġ q (t) = 2γR q (t) ,ṙ q (t) = B q (t)R q (t) , R q (t) = −γ[f q (t) − g q (t)] − B q (t)r q (t) ,(5)with ε q = −2 cos q, r q (t) = Re ρ q (t), R q (t) = Im ρ q (t), and B q (t) = −V s − ε q + 2V s /(L + 1) cos 2 (q)g π−q (t) − sin 2 (q)g q (t) + k 1 − cos k cos q g k (t) ≈ −V s − ε q ≡ B q .
We solve the set of Eqs. (5) numerically, and the results up to intermediate times are in excellent agreement with the DMRG data, see Fig. 1. Using further approximations, we analytically find that the oscillation frequency is given by 2 and depends only weakly on q [33]. This means that the dephasing process is very slow. For longer times and short distances we see that the amplitude of the oscillations in the DMRG data is decaying faster than predicted by our equations of motion approach. Here it is important to realize that due to the Hartree-Fock decoupling the equations of motion effectively describe the time evolution under a free particle Hamiltonian. This approach therefore takes only the slow dephasing process discussed above into account. The additional decay seen in the DMRG data is due to slow relaxation processes involving energymomentum transfer between interacting particles which are not captured in our equations of motion approach.
Ω 2 q = B 2 q + (2γ) 2 with Ω 2 q→0 ≈ 1 + (2γ)
A much faster relaxation occurs if we increase the particle density with a maximum in the relaxation rate at half filing. The DMRG data for a quench in the halffilled case in Fig. 2 show indeed that the system almost completely equilibrates within the simulation time t max . Due to the particle-hole symmetry of the Hamiltonian and the initial state we have C 0 (t) ≡ 1/2 and C 2j (t) ≡ 0. For odd distances we now see, instead of long-time oscillations, an exponential damping which allows us to extrapolate the correlation functions and to read off the value for C j (t → ∞) [33]. Due to the lightcone-like spreading of the correlations [17,34], the short-range correlation functions in the middle of the chain are, for the time range shown in Fig. 2, not affected by the boundaries and are almost indistinguishable from those for an infinite system. By extrapolating our numerical data we thus approximately obtain C j (t → ∞) for a system in the thermodynamic limit.
The corresponding distribution function f q (t), shown in Fig. 3(a), has already become completely smooth after a short time, Jt = 5, and can be well fitted by a free fermion distribution function f q T = 1/(e εq/T + 1). However, the system has not fully equilibrated yet. Fig. 3(c) shows that we have two distinct relaxation regimes. In regime R I we have a relatively quick reshuffling in the distribution leading to a prethermalized state [ 6,35]. This is followed by a slow drift of the occupation numbers in regime R II which, when extrapolated in time, leads to the final distribution for the equilibrated state. While both distributions can be well fitted by f q T , the temperature should not be used as a fitting parameter but should rather be determined by energy conservation. We therefore expect that the equilibrated system is described by the ensemble, ρ = exp(−H/T )/Z, with the chemical potential µ = 0 due to particle-hole symmetry. The temperature T is determined by Eq. (2) with Q n replaced by H. The lhs of Eq. (2) is now an expectation value for a noninteracting system and can be obtained analytically. The thermal average on the rhs is calculated using a static DMRG calculation [31]. For the particular quench in Fig. 3 we find T /J = 0.54. This then allows the calculation of C j T ≡ Tr{Ĉ j e −H/T }/Z by the DMRG algorithm as shown in Fig. 2. The results for the corresponding distribution function are shown as a solid line in Fig. 3(b) and agree well with the time extrapolated values, demonstrating a local thermalization. If the additional sites are to represent an effective bath, the distribution function in the chain should become a Fermi distribution. However, as can be seen in Fig. 3(b), f q T =0.54J differs significantly from the equilibrium distribution. One obvious reason is that the effective coupling between the chain and bath in the thermal state ∼ γ s † i c i T =0.54J ≈ 0.28γ is not small. Next, we therefore consider cases where we successively reduce the coupling γ. In order to be able to still find the equilibrated state within the limited simulation time we now use as the initial state |Ψ I 0 (1, γ) which yields a much smoother initial distribution. Results for different coupling strengths γ are shown in Fig. 4. We indeed find that the momen-tum distribution in equilibrium now approaches the free fermion distribution with the temperature determined by Eq. (2). At γ = 0.2 the effective coupling between the chain and bath ∝ γ s † i c i T =0.33J ≈ 0.06γ is very small. Apart from the usual Pauli blocking there is another mechanism which explains the very weak coupling between the subsystems. Because the nearest neighbor occupation n B j n B j+1 T =0.33J = 0.1 is also small we can approximately project out all states where nearestneighbor sites in the bath are occupied. This leads to an effective density-density interaction ∝ (γ 2 /V s )n A j n B j between the subsystems, leading to a slow relaxation for small γ [33]. In this strong coupling limit, the hybridization part of the Hamiltonian Eq. (3) also gets projected
∝ γ(s † j c j + h.c.)(1 − n B j−1 )(1 − n B j+1
) explaining the small value for the effective coupling given above. Thus the interactions help to decouple the two subsystems explaining the almost perfect free Fermi distribution in the chain for γ = 0.2.
Conclusions.
We have studied thermalization in a strongly correlated model which can be viewed either as a closed quantum system or as a free fermionic chain coupled to a bath. Contrary to the common approach of using a Lindblad equation to study open quantum systems, our model has a microscopically specified bath. Therefore we can simulate the nonequilibrium dynamics of the system and bath, and directly compare the two different viewpoints. For low particle densities we have shown that an equation of motion approach on the Hartree-Fock level is sufficient to quantitatively describe the intermediate time dynamics. At this level only slow dephasing processes are captured. For the future it seems promising to use a higher order decoupling which might also capture the faster relaxation processes which we observe in the numerical simulations. While the relaxation rate Γ ∼ V s n B j n B j+1 at small interactions or low densities is too small to observe equilibration within the limited numerical simulation time we do observe thermalization at stronger interactions near half filing where Γ is larger. We note that the relaxation rate changes continuously with the microscopic parameters of the model so that the definition of a 'nonequilibrium phase transition' based on the accessible simulation time t max seems problematic [21]. Most interestingly, we find that strong interactions lead to an effective disentanglement between the subsystems and therefore increase the decoherence times. Furthermore, even an extremely simple bath where Markovian dynamics cannot be taken for granted can be sufficient to fully equilibrate a subsystem without intrinsic relaxation processes.
The authors thank S. Manmana, F.H.L. Essler, and L. Santos for discussions. J.S. and N.S. acknowledge support by the Collaborative Research Centre SFB/TR49 and the graduate school of excellence MAINZ and J.R. acknowledges support by the National Natural Sci-
Equation of motion approach
We set up a set of equations for the following three time-dependent expectation values f q (t) = Ψ(t)|c † q c q |Ψ(t) , g q (t) = Ψ(t)|s † q s q |Ψ(t) , and (6)
ρ q (t) = Ψ(t)|c † q s q |Ψ(t) .
Using Heisenberg's equation of motion,
O = i[H, O] ,(7)
with the Hamiltonian given by Eq. (1) of the main text and the standard fermionic commutation relations, one finds, with n B j = s † j s j ,
f q (t) = 2γR q (t) ,(8)g q (t) = −2γR q (t) + 2V s L + 1 i =j sin [qi] sin [qj] Ψ(t)| s † i s j − s † j s i n B j+1 (1 − δ i,j+1 ) + n B j−1 (1 − δ i,j−1 ) |Ψ(t) , iρ q (t) = i[ṙ q (t) + iṘ q (t)] = − [ε q + V s ] ρ q (t) + γ [f q (t) − g q (t)] +V s 2 L + 1 L−1 j=1 sin [qj] Ψ(t)|c † q s j n B j+1 |Ψ(t) + L j=2 sin [qj] Ψ(t)|c † q s j n B j−1 |Ψ(t) .
We have introduced the real functions r q (t) = Re ρ q (t) and R q (t) = Im ρ q (t). This set of coupled equations is exact.
To solve this set of equations we apply two approximations. Firstly a Hartree-Fock decoupling is used, i.e. we apply Wick's theorem so that we have only two point correlation functions present, for example
c † q s j n B j−1 = c † q s j n B j−1 − c † q s j−1 s † j−1 s j . (9)
One should note that, amongst other effects, this approximation leavesġ
q (t) = −2γR q (t) = −ḟ q (t)(10)
and hence g q (t) + f q (t) ≡ N q becomes independent of time. Therefore, by performing the Hartree-Fock decoupling, we lose all relaxation processes which can reshuffle the occupation of the momenta. Nonetheless, as demonstrated in Fig. 1 of the main text, for small densities this approximation is sufficient to get good quantitative agreement with DMRG calculations for intermediate times. The second approximation is "instantaneous dephasing", which means that all off-diagonal elements of Ψ(t)|c † q c k |Ψ(t) , etc., are taken to 'instantaneously dephase' and we keep only diagonal terms. For a periodic system this would be guaranteed by translational invariance, here it amounts to disregarding finite size effects from the boundaries. This means that all two point correlation functions are diagonal in momentum space. Following this we havė with
f q (t) = −ġ q (t) = 2γR q (t) ,ṙ q (t) = B q (t)R q (t) , R q (t) = −γ[f q (t) − g q (t)] − B q (t)r q (t) ,(11)B q (t) = 2V s L + 1 cos 2 [q] g π−q (t) − sin 2 [q] g q (t) (12) + k (1 − cos [k] cos [q]) g k (t) − V s − ε q .
The coupled first order differential equations, given by Eq. (11), can then be solved iteratively. The first line in Eq. (12) is a O(1/L) finite size correction. The oscillation frequency of R q (t) is the same as that of f q (t) and g q (t) and can be extracted from these equations analytically. We can write a second order differential equation for R q (t):
R q (t) + 4γ 2 + B 2 q (t) R q (t) = −Ḃ q (t)r q (t) . (13)
One finds, with a weakly time dependent bath occupation, such thatḂ q ≈ 0,
R q (t) + 4γ 2 + B 2 q (t) ≡Ω 2 q R q (t) = 0 .(14)
For small particle densities in the bath we can approxi-
mate B q (t) ≈ −V s − ε q ≡ B q which
gives
Ω q ≈ (ε q + V s ) 2 + 4γ 2 .(15)
The Hartree-Fock decoupled solutions oscillate with the frequency Ω q which is only weakly q-dependent, see inset of Fig. 5. This explains why no dephasing effects are seen in the Hartree-Fock solution on the timescales that we consider, see Fig. 1 in the main text. The approximation B q (t) ≈ B q is ensured in our case by the small bath occupation. For example, in the initial state |Ψ I 0 (1, 1) , at a density of 0.11 particles per site, we have
1 L + 1 q g q (t = 0) = 0.0344 |B q |(16)
and 1 L + 1 q g q (t = 0) cos q = 0.0314 |B q | .
Contrary to the Hartree-Fock results, the DMRG data show an additional relaxation, see Fig. 1 of the main text. A signature of the beginning of this relaxation can also be seen in the long time mean of the distribution function, f q , see Fig. 5 which shows a redistribution of the occupation of quasi-momenta around the Fermi momentum. The low density relaxation rate Γ ∼ V s j n B j n B j+1 /L, however, is small so that we can not see full thermalization within the DMRG simulation time t max .
Particle-hole symmetry
We define H as the Hamiltonian given by Eq. (1) in the main text, but with hopping in the bath included:
H = H − J L−1 j=1 s † j s j+1 + h.c. .(18)
H , and therefore also all half-filled groundstates, has particle-hole symmetry. The Hamiltonian H is invariant under the mapping T ph :
c j → (−1) j c † j c † j → (−1) j c j s j → (−1) j+1 s † j s † j → (−1) j+1 s j(19)
which exactly describes particle-hole inversion.
In our analysis we have considered two correlation functions. Firstly the real space two point correlation function C j (t), defined by for time evolution with H given by Eq. (1) in the main text. Under the mapping given by Eq. (19) H → H and |Ψ 0 → |Ψ 0 and one finds that C j=0 (t) = 1/2, and C 2j (t) = 0 for non-zero j. Secondly we analyzed the momentum distribution in the chain, f q (t). For the initial states under consideration, and therefore for all times, this can be shown to satisfy f q (t) + f π−q (t) = 1 by using the same mapping.
C j (t) = Ψ 0 |e iHt c † (L+1)/2 c (L+1)/2+j e −iHt |Ψ 0 ,(20)
Fitting and extrapolation
In this section we explain the fitting and extrapolation techniques we used to find our long time data at halffilling. For quench I we extrapolated in momentum space. The long time limit for f q and γ = 1 and γ = 0.6 can be found directly by time averaging the data or by fitting to
f q (Jt 1) f q (t → ∞) + ae −Γt cos [Ωt − φ] .(21)
This functional form takes into account exponential relaxation and a simple oscillation, see Fig. 6(a). Note that the fitting is only performed right of the solid line at Jt ≈ 5, to ignore the effect of the short time dynamics. A Fourier analysis confirms that there is one dominant frequency in the dynamics of f q (t). However, in contrast to the low density case this frequency is momentum dependent. In particular for the case γ = 1 shown in Fig. 6(a), Ω(q = 11π/(L + 1)) ≈ 2.7 and Ω(q = 21π/(L + 1)) ≈ 1.9. The relaxation rate, Γ(q = 11π/(L + 1)) ≈ 6.7 · 10 −2 and Γ(q = 21π/(L + 1)) ≈ 2.7 · 10 −2 , is of the same order of magnitude for all momenta hinting at one dominant relaxation process. Let us stress that in this case the result for f q (t → ∞) depends only weakly on the extrapolation procedure used, i.e. time averaging or fitting with different fit intervals, with a variation in f q (t → ∞) which is about the symbol size used in the corresponding plots in the main text. For γ = 0.2, see Fig. 6(b), such a simple fitting function will no longer work due to the presence of various oscillation frequencies. A Fourier analysis confirms that there is more than one oscillation frequency involved, but the times are not sufficient to extract how many there are and what their magnitudes may be. Instead we trace out the overall trend by fitting to
f q (Jt 1) f q (t → ∞) + ae −Γt .(22)
Γ captures the gradual drift of the oscillations which can also be seen by using running averages. This procedure is robust when choosing a variety of different time regions over which to perform the fitting, and gives again errors smaller than the symbol sizes used in the plots of the main text.
For quench II we extrapolate in real space. Fig. 7 shows the fitting for C 1 (t) and C 3 (t). We fit the dynamics with C j (Jt 1) C j (t → ∞) + ae −Γt (cos [Ωt − φ] + b) .
As for quench I, only times right of the vertical lines in Fig. 7 are used for fitting.
Initial state independence
After relaxation the equilibrium state should depend only on the energy in the system. As example, we take two initial states, |Ψ I 0 and |Ψ II 0 , constructed to have the same energy after a quench
E = Ψ II 0 |H|Ψ II 0 = Ψ I 0 |H|Ψ I 0 .(24)
The two different initial states are time evolved with the same Hamiltonian, with J = 1, γ = 1, and V s = 1. We use the initial states |Ψ II 0 (1, 0.6, 1) (same as in Fig. 3 of the main text) and |Ψ I 0 (4.48, 0.8) with energy E = −51.19. Fig. 8 demonstrates that both states evolve towards the same equilibrium state, well described by the grand canonical ensemble Tr{f q e −H/T }/Z with the temperature T /J = 0.54 fixed by E = Tr{He −H/T }/Z and µ = 0 due to particle-hole symmetry.
FIG. 1 :
1(Color online) Cj(t) from Eq. (4) for j = 0, . . . , 5 (top to bottom) for a quench with initial state |Ψ I 0 (1, 1) , and Hamiltonian H with J = 1, γ = 1, and Vs = 1 at low densities. DMRG results (symbols) are compared to the solution of Eq. (5) (lines).
online) DMRG data (symbols) for Cj(t) at half filing for a quench with initial state |Ψ II 0 (1, 0.6, 1) , and Hamiltonian H with J = 1, γ = 1, and Vs = 1. The lines are the thermal expectation values Cj T .FIG. 3: (Color online) (a) fq(t = 0) for |Ψ II 0 (1, 0.6, 1) (circles), fq(t = 5) (triangles) and Fermi function fit fq T =0.7J , and the extrapolated distribution fq(t → ∞) (diamonds) with a fit fq T =J . (b) fq(t → ∞) (diamonds) compared to the thermal average Tr{fqe −H/T }/Z (solid line) and fq T (dashed line) where T /J = 0.54 is fixed by Eq. (2). (c) fq(t) for q = 13π/52 (solid line) and q = 16π/52 (dashed line).
FIG. 4 :
4(Color online) The initial distribution (circles), fq(t → ∞) from DMRG (diamonds), the thermal distribution Tr{fqe −H/T }/Z (solid line), and the free fermion distribution fq T (dashed line) with: (a) γ = 1, T /J = 0.19, (b) γ = 0.6, T /J = 0.18, and (c) γ = 0.2, T /J = 0.33.
FIG. 5 :
5Main: A quench with initial state |Ψ I 0 (1, 1) , and Hamiltonian H with J = 1, γ = 1, and Vs = 1 at low densities. Shown is fq(t = 0) (circles),fq within Hartree-Fock (squares), andfq obtained by DMRG (diamonds). Inset: Ωq obtained by DMRG (symbols) and within Hartree-Fock (line).
FIG. 6 :FIG. 7 :
67Momentum space extrapolation for a quench with initial state |Ψ I 0 (1, γ) , and Hamiltonian H with J = 1 and Vs = 1 where (a) γ = 1, and (b) γ = 0.2. Shown are the momenta q = 11π/(L + 1) (upper curves) and q = 21π/(L + 1) (lower curves). The dynamics (symbols) are compared with the fit (dashed line), and the thermal average at the appropriate temperature (solid line). Fitting is performed for times greater than the solid vertical line. The arrows on the right hand side show fq(t → ∞) from (a) Eq. (21), and (b) Eq. (22). Real space extrapolation for a quench with initial state |Ψ II 0 (1, 0.6, 1) , and Hamiltonian H with J = 1, γ = 1, and Vs = 1. The DMRG data (symbols) are compared with the fit (dashed line), and the thermal average at the appropriate temperature (solid line). Plotted is Cj(t) with (a) j = 1, and (b) j = 3. Fitting is performed for times greater than the solid vertical line. The arrows on the right hand side show Cj(t → ∞) from Eq. (23).
FIG. 8 :
8Distribution function fq in the chain for a system of size L = 51. fq(t = 0) is shown for the initial state |Ψ II 0 (1, 0.6, 1) (circles), and the initial state |Ψ I 0 (4.48, 0.8) (squares). In both cases we time evolve with the same Hamiltonian H with J = 1, γ = 1, and Vs = 1. fq(t → ∞) after time evolving |Ψ II 0 (diamonds) and |Ψ II 0 (triangles) are compared with the thermal average Tr{fqe −H/T }/Z (solid line) where T /J = 0.54 is fixed by the initial energy, see text. Alternate q points are plotted for the two quenches to aid clarity.
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Here we want to also include the case of integrable models in one dimension where this number has to increase linearly with system size. Generically this number is finiteGenerically this number is finite. Here we want to also include the case of integrable models in one dimension where this number has to increase linearly with system size.
If this condition is not fulfilled we cannot expect, in general, that time averages become independent of the microscopic details of the initial state. If this condition is not fulfilled we cannot expect, in gen- eral, that time averages become independent of the mi- croscopic details of the initial state.
| []
|
[
"Circuit Complexity and Decompositions of Global Constraints",
"Circuit Complexity and Decompositions of Global Constraints"
]
| [
"Christian Bessiere [email protected] \nNICTA and UNSW Sydney\nNICTA and UNSW Sydney\nLIRMM\nCNRS Montpellier\nNICTA Sydney\n\n",
"George Katsirelos \nNICTA and UNSW Sydney\nNICTA and UNSW Sydney\nLIRMM\nCNRS Montpellier\nNICTA Sydney\n\n",
"Nina Narodytska [email protected] \nNICTA and UNSW Sydney\nNICTA and UNSW Sydney\nLIRMM\nCNRS Montpellier\nNICTA Sydney\n\n",
"Toby Walsh [email protected] \nNICTA and UNSW Sydney\nNICTA and UNSW Sydney\nLIRMM\nCNRS Montpellier\nNICTA Sydney\n\n"
]
| [
"NICTA and UNSW Sydney\nNICTA and UNSW Sydney\nLIRMM\nCNRS Montpellier\nNICTA Sydney\n",
"NICTA and UNSW Sydney\nNICTA and UNSW Sydney\nLIRMM\nCNRS Montpellier\nNICTA Sydney\n",
"NICTA and UNSW Sydney\nNICTA and UNSW Sydney\nLIRMM\nCNRS Montpellier\nNICTA Sydney\n",
"NICTA and UNSW Sydney\nNICTA and UNSW Sydney\nLIRMM\nCNRS Montpellier\nNICTA Sydney\n"
]
| []
| We show that tools from circuit complexity can be used to study decompositions of global constraints. In particular, we study decompositions of global constraints into conjunctive normal form with the property that unit propagation on the decomposition enforces the same level of consistency as a specialized propagation algorithm. We prove that a constraint propagator has a a polynomial size decomposition if and only if it can be computed by a polynomial size monotone Boolean circuit. Lower bounds on the size of monotone Boolean circuits thus translate to lower bounds on the size of decompositions of global constraints. For instance, we prove that there is no polynomial sized decomposition of the domain consistency propagator for the ALLDIFFERENT constraint. | null | [
"https://arxiv.org/pdf/0905.3757v1.pdf"
]
| 10,604,876 | 0905.3757 | 8ae8533b051d66ba9b642759763c822413bae473 |
Circuit Complexity and Decompositions of Global Constraints
Christian Bessiere [email protected]
NICTA and UNSW Sydney
NICTA and UNSW Sydney
LIRMM
CNRS Montpellier
NICTA Sydney
George Katsirelos
NICTA and UNSW Sydney
NICTA and UNSW Sydney
LIRMM
CNRS Montpellier
NICTA Sydney
Nina Narodytska [email protected]
NICTA and UNSW Sydney
NICTA and UNSW Sydney
LIRMM
CNRS Montpellier
NICTA Sydney
Toby Walsh [email protected]
NICTA and UNSW Sydney
NICTA and UNSW Sydney
LIRMM
CNRS Montpellier
NICTA Sydney
Circuit Complexity and Decompositions of Global Constraints
We show that tools from circuit complexity can be used to study decompositions of global constraints. In particular, we study decompositions of global constraints into conjunctive normal form with the property that unit propagation on the decomposition enforces the same level of consistency as a specialized propagation algorithm. We prove that a constraint propagator has a a polynomial size decomposition if and only if it can be computed by a polynomial size monotone Boolean circuit. Lower bounds on the size of monotone Boolean circuits thus translate to lower bounds on the size of decompositions of global constraints. For instance, we prove that there is no polynomial sized decomposition of the domain consistency propagator for the ALLDIFFERENT constraint.
Introduction
Global constraints are a vital component of constraint toolkits. They permit users to model common patterns and to exploit efficient propagation algorithms to reason about these patterns. A promising mechanism to implement such global constraints is to develop decompositions into sets of primitive constraints that do not hinder propagation. For example, Bacchus has shown how to decompose global propagators for the generic TABLE constraint, as well as for the REGULAR, AMONG and SEQUENCE constraints into conjunctive normal form (CNF) [Bacchus, 2007]. Such decompositions can then be used in SAT solvers, allowing us to profit from techniques like clause learning and backjumping. In recent years, many other decompositions have been proposed for a wide range of global constraints including REGULAR and GRAMMAR [Quimper and Walsh, 2006;, SEQUENCE [Brand et al., 2007], PRECEDENCE , CARDPATH and SLIDE [Bessiere et al., 2008]. Many other global constraints can be decomposed using ROOTS and RANGE, which can themselves be propagated effectively using some simple decompositions [Bessiere et al., 2005;Bessiere et al., 2006a;Bessiere et al., 2006b]. Finally, many global constraints specified by automata can be decomposed into signature and transition constraints without hindering propagation [Beldiceanu et al., 2005].
This raises the important open question of which global constraints can be effectively propagated using simple encodings [Bessiere and Van Hentenryck, 2003]. We show that circuit complexity can be used to resolve this question. Our main result is that there is a polynomial sized decomposition of a constraint propagator into CNF if and only if the propagator can be computed by a polynomial size monotone Boolean circuit. It follows therefore that bounds on the size of monotone Boolean circuits give bounds on the size of decompositions of global constraints into CNF. For instance, a super-polynomial lower bound on the size of a Boolean circuit for perfect matching in a bipartite graph gives a super-polynomial lower bound on the size of a CNF decomposition of the domain consistency propagator for the ALLDIFFERENT constraint. Our results directly extend to decompositions into CSP constraints of bounded arity with domains given in extension since such decompositions can be translated into clauses of polynomial size [Bessiere et al., 2003]. The tools of circuit complexity are thus useful in understanding the limits of what we can achieve with decompositions.
2 Background CSP. A constraint satisfaction problem (CSP) P consists of a set of variables X, each of which has a finite domain D(X i ), and a set of constraints C. An assignment to a variable X i is a mapping of X i to a value j ∈ D(X i ), called literal, and written X i = j. We write D(X) (resp. D (X)) for sets of
literals {X i = j | X i ∈ X ∧ j ∈ D(X i )} (resp. {X i = j | X i ∈ X∧j ∈ D (X i )})
and P(D) for the set of all such sets. An assignment to a set of variables X is a set that contains exactly one assignment to each variable in X. A constraint C ∈ C has a scope, denoted scope(C) ⊆ X and allows a subset of the possible assignments to the variables scope(C), called solutions of C. A solution of P is an assignment of one value to each variable such that all constraints are satisfied.
A propagator for a constraint C is an algorithm which takes as input the domains of the variables in scope(C) and re-turns restrictions of these domains. Following [Schulte and Stuckey, 2004], we can formally define a propagation algorithm as a function:
Definition 1 (Propagator) A propagator f for a constraint C is a polynomial time computable function f : P(D) → P(D), such that f is monotone, i.e., D (X) ⊆ D(X) =⇒ f (D (X)) ⊆ f (D(X)), contracting, i.e., f (D(X)) ⊆ D(X), and idempotent, i.e., f (f (D(X))) = f (D(X)). If a literal X i = j is in D(X) \ f (D(X)) then X i = j does not belong to any solution of C given D(X). If f detects that C has no solutions under D(X) then f (D(X)) = ∅.
A propagator detects dis-entailment if when no possible assignment is a solution of C then f (D(X)) = ∅. A propagator enforces domain consistency (DC) when X i = j ∈ f (D(X)) implies that there exists a solution of C that contains X i = j.
We also define the consistency checker for a constraint C as a function that returns 0 when it detects that no possible assignment is a solution of the constraint and 1 otherwise, rather than restricting domains.
Definition 2 (Consistency checker) A consistency checker f for a constraint C is a polynomial time computable function f :
P(D) → {0, 1} such that f is monotone, i.e., D (X) ⊆ D(X) =⇒ f (D (X)) ≤ f (D(X)
). If f (D(X)) = 0 then no possible assignment under D(X) is a solution of C.
We can obtain a polynomial time consistency checker f C of a constraint C from a polynomial time propagator f P for C and vice versa [Bessiere et al., 2007]. Given the propagator f P , the corresponding consistency checker f C is defined as:
f C (D(X)) = 0 f P (D(X)) = ∅ 1 otherwise(1)
Conversely, given f C , the propagator f P is
f P (D(X)) = D(X) \ {X i = j | f C (D(X)| Xi=j ) = 0} (2) where D(X)| Xi=j = D(X) \ {X i = k|k = j}.
SAT. The Boolean satisfiability problem (SAT) is a special case of the CSP where variables are Boolean. For each Boolean variable x i there exist two literals x i and x i . Constraints in conjunctive normal form (CNF) are disjunctions of literals, called clauses and sometimes written simply as tuples of literals. Unit propagation forces a literal to TRUE if it appears in a clause where all other literals are FALSE and continues until a fix-point is reached. If all literals in a clause are made FALSE, we say that the empty clause is produced. A stronger form of inference is the failed literal test [Freeman, 1995]. For each literal l of an unset variable x, the failed literal test sets l to TRUE, performs unit propagation, checks whether the empty clause was produced and retracts l and its consequences. If the empty clause was produced, l is set to FALSE. A CSP instance can be encoded as a SAT instance. The most widely used mapping of CSP variables to Boolean variables is the direct encoding. Each CSP variable X i with domain D(X i ) is encoded in SAT as a set of propositions x i,j , X i ∈ X, j ∈ D(X i ) such that X i = j ⇐⇒ x i,j . The property that each CSP variable has at most one value is enforced by the set of clauses (x i,j , x i,k ) for all k ∈ D(X i ), k = j and the property that each CSP variable has at least one value is enforced by the set of clauses j∈D(Xi) x i,j . We denote this propositional representation of D(X) as D sat (X).
Note that the propositional representation D sat (X) represents the current state of the domains D(X) during search. This means that when the domains change, we need to be able to make the corresponding change in the direct encoding. Consequently, the fact (X i = j) ∈ D(X) is represented by x i,j being unset, rather than TRUE. When the value X i = j is pruned, then x i,j is set to FALSE. Only when X i = j is the only possible assignment for X i is x i,j set to TRUE. This means that the same domain can be represented by different partial instantiations of the direct encoding. For example, given the CSP variable X 1 with initial domain {1, 2, 3}, the instantiation D sat ({X 1 }) = {x 1,2 , x 1,3 } (with x 1,1 unset) corresponds to the same domain as
D sat ({X 1 }) = {x 1,1 , x 1,2 , x 1,3 }, which is D({X 1 }) = {X 1 = 1}.
Boolean Circuits. A Boolean circuit S is a directed acyclic graph (DAG). Each source vertex of the DAG is an input gate and the unique sink of the DAG is the output gate. Each noninput vertex is labelled with a logical connective, such as and (∧), or (∨) and not (¬). An input b to the circuit is an assignment of a value 0 or 1 to each input gate. 1 The value of a non-input gate is computed by applying the connective that it is labelled with to the values of its ancestor gates. The value of the circuit S(b) is the value of its output gate.
Any polynomial time decision algorithm can be encoded as a Boolean circuit of polynomial size for a fixed length input [Papadimitriou and Steiglitz, 1982].
In this paper, we will use a restriction of Boolean circuits to ∧-gates and ∨-gates, called monotone circuits. The family of functions that are computable by monotone circuits is exactly all the monotone Boolean functions. Note that there exist families of polynomial time computable monotone Boolean functions such that the smallest monotone circuit that computes them is super-polynomial in size [Razborov, 1985].
Definition 3 (Monotone Boolean function)
A Boolean function f is monotone iff f (b) = 0 implies f (b ) = 0 for all b ≤ b, where ≤ is the pairwise vector comparison, i.e., b i ≤ b i for all i.
A consistency checker f C , previously defined as a monotone function over sets, can also be formalised as a monotone Boolean function whose input is the characteristic function of the set D(X). Literals X i = j are mapped to arguments b i,j of the function, with b i,j = 1 iff X i = j ∈ D(X). We use D b (X) to denote the setting of the b i,j inputs for a given set of domains D(X).
Properties of CNF decompositions
In this section, we define formally a CNF decomposition of a propagator and of a consistency checker. As with propagators and consistency checkers [Bessiere et al., 2007], we show that there exists a polynomial time conversion between the CNF decompositions of a propagator and of the corresponding consistency checker. Definition 4 (CNF Decomposition of a propagator) A CNF decomposition of a propagation algorithm f P is a formula in CNF C P over variables x ∪ y such that
• The input variables x are the propositional representation D sat (X) of D(X) and y is a set of auxiliary variables whose size is polynomial in |x|. • x i,j is set to FALSE by unit propagation if and only if X i = j / ∈ f P (D(X)). • Unit propagation on C P produces the empty clause when f P (D(X)) = ∅. Example 1 To illustrate Definition 4, consider a TABLE constraint over the variables X 1 , X 2 with D(X 1 ) = D(X 2 ) = {a, b} and the satisfying assignments: { a, a , b, b a, b }. [Bacchus, 2007] decomposes such a TABLE constraint into CNF using the following set of clauses:
x 1a ⇒ y 1 ∨ y 3 x 2a ⇒ y 1 y 1 ⇒ x 1a y 1 ⇒ x 2a x 1b ⇒ y 2 x 2b ⇒ y 2 ∨ y 3 y 2 ⇒ x 1b y 2 ⇒ x 2b y 3 ⇒ x 1a y 3 ⇒ x 2b y 1 ∨ y 2 ∨ y 3 where x = {x i,j }, i ∈ {1, 2}, j ∈ {a, b} is the propo- sitional representation D sat (X) of D(X) and y = {y i }, i ∈ {1
, 2, 3} are auxiliary variables that correspond to satisfying tuples. Note that we have extended Bacchus's encoding with the clause (y 1 ∨ y 2 ∨ y 3 ) to detect failure. Suppose the value a is removed from the domain of X 1 . The assignment x 1a = FALSE forces the variable y 1 to FALSE, which in turn causes the variable x 2a to FALSE, removing the value a from the domain of X 2 as well.
In example 1, we have decomposed a constraint into clauses by introducing variables. In general, an encoding might be exponentially bigger if auxiliary variables are not used (e.g., the parity function [Darwiche and Marquis, 2002]). Definition 5 (CNF Decomposition of a consistency checker) A CNF decomposition of a consistency checker f C is a CNF C C over variables x ∪ y ∪ {z} such that
• The input variables x are the propositional representation D sat (X) of D(X) and y is a set of auxiliary variables whose size is polynomial in |x|. The variable z is the output variable. • Unit propagation on C C never forces any variable from
x or generates the empty clause if no variable in y is set externally to C C , i.e., every variable y ∈ y is either unset or forced by a clause in C C . • z is set to FALSE by unit propagation if and only if f C (D(X)) = 0. Example 2 Consider the TABLE constraint from Example 1. We construct a CNF decomposition of a consistency checker using the CNF decomposition of a propagator. The clauses that cause pruning of input variables domains are removed and the last clause is augmented with the output variable z to avoid generation of the empty clause in the case of failure:
y 1 ⇒ x 1a y 1 ⇒ x 2a y 2 ⇒ x 1b y 2 ⇒ x 2b y 3 ⇒ x 1a y 3 ⇒ x 2b y 1 ∧ y 2 ∧ y 3 ⇒ z
In this case, if the value a is removed from the domain of X 1 , unit propagation will not deduce that a has to be removed from the domain of X 2 . Consider instead the case when the values a and b are removed from the domains of X 1 and X 2 , respectively. The literals x 1a = FALSE and x 2b = FALSE force the auxiliary variables y 1 , y 2 and y 3 to be FALSE. Therefore, the output variable z is forced to FALSE, signalling that the TABLE constraint does not have a solution under D(X).
In example 2, we transformed the propagator of example 1 into a consistency checker in an ad-hoc manner. The next theorem shows that this can be done in a generic way. We give a polynomial transformation of CNF decompositions of a propagator into consistency checkers This mirrors the results of [Bessiere et al., 2007] for CNF decompositions.
Theorem 1 There exists a polynomial time and space conversion between the CNF decomposition of a propagator f P and that of the corresponding consistency checker f C .
Proof: (→) We construct C C as a transformation of C P such that the output variable z of C C is FALSE iff unit propagation on C P produces the empty clause.
Let the set of clauses of C P be c 1 . . . c m . For each variable p ∈ x∪y, we introduce 2 variables p t and p f in C C so that p t and p f are true if p is forced to TRUE or FALSE, respectively:
p =⇒ p t p =⇒ p f(3)
Then, we simulate unit propagation for each clause c k by replacing it with 3 implications 2 that contain the variables p t and p f rather than p. For example, to simulate unit propagation for the clause c 1 = (p, q, r), we replace it with
p f ∧ q f =⇒ r f p f ∧ r t =⇒ q t q f ∧ r t =⇒ p t (4)
Unit propagation on (4) can never derive the empty clause, because the true and false values of p are encoded in different variables p t and p f , which may be true simultaneously. When this happens, unit propagation on C P would generate the empty clause, therefore we must set the output variable z to FALSE, using the following clauses:
p t ∧ p f =⇒ z(5)
The union of the clauses (3), (4) and (5) is a CNF decomposition of f C with size O(|x∪y|+|C P |) = O(|C P |), therefore the transformation is polynomial.
(←) We outline the proof here. We replicate the equation (2) by simulating the failed literal test on C C ∪{(z)}. For each literal x i,j we create a copy of C C , denoted by C C | xi,j , in which all literals x i,k , k = j are FALSE. We use C C | xi,j to record the results of unit propagation when X i = j. When unit propagation sets the output variable z xi,j of the copy C C | xi,j to FALSE then the propositional literal x i,j is made FALSE by the additional clause (z xi,j =⇒ x i,j ).
The decomposition C P is then the union of the copies of C C and the clauses (z xi,j =⇒ x i,j ):
C P = xi,j ∈x (C C | xi,j ∪ (z xi,j , x i,j ))(6)
The size of C P is O(|x| · |C C |), therefore the transformation is polynomial. 2 Using the encoding of theorem 1, a CNF decomposition of a consistency checker that detects dis-entailment can be made into a propagator that enforces domain consistency. As an example, consider the CNF decomposition of a propagator that detects dis-entailment for the SEQUENCE constraint, proposed in [Bacchus, 2007]. The size of this decomposition is O(n 2 ), where n is the number of variables in the SEQUENCE constraint. These variables are binary, hence the transformation of theorem 1 yields a decomposition of a DC propagator with size O(n 3 ). This is also the complexity of the DC propagator proposed in [van Hoeve et al., 2006].
Since all definitions of CNF decompositions that we introduced in this section are polynomially equivalent, in the remainder of this paper we only prove results for CNF decompositions of consistency checkers.
Equivalence to monotone circuits
In this section, we show our main result, which establishes a connection between CNF decompositions of constraints and circuit complexity.
Theorem 2 A consistency checker f C can be decomposed to a CNF of polynomial size if and only if it can be computed by a monotone circuit of polynomial size.
The proof of theorem 2 is constructive. We will first show the reverse direction, using the Tseitin encoding [Tseitin, 1983] of a monotone circuit.
Definition 6 (Tseitin encoding of a Boolean circuit) The Tseitin encoding of a circuit S into clausal form has one propositional variable for each input of S and for each gate of S. W.l.o.g, we assume all gates have fan-in 2. For each ∧-gate g with inputs x 1 , x 2 , the Tseitin encoding contains the clauses (x 1 , g), (x 2 , g), (x 1 , x 2 , g) and for each ∨-gate it contains the clauses (x 1 , g), (x 2 , g), (x 1 , x 2 , g). Given any complete instantiation of the input variables, unit propagation on the Tseitin encoding sets the variable corresponding to the output gate of S to TRUE if the circuit computes 1 and to FALSE otherwise.
Suppose that a consistency checker f C can be encoded into a monotone circuit S C of polynomial size. The Tseitin encoding of S C turns out to be a CNF decomposition of f C . This is a direct consequence of the following lemma.
Lemma 1 Let S C be a monotone circuit and C C be its Tseitin encoding. Let I be a partial instantiation of the input variables x of C C and b be the corresponding input to S C , where b i = 0 iff x i ∈ I. Then, unit propagation on C C with I forces the output variable z to FALSE if and only if S C (b) = 0.
Proof: (→) This follows from the correctness of the Tseitin encoding.
(←). Suppose that S C (b) = 0, but the output variable z is not forced to FALSE by unit propagation under I. Consider an instantiation I of the input variables of C C , which is the same as I with unset variables fixed to TRUE. Let y ∈ y ∪ {z} be an auxiliary variable that is unset under I. All such variables correspond to a gate in S C . Since C C is an encoding of the monotone circuit S C , y will be set to TRUE under I . This means that the output variable z is also set to TRUE. By the correctness of the Tseitin encoding, S C (b) = 1, a contradiction. 2
Corollary 1 Let S C be a monotone circuit and C C be its Tseitin encoding. Let I be a partial instantiation of the input variables x of C C . Then, unit propagation on C C with I forces the output variable z to FALSE if and only if S C (b) = 0, for all b where b is the input to S C that corresponds to any extension of I to a complete instantiation.
Proof: This follows from lemma 1 and the fact that S C is a monotone circuit. 2
Interestingly, lemma 1 cannot be generalised to nonmonotone Boolean circuits. The next example shows that there exists a non-monotone Boolean circuit S that computes a monotone function, and a partial instantiation I with b the corresponding input to S, such that S(b) = 0 but unit propagation on the Tseitin encoding of S under the instantiation I does not set the output variable to FALSE.
Figure 1
A circuit whose Tseitin encoding is incomplete. Example 3 Consider the non-monotone circuit S shown in figure 1. Note that S computes a monotone function. The Tseitin encoding of S introduces three Boolean variables g 1 , g 2 and g 3 for the gates OR 1 , OR 2 and AN D 3 , respectively, and the clauses (x 1 , g 1 ), (x 2 , g 1 ), (g 1 , x 1 , x 2 ), (x 1 , g 2 ), (x 2 , g 2 ), (g 2 , x 1 , x 2 ), (g 3 , g 1 ), (g 3 , g 2 ), (g 1 , g 2 , g 3 ).
Now suppose that I = {x 1 }. Then, b = {x 1 = 0, x 2 = 1} and S(b) = 0. Since S computes a monotone function, all possible extensions of x evaluate to 0. But in the Tseitin encoding, setting x 1 to FALSE does not make any clauses unit, therefore unit propagation does not set g 3 to FALSE. 2
We now show the forward direction of theorem 2: every CNF decomposition C C of a consistency checker f C can be converted to a monotone circuit that computes f C with at most a polynomial increase in size.
This transformation exploits two properties of CNF decompositions, namely, that only positive literals of input variables appear in C C , and that unit propagation only makes auxiliary variables FALSE. We show the former property in lemma 2 and the latter in lemma 3.
Lemma 2 Let C C be the CNF decomposition of a consistency checker f C . There exists a polynomial size CNF decomposition C C of f C such that negative literals of the input variables do not appear in any clause in C C .
Proof: We construct C C by removing from C C all clauses that contain a negative literal of an input variable. We show by contradiction that unit propagation on C C and C C produces identical results for the output variable z.
Let I be a partial instantiation of the input variables such that unit propagation on C C under I sets z to FALSE but leaves z unset on C C . Since unit propagation on C C and C C produces different results, at least one of the removed clauses becomes unit under I in C C . By definition, C C never forces any literal of an input variable, so for any removed clause to become unit, all the literals of input variables in it have to be FALSE. Since at least one of these literals is negative, at least one input variable has to be set to TRUE in I.
We construct another partial instantiation I from I by setting the same literals to FALSE as I and leaving the rest unset, i.e., I = {x i,j |x i,j ∈ I}. The partial instantiations I and I represent the same domains D(X), because the mapping from partial instantiation to domain depends only on the literals that are FALSE. By this and the fact that C C is a decomposition of f C , unit propagation on C C under I forces the output variable z to the same value as under I, FALSE. Consider the result of unit propagation on C C under I . Recall that by definition C C does not modify input variables and I does not have literal set to TRUE by construction. Hence, none of the clauses that we remove from C C to get C C can become unit after performing UP on C C under I . Hence, unit propagation in C C under I sets z to FALSE as in C C . On the other hand, I sets a superset of the literals that I sets, so unit propagation on C C under I also sets z to FALSE, a contradiction, since we assumed that C C leaves z unset under I. 2
In practice, a CNF decomposition of a consistency checker may not be self contained and may depend on the existence of clauses in the direct encoding of variable domains. In this case, we cannot just remove clauses that contain negative literals of input variables, as lemma 2 suggests. However, using the clauses of the direct encoding, we can substitute negative literals with the disjunction of positive literals. For instance, consider a variable X 2 with the domain {1, 2, 3} and a clause (x 1,1 , x 2,2 , y) in C C . The literal x 2,2 can make this clause unit. The direct encoding of D(X 2 ) includes a clause (x 2,1 , x 2,2 , x 2,3 ). Note that the literal x 2,2 is TRUE if and only if literals x 2,1 and x 2,3 are FALSE. Therefore, the literal x 2,2 can be replaced with the disjunction (x 2,1 , x 2,3 ) and the clause (x 1,1 , x 2,2 , y) is transformed to the clause (x 1,1 , x 2,1 , x 2,3 , y).
The next step is to show that we can transform a CNF decomposition so that each auxiliary variable is unset or FALSE for all inputs that make the output variable FALSE. The transformation is a renaming of the auxiliary variables. Lemma 3 describes the property that allows this transformation.
Lemma 3 Let C C be a CNF decomposition of a consistency checker f C over the variables x ∪ y ∪ {z}, I 1 = D sat 1 (X), I 2 = D sat 2 (X) be the propositional representa-tions of any two domain settings such that unit propagation on C C forces z to FALSE under both I 1 and I 2 . For any variable y ∈ y, if y is forced to FALSE (TRUE) by unit propagation under I 1 then it is not forced to TRUE (FALSE) by unit propagation under I 2 .
Proof: Let a variable y be forced to TRUE by unit propagation under I 1 and to FALSE under I 2 , but z is FALSE under both I 1 and I 2 . Consider the partial instantiation I such that if a variable x ∈ x is FALSE in either I 1 or I 2 , it is also FALSE in I, otherwise it is unset. Since I fixes a superset of the literals that are fixed in either I 1 or I 2 , all clauses that became unit by either I 1 or I 2 will also be unit in I. Therefore, unit propagation under I will force at least the union of the sets of literals forced by I 1 and I 2 . This means that unit propagation under I will make both y and y TRUE, which generates the empty clause. This is a contradiction, as C C can never produce the empty clause. 2
Corollary 2 A CNF decomposition C C of a consistency checker f C over variables x ∪ y ∪ {z}, can be polynomially converted into a decomposition C C of f C such that every variable in y is either unset or FALSE when z is FALSE.
Proof:
We construct C C from C C by flipping the polarity of those variables that are set to TRUE when z is FALSE. 2 Lemma 2 and corollary 2 allow us to precisely characterize the form of the clauses in a CNF decomposition.
Corollary 3 Let C C be a CNF decomposition of a consistency checker f C . The variables of C C can be renamed so that each clause has exactly one negative literal.
Proof: By lemma 2, all input variables are positive literals in the decomposition and by definition 5 they are never forced by unit propagation on C C . In addition, by corollary 2, we can rename the auxiliary variables so that unit propagation on C C may only ever set them to FALSE. Then, in any clause that consists of input variables and one auxiliary variable y, y must be negative, otherwise it may be set to TRUE, a contradiction.
Suppose there exists a clause c with two auxiliary variables y 1 and y 2 and both are negative in c. Since neither y 1 nor y 2 can ever be made TRUE, this clause can never become unit and can be ignored. Suppose the literals of both y 1 and y 2 are positive in c. Then, if c becomes unit, it makes one of the auxiliary variables TRUE, a contradiction. Thus, exactly one of the literals of y 1 and y 2 is negative in c. The same reasoning can be extended to clauses with more than two auxiliary variables. 2
The condition described by corollary 3 is similar to C C being re-nameable anti-Horn, but is stronger as it requires exactly one negative literal in each clause, rather than at most one. This condition allows us to build a monotone circuit from a decomposition, using the construction of the next lemma.
Lemma 4 Let C C be a CNF decomposition of a consistency checker f C . Then, there exists a monotone circuit S C of size O(n|C C |) that computes f C .
Proof: We assume that C C is in the form described in corollary 3.
Figure 2
Conversion of a CNF decomposition of a consistency checker into a monotone Boolean circuit. x 5 x 6
x 3
x 4
x 7
x 7 x 7
x 1 Layer 1 Layer 2 Layer 3 c 1 =(x 1 ,x 2 ,y 1 ) c 2 =(x 5 ,x 6 ,y 2 ) c 4 =(x 3 ,y 2 ,y 1 ) c 5 =(y 1 ,y 2 ,x 7 ,z) c 3 =(x 4 ,y 1 ,y 2 )
The inputs of the circuit correspond to the input variables of C C . For each input variable x i,j of C C , there exists an input b i,j of S C which is 0 if x i,j is FALSE and 1 otherwise. Internal gates of the circuit correspond to auxiliary variables after a certain number of unit propagation steps, using the same mapping.
We create a circuit with |y| layers 1 . . . |y|. Let c 1 , . . . , c m be the clauses of C C . The i th layer of the circuit contains an ∨-gate c i j for each clause c j , called clause gates and an ∧-gate y i k for each auxiliary variable y k , called variable gates. Consider a clause c j which contains y as the sole negative literal (recall that corollary 3 ensures that this is the case), the positive literals of input variables x j1 , . . . , x jq and the positive literals of auxiliary variables y jq+1 , . . . , y jq+r . The inputs of each gate c i j are b j1 , . . . , b jq and y i−1 jq+1 , . . . , y i−1 jq+r . Let the clauses with y k as the sole negative literal be c k1 , . . . , c ks . Then, the inputs of each gate y i k are c i k1 , . . . , c i ks . The output of the circuit is z |y| . Note that in this construction the inputs of some the gates may not be defined. This is the case, for example, for the gate c 1 i , where the clause c i contains the positive literals of some auxiliary variables. If this happens for a clause gate, we omit it, while if it happens for a variable gate, we omit the undefined input. If all the inputs of a variable gate are undefined, we omit the gate.
This construction computes one breadth first application of unit propagation at each layer. Specifically, the gate y i k is 0 iff y k is forced to FALSE after i or fewer breadth first steps of unit propagation, while the gate c i j is 0 iff the negated variable in c j is forced to FALSE after i or fewer breadth first steps of unit propagation. We show this by induction. For the first layer, there exist gates only for clauses with no positive literals of auxiliary variables. Consider any such gate c j which contains the negative literal y k . All the propositional variables in c j except y k are FALSE iff the corresponding inputs are 0. Thus c 1 j is 0 iff y k is FALSE after unit propagation of c j . If many clauses contain the negative literal y k , then at least one of them sets y k to FALSE in one breadth first step iff there exists a clause gate that is 0 and is an input to the variable gate y 1 k , which is an ∧-gate and is thus 0. For the inductive step, assume that the layers 1 . . . k − 1 compute k − 1 breadth first steps of unit propagation. The same reasoning as for the base case shows that the results of unit propagation are correctly computed for the k th layer. Note that the k th layer may also contain gates that were omitted at previous levels. Since the inputs of these gates are correctly computed by the inductive hypothesis, the gates that are new to the k th layer are also correctly computed.
To conclude the proof, observe that in the extreme case, unit propagation will set one more literal at every breadth first step, thus after |y| steps it must either arrive at a fixpoint or set all literals. Since the circuit has |y| layers, it will correctly compute the result of unit propagation on C C . 2
We illustrate the construction of lemma 4 with an example.
Example 4 Consider the CNF decomposition C C = {c 1 , c 2 , c 3 , c 4 , c 5 }, where c 1 = (x 1 , x 2 , y 1 ), c 2 = (x 5 , x 6 , y 2 ), c 3 = (x 4 , y 1 , y 2 ), c 4 = (x 3 , y 2 , y 1 ), c 5 = (y 1 , y 2 , x 7 , z).
We construct a monotone circuit S C from C C , (figure 2). For a given instantiation of the input variables, this circuit computes 0 for the corresponding Boolean inputs if and only if unit propagation on C C forces the output variable to FALSE.
The circuit consists of 3 layers, with gates 1 and 2 in the first layer, 3-8 in the second and gate 9 in the third. The gates 1-6 and 9 are clause gates, while gates 7 and 8 are variable gates. A strict application of the construction of lemma 4 would also have variable gates in layers 1 and 3, but we omit them here as they would be single-input gates. Note that in figure 2, inputs are replicated at each layer to reduce clutter.
We note also that the layered construction of lemma 4 is necessary. A circuit that attempts to capture unit propagation on all clauses without using layers would have to contain a cycle between the gates that compute y 1 and y 2 , because y 1 would need to be an input of the clause gate c 3 that computes y 2 and y 2 would need to an input of the clause gate c 4 that computes y 1 . Constructing a layered circuit allows us to remove such cycles. 2
The proof of theorem 2 is now immediate from lemmas 1 and 4. Since CNF decompositions of consistency checkers can be converted in polynomial time to and from CNF decompositions of propagators, theorem 2 also holds for propagators.
Non decomposable global constraints
Corollary 4 now uses an existing circuit complexity result to show that, unsurprisingly, there is no polynomial size CNF decomposition of the domain consistency propagator for the ALLDIFFERENT constraint. This also applies to generalizations of ALLDIFFERENT, such as GCC. Corollary 4 There is no polynomial sized CNF decomposition of the ALLDIFFERENT domain consistency propagator. Proof: Régin [Régin, 1994] showed that an ALLDIFFERENT constraint has a solution iff the corresponding bipartite value graph (i.e., the graph where the node representing a variable has an edge to every node that represents a value in its domain) has a perfect matching. In addition, every bipartite graph corresponds to the value graph of an ALLDIFFERENT constraint and DC propagators detect dis-entailment. Thus, if there exists a polynomial size CNF decomposition of the ALLDIFFERENT DC propagator, we can construct a monotone circuit that computes whether a bipartite graph has a perfect matching. But Razborov [Razborov, 1985] showed that the smallest monotone circuit that computes whether there exists a perfect matching for a bipartite graph is superpolynomial in the number of vertices in the graph. Therefore, the smallest CNF decomposition of the ALLDIFFERENT DC propagator is super-polynomial in size. 2
On the other hand, bound and range consistency propagators of ALLDIFFERENT can be decomposed, as we argue in [Bessiere et al., 2009].
Conclusions and Future Work
In this paper we have shown how the tools of circuit complexity can be used to study decompositions of global propagators into CNF. Our results directly extend to decompositions into CSP constraints of bounded arity with domains given in extension since such decompositions can be translated into clauses of polynomial size. An interesting next step is to consider the decomposability of constraint propagators into more expressive primitive constraints where domains are represented in logarithmic space via their bounds. CSP solvers provide this feature which is missing in CNF. We conjecture that there exists an equivalence between such CSP decompositions of constraint propagators and monotone arithmetic circuits that are generalizations of Boolean monotone circuits to real numbers and gates for addition and multiplication. Since lower bound results on monotone circuits usually transfer to monotone arithmetic circuits, this would imply that the domain consistency propagator for ALLDIFFERENT cannot be decomposed to constraints that exploit (exponentially) large domains.
This is in contrast to TRUE and FALSE for SAT variables.
We assume that formulas are given in 3-CNF form. We can convert any CNF formula to 3-CNF, increasing its size by at most a constant factor and without hindering unit propagation[Garey and Johnson, 1979, section 3.1.1].
Reformulation of Global Constraints Based on Constraints Checkers Filtering algorithms for the same constraint. F Bacchus, Beldiceanu, 6th Int. Conf. on Theory and Applications of Satisfiability Testing. C. Bessiere and P. Van Hentenryck10Constraints13th Int. Conf. on Principles and Practices of CP (CP2007)References [Bacchus, 2007] F. Bacchus. GAC via unit propagation. In 13th Int. Conf. on Principles and Practices of CP (CP2007), 133-147. 2007. [Beldiceanu et al., 2005] N. Beldiceanu, I. Katriel, and S. Thiel. Reformulation of Global Constraints Based on Constraints Checkers Filtering algorithms for the same constraint. Con- straints, 10(4): 339-362, 2005. [Bessiere and Van Hentenryck, 2003] C. Bessiere and P. Van Hen- tenryck. To be or not to be ... a global constraint. In 9th Int. Conf. on Principles and Practices of CP (CP2007), 789-794. 2003 [Bessiere et al., 2003] C. Bessiere, E. Hebrard, and T. Walsh. Local consistencies in SAT. In 6th Int. Conf. on Theory and Applica- tions of Satisfiability Testing, 299-314. 2003
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A filtering algorithm for constraints of difference in CSP. J-C Régin, ; C Schulte, P J Stuckey ; G. Tseitin ; Van Hoeve, Automation of Reasoning: Classical Papers in Comp. Logic. 2CP-2006, 1994] J-C. Régin. A filtering algorithm for constraints of difference in CSP. In AAAI, p. 362-367, 1994. [Schulte and Stuckey, 2004] C. Schulte and P. J. Stuckey. Speeding up constraint propagation. In CP-2004, pages 619-633, 2004. [Tseitin, 1983] G. Tseitin. On the complexity of proofs in proposi- tional logics. In Automation of Reasoning: Classical Papers in Comp. Logic 1967-1970, vol.2. 1983. [van Hoeve et al., 2006] W. J. van Hoeve, G. Pesant, L. M. Rousseau, and A. Sabharwal. Revisiting the sequence constraint. In CP-2006, pages 620-634, 2006.
Symmetry Breaking using Value Precedence. T Walsh, 17th European Conf. on AI. , 2006] T. Walsh. Symmetry Breaking using Value Prece- dence. In 17th European Conf. on AI, 168-172. 2006.
| []
|
[
"ON EVENTUAL COMPACTNESS OF COLLISIONLESS KINETIC SEMIGROUPS WITH VELOCITIES BOUNDED AWAY FROM ZERO",
"ON EVENTUAL COMPACTNESS OF COLLISIONLESS KINETIC SEMIGROUPS WITH VELOCITIES BOUNDED AWAY FROM ZERO"
]
| [
"B Lods ",
"M Mokhtar-Kharroubi "
]
| []
| []
| A. In this paper, we consider the long time behaviour of collisionless kinetic equation with stochastic diffuse boundary operators for velocities bounded away from zero. We show that under suitable reasonable conditions, the semigroup is eventually compact. In particular, without any irreducibility assumption, the semigroup converges exponentially to the spectral projection associated to the zero eigenvalue as t → ∞. This contrasts drastically to the case allowing arbitrarily slow velocities for which the absence of a spectral gap yields at most algebraic rate of convergence to equilibrium. Some open questions are also mentioned. This implies of course that also V is an orthogonally invariant subset of R d | 10.1007/s00028-022-00777-8 | [
"https://arxiv.org/pdf/2105.09662v2.pdf"
]
| 234,789,991 | 2105.09662 | 326a3cae8d8b5df6e936076579d9cd2362a043dd |
ON EVENTUAL COMPACTNESS OF COLLISIONLESS KINETIC SEMIGROUPS WITH VELOCITIES BOUNDED AWAY FROM ZERO
30 Sep 2021
B Lods
M Mokhtar-Kharroubi
ON EVENTUAL COMPACTNESS OF COLLISIONLESS KINETIC SEMIGROUPS WITH VELOCITIES BOUNDED AWAY FROM ZERO
30 Sep 2021Kinetic equationBoundary operatorsNon-zero velocitiesConvergence to equilib- rium 1 I
A. In this paper, we consider the long time behaviour of collisionless kinetic equation with stochastic diffuse boundary operators for velocities bounded away from zero. We show that under suitable reasonable conditions, the semigroup is eventually compact. In particular, without any irreducibility assumption, the semigroup converges exponentially to the spectral projection associated to the zero eigenvalue as t → ∞. This contrasts drastically to the case allowing arbitrarily slow velocities for which the absence of a spectral gap yields at most algebraic rate of convergence to equilibrium. Some open questions are also mentioned. This implies of course that also V is an orthogonally invariant subset of R d
I
The present paper is the third of a program initiated in [23] and pursued in [24] on the systematic study of L 1 -solutions ψ(t) to the transport equation
∂ t ψ(x, v, t) + v · ∇ x ψ(x, v, t) = 0, (x, v) ∈ Ω × V, t 0 (1.1a)
with initial data ψ(x, v, 0) = ψ 0 (x, v), (x, v) ∈ Ω × V, (1.1b) under diffuse boundary conditions ψ |Γ − = H(ψ |Γ + ), (1.1c) where Ω is a bounded open subset of R d and V is a given closed subset of R d (see Assumptions 1.1 for major details), Γ ± = {(x, v) ∈ ∂Ω × V ; ±v · n(x) > 0} (n(x) being the outward unit normal at x ∈ ∂Ω) and H is a linear boundary operator relating the outgoing and incoming fluxes ψ |Γ + and ψ |Γ − in the domain Ω.
Our main assumption on the phase space is summarized in the following
Assumptions 1.1. The phase space Ω × V is such that (1) Ω ⊂ R d (d 2)
is an open and bounded subset with C 1 boundary ∂Ω.
(2) V is the support of a nonnegative locally finite Borel measure m and there exists some r 0 > 0 such that |v| r 0 ∀v ∈ V. In the sequel, we denote by X := L 1 (Ω × V , dx ⊗ m(dv)) endowed with its usual norm · X .
With respect to our previous contributions, the main novelty of the present paper lies in assumption (1.2) which, since V is a closed subset of R d , is equivalent to 0 / ∈ V . This corresponds to the physical situation of a gas in a vessel for which particle velocities are bounded away from zero as it occurs for instance in the study of kinetic neutron transport in nuclear reactors [26]. Heuristically, a particle starting from Ω with given velocity v will reach the boundary ∂Ω in some finite time and suffer collision with the boundary which will induce a very fast thermalization of the gas.
Our main scope for the present paper is to give a rigorous justification of this heuristic consideration and show that, under suitable assumptions on the boundary operator H, the convergence to equilibrium for solution to (1.1) is exponential. This will be done by a careful spectral analysis of the transport operator T H associated to (1.1) (see Section 2 for precise functional setting and definitions) combined with some compactness properties of the C 0 -semigroup associated to (1.1). It is important to emphasize already that our approach does not resort to any kind of irreducibility properties of the semigroup. This is in contrast with the framework adopted in our previous contributions. In particular, our result covers situations more general than the mere return to equilibrium but deals rather with the general asymptotic properties of the C 0 -semigroup governing (1.1).
1.1. Related literature. Deriving the precise rate of convergence to equilibrium for linear or nonlinear kinetic equations is of course a problem of paramount importance for both theoretical and applied study of kinetic models. This problem has a long history for collisional models for which both qualitative and quantitative approaches have been proposed (see [15,16,26,28]).
For collisionless kinetic equations for which thermalization is driven by boundary effects, the literature on the topic is more recent. We refer the reader to [6,7,8,20,24] for a complete overview of the literature on the topic and mention here only the pioneering works [1,22].
For general domains, a general theory on the existence of an invariant density and its asymptotic stability (i.e. convergence to equilibrium) has been obtained recently [23] (see also earlier one-dimensional results [27]). More precisely, whenever the C 0 -semigroup (U H (t)) t 0 associated to T H is irreducible we proved in [23] that there exists a unique invariant density
Ψ H ∈ D(T H ) with Ψ H (x, v) > 0 for a. e. (x, v) ∈ Ω × V, Ω×V Ψ H (x, v)dx ⊗ m(dv) = 1 and lim t→∞ U H (t)f − P 0 f X = 0, ∀f ∈ X (1.3)
where P 0 denotes the ergodic projection (see (1.5) for the precise definition). In our contribution [24], using an explicit representation of the semigroup (U H (t)) t 0 obtained recently in [3] as well as some involved tauberian approach, we obtain explicit rates of convergence to equilibrium for solutions to (1.1) under mild assumptions on the initial datum ψ 0 . The ideas introduced in [24] are applied in the present contribution to deal with non zero velocities.
In most of the existing literature, arbitrarily slow particles are taken into account. In particular, the return to equilibrium can be made arbitrarily slow. The existence of too many slow particles is the reason for the slow return to equilibrium in the case of a collisionless gas in a container with constant wall temperature as numerically observed in particular in [31]. More specifically, quoting from [31], "fast molecules hit the boundary and are thermalized quickly, whereas it takes a long time for slow molecules to interact with the boundary. " For the specific case studied in this paper, i.e. |v| r 0 ∀v ∈ V slow particles are clearly not taken into account. For this case, the literature is scarce. We mention, for collisional linear kinetic equation, the pioneering work [21] which obtains also the eventual compactness of the semigroup governing the collisional transport equation with absorbing boundary conditions. For the collisionless model (1.1) studied here, we mention that an exponential convergence to equilibrium has been obtained in [1] for a model of radiative transfer (corresponding to unitary velocities, i.e. V is the unit sphere of R d ). The very elegant proof of [1] consists in reducing the problem to the study of a renewal integral equation for a scalar unknown quantity. Such a method exploits extensively several symmetry properties of the domain Ω and seems to apply only for spherically symmetric domain under some isotropy of the initial condition ψ 0 in (1.1b). We also wish to point out that related mono-energetic models (for which V is the unit sphere) have been extensively studied in the probability literature in which they are referred to as "stochastic billiards". The speed of convergence of such stochastic process towards its invariant distribution have been established, for various geometry of Ω in a seminal paper [17] and in the more recent contributions [13,14,18].
1.2. Our contribution. Let us make our assumptions more precise together with our main result. With respect to our previous contribution [23], we do not consider abstract and general boundary operator here but focus our attention on the specific case of a diffuse boundary operator of the following type:
Assumptions 1.2. The boundary operator H : L 1 (Γ + , dµ + ) → L 1 (Γ − , dµ − )
is an isotropic diffuse operator (dµ ± are positive measures on Γ ± see Section 2), i.e. it is given by
Hψ(x, v) = v ′ ·n(x)>0 k(x, |v|, |v ′ |)ψ(x, v ′ )|v ′ · n(x)|m(dv ′ ), (x, v) ∈ Γ − where the kernel k(x, |v|, |v ′ |) is nonnegative and measurable with v·n(x)<0 k(x, |v|, |v ′ |)|v · n(x)|m(dv) = 1, ∀(x, v ′ ) ∈ Γ + . (1.4)
We refer to Section 5 for various examples of diffuse boundary operators of physical interest covered by our results. We will often use the abuse of notation k(x, v, v ′ ) = k(x, |v|, |v ′ |), keeping in mind that the kernel is isotropic with respect to each velocity variables. This isotropy is simplifying assumption but more general kernels can be handled by our approach as illustrated in [24]. We preferred here to adopt this simplified framework avoiding too technical computations.
As already said, our approach does not require any irreducibility properties, and in particular, covers situation more general than those studied usually where the existence (and uniqueness) of some normalized steady solution to (1.1) is assumed yielding to the convergence (1.3).
Besides a new simplified proof of a weak compactness result given in [23], we extend the convergence in (1.3) into two directions:
• First, we get rid of the irreducibility assumption and study the long-time asymptotics of the C 0 -semigroup (U H (t)) t 0 also in the case in which there is more than one steady solution to (1.1). • Second, we make the convergence (1.3) quantitative by showing that the semigroup (U H (t)) t 0 is eventually compact. Besides its own interest, such a compactness result implies that the convergence in (1.3) is exponentially fast. Moreover, it implies that 0 is a semi-simple eigenvalue of T H (this is the main tool which allows us to avoid any irreducibilty assumption for the long-time asymptotics). More precisely, our main result can be stated as follows Theorem 1.3. Let Assumptions 1.1 and 1.2 be in force. Assume that ∂Ω is of class C 1,α for some α >
-semigroup (U H (t)) t 0 governing equation (1.1) is eventually compact in X, i.e. there exists some τ ⋆ > 0 such that U H (t) is a compact operator in X for any t > τ ⋆ .
Moreover, there exists λ ⋆ > 0 such that
S(T H ) ∩ {λ ∈ C ; Reλ > −λ ⋆ } = {0}
where 0 is an eigenvalue of T H which is a first order pole of the resolvent R(·, T H ). In particular, for any λ 0 ∈ (0, λ ⋆ ) there is C > 0 such that
U H (t)f − P 0 f X C exp (−λ 0 t) f X
for any t 0, and any f ∈ X where P 0 is the spectral projection associated to the zero eigenvalue. Remark 1.4. Whenever the semigroup (U H (t)) t 0 is irreducible, one has
P 0 f = ̺ f Ψ H , with ̺ f = Ω×V f (x, v)dx ⊗ m(dv), (1.5)
for any f ∈ X where Ψ H is the unique positive invariant density of T H with unit mass. In this case, like in (1.3), P 0 is the so-called ergodic projection of T H .
The proof of the above result is based upon suitable compactness properties of some boundary operators which have been studied already in our contributions [23,24] and made precise in the situation considered here. We recall that these operators, already studied in [23], are the fundamental bricks on which the resolvent of T H is constructed, in particular, for λ > 0, it is known that the resolvent R(λ, T H ) is given by
R(λ, T H ) = R(λ, T 0 ) + ∞ n=0 Ξ λ H (M λ H) n G λ
where the operators Ξ λ , M λ , G λ are precisely defined in Section 2 while R(λ, T 0 ) is the resolvent of the transport operator associated to absorbing boundary conditions (corresponding to H = 0).
Under the assumption 0 / ∈ V (and in contrast with what happens in the general case 0 ∈ V ), the spectrum of T 0 is empty and the various operators are defined and bounded for any λ ∈ C and depend on λ in an analytic way. Moreover,
(M λ H) 2 is a weakly compact operator in L 1 (Γ + , dµ + )
We give here a new simplified proof of this weak-compactness property which was obtained in [23, Theorem 5.1] by highly technical means. The simplified proof presented here is based on an important change of variables for boundary operators introduced in [24]. Such compactness induces naturally a complete picture of the asymptotic spectrum of the generator T H : the spectrum
S(T H ) of T H in L 1 (Ω × V dx ⊗ m(dv))
consists of isolated eigenvalues with finite algebraic multiplicities and there is λ ⋆ > 0 such that
S(T H ) ∩ {λ ∈ C ; Reλ > −λ ⋆ } = {0}.
Moreover, using a suitable change of variable introduced in [24], one can also prove an explicit decay of (M λ H) 2 of the form
(M λ H) 2 B(L 1 (Γ + ,dµ + )) C |λ| ∀λ ∈ C Reλ > 0. (1.6)
This allows to transfer the weak compactness of the (M λ H) 2 into some compactness of the semigroup U H (t) for t large enough. Indeed, thanks to a representation of the semigroup (U H (t)) t 0 as a series of operators, reminiscent of Dyson-Phillips expansion series and derived in [3],
U H (t)f = ∞ n=0 U n (t)f, t > 0, f ∈ L 1 (Ω × V, dx ⊗ m(dv)) our assumption 0 / ∈ V implies that, for any N > 0, there is τ N > 0 such that U n (t) = 0 ∀t > τ N , n < N,
i.e. the first terms of the representation series vanish for t large enough. We wish to emphasize here that such a representation series is a very natural representation of the solution to (1.1) which consists in following the trajectories of particles inside the domain Ω and for which change of velocities occur only due to the interaction with the boundary ∂Ω. Roughly speaking, for each n ∈ N, the term U n (t) takes into account the n-th rebound on the particles on ∂Ω.
From the above considerations, we can deduce by complex Laplace inversion formula [2] that
U H (t)f = 1 2π ∞ −∞ ∞ n=N Ξ ε+iη H (M ε+iη H) n G ε+iη f dη, ε > 0
where, thanks to the estimate (1.6), the convergence actually holds in operator norm yielding the
compactness of U H (t) for t > τ N if N is large enough.
We believe that the approach adopted here is robust enough to be applied also to more general problems (including collisional models with general boundary conditions) as well as the study of (1.1) in more general L p (Ω × V, dx ⊗ m(dv)), 1 p < ∞. Moreover, even though our analysis is restricted, for technical reasons, to the case of a diffuse boundary operator satisfying Assumptions 1.2, we are convinced that our method could also be adapted to deal with more general partly diffusive boundary operators (of Maxwell-type) as those considered in [23,6] (see Appendix A for partial results in that direction). For
any E ⊂ R d , we denote with 1 E the indicator function of E defined as 1 E (x) = 1 if x ∈ E and 1 E (x) = 0 if x / ∈ E.
1.4. Organization of the paper. After this Introduction, Section 2 presents several technical known results and the functional setting introduced in [23]. In Section 3, we recall the fundamental change of variable obtained in [24] as well as the weak compactness of (M λ H) 2 together with the full proof of Estimate (1.6). In Section 4 we apply this estimate to derive the eventual compactness of the semigroup (U H (t)) t 0 (Theorem 4.12) yielding to our main result Theorem Part of this research was performed while the second author was visiting the "Laboratoire de Mathématiques CNRS UMR 6623" at Université de Franche-Comté in February 2020. He wishes to express his gratitude for the financial support and warm hospitality offered by this Institution.
We are grateful to both the anonymous referees for their careful readings and observations which contribute to improve the overall presentation of the paper.
P
We collect here several preliminary and known results scattered in the literature. Notice that, in this Section, we will make no use of our fundamental assumption 0 / ∈ V . In particular, the results quoted in this Section remain valid in the case in which 0 ∈ V . We will see in the subsequent Sections that several of the results presented here can be drastically improved under (1.2).
2.1. Functional setting. We introduce in this subsection the various mathematical tools and functional spaces used in the rest of the paper. Let us begin with introducing the travel time of particles in Ω, defined as:
Definition 2.1. For any (x, v) ∈ Ω × V, define t ± (x, v) = inf{ s > 0 ; x ± sv / ∈ Ω}.
To avoid confusion, we will set τ ± (x, v)
:= t ± (x, v) if (x, v) ∈ ∂Ω × V.
Under the assumption (1.2), the travel time is actually bounded, since
t ± (x, v) D |v| D r 0 , ∀v ∈ V (2.1)
where D denotes the diameter of Ω, D = sup{|x − y| , x, y ∈Ω}.
In order to exploit this local nature of the boundary conditions, we introduce the following notations. For any x ∈ ∂Ω, we define
Γ ± (x) = {v ∈ V ; ±v · n(x) > 0}, Γ 0 (x) = {v ∈ V ; v · n(x) = 0}
and we define the measure µ x (dv) on Γ ± (x) given by
µ x (dv) = |v · n(x)|m(dv).
We introduce the partial Sobolev space
W 1 = {ψ ∈ X ; v · ∇ x ψ ∈ X}.
It is known [10,11] that any ψ ∈ W 1 admits traces ψ |Γ ± on Γ ± such that
ψ |Γ ± ∈ L 1 loc (Γ ± ; dµ ± (x, v)) where dµ ± (x, v) = |v · n(x)|π(dx) ⊗ m(dv),
denotes the "natural" measure on Γ ± . Here, π(dx) denotes the surface Lebesgue measure on ∂Ω.
Notice that, since dµ + and dµ − share the same expression, we will often simply denote it by
dµ(x, v) = |v · n(x)|π(dx) ⊗ m(dv),
the fact that it acts on Γ − or Γ + being clear from the context. Note that
∂Ω × V := Γ − ∪ Γ + ∪ Γ 0 , where Γ 0 := {(x, v) ∈ ∂Ω × V ; v · n(x) = 0}.
We introduce the set
W = ψ ∈ W 1 ; ψ |Γ ± ∈ L 1 ± where we recall that L 1 ± = L 1 (Γ ± , dµ ± )
. One can show [10,11] that
W = ψ ∈ W 1 ; ψ |Γ + ∈ L 1 + = ψ ∈ W 1 ; ψ |Γ − ∈ L 1 − .
Then, the trace operators B ± :
B ± : W 1 ⊂ X → L 1 loc (Γ ± ; dµ ± ) ψ −→ B ± ψ = ψ |Γ ± , are such that B ± (W ) ⊆ L 1
± . Let us define the maximal transport operator T max as follows:
T max : D(T max ) ⊂ X → X ψ → T max ψ(x, v) = −v · ∇ x ψ(x, v), with domain D(T max ) = W 1 . Now, for any bounded boundary operator H ∈ B(L 1 + , L 1 − ), define T H as T H ϕ = T max ϕ for any ϕ ∈ D(T H ) := {ψ ∈ W ; ψ |Γ − = H(ψ |Γ + )}.
In particular, the transport operator with absorbing conditions (i.e. corresponding to H = 0) will be denoted by T 0 .
2.2.
About the resolvent of T H . We can now describe the resolvent of the operator T H introducing first a series of useful operators. For any λ ∈ C such that Reλ > 0, define
M λ : L 1 − −→ L 1 + u −→ M λ u(x, v) = u(x − τ − (x, v)v, v)e −λτ − (x,v) , (x, v) ∈ Γ + ; Ξ λ : L 1 − −→ X u −→ Ξ λ u(x, v) = u(x − t − (x, v)v, v)e −λt − (x,v) 1 {t − (x,v)<∞} , (x, v) ∈ Ω × V ; G λ : X −→ L 1 + ϕ −→ G λ ϕ(x, v) = τ − (x,v) 0 ϕ(x − sv, v)e −λs ds, (x, v) ∈ Γ + ; and R λ : X −→ X ϕ −→ R λ ϕ(x, v) = t − (x,v) 0 ϕ(x − tv, v)e −λt dt, (x, v) ∈ Ω × V .
The interest of these operators is related to the resolution of the boundary value problem:
(λ − T max )ϕ = g, B − ϕ = u, (2.2)
where λ > 0, g ∈ X and u is a given function over Γ − . Such a boundary value problem, with u ∈ L 1 − and g ∈ X can be uniquely solved and its unique solution ϕ ∈ D(T max ) is given by
ϕ = R λ g + Ξ λ u (2.3) with B + f ∈ L 1 + and B + ϕ L 1 + + λ ϕ X u L 1 − + g X . (2.4)
We refer to [5, Theorem 2.1] for more details on the boundary value problem (2.2). In particular, for any λ > 0,
Ξ λ B(L 1 − , X) λ −1 R λ B(X) λ −1 , G λ B(X,L 1 + ) 1 (2.5)
where the first inequality is established in [5, Remark 3.2] while the second and third ones are deduced from (2.4) for u = 0 so that ϕ = R λ g and B + ϕ = G λ g. Moreover, one has
M λ B(L 1 − ,L 1 + ) 1 ∀λ ∈ C + (2.6)
which can be easily deduced from the identity
Γ − ψ(z, v)dµ − (z, v) = Γ + ψ(x − τ − (x, v)v, v)dµ + (x, v), ∀ψ ∈ L 1 − (2.7) established in [4, Proposition 2.11].
Actually, for λ = 0, we can extend the definition of these operators in an obvious way and, in contrast with what happens in the general case in which 0 ∈ V (see [24,Section 2.4]), the fact that velocities are bounded away from zero implies here that all the resulting operators remain bounded for λ = 0. Indeed, when 0 ∈ V , the operators Ξ 0 and R 0 are not necessarily bounded (the estimates (2.5) clearly deteriorate when λ → 0), see [24, Section 2.4] for a thorough description of these operators. We will see in Section 4 that the situation is much more favourable whenever 0 / ∈ V .
We can complement the above result with the following Proposition 2.2. Let Assumptions 1.1 and 1.2 be in force. Introduce the half-plane
C + = {z ∈ C ; Rez > 0}.
Then, for any λ ∈ C + one has r σ (M λ H) < 1 and
R(λ, T H ) = R λ + Ξ λ HR(1, M λ H)G λ = R(λ, T 0 ) + ∞ n=0 Ξ λ H (M λ H) n G λ (2.8)
where the series converges in B(X).
Proof. The fact that r σ (M λ H) < 1 for λ ∈ C + is given in [
S + (x) = σ ∈ S d−1 ; σ · n(x) > 0 = Γ + (x) ∩ S d−1 .
Then, for any nonnegative measurable mapping g : S d−1 → R, one has,
S + (x) g(σ) |σ · n(x)|dσ = ∂Ω g x − y |x − y| J (x, y)π(dy), and J (x, y) = 1 Σ + (x) (y) |(x − y) · n(x)| |x − y| d+1 |(x − y) · n(y)|, ∀y ∈ Σ + (x) (3.1) with Σ + (x) = {y ∈ ∂Ω : ]x, y[ ⊂ Ω ; (x − y) · n(x) > 0 ; n(x − y) · n(y) < 0} where ]x, y[ = {tx + (1 − t)y ; 0 < t < 1}
is the open segment joining x and y.
It is easy to deduce from the above expression of J (x, y), that J (x, y) |x − y| 1−d for any (x, y) ∈ ∂Ω × ∂Ω, x = y. Whenever the boundary ∂Ω is more regular than the mere class C 1 one can strengthen this estimate to get the following Lemma 3.2. [24, Lemma 6.5] Assume that ∂Ω is of class C 1,α , α ∈ (0, 1) then, there exists a positive constant C Ω > 0 such that
|(x − y) · n(x)| C Ω |x − y| 1+α , ∀x, y ∈ ∂Ω.
Consequently, with the notations of Lemma 3.1, there is a positive constant C > 0 such that
J (x, y) C |x − y| d−1−2α , ∀x, y ∈ ∂Ω, x = y.
We recall then the following generalization of the polar decomposition theorem (see [32, Lemma 6.13, p.113]):
Lemma 3.3. Let m 0 be the image of the measure m under the transformation v ∈ R d → |v| ∈ [0, ∞), i.e. m 0 (I) = m {v ∈ R d ; |v| ∈ I} for any Borel subset I ⊂ R + . Then, for any ψ ∈ L 1 (R d , m) it holds R d ψ(v)m(dv) = 1 |S d−1 | ∞ 0 m 0 (d̺) S d−1 ψ(̺ σ)dσ where dσ denotes the Lebesgue measure on S d−1 with surface |S d−1 |. Remark 3.4. Notice that, under the assumption 0 / ∈ V , one sees that the measure m 0 is supported on [r 0 , ∞) where r 0 is defined in (1.2).
We can deduce from the above change of variables the following useful expression for HM λ H (see [24,Proposition 6.8]
HM λ Hϕ(x, v) = Γ + J λ (x, v, y, w)ϕ(y, w) |w · n(y)|m(dw)π(dy) (3.2) where J λ (x, v, y, w) = J (x, y) ∞ 0 ̺ k(x, |v|, ̺)k(y, ̺, |w|) exp −λ |x − y| ̺ m 0 (d̺) |S d−1 | (3.3) for any (x, v) ∈ Γ − , (y, w) ∈ Γ + .
3.2.
Weak-compactness. In [23, Section 5], we derived in a broad generality the weak-compactness of HM 0 H for a general class of diffuse boundary operator H (see [23,Theorem 5.1] for a precise statement). For a given x ∈ ∂Ω, we introduce the bounded operator
H(x) ∈ B(L 1 (Γ + (x)), L 1 (Γ − (x)))
with kernel k(x, ·, ·). We introduce the following definition Definition 3.6. We say that the family
H(x) ∈ B(L 1 (Γ + (x)), L 1 (Γ − (x))), x ∈ ∂Ω
is collectively weakly compact if, for any x ∈ ∂Ω, H(x) is weakly-compact and
lim m→∞ sup x∈∂Ω sup v ′ ∈Γ + (x) Sm(x,v ′ ) k(x, v, v ′ ) µ x (dv) = 0
where, for any m ∈ N and any (
x, v ′ ) ∈ Γ + S m (x, v ′ ) = {v ∈ Γ − (x) ; |v| m} ∪ {v ∈ Γ − (x) ; k(x, v, v ′ ) m}.
We recall a key weak compactness result from [23] which holds for ∂Ω of class C 1 . The proof established therein is very long and highly technical but, thanks to Proposition 3.5, we are able to provide a new and much shorter proof for ∂Ω of class C 1,α (α > 0), see Appendix B:
Theorem 3.7. Under Assumptions 1.2, assume that the family
H(x) ∈ B(L 1 (Γ + (x)), L 1 (Γ − (x))),
x ∈ ∂Ω is collectively weakly compact. Then, HM 0 H : L 1 + → L 1 − is weakly-compact.
M
In all this Section, we will always assume that Assumptions 1.1 and 1.2 hold true together with the conclusion of Theorem 3.7, i.e.
HM 0 H : L 1 + → L 1 − is weakly-compact.
It will be assumed implicitly in all the next statements without further mention.
4.1.
Fine properties of T H . We begin with a full description of the spectrum of the transport operator T H under our main assumption about the velocity space V which we recall is
0 / ∈ V.
Thus, (1.2) holds true. In this case, one sees that the measure m 0 appearing in Lemma 3.3 is supported on a subset of [r 0 , ∞) and, as already mentioned,
t − (x, v) D r 0 , ∀(x, v) ∈ Ω × V . (4.1)
This results readily in the following properties of the operators introduced in Section 2.2
Lemma 4.1. The mappings
λ ∈ C −→ Ξ λ ∈ B(L 1 − , X), λ ∈ C −→ M λ ∈ B(L 1 − , L 1 + ) λ ∈ C −→ G λ ∈ B(X, L 1 + ), λ ∈ C −→ R λ ∈ B(X)
are all well-defined and analytic (i.e. there are entire mappings). In particular, S(T 0 ) = ∅.
Proof. The proof of the result is straightforward. For instance, one can check easily that, from (4.1) and (2.6),
M λ B(L 1 − ,L 1 + ) exp (Reλ) − Dr −1 0 M 0 B(L 1 − ,L 1 + ) = exp (Reλ) − Dr −1 0 (4.2)
where (Reλ) − = max(0, −Reλ) is the negative part of Reλ. One argues in the same way for the other operators to prove they are bounded operators. As far as analyticity is concerned, let us for instance focus on Ξ λ . For any f ∈ L 1 − and g ∈ X ⋆ (the dual of X) the mapping λ ∈ C → g, Ξ λ f ∈ C is analytic (where ·, · is the duality bracket between X ⋆ and X). This proves that
λ ∈ C −→ Ξ λ ∈ B(L 1 − , X) is analytic (see [2, Proposition A.3, Appendix A])
. One argues in the same way for the other operators.
A first result about the spectrum of T H is the following Proof. We first notice that, thanks to Lemma 4.1, it is straightforward that, if 1 / ∈ S(M λ H) then
(λ − T H ) is invertible with
This proves that, if λ ∈ S(T H ) then 1 ∈ S(M λ H). Conversely, assume that 1 ∈ S(M λ H). Since
|M λ ϕ| M Reλ |ϕ| M 0 |ϕ| if Reλ 0 exp −Reλ D r −1 0 M 0 |ϕ| if Reλ < 0. (4.3) Because HM 0 H ∈ B(L 1 + , L 1 − ) is weakly-compact, so is HM λ H (λ ∈ C)
by a domination argument. Thus, for λ ∈ C, (M λ H) 2 ∈ B(L 1 + ) is weakly-compact and (M λ H) 4 is compact by the Dunford-Pettis property and therefore S(M λ H) = S p (M λ H). Let then ψ ∈ L 1 + be such that ψ = M λ Hψ, setting u = Hψ and ϕ = Ξ λ u one sees that ϕ = 0, ϕ ∈ D(T max ) with T max ϕ = λΞ λ u = λϕ since ϕ is the unique solution to (2.2) (with g = 0) according to (2.3). Moreover, by construction,
B − ϕ = u and B + ϕ = B + Ξ λ u = M λ u = M λ Hψ = ψ so that HB + ϕ = Hψ = u = B − ϕ which implies ϕ ∈ D(T H )
. This proves that λ ∈ S p (T H ).
4.2.
Useful decay estimates. The scope of this technical subsection is to establish the decay, as |Imλ| → ∞, of (M λ H) 2 B(L 1 + ) , which, in turn, will yield some quantitative decay estimates for some remainders of the series (2.8). It will be obtained under the following technical assumptions Assume moreover that, for almost every (x, v) ∈ Γ + and almost every (y, w) ∈ Γ + , the mappings
̺ ∈ (r 0 , ∞) −→ k(x, |v|, ̺) ∈ R + , and ̺ ∈ (r 0 , ∞) −→ k(y, ̺, |w|) ∈ R + are differentiable with sup (y,w)∈Γ + ∞ r 0 ̺ d+1 ̺ k(y, ̺, |w|) ̟ ′ (̺) + ̺ ̟(̺) |∂ ̺ k(y, ̺, |w|)| + k(y, ̺, |w|)̟(̺) d̺ < ∞;
(4.6) and
sup x∈∂Ω sup (y,w)∈Γ + ∞ r 0 ̺ d+2 ̟(̺)k(y, ̺, |w|)d̺ Γ − (x) |∂ ̺ k(x, |v|, ̺)| µ x (dv) < ∞. (4.7)
The role of Assumptions 4.3 is mainly technical to ensure the following Lemma to hold and we will prove in Section 5 that it can be checked for several models of physical interest: Proof. A more general proof has been given in [24, Proposition 6.8] to get a decay of order 1/|λ|. We repeat the proof here to emphasize the difference and the emergence of the additional exponential term. From (3.3) and Lemma 3.2, one has for all (
x, v) ∈ Γ − , (y, w) ∈ Γ + |J λ (x, v, y, w)| C Ω |x − y| d−1−2α ∞ r 0 ̺ k(x, |v|, ̺)k(y, ̺, |w|) exp −λ|x − y|̺ −1 m 0 (d̺) |S d−1 | .
for some positive constant C Ω . We compute this last integral as follows:
∞ r 0 ̺ k(x, |v|, ̺)k(y, ̺, |w|) exp −λ|x − y|̺ −1 m 0 (d̺) |S d−1 | = 1 λ|x − y| ∞ r 0 ̺ d+2 ̟(̺)k(x, |v|, ̺)k(y, ̺, |w|) λ|x − y| ̺ 2 exp −λ|x − y|̺ −1 d̺
which, after integration by parts and using (4.4) yields
∞ r 0 ̺ k(x, |v|, ̺)k(y, ̺, |w|) exp −λ|x − y|̺ −1 m 0 (d̺) |S d−1 | = − 1 λ|x − y| ∞ r 0 d d̺ ̺ d+2 ̟(̺)k(x, |v|, ̺)k(y, ̺, |w|) exp −λ|x − y|̺ −1 d̺ − 1 λ|x − y| r d+2 0 ̟(r 0 )k(x, |v|, r 0 )k(y, r 0 , |w|) exp −λ|x − y|r −1 0 .
This results in the following estimate for the kernel J λ (x, v, y, w):
|J λ (x, v, y, w)| C Ω |λ| |x − y| d−2α (|I 1 (λ, x, y, v, w)| + I 2 (λ, x, v, y, w)) with I 1 (λ, x, v, y, w) = ∞ r 0 d d̺ ̺ d+2 ̟(̺)k(x, |v|, ̺)k(y, ̺, |w|) exp −λ|x − y|̺ −1 d̺ and I 2 (λ, x, v, y, w) = r d+2 0 ̟(r 0 )k(x, |v|, r 0 )k(y, r 0 , |w|) exp −Reλ|x − y|r −1 0 for any λ = 0, (x, v) ∈ Γ − , (y, w) ∈ Γ + .
Notice that, for any (y, w) ∈ Γ + and x ∈ ∂Ω
Γ − (x) I 2 (λ, x, v, y, w)|v · n(x)|m(dv) = r d+2 0 ̟(r 0 ) exp −Reλ|x − y|r −1 0 k(y, r 0 , |w|)
using the normalization (1.4). Thus Evaluating the derivative with respect to ̺ thanks to Leibniz rule, one writes
I 1 (λ, x, v, y, w) = 4 j=1 I 1,j (λ, x, v, y, w) where I 1,1 (λ, x, v, y, w) = ∞ r 0 ̺ d+2 ̟(̺)k(x, |v|, ̺) ∂ ̺ k(y, ̺, |w|) exp −λ|x − y|̺ −1 d̺ I 1,2 (λ, x, v, y, w) = ∞ r 0 ̺ d+2 ̟(̺)∂ ̺ k(x, |v|, ̺) k(y, ̺, |w|) exp −λ|x − y|̺ −1 d̺ I 1,3 (λ, x, v, y, w) = ∞ r 0 ̺ d+2 ̟ ′ (̺)k(x, |v|, ̺) k(y, ̺, |w|) exp −λ|x − y|̺ −1 d̺ I 1,4 (λ, x, v, y, w) = (d + 2) ∞ r 0 ̺ d+1 ̟(̺)k(x, |v|, ̺) k(y, ̺, |w|) exp −λ|x − y|̺ −1 d̺.
Using the normalisation condition (1.4), one has
Γ − (x) |I 1,1 (λ, x, v, y, w)| |v · n(x)|m(dv) ∞ r 0 ̺ d+2 ̟(̺) |∂ ̺ k(y, ̺, |w|)| exp (Reλ) − |x − y|̺ −1 d̺ exp (Reλ) − Dr −1 0 ∞ r 0 ̺ d+2 ̟(̺) |∂ ̺ k(y, ̺, |w|)| d̺.
Thus, assumption (4.6) yields sup (y,w)∈Γ + Γ − (x)
|I 1,1 (λ, x, v, y, w)| |v · n(x)|m(dv) C exp (Reλ) − Dr −1 0 .
In the same way, one sees easily that (4.6) implies that
sup (y,w)∈Γ + Γ − (x) (|I 1,3 (λ, x, v, y, w)| + |I 1,4 (λ, x, v, y, w)|) |v · n(x)|m(dv) C exp (Reλ) − Dr −1 0 .
Finally, one checks easily that (4.7) implies
sup x∈∂Ω sup (y,w)∈Γ + Γ − (x) |I 1,2 (λ, x, v, y, w)| |v · n(x)|m(dv) C exp (Reλ) − Dr −1 0 .
Combining all these estimates, we finally obtain that there exists some positive constant C (depending only on r 0 ) such that
Γ − (x) |J λ (x, v, y, w)| |v · n(x)|m(dv) C |λ||x − y| d−2α exp (Reλ) − Dr −1 0 ∀x ∈ ∂Ω, ∀(y, w) ∈ Γ + .
We get the result since, for α > 1 2 ,
sup y∈∂Ω ∂Ω π(dx) |x − y| d−2α < ∞ ,
the kernel |x − y| 2α−d being of order strictly less than d − 1 (see [19,Prop. 3.11]).
The above, combined with Proposition 3.5 yields the following Lemma 4.5. Assume that Assumptions 4.3 are in force and ∂Ω is of class C 1,α with α > 1 2 . There exists a positive constant C such that
(M λ H) 2 B(L 1 + ) C |λ| exp 2r −1 0 (Reλ) − D holds for any λ ∈ C, λ = 0.
Proof. It is clear from Proposition 3.5 that, for any ψ ∈ L 1 + ,
(M λ H) 2 ψ L 1 + M λ B(L 1 − ,L 1 + ) HM λ Hψ L 1 − M λ B(L 1 − ,L 1 + ) Γ + |ψ(y, w)|dµ + (y, w) Γ − |J λ (x, v, y, w)| dµ − (x, v) so that, using that M λ B(L 1 − ,L 1 + ) exp (Reλ) − Dr −1 0 (see (4.2)) we get (M λ H) 2 ψ L 1 + exp (Reλ) − Dr −1 0 sup (y,w)∈Γ + Γ − |J λ (x, v, y, w)| dµ − (x, v)
and we conclude then with Lemma 4.4.
We also establish here a simple consequence of Lemma 4.5:
Lemma 4.6. Assume that Assumptions 4.3 are in force and ∂Ω is of class C 1,α with α > 1 2 . For any N 2, there exists some positive constant C N > 0 depending on N and such that, for any λ ∈ C + it holds
∞ n=N Ξ λ H (M λ H) n G λ B(X) C N |λ| −⌊ N 2 ⌋ 1 Reλ 1 − exp −Dr −1 0 Reλ (4.9)
where N 2 denotes the integer part of N 2 . In particular, for any N 4,
∞ −∞ ∞ n=N Ξ ε+iη H (M ε+iη H) n G ε+iη B(X) dη < ∞ , ∀ε > 0. (4.10)
Proof. Since r σ (M λ H) < 1 for any Reλ > 0 (see Proposition 2.2), one has
∞ n=N Ξ λ H (M λ H) n G λ = Ξ λ H (M λ H) N R (1, M λ H) G λ .
One notices that, for any λ ∈ C, Reλ > 0, one has
Ξ λ H B(L 1 + ,X) 1 Reλ , G λ B(X,L 1 + ) 1 so that, for any N 2 ∞ n=N Ξ λ H (M λ H) n G λ B(X) 1 Reλ (M λ H) N B(L 1 + ) R (1, M λ H) B(L 1 + ) .
Since, for Reλ > 0 and using (4.2) to estimate M λ H B(L 1 + ) ,
R (1, M λ H) B(L 1 + ) 1 1 − M λ H B(L 1 + ) 1 1 − exp −Dr −1 0 Reλ , one deduces that ∞ n=N Ξ λ H (M λ H) n G λ B(X) 1 Reλ 1 − exp −Dr −1 0 Reλ (M λ H) N B(X)
.
Now, since M λ H B(L 1 + )
1 according to (4.2) (recall that Reλ > 0), one deduces easily from Lemma 4.5 that
(M λ H) N B(L 1 + ) C |λ| ⌊ N 2 ⌋
from which (4.9) follows. One deduces then, for any ε > 0 that
∞ n=N Ξ ε+iη H (M ε+iη H) n G ε+iη B(X) C N ε 1 − exp −Dr −1 0 ε |ε + iη| −⌊ N 2 ⌋
and, for N 4, (4.10) follows since N 2 > 1.
Semigroup decay.
We aim now to prove that the semigroup (U H (t)) t 0 generated by T H converges exponentially fast to equilibrium. We will use here the following representation of the semigroup in terms of a Dyson-Phillips obtained in [3]. First, recall the definition of the C 0 -semigroup generated by T 0 :
U 0 (t)f (x, v) = f (x − tv, v)1 {t<t − (x,v)} , f ∈ X, t 0.
We begin with the following definition where D 0 = {f ∈ D(T max ) ; B − f = 0 = B + f }:
Definition 4.7. Let t 0, k 1 and f ∈ D 0 be given. For (x, v) ∈ Ω × V with t − (x, v) t,
there exists a unique y ∈ ∂Ω with (y, v) ∈ Γ − and a unique 0 < s < min(t, τ + (y, v)) such that x = y + sv and then one sets
[U k (t)f ](x, v) = HB + U k−1 (t − s)f (y, v). We set [U k (t)f ](x, v) = 0 if t − (x, v) t and U k (0)f = 0. Remark 4.8. Notice that, for any (x, v) ∈ Ω × V and t > τ − (x, v) one has y = x − τ − (x, v), s = τ − (x, v).
Then, one has the following extracted from [3]:
Theorem 4.9. For any k 1, f ∈ D 0 one has U k (t)f ∈ X for any t 0 with
U k (t)f X f X .
In particular, U k (t) can be extended to be a bounded linear operator, still denoted U k (t) ∈ B(X) with U k (t) B(X) 1 ∀t 0, k 1.
Moreover, the following holds for any k 1 (1) (U k (t)) t 0 is a strongly continuous family of B(X).
(2) For any f ∈ X and λ > 0, setting
L k (λ)f = ∞ 0 exp(−λt)U k (t)f dt one has, for k 1, L k (λ)f ∈ D(T max ) with T max L k (λ)f = λ L k (λ)f and B ± L k (λ)f ∈ L 1 ± with B − L k (λ)f = HB + L k−1 (λ)f B + L k (λ)f = (M λ H) k G λ f.
(3) For any f ∈ X, the series ∞ k=0 U k (t)f is strongly convergent and it holds
U H (t)f = ∞ k=0 U k (t)fL k (λ)f = Ξ λ HB + L k−1 (λ)f. Since L 0 (λ)f = R λ f we deduce that, for any k 1, L k (λ) = Ξ λ H (M λ H) k−1 G λ .
In particular, one sees that, in the representation series (2.8) that, for any n 0
Ξ λ H (M λ H) n G λ f = ∞ 0 exp(−λ t)U n+1 (t)f dt (4.11)
for any λ > 0 which is of course coherent with the above point (3) and the representation of the resolvent of T H .
The exact expression of the iterated U k (t) allows to prove the following which is the crucial point for our analysis here, namely, under the assumption |v| r 0 , ∀v ∈ V each term of the above series is vanishing for large time:
Lemma 4.11. Let (U k (t)) k 0,t 0 be the family of operators defined in Definition 4.7. Then, under assumption (1.2), for any n 0,
U n (t) ≡ 0 ∀t τ n := (n+1)D r 0 .
Proof. Once noticed that, for t τ 0 , U 0 (t) = 0, the proof is a simple induction using the Definition 4.7. Indeed, assuming U k−1 (t) = 0 for t τ k−1 = kτ 0 , one recalls that
U k (t)f (x, v) = HB + U k−1 (t − s)f (y, v), (x, v) ∈ Ω × V, y = x − t − (x, v)v with s = t − (x, v), we get that, if t − s τ k−1 then U k (t)f (x, v) = 0. Being s = t − (x, v) τ 0 ,
we have that t − s τ k−1 and U k (t)f (x, v) = 0 for any (x, v) as soon as t τ k−1 + τ 0 . This means that U k (t) = 0 for t τ k = τ k−1 + τ 0 = (k + 1)τ 0 .
We are in position to prove the main result of this paper where we recall that D = diam(Ω) and r 0 := inf{|v| ; v ∈ V }.
Proof. From Lemma 4.11 and Theorem 4.9, for any N 0 and any f ∈ X, one has
U H (t)f = ∞ n=N +1 U n (t)f ∀t τ N .
Notice also that, since U H (t) B(X) = 1 for any t 0, the type ω 0 (U H ) of the semigroup (U H (t)) t 0 is equal to zero, i.e. ω 0 (U H ) = 0.
According to the Laplace inversion formula [2, Proposition 3.12.1], for any ε > 0 and any t 0 one has
U H (t)f = lim ℓ→∞ 1 2π ℓ −ℓ exp ((ε + iη) t) R(ε + iη, T H )f dη, = lim ℓ→∞ 1 2π ℓ −ℓ exp ((ε + iη) t) ∞ n=0 Ξ ε+iη H (M ε+iη H) n G ε+iη f dη, ∀f ∈ D(T H ).
Then, one deduces easily from (4.11) that, for any f ∈ D(T H ) it holds, for ε > 0,
lim ℓ→∞ 1 2π ℓ −ℓ exp ((ε + iη) t) N −1 n=0 Ξ ε+iη H (M ε+iη H) n G ε+iη f dη = N −1 n=0 U n+1 (t)f = 0, if t τ N .
Therefore, for any f ∈ D(T H ) and any t τ N ,
U H (t)f = ∞ n=N U n+1 (t)f = 1 2π lim ℓ→∞ ℓ −ℓ exp (ε + iη) t ∞ n=N Ξ ε+iη H (M ε+iη H) n G ε+iη f dη ε > 0 (4.12)
where the convergence holds in X. Recall that r σ (M ε+iη H) < 1 for any η ∈ R and therefore
∞ n=N Ξ ε+iη H (M ε+iη H) n G ε+iη = Ξ ε+iη H (M ε+iη H) N R (1, M ε+iη H) G ε+iη
is a compact operator for any N 4. Consequently, for any ℓ ∈ R,
1 2π ℓ −ℓ ∞ n=N Ξ ε+iη H (M ε+iη H) n G ε+iη dη
is a compact operator as soon as N 4. Since moreover, Lemma 4.6 implies that the integral
∞ −∞ ∞ n=N Ξ ε+iη H (M ε+iη H) n G ε+iη B(X) dη < ∞
one sees that the convergence in (4.12) actually holds in operator norm and, as such, U H (t) is the limit of compact operators which proves the compactness of U H (t) for any t τ N and N 4.
The role of the zero eigenvalue of T H can be made more precise here and the asymptotic behaviour of (U H (t)) t 0 follows, yielding a full proof of Theorem 1.3 in the Introduction: and, for any a ∈ (0, λ ⋆ ), there exists a positive constant C a > 0 such that, for any f ∈ X, it holds
U H (t)f − P 0 f X C a exp(−a t) f X ∀t 0
where P 0 denotes the spectral projection associated to the zero eigenvalue.
Proof. With the terminology of [12], Theorem 4.12 asserts that (U H (t)) t 0 is eventually compact. Therefore, from [12, Proposition 9.2], its type ω 0 (U H ) coincide with the spectral bound s(T H ) of its generator ‡ . Because U H (t) B(X) = 1, one has ω 0 (U H ) = 0 = s(T H ).
Due to the eventual compactness of (U H (t)), its essential type ω ess (U H ) is such that
−∞ = ω ess (U H ) < ω 0 (U H ) = 0 = s(T H ).
In particular, 0 is an isolated eigenvalue of T H with finite algebraic multiplicity and there is λ ⋆ > 0 such that
S(T H ) ∩ {λ ∈ C ; Reλ −λ ⋆ } = {0}.
(4.13) Moreover (see [12,Theorem 9.11]), for any a ∈ (0, λ ⋆ ), there is C a > 0 such that
U H (t) (I − P 0 ) f X = U H (t)f − exp (tN 0 ) P 0 f X C a exp (−at) f X (4.14)
for any t 0 and any f ∈ X where P 0 is the spectral projection associated to the zero eigenvalue and N 0 = T H P 0 is a nilpotent bounded operator. Precisely, if m denotes the order of the pole 0 of the resolvent R(·, T H ), one has N m 0 = 0, N j 0 = 0 with j < m and consequently,
exp (tN 0 ) = m−1 k=0 t k k! N k 0 .
Since the semigroup (U H (t)) t 0 is bounded, we deduce that the mapping
t 0 −→ m−1 k=0 t k k! N k 0 P 0 B(X) ‡
This can also be deduced from the fact that (U H (t)) t 0 is a positive C0-semigroup on X = L 1 (Ω × V ), see [12,Theorem 9.5] is bounded. The only way for this to be true is that
N k 0 P 0 = 0 ∀k = 1, . . . , m − 1
which, since N 0 = T H P 0 , implies in particular that T H P 2 0 = 0. Because P 0 is a projection, one has N 0 = 0, i.e. m = 1 which proves the first part of the result. The second part has been established in (4.14) (see also [12,Theorem 9.11]). where the range of T H is closed.
Remark 4.15. Whenever the C 0 -semigroup (U H (t)) t 0 is irreducible, the expression of the spectral projection is more explicit. We recall here that, if one assumes, besides Assumptions 1.2, that In this case, the projection P 0 is given by (1.5), i.e.
k(x, v, v ′ ) > 0 for µ x -a.e. v ∈ Γ − (x), v ′ ∈ Γ + (x).P 0 f = ̺ f Ψ H , with ̺ f = Ω×V f (x, v)dx ⊗ m(dv).
More generally, such an expression of P 0 is true if dimKer(T H ) = 1 (independently of the irreducibility assumption).
E
In this Section, we briefly illustrate the main results established so far for several examples of particular relevance. We also propose several open problems that we believe are of interest for the study of linear transport equations.
We begin with the following example:
Example 5.1. We consider the case in which
k(x, v, v ′ ) = γ −1 (x)G(x, v)
where G : ∂Ω × V → R + is a measurable and nonnegative mapping such that (i) G(x, ·) is radially symmetric and differentiable for π-almost every x ∈ ∂Ω; (ii) G(·, v) ∈ L ∞ (∂Ω) for almost every v ∈ V ; (iii) the mapping x ∈ ∂Ω → G(x, ·) ∈ L 1 (V, |v|m(dv)) is piecewise continuous, (iv) the mapping x ∈ ∂Ω → γ(x) is bounded away from zero where
γ(x) := Γ − (x) G(x, v)|v · n(x)|m(dv) ∀x ∈ ∂Ω,
i.e. there exist γ 0 > 0 such that γ(x) γ 0 for π-almost every x ∈ ∂Ω. In that case, it is easy to show that the associated boundary operator H is satisfying Assumptions 1.2 and, whenever m(dv) = ̟(|v|)dv for some radially symmetric and nonnegative function ̟(|v|), one checks without difficulty that
Assumptions 4.3 are met if lim ̺→∞ ̺ d+2 G(y, ̺)̟(̺) = 0, ∀y ∈ ∂Ω sup y∈∂Ω ∞ r 0 G(y, ̺) ̟ ′ (̺) + ̟(̺) ̺ + |∂ ̺ G(y, ̺)| ̟(̺) ̺ d+2 d̺ < ∞. (5.1)
Under such assumption, the existence of an invariant density Ψ H has been derived in [23,Theorem 6.7] and, for ∂Ω of class C 1,α (α > 1 2 ), the conclusions of Theorem 4.12 and Corollary 4.13 hold true. Notice that, in this case, the zero eigenvalue is simple.
Example 5.2. A more specific case can be considered here which corresponds to the previous Example with
G(x, v) = M θ(x) (v), M θ (v) = (2πθ) −d/2 exp − |v| 2 2θ , x ∈ ∂Ω, v ∈ V = {w ∈ R d ; |w| > r 0 }.
for some r 0 > 0 given. Then,
γ(x) = κ d θ(x) V |w|M 1 (w)dw, x ∈ ∂Ω
for some positive constant κ d depending only on the dimension. Assume the mapping θ : ∂Ω → θ(x) ∈ R + to be continuous and bounded from below by some positive constant,
inf x∈∂Ω θ(x) = θ 0 > 0.
Then, for the special choice
̟(̺) = ̺ m , m 0 exp (α̺ s ) , α > 0, s ∈ (0, 2), exp β̺ 2 ,
β ∈ (0, 1 2θ∞ ) where θ ∞ = sup x∈∂Ω θ(x), one sees that Assumptions 4.3 are met (see (5.1)). Therefore, for ∂Ω of class C 1,α (α > 1 2 ), the conclusions of Theorem 4.12 and Corollary 4.13 hold true. Even if the two previous examples are such that k(x, v, v ′ ) is actually independent of v ′ , our method applies to more general situation since Assumptions 4.3 which provide some practical conditions ensuring the validity of our results is covering, in full generality, the case of a kernel k depending on both v and v ′ . The most physically relevant model of boundary conditions for which the kernel k(x, v, v ′ ) is really depending on the velocity v ′ is the so-called Cercignani-Lampis boundary conditions [9]. Such a model has been thoroughly studied in a recent contribution [7] and, unfortunately, it seems that such a model does not fall into the framework described in the present paper in full generality since the conclusion of Theorem 3.7 does not seem to apply for such a model, see [7,Proposition 13].
We conclude this Section with the following open problems. The first one regards the case in which the 0 is not a simple eigenvalue A second open problem regards the role of the regularity of ∂Ω Open Problem 2. We may wonder if the assumption that ∂Ω is of class C 1,α with α > 1 2 is really necessary. Such an assumption plays a role only in the proof of Lemma 4.4 thanks to Lemma 3.2 but seems only technical and, under the mere assumption ∂Ω of class C 1 , we conjecture that U H (t) is compact for t large enough. Notice also that it would be interesting to extend our results to the case in which ∂Ω is piecewise of class C 1 which would allow to cover also the case stochastic billiards on polygonal tables studied in [17]. A A. T
In this appendix, we provide some insights about the generalisation of the results obtained so far to the general case of partly diffuse boundary operators as introduced in our first contribution [23]. We describe the asymptotic spectrum of the generator and give a conjecture on the quasicompactness of the semigroup. We begin with recalling the definition from [23] adapted to our context (see also [32]): Definition A.1. We shall say that a boundary operator H ∈ B(L 1
+ , L 1 − ) is stochastic partly diffuse if it writes Hψ(x, v) = α(x) Rψ(x, v) + (1 − α(x)) Kψ(x, v), (x, v) ∈ Γ − , ψ ∈ L 1 + (A.1) where α(·) : ∂Ω → [0, 1] is measurable, K ∈ B(L 1 + , L 1 − )
is a stochastic diffuse boundary operator satisfying Assumptions 1.2 and R is a reflection operator
R(ϕ)(x, v) = ϕ(x, V(x, v)) ∀(x, v) ∈ Γ − , ϕ ∈ L 1 + where V : x ∈ ∂Ω → V(x, ·)
is a field of bijective bi-measurable and µ x -preserving mappings
V(x, ·) : Γ − (x) ∪ Γ 0 (x) → Γ + (x) ∪ Γ 0 (x) such that i) |V(x, v)| = |v| for any (x, v) ∈ Γ − . ii) If (x, v) ∈ Γ 0 then (x, V(x, v)) ∈ Γ 0 , i.e. V(x, ·) maps Γ 0 (x) in Γ 0 (x).
iii) The mapping
(x, v) ∈ Γ − → (x, V(x, v)) ∈ Γ + is a C 1 diffeomorphism. Example A.2.
In practical situations, the most frequently used pure reflection conditions are (a) the specular reflection boundary conditions for which
V(x, v) = v − 2(v · n(x)) n(x) (x, v) ∈ Γ − .
Notice that, for V to be a C 1 diffeormorphism, we need ∂Ω to be of class
C 2 . (b) The bounce-back reflection conditions for which V(x, v) = −v, (x, v) ∈ Γ − .
With the classical terminology used in kinetic theory of gases, the parameter
β(x) = 1 − α(x),
x ∈ ∂Ω is referred to as the accomodation coefficient. It has been shown in [23] that
(M 0 H) 2 = (M 0 (βK)) 2 + (M 0 (αR)) 2 + M 0 (αR)M 0 (βK) + M 0 (βK)M 0 (αR) Setting β ∞ := ess sup x∈∂Ω β(x)
one has (M 0 (βK)) 2 is weakly compact and
(M 0 (αR)) 2 + M 0 (αR)M 0 (βK) + M 0 (βK)M 0 (αR) B(L 1 + ) (1 + osc(β)) 2 − β 2 ∞ where osc(β) = esssup x∈∂Ω β(x)−essinf x∈∂Ω β(x)
is the oscillation of β(·). As in [23, Theorem 5.6], we assume that
c β := (1 + osc(β)) 2 − β 2 ∞ < 1. (A.
2) We point out that such an assumption of course excludes the case of pure reflection boundary conditions, corresponding to α ≡ 1. We set
λ β := − r 0 2D log c β > 0
and have the following
Lemma A.3.
Assume that H is a partly diffuse operator in the sense of the above Definition A.1 satisfying (A.2). Then, there is a discrete set Θ ⊂ C such that, for any λ ∈ C with Reλ > −λ β the following alternative holds: i) either 1 is the resolvent set of M λ H ii) or 1 ∈ S p (M λ H) and then λ ∈ Θ.
Proof. Notice that
(M λ H) 2 = (M λ (βK)) 2 + (M λ (αR)) 2 + M λ (αR)M λ (βK) + M λ (βK)M λ (αR) =: (M λ (βK)) 2 + L λ (A.3)
where (M λ (βK)) 2 is a weakly-compact operator (by a simple domination argument). Invoking (4.3), one sees that, for Reλ 0, it holds
L λ B(L 1 + ) L 0 B(L 1 + ) = (M 0 (αR)) 2 + M 0 (αR)M 0 (βK) + M 0 (βK)M 0 (αR) B(L 1 + ) < 1
whereas, for Reλ < 0
L λ B(L 1 + ) exp −2 DReλ r 0 L 0 B(L 1 + ) exp −2 DReλ r 0 c β < 1
as soon as Reλ > r 0 2D log c β . Consequently, r ess (M λ H) 2 < 1 for any Reλ > −λ β . From the spectral mapping theorem, we deduce then that
r ess (M λ H) < 1 ∀Reλ > −λ β .
As a consequence, for Reλ > −λ β ,
1 ∈ S (M λ H) ⇐⇒ 1 ∈ S p (M λ H)
and in particular, if 1 ∈ S (M λ H) then 1 ∈ S p (M λ H) 2 . Let us therefore investigate the spectral problem
g − (M λ H) 2 g = h which, thanks to (A.3) is equivalent to g − L λ g − (M λ (βK)) 2 g = h, i.e. g − R (1, L λ ) (M λ (βK)) 2 g = R (1, L λ ) h.
Since (M λ (βK)) 2 is weakly-compact we deduce from the analytic Fredholm alternative that the set Θ := {λ ∈ C ; Reλ > −λ β and 1 ∈ S(R (1, L λ ) (M λ (βK)) 2 )} is discrete. This in particular implies that the set {λ ∈ C ; Reλ > −λ β and 1 ∈ S p (M λ H) 2 } is discrete. If now Reλ > −λ β and λ ∈ C\Θ, then 1 belongs to the resolvent set of R (1, L λ ) (M λ (βK) 2 ) which implies that 1 is the resolvent set of (M λ H) 2 . This proves the Lemma.
This leads then to the following Proposition A.4. Assume that H is a partly diffuse operator in the sense of the above Definition A.1 which satisfies (A.2). Setting
λ β := − r 0 2D log c β > 0,
for any η ∈ (0, λ β ), S(T H ) ∩ {λ ∈ C ; Reλ −η} consists at most in a finite number of eigenvalues of T H with finite algebraic multiplicities.
Proof. Recall that, for Reλ > −λ β ,
λ ∈ S(T H ) ⇐⇒ 1 ∈ S (M λ H) ⇐⇒ 1 ∈ S p (M λ H) and
λ ∈ S p (T H ) ⇐⇒ 1 ∈ S p (M λ H) . Therefore, from the previous Lemma, S(T H ) ∩ {λ ∈ C ; Reλ > −λ β } consists at most in a discrete set of eigenvalues with finite algebraic multiplicity. Now, if 1 ∈ S p (M λ H) then 1 ∈ S (M λ H) 2 . Since, for any η ∈ (0, λ β ),
lim R→∞ sup |Imλ| R sup Reλ −η (M λ H) 2 B(L 1 + ) = 0
one sees that, for η ∈ (0, λ β ), the set {λ ∈ C ; Reλ −η} ∩ {λ ∈ C ; 1 ∈ S p (M λ H)} is at most finite which proves the result.
We can complement the above with the following Proof. For λ ∈ C, Reλ > −λ β , one has
R(λ, T H ) = R(λ, T 0 ) + Ξ λ HR(1, M λ H)G λ .
One observes that, for any η ∈ (0, λ β ) and any Reλ −η, it holds for any λ ∈ ∆ M,η which achieves the proof.
The spectral structure of T H together with Lemma A.5 allow to show in a standard way that, for any f ∈ D(T H ), one can prove that there is η > 0 and C f 0 such that
U H (t) (I − P 0 ) f X C f exp (−ηt) t 0 (A.4)
where C f actually depends on f and T H f . Such an estimate is a general consequence of an abstract result from [33] which asserts that, for general C 0 -semigroup (V (t)) t 0 on X with for some λ ∈ C \ S (A). The resolvent identity shows that ω m (V ) is independent of λ. In the present situation, once we notice that 0 is an isolated and dominant eigenvalue of T H with finite algebraic multiplicity and denoting by P 0 the associated spectral projection, one can apply the inequality (A.5) with
A = T H (I − P 0 ) , V (t) = U H (t) (I − P 0 ) , t 0
where Lemma A.5 exactly means that s 0 (A) < 0 proving the inequality (A.4). We refer the reader to [30,Section 2] for full details on this approach for similar kind of results for collisional kinetic theory (see also [25]). This leads to the following conjecture We give here a simple proof of Theorem 3.7 in the case in which Ω is of class C 1,α with α > 0. We actually prove that ϕ(x, v ′ )µ x (dv ′ ), ϕ ∈ L 1 + , where R 0 > 0. This of course corresponds to the case k(x, v, v ′ ) ≡ 1. Notice that, being m a locally finite Borel measure over R d , one has m(V ) < ∞. In such a case, Proposition 3.5 asserts that HM λ Hϕ(x, v) = Γ + J λ (x, v, y, w)ϕ(y, w) |w · n(y)|m(dw)π(dy) with J λ (x, v, y, w) = J (x, y)
R 0 r 0 ̺ exp −λ |x − y| ̺ m 0 (d̺) |S d−1 | for any (x, v) ∈ Γ − , (y, w) ∈ Γ + . Thus, |J λ (x, v, y, w)| J (x, y) R 0 r 0 ̺ m 0 (d̺) |S d−1 | C 0 J (x, y)
since m 0 ([r 0 , R 0 ]) < ∞ (recall that m(V ) < ∞). By a domination argument, it is enough to prove the weak compactness of the operator K ∈ B(L 1 + , L 1 (∂Ω)) given by
Kϕ(x) = Γ + J (x, y)ϕ(y, w) |w · n(y)|m(dw)π(dy), ϕ ∈ L 1 + , x ∈ ∂Ω.
This operator can be written as K = J 0 P where P ∈ B(L 1 + , L 1 (∂Ω)) is the projection operator
Pϕ(x) = Γ + (x)
ϕ(x, w)µ x (dw) , x ∈ ∂Ω , ϕ ∈ L 1 + and J 0 ∈ B(L 1 (∂Ω)) is given by
J 0 ψ(x) = ∂Ω
J (x, y)ψ(y)π(dy), x ∈ ∂Ω , ψ ∈ L 1 (∂Ω).
Let us now show that J 0 ∈ B(L 1 (∂Ω)) is weakly compact which will give the result. Again, using Lemma 3.2 together with a domination argument, it is enough to prove the weak compactness of the operator J 1 ∈ B(L 1 (∂Ω)) given by
J 1 ψ(x) = ∂Ω
|x − y| 1+2α−d ψ(y)π(dy), x ∈ ∂Ω , ψ ∈ L 1 (∂Ω).
We note that its kernel is of order strictly less than d − 1 since α > 0. This is done by an approximation argument introducing, for any ε > 0, J ε 1 ψ(x) = ∂Ω 1 |x−y| ε |x − y| 1+2α−d ψ(y)π(dy), x ∈ ∂Ω , ψ ∈ L 1 (∂Ω).
For any ε > 0, J ε 1 has a bounded kernel and is clearly weakly compact while lim
measure m is absolutely continuous with respect to the Lebesgue measure over R d and is orthogonally invariant (i.e. invariant under the action of the orthogonal group of matrices in R d ) * , i.e. there exists a radially symmetric function ̟(v) = ̟(|v|) such that m(dv) = ̟(|v|)dv.
1 2
1and H satisfies 4.3. Then, the C 0
1. 3 .
3Notations. In all the sequel, for any Banach space Y , if A : D(A) ⊂ Y → Y is a given closed and densely defined linear operator, the spectrum of A is denoted by S(A) whereas its point spectrum, i.e. the set of eigenvalues of A, is denoted by S p (A). The spectral bound s(A) of A is defined as s(A) = sup{Reλ , λ ∈ S(A)}.
For any bounded operator B ∈ B(Y ), r σ (B) denotes the spectral radius of B defined as r σ (B) = sup{|λ| ; λ ∈ S(B)} and we recall Gelfand's formula which provides an alternative formulation as
1. 3 .
3Section 5 exhibits several examples of applications of our results as well as some open problems and conjectures about related questions. The paper ends with two Appendices. Appendix A gives a description of the asymptotic spectrum of T H in the more general case of partly diffuse boundary operators and discusses in an informal way the quasi-compactness of (U H (t)) t 0 . Appendix B gives a short proof of the weak compactness of HM λ H. Ministry of Education, University and Research (MIUR), "Dipartimenti di Eccellenza" grant 2018-2022 as well as the support from the de Castro Statistics Initiative, Collegio Carlo Alberto (Torino).
Lemma 4. 2 .
2Let λ ∈ C. Then, λ ∈ S(T H ) if and only if 1 ∈ S(M λ H). In particular S(T H ) = S p (T H ).
Assumptions 4. 3 .
3Assume that m 0 is given by † m 0 (d̺) = |S d−1 |̺ d−1 ̟(̺)d̺ for some positive and differentiable mapping ̟ : [r 0 , ∞) → (0, ∞) with lim ̺→∞ ̺ d+2 k(x, |v|, ̺)k(y, ̺, |w|)̟(̺) = 0, ∀(x, v) ∈ Γ − , (y, w) ∈ Γ + ; (4.4) sup (y,w)∈Γ + k(y, r 0 , |w|) < ∞ . (4.5)
Lemma 4 . 4 .
44Under Assumptions 4.3 and if ∂Ω is of class C 1,α with α > 1 2 , then for any λ ∈ C, λ = 0, it holdssup (y,w)∈Γ + Γ − |J λ (x, v, y, w)| dµ − (x, v) C |λ| exp Dr −1 0 (Reλ) −for some positive C > 0 where (Reλ) − = −min(0, Reλ) denotes the negative part of Reλ. † This means that the measure m is absolutely continuous with respect to the Lebesgue measure over R d with m(dv) = ̟(|v|)dv.
I 2
2(λ, x, v, y, w)|v · n(x)|m(dv) C exp (Reλ) − Dr −1 0 k(y, r 0 , |w|) for some positive constant C > 0 depending only on r 0 . Using (4.5) we get then sup (y,w)∈Γ + Γ − (x) I 2 (λ, x, v, y, w)|v · n(x)|m(dv) C k(·, r 0 , ·) L ∞ (Γ + ) exp (Reλ) − Dr −1 0 . (4.8)
Remark 4 . 10 .
410One sees from the point (2) together with [23, Theorem 2.4] that, for any k 1,
Corollary 4 . 13 .
413Assume that Assumptions 4.3 are in force and ∂Ω is of class C 1,α with α > 1 2 . Then, 0 is a simple pole of the resolvent of T H
Remark 4 . 14 .
414Notice that, since 0 is a simple pole of the resolvent R(·, T H ), its geometrical and algebraic multiplicity (as an eigenvalue of T H ) coincide, i.e. dimKer(T H ) = dimRange(P 0 ) = n ∈ N and X = Ker(T H ) ⊕ Range(T H )
( 4 . 15 )
415Then (see[23, Section 4]) the operator M 0 H is irreducible as well as the C 0 -semigroup (U H (t)) t 0 .The semigroup admits a unique invariant density Ψ H ∈ D(T H ) withΨ H (x, v) > 0 for a. e. (x, v) ∈ Ω × R d , Ψ H X = 1,and Ker(T H ) = Span(Ψ H ).
Open Problem 1 .
1If 0 is not a simple eigenvalue, i.e. if dimKer(T H ) = dim (Range P 0 ) = n > 1 , then one may wonder what is exactly the form of the spectral projection P 0 . We conjecture that, in this case, there exist exactly n distinct nonnegative eigenfunctions Ψ 1 , . . . , Ψ n with pairwise disjoint supports associated to the zero eigenvalue of T H .
Lemma A. 5 .
5Under the Assumption of Proposition A.4, for any η ∈ (0, λ β ), there is M > 0 such that sup R(λ, T H ) B(X) ; Reλ −η , |Imλ| M < ∞.
R
(λ, T 0 ) B(X) R(−η, T 0 ) B(X) . Since lim |Imλ|→∞ (M λ H) on {λ ∈ C ; Reλ −η}, for any c < 1, there is M > 0 such that (M λ H) set ∆ M,η := {λ ∈ C ; Reλ −η ; |Imλ| M }.In particular, r σ (M λ H) < 1 for any λ ∈ ∆ M,η andR(1, M λ H) = ∞ n=0 (M λ H) n λ ∈ ∆ M,ηWriting n = 2k + s with k ∈ N and s ∈ {0, 1}, one sees that, for λ ∈ ∆ M,η ,R(1, M λ H)
m (A) := inf s > s(A) ; R(α + iβ, A) B(X) = O (|β| m ) as |β| → ∞, α s and ω m (V ) = inf{ω ∈ R ; sup t 0 e −ωt V (t)R(λ, A) m B(X) < ∞}
Conjecture A. 6 .
6We conjecture that, under the Assumption of Proposition A.4, the C 0 -semigroup (U H (t)) t 0 admits a positive spectral gap λ 0 ∈ (0, λ β ) such thatU H (t) (I − P 0 ) B(X) = O (exp (−λ 0 t)
HM λ H : L 1 + → L 1 − is weakly-compact for any Reλ 0. (B.1)As in the proof of [23, Theorem 5.1] by approximation and domination arguments, to prove the result, we can restrict ourselves without loss of generality to the case in whichV := {v ∈ R d ; r 0 |v| R 0 }, Hϕ(x, v) = Γ + (x)
J 1 B(L 1 (∂Ω)) = 0 because the kernel of J ε 1 − J 1 is supported on {|x − y| < ε} (see [19,Proposition 3.11 & Exercise 1, page 121-123]). This proves (B.1) and achieves the proof of Theorem 3.7.
U
23, Proof of Theorem 6.7]. We notice here that Assumptions 6.1 and 4.4 of [23] are satisfied under our Assumptions 1.1 and 1.2. For this one deduces that I − M λ H ∈ B(L 1 + ) is invertible and the expression of the resolvent (2.8) is then easy to deduce (see e.g. [5, Theorem 4.2]). Remark 2.3. As already mentioned, the previous result holds true in a more general situation, in particular, it still holds whenever 0 ∈ V . 3. G H 3.1. Useful change of variables from [24]. We begin this section with a very useful change of variables, derived in our previous contribution [24, Section 6] (in particular, it still holds true if 0 ∈ V ), which can be formulated as follows Proposition 3.1. Assume that ∂Ω satisfies Assumptions 1.1. For any x ∈ ∂Ω, we set
) .
)Proposition 3.5. Assume that H satisfy Assumptions 1.2. For any λ ∈ C + , it holds
Email address: [email protected] Email address: [email protected]
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Acknowledgments. B. Lods gratefully acknowledges the financial support from the Italian
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| []
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[]
| [
"\n‡Department of Mathematics\nLINGLING FAN † AND XIANDE YANG ‡ †Department of Mathematics and Statistics\nMemorial University of Newfoundland\nA1C 5S7St.John'sCanada\n",
"\nHarbin Institute of Technology\n150001HarbinChina\n"
]
| [
"‡Department of Mathematics\nLINGLING FAN † AND XIANDE YANG ‡ †Department of Mathematics and Statistics\nMemorial University of Newfoundland\nA1C 5S7St.John'sCanada",
"Harbin Institute of Technology\n150001HarbinChina"
]
| []
| Let R be an associative ring with identity, C(R) denote the center of R, and g(x) be a polynomial in the polynomial ring C(R)[x]. R is called strongly g(x)-clean if every element r ∈ R can be written as r = s + u with g(s) = 0, u a unit of R, and su = us. The relation between strongly g(x)-clean rings and strongly clean rings is determined, some general properties of strongly g(x)-clean rings are given, and strongly g(x)-clean rings generated by units are discussed. | null | [
"https://arxiv.org/pdf/0803.3353v1.pdf"
]
| 18,300,046 | 0803.3353 | 03a27ca150d523c208655983bddc6e642b40ce52 |
24 Mar 2008
‡Department of Mathematics
LINGLING FAN † AND XIANDE YANG ‡ †Department of Mathematics and Statistics
Memorial University of Newfoundland
A1C 5S7St.John'sCanada
Harbin Institute of Technology
150001HarbinChina
24 Mar 2008ON STRONGLY g(x)-CLEAN RINGS 1strongly g(x)-clean ringsstrongly clean ringsrings generated by units Mathematics Subject Classification: 16U6016U99
Let R be an associative ring with identity, C(R) denote the center of R, and g(x) be a polynomial in the polynomial ring C(R)[x]. R is called strongly g(x)-clean if every element r ∈ R can be written as r = s + u with g(s) = 0, u a unit of R, and su = us. The relation between strongly g(x)-clean rings and strongly clean rings is determined, some general properties of strongly g(x)-clean rings are given, and strongly g(x)-clean rings generated by units are discussed.
Introduction
Let R be an associative ring with the group of units U (R). R is called clean if for every element r ∈ R, r = e + u with e 2 = e ∈ R and u ∈ U (R) [10] and R is called strongly clean if in addition, eu = ue [11].
Let C(R) denote the center of a ring R and g(x) be a polynomial in C(R) [x]. Camillo and Simón [2] say R is g(x)-clean if for every element r ∈ R, r = s + u with g(s) = 0 and u ∈ U (R). If V is a countable dimensional vector space over a division ring D, Camillo and Simón proved that End( D V ) is g(x)-clean if g(x) has two distinct roots in C(D) [2].
Nicholson and Zhou generalized Camillo and Simón's result by proving that
End( R M ) is g(x)-clean if R M is a semisimple R-module and g(x) ∈ (x − a)(x − b)C(R)[x]
where a, b ∈ C(R) and b, b − a ∈ U (R) [12]. [5,14] completely determined the relation between clean rings and g(x)-clean rings independently. What is the relation between strongly clean rings and g(x)-clean rings? In this paper, we continue this topic. In Section 2, we define strongly g(x)-clean rings and determine the relation between strongly g(x)-clean rings and strongly clean rings; in Section 3, some general properties of strongly g(x)-clean rings are given; and in Section 4, some classes of strongly g(x)-clean rings generated by units are discussed.
Throughout the paper, T n (R) denotes the upper triangular matrix ring of order n over R, N denotes the set of all positive integers, and Z represents the ring of integers.
2. strongly g(x)-clean rings vs strongly clean rings
Definition 2.1. Let g(x) ∈ C(R)[x]
be a fixed polynomial. An element r ∈ R is strongly g(x)-clean if r = s + u with g(s) = 0, u ∈ U (R), and su = us. R is strongly g(x)-clean if every element of R is strongly g(x)-clean.
Strongly clean rings are exactly strongly (x 2 − x)-clean rings. However, there are strongly g(x)-clean rings which are not strongly clean and vice versa:
Let Z (p) = { m n ∈ Q : gcd(p, n) = 1 and p prime } be the localization of Z at the prime ideal pZ and C 3 be the cyclic group of order 3.
Example 2.3. Let R = Z (p) and g(x) = (x − a)(x 2 + 1) ∈ C(R)[x]
. Then R is strongly clean but by a easy verification we know R is not strongly g(x)-clean. Let R be a boolean ring with more than two elements with c = 0, 1. Then R is strongly clean but R is not
strongly g(x) = (x + 1)(x + c)-clean by [5, Example 2.3].
However, for some type of polynomials, strong cleanness and strong g(x)-cleanness are equivalent.
Theorem 2.4. Let R be a ring and g(x) ∈ (x − a)(x − b)C(R)[x] with a, b ∈ C(R)
. Then the following hold:
(1) R is strongly (x − a)(x − b)-clean if and only if R is strongly clean and (b − a) ∈ U (R). (2) If R is strongly clean and (b − a) ∈ U (R), then R is strongly g(x)-clean. Proof. (1). "⇐". Let r ∈ R. Since R is strongly clean and (b − a) ∈ U (R), r−a b−a = e + u where e 2 = e ∈ R, u ∈ U (R), and eu = ue. Thus, r = [e(b − a) + a] + u(b − a) where u(b − a) ∈ U (R), [e(b − a) + a − a][e(b − a) + a − b] = 0, and [e(b − a) + a]u(b − a) = u(b − a)[e(b − a) + a]. Hence, R is strongly (x − a)(x − b)-clean. "⇒". Since a is strongly (x − a)(x − b)-clean, there exist u ∈ U (R) and s ∈ R such that a = s + u with (s − a)(s − b) = 0 and su = us. Hence, s = b. So (b − a) ∈ U (R). Let r ∈ R. Since R is strongly (x − a)(x − b)-clean, r(b − a) + a = s + u where (s − a)(s − b) = 0, u ∈ U (R), and su = us. Thus, r = s−a b−a + u b−a where u b−a ∈ U (R), ( s−a b−a ) 2 = (s−a)(s−b+b−a) (b−a) 2 = (s−a)(b−a) (b−a) 2 = s−a b−a , and s−a b−a · u b−a = u b−a · s−a b−a .
So R is strongly clean.
(2). By (1).
Corollary 2.5. For a ring R, R is strongly clean if and only if R is strongly
(x 2 + x)- clean.
Proof. It follows from Theorem 2.4 by letting a = 0 and b = −1.
Remark 2.6. The equivalence of strong (x 2 + x)-cleanness and strong cleanness is a ring property since it holds for a ring R but it may fail for a single element. For example, 1 + 1 = 2 ∈ Z is strongly clean but 2 is not strongly (x 2 + x)-clean in Z.
(x − ea)(x − eb)C(R)[x] and R is strongly (x − a)(x − b)-clean with a, b ∈ C(R), then eRe is strongly g(x) -clean.
Proof. By Theorem 2.4, R is strongly (x − a)(x − b)-clean if and only if R is strongly clean and b − a ∈ U (R). If R is strongly clean, then eRe is strongly clean by [13]. Again by Theorem 2.4, eRe is strongly (x − ea)(x − eb)-clean.
However, generally, strongly g(x)-clean property is not a Morita invariant: When
g(x) = (x − a)(x − b) where a, b ∈ C(R) with b − a ∈ U (R)
, the matrix ring over the local ring Z (p) is not strongly clean [3] (hence, not strongly g(x)-clean).
strongly g(x)-clean rings vs rings generated by units and roots of 1
For any n ∈ N, U n (R) denotes the set of elements of R which can be written as a sum of no more than n units of R [8]. A ring R is called generated by its units if R = ∞ n=1 U n (R). We use strong g(x)-cleanness to characterize some rings in which every element can be written as the sum of unit and a root of 1 which commute. (1) R is strongly (x 2 − 2 n x)-clean.
(2) R is strongly (x 2 − 1)-clean.
(3) R is strongly clean and 2 ∈ U (R).
(4) R = U 2 (R) and for any a ∈ R, a can be expressed as a = u + v with some u, v ∈ U (R), uv = vu, and v 2 = 1. Thus, a = (−u)
+ (1 − v) with −u ∈ U (R), (1 − v) 2 = 2(1 − v), and (−u)(1 − v) = (1 − v)(−u)
. By "(1) ⇔ (3)" and n = 1, we proved that (4) implies (3).
(2) ⇒ (4). If R is strongly (x 2 − 1)-clean, then for any r ∈ R, there exist v, u ∈ U (R) such that r = v + u with v 2 = 1 and uv = vu. Proof. The first statement is clear. Let r ∈ R. Then r = s + u with u ∈ U (R), s 2 + s + 1 = 0, and su = us. So s 4 − s = 0. Thus, R is strongly (x 4 − x)-clean. Moreover, ever element in strongly (x 2 + x + 1)-clean ring R can be written as the sum of a unit and a cubic root of 1 which commute with each other.
(x 2 + cx + d)-clean, then R = U 2 (R). In particular, if R is strongly (x 2 + x + 1)-clean, then R = U 2 (R) is strongly (x 4 − x)-clean
Lemma 4.6.
[5] Let a ∈ R. The following are equivalent for n ∈ N:
(1) a = a(ua) n for some u ∈ U (R).
(2) a = ve for some e n+1 = e and some v ∈ U (R).
(3) a = f w for some f n+1 = f and some w ∈ U (R). Finally, we give a property which has nothing to do with rings generated by units but it relates to strongly (x n − x)-clean rings.
Proposition 4.8. Let R be a ring and n ∈ N. Then R is strongly (ax 2n − bx)-clean if and only if R is strongly (ax 2n + bx)-clean.
Proof. " ⇒ ". Suppose R is strongly (ax 2n − bx)-clean. Then for any r ∈ R, −r = s + u, as 2n −bs = 0, u ∈ U (R), and su = us. So r = (−s)+(−u) where (−u) ∈ U (R), a(−s) 2n + b(−s) = 0, and (−s)(−u) = (−u)(−s). Hence, r is strongly (ax 2n + bx)-clean. Therefore, R is strongly (ax 2n + bx)-clean.
" ⇐ ". Suppose R is strongly (ax 2n + bx)-clean. Let r ∈ R. Then there exist s and u such that −r = s + u, as 2n + bs = 0, u ∈ U (R), and su = us. So r = (−s) + (−u) satisfies a(−s) 2n − b(−s) = 0, −u ∈ U (R), and (−s)(−u) = (−u)(−s). Hence, R is strongly (ax 2n − bx)-clean.
For 2n + 1 ∈ N, we do not know if the strong (x 2n+1 − x)-cleanness of R is equivalent to the strong (x 2n+1 + x)-cleanness of R.
Example 2 . 2 .
22Let R be a commutative local or commutative semiperfect ring with 2 ∈ U (R). By the proof of [14, Theorem 2.7], RC 3 is strongly (x 6 − 1)-clean. In particular, Z (7) C 3 is a strongly (x 6 − 1)-clean ring. Furthermore, by [5, Example 2.2], Z (7) C 3 is strongly (x 4 − x)-clean. However, Z (7) C 3 is not strongly clean [7, Example 1].
Theorem 4 . 1 .
41Let R be a ring and n ∈ N. Then the following are equivalent:
Proof. (1) ⇒ (3). To prove 2 ∈ U (R). Suppose 2 / ∈ U (R), then R = R/(2 n R) = 0. Let 2 n = s + u with s 2 − 2 n s = 0, u ∈ U (R), and su = us. 0 = 2 n = s + u implies that s = −u ∈ U (R). But s 2 = s 2 = 2 n s = 0, a contradiction. So 2 ∈ U (R). Let a = 0 and b = 2 n . Then by (1) of Theorem 2.4, R is strongly clean.
( 3 )
3⇒ (1). By (1) of Theorem 2.4, R is strongly (x 2 − 2 n x)-clean.
( 3 )
3⇒ (4). Let a ∈ R. By "(1) ⇔ (3)", let n = 1. Then 1 − a = s + u where s 2 = 2s, u ∈ U (R), and su = us. Then a = (−u) + (1 − s) with −u ∈ U (R), (1 − s) 2 = 1, and (−u)(1 − s) = (1 − s)(−u).
( 4 )
4⇒ (3). Let a ∈ R. By (4), 1 − a = u + v where u ∈ U (R), v 2 = 1, and uv = vu.
( 4 )
4⇒(2). Let a ∈ R. Then a can be expressed as a = u + v with u, v ∈ U (R), v 2 = 1, and uv = vu. So v is the root of x 2 − 1. Hence, R is strongly (x 2 − 1)-clean.
Example 4. 2 .
2Rings in Example 2.7 are strongly (x 2 − nx)-clean. In particular, they are strongly (x 2 − 2 n x)-clean rings in which every element can be written as the sum of a unit and a square root of 1 which commute.
Example 4. 3 .
3Let F be a field and V be a vector space over F of infinite dimension, and let R be the subring of E = End F (V ) generated by the identity and the finite rank transformations. Then R is strongly clean [13, Example 7]. In fact, E is locally Artinian. So the matrix ring M k (E) is strongly (x 2 − nx)-clean with charF |n. If charF = 2, then every element in the matrix ring can be written as the sum of a unit and a square root of 1 which commute.
Example 4. 4 .
4Let A = F [x 1 , x 2 , · · · ] be the polynomial ring in a countably infinite set of indeterminates (x 1 , x 2 , · · · ) over a field F , and let I = (x k1 1 , x k2 2 , x k3 3 , ...) with k i > 0. Then R = A/I is a local ring of dimension 0 which is not Noetherian. But R is locally Artinian. So the matrix ring M k (R) is strongly (x 2 − nx)-clean with charF |n.If charF = 2, then every element in the matrix ring can be written as the sum of a unit and a square root of 1 which commute.
Proposition 4. 5 .
5Let R be a ring with c, d ∈ C(R) and d ∈ U (R). If R is strongly
with every element is the sum of a unit and a cubic root of 1 which commute with each other.
Proposition 4 . 7 .
47Let R be an strongly (x n − x)-clean ring where n ≥ 2 and a ∈ R.Then either (i) a = u + v where u ∈ U (R), v n−1 = 1, and uv = vu or (ii) both aR and Ra contain non-trivial idempotents.Proof. Since R is strongly (x n − x)-clean, a = s + u with u ∈ U (R),s n = s, and su = us. Then s n−1 a = s n−1 u + s. So (1 − s n−1 )a = (1 − s n−1 )u. Since 1 − s n−1 is an idempotent, by Lemma 4.6, (1 − s n−1 )u = vg where v ∈ U (R) and g 2 = g ∈ R. So g = v −1 (1 − s n−1 )a ∈ Ra. Suppose (i) does not hold, then 1 − s n−1 = 0, this implies g = 0. Thus, Ra contains a non-trivial idempotent. Similarly, aR contains a non-trivial idempotent.
AcknowledgmentsThe first author is partially supported by NSERC, Canada and the second is supported by the Research Grant from Harbin Institute of Technology.
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The Nicholson-Varadarajan theorem on clean linear transformations. V P Camillo, J J Simón, Glasgow Math. J. 44V. P. Camillo and J. J. Simón, The Nicholson-Varadarajan theorem on clean linear transformations, Glasgow Math. J., 44 (2002): 365-369.
On strongly clean matrix and trianglular matrix rings. J Chen, X Yang, Y Zhou, Comm. Algebra. 3410J. Chen, X. Yang, and Y. Zhou, On strongly clean matrix and trianglular matrix rings, Comm. Algebra, 34 (10) (2006): 3659-3674.
A question on strongly clean rings. W Chen, Comm. Algebra. 347W. Chen, A question on strongly clean rings, Comm. Algebra, 34 (7) (2006): 2374-2350.
On rings whose elements are the sum of a unit and a root of a fixed polynomial. L Fan, X Yang, Comm. Algebra. 361L. Fan and X. Yang, On rings whose elements are the sum of a unit and a root of a fixed polynomial, Comm. Algebra, 36 (1) 2008: 269-278.
Strongly clean property and stable range one of some rings. L Fan, X Yang, preprintL. Fan and X. Yang, Strongly clean property and stable range one of some rings, preprint.
Extensions of clean rings. J Han, W K Nicholson, Comm. Algebra. 20J. Han and W. K. Nicholson, Extensions of clean rings, Comm. Algebra, 20 (2001): 2589-2596.
Two classes of rings generated by their units. M Henriksen, J. Algebra. 31M. Henriksen, Two classes of rings generated by their units, J. Algebra, 31 (1974): 182-193.
Clean semiprime f -rings with bounded inversion. W W Mcgovern, Comm. Algebra. 7W. W. McGovern, Clean semiprime f -rings with bounded inversion, Comm. Algebra, 31 (7) (2003): 3295-3304.
Lifting idempotents and exchange rings. W K Nicholson, Trans. Amer. Math. Soc. 229W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc., 229 (1977): 269-278.
Strongly clean rings and Fitting's lemma. W K Nicholson, Comm. Algebra. 27W. K. Nicholson, Strongly clean rings and Fitting's lemma, Comm. Algebra, 27 (1999): 3583-3592.
Endomorphisms that are the sum of a unit and a root of a fixed polynomial. W K Nicholson, Y Zhou, Canad. Math. Bull. 49W. K. Nicholson and Y. Zhou, Endomorphisms that are the sum of a unit and a root of a fixed polynomial, Canad. Math. Bull., 49 (2006): 265-269.
Sánchez Campos, On strongly clean rings. unpublishedSánchez Campos, On strongly clean rings, 2002, unpublished.
A note on clean rings. Z Wang, J Chen, Algebra Colloquium. 143Z. Wang and J. Chen, A note on clean rings, Algebra Colloquium, 14 (3) (2007): 537-540.
| []
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[
"COMPARING THE BENEFIT OF SYNTHETIC TRAINING DATA FOR VARIOUS AUTOMATIC SPEECH RECOGNITION ARCHITECTURES",
"COMPARING THE BENEFIT OF SYNTHETIC TRAINING DATA FOR VARIOUS AUTOMATIC SPEECH RECOGNITION ARCHITECTURES"
]
| [
"Nick Rossenbach \nHuman Language Technology and Pattern Recognition\nComputer Science Department\nRWTH Aachen University\n52074AachenGermany\n\nAppTek GmbH\n52062AachenGermany\n",
"Mohammad Zeineldeen \nHuman Language Technology and Pattern Recognition\nComputer Science Department\nRWTH Aachen University\n52074AachenGermany\n\nAppTek GmbH\n52062AachenGermany\n",
"Benedikt Hilmes \nHuman Language Technology and Pattern Recognition\nComputer Science Department\nRWTH Aachen University\n52074AachenGermany\n",
"Ralf Schlüter \nHuman Language Technology and Pattern Recognition\nComputer Science Department\nRWTH Aachen University\n52074AachenGermany\n\nAppTek GmbH\n52062AachenGermany\n",
"Hermann Ney \nHuman Language Technology and Pattern Recognition\nComputer Science Department\nRWTH Aachen University\n52074AachenGermany\n\nAppTek GmbH\n52062AachenGermany\n"
]
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"Human Language Technology and Pattern Recognition\nComputer Science Department\nRWTH Aachen University\n52074AachenGermany",
"AppTek GmbH\n52062AachenGermany",
"Human Language Technology and Pattern Recognition\nComputer Science Department\nRWTH Aachen University\n52074AachenGermany",
"AppTek GmbH\n52062AachenGermany",
"Human Language Technology and Pattern Recognition\nComputer Science Department\nRWTH Aachen University\n52074AachenGermany",
"Human Language Technology and Pattern Recognition\nComputer Science Department\nRWTH Aachen University\n52074AachenGermany",
"AppTek GmbH\n52062AachenGermany",
"Human Language Technology and Pattern Recognition\nComputer Science Department\nRWTH Aachen University\n52074AachenGermany",
"AppTek GmbH\n52062AachenGermany"
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| Recent publications on automatic-speech-recognition (ASR) have a strong focus on attention encoder-decoder (AED) architectures which tend to suffer from over-fitting in low resource scenarios. One solution to tackle this issue is to generate synthetic data with a trained text-to-speech system (TTS) if additional text is available. This was successfully applied in many publications with AED systems, but only very limited in the context of other ASR architectures. We investigate the effect of varying pre-processing, the speaker embedding and input encoding of the TTS system w.r.t. the effectiveness of the synthesized data for AED-ASR training. Additionally, we also consider internal language model subtraction for the first time, resulting in up to 38% relative improvement. We compare the AED results to a state-of-theart hybrid ASR system, a monophone based system using connectionist-temporal-classification (CTC) and a monotonic transducer based system. We show that for the later systems the addition of synthetic data has no relevant effect, but they still outperform the AED systems on LibriSpeech-100h. We achieve a final word-error-rate of 3.3%/10.0% with a hybrid system on the clean/noisy test-sets, surpassing any previous state-of-the-art systems on Librispeech-100h that do not include unlabeled audio data. | 10.1109/asru51503.2021.9688255 | [
"https://arxiv.org/pdf/2104.05379v3.pdf"
]
| 233,210,536 | 2104.05379 | e9a2c9f136bb39305a4654cf23fc91746b831ce4 |
COMPARING THE BENEFIT OF SYNTHETIC TRAINING DATA FOR VARIOUS AUTOMATIC SPEECH RECOGNITION ARCHITECTURES
Nick Rossenbach
Human Language Technology and Pattern Recognition
Computer Science Department
RWTH Aachen University
52074AachenGermany
AppTek GmbH
52062AachenGermany
Mohammad Zeineldeen
Human Language Technology and Pattern Recognition
Computer Science Department
RWTH Aachen University
52074AachenGermany
AppTek GmbH
52062AachenGermany
Benedikt Hilmes
Human Language Technology and Pattern Recognition
Computer Science Department
RWTH Aachen University
52074AachenGermany
Ralf Schlüter
Human Language Technology and Pattern Recognition
Computer Science Department
RWTH Aachen University
52074AachenGermany
AppTek GmbH
52062AachenGermany
Hermann Ney
Human Language Technology and Pattern Recognition
Computer Science Department
RWTH Aachen University
52074AachenGermany
AppTek GmbH
52062AachenGermany
COMPARING THE BENEFIT OF SYNTHETIC TRAINING DATA FOR VARIOUS AUTOMATIC SPEECH RECOGNITION ARCHITECTURES
Index Terms-speech recognitiontext-to-speechsemi- supervised trainingarchitecture comparison
Recent publications on automatic-speech-recognition (ASR) have a strong focus on attention encoder-decoder (AED) architectures which tend to suffer from over-fitting in low resource scenarios. One solution to tackle this issue is to generate synthetic data with a trained text-to-speech system (TTS) if additional text is available. This was successfully applied in many publications with AED systems, but only very limited in the context of other ASR architectures. We investigate the effect of varying pre-processing, the speaker embedding and input encoding of the TTS system w.r.t. the effectiveness of the synthesized data for AED-ASR training. Additionally, we also consider internal language model subtraction for the first time, resulting in up to 38% relative improvement. We compare the AED results to a state-of-theart hybrid ASR system, a monophone based system using connectionist-temporal-classification (CTC) and a monotonic transducer based system. We show that for the later systems the addition of synthetic data has no relevant effect, but they still outperform the AED systems on LibriSpeech-100h. We achieve a final word-error-rate of 3.3%/10.0% with a hybrid system on the clean/noisy test-sets, surpassing any previous state-of-the-art systems on Librispeech-100h that do not include unlabeled audio data.
INTRODUCTION
Many publications [1,2,3] have shown that ASR systems using an AED architecture can achieve similar performance on large corpora compared to other methods such as hiddenmarkov-model (HMM) based hybrid models [4], CTC models [5,6] or recurrent neural network Transducer (RNN-T) models [7,8]. For smaller corpora however, they usually suffer from a much stronger performance loss [4,9]. To increase the effectiveness of attention-based ASR systems different methods were proposed, such as data augmentation techniques like SpecAugment [10], various regularization techniques [11], generating synthetic data from additional text [12,13,14,15,16] or using unlabeled speech data [9,16,17].
In this work we will make use of SpecAugment and a Tacotron-2 [18] style TTS system to boost the performance of ASR systems on LibriSpeech-100h [19], which is a quite common task to test the performance of synthetic data or semi-supervised training approaches. LibriSpeech-360h and LibriSpeech-500h are then used as text-only or audio-only data. In this context we experiment with different text encoding, speaker encoding and a novel approach to improve on the stability issues in autoregressive TTS systems. These issues are caused by the nature of the ASR data in contrast to TTS targeted data [13,20]. It was shown in [20] that the stability does not only depend on the used TTS system, but also on the quality and processing style of the data. LibriSpeech contains many utterances with unnaturally long pauses, which are causing stability issues in TTS, also referred to as "biasproblem" [21]. While it is impossible to have an objective metric for TTS quality without using human ratings, [22] presented two objective metrics which give an indication for stability issues.
Most previous publications on generating synthetic data for LibriSpeech-100h only aimed at improving a single AED system [12,14,15]. In this work, we will use four fundamentally different architectures to show the effects of synthetic data. For RNN-T systems, there is prior work on domain adaptation with TTS [23] and a recent publication on improving LibriSpeech [24] by using synthetic data. The four state-of-the-art baselines we present are an AED system, a hybrid ASR system, a CTC system and a monotonic RNN-T system. One objective of this paper is to determine how much we can close the gap between an AED system and a hybrid system by adding synthetic data under fair conditions. To the best of our knowledge no recent publication shows strong results with a hybrid ASR system on LibriSpeech-100h, with the previous best results being reported in [4]. We wanted to see if models without label context such as the hybrid and the CTC acoustic models can improve with synthetic data, and if yes, how much. In [14] it was shown that synthetic data and language model fusion have orthogonal effects. As ASR systems with label context such as AED and RNN-T benefit from subtracting an estimated internal language model (ILM) [25,26,27,28,29], we investigate if the benefits of synthetic data from additional text are retained when using ILM subtraction methods for AED systems.
We make general versions of the used RETURNN [30] and RASR [31] toolkit configuration files available online 1 . All experiments in this work were managed with Sisyphus [32].
SPEECH RECOGNITION
Attention-based ASR
The AED system is inspired by previous work on attentionbased ASR systems using bidirectional long-short-termmemory (BLSTM) [33] layers as encoder and decoder [34,35]. Our encoder consists of 2 convolutional layers followed by 6 BLSTM layers with 1024 dimensions per direction. We use time downsampling via max-pooling layers with a factor of 6. We apply regularization techniques for both encoder and decoder similar to [11]. We apply a dropout [36] of 30% to the LSTM input and 30% drop-connect to the recurrent hidden-to-hidden weight matrices. Our decoder consists of a Zoneout-LSTM layer [37] with a dimension of 1000. We apply 30% attention dropout and embedding dropout, as well as using 10 −3 weight decay and 0.1 label smoothing. We use byte-pair-encoding (BPE) [38] as output labels with a vocabulary size of 2k. We add CTC as additional loss on top of the encoder for training stability.
Hybrid ASR
The hybrid ASR system used in this paper follows the trainclean-100 model presented in [4]. The initial alignment is created by using a Gaussian-mixture-model (GMM) that is as described in Section 2.1.1 of the aforementioned paper. The hybrid BLSTM acoustic model (AM) consists of 8 BLSTMlayers with 1024 hidden units per direction. The model uses 12k classification and regression tree (CART) [39] based labels and is trained with standard cross-entropy loss. We do not use dropout or L2 constraints for the hybrid training. The BLSTM network is implemented in RETURNN, the GMMtraining and the decoder in RASR.
CTC ASR
Similar to the monotonic RNN-T model presented in [40], our CTC model uses a 6-Layer BLSTM network, with a factor 2 max-pooling layer applied between the third and fourth layer. Each LSTM layer has a hidden dimension of 512 per direction and includes an L2 loss and dropout of 0.1 for regularization. The CTC labels consists of the CMUDict 2 phonemes with and 1 https://github.com/rwth-i6/returnn-experiments/tree/ master/2021-tts-asr-librispeech100h 2 http://www.speech.cs.cmu.edu/cgi-bin/cmudict/ without an additional end-of-word labeling symbol "#", and a unified silence/blank symbol. This results in a total number of 139 labels. The BLSTM network is implemented in RETURNN. The state machine for Baum-Welch loss computation used during training and the decoder are implemented in RASR.
RNN-Transducer ASR
The monotonic RNN-T model also follows [40], using the same encoder model and labels as the CTC model. The prediction network is a 2-layer LSTM with 1024 dimensions each, so other than in [40] the model uses the complete history for the prediction network. During training three different losses are used. First, the model includes a CTC loss on the encoder output states which is the same as in the CTC model. Then, there is both a normal CE loss with the aligned labels as target, but also a segmental loss which excludes blank positions.
Language Modeling
For the attention-based setup we trained a 24-layer Transformer language model (LM) following the setup described in [41]. The BPE-perplexity of the final model on the dev-sets is 18.6, which would be equivalent to 73.6 on word-level. For the hybrid and CTC setup, we use a 2-layer LSTM network with a hidden dimension of 4096 and an output vocabulary of 200k, as a Transformer network is computationally too expensive in these scenarios. The word-level perplexity on the dev-sets is 60.
SPEECH SYNTHESIS
The synthesis system consists of an LSTM-based encoderdecoder architecture with a location-sensitive attention. This architecture is commonly referred to as "Tacotron-2" [18]. The setup closely follows [14], with a different variation of input and speaker embedding as well as minor changes. The Zoneout-LSTM in the encoder was replaced by our native CUDA LSTM implementation and a dropout layer for increased training and decoding speed. For the decoder, we kept the 2-layer Zoneout-LSTMs, but used a hidden dimension of 768. To convert the log-mel features of the TTS system into audio-signals, we use a separately trained 2-layer BLSTM to convert the features first back into linear STFT features [14]. The linear features are transformed into raw-waveforms with the Griffin & Lim [42] algorithm and stored as .ogg files. We found out that the usage of .ogg files has no effect on the ASR training. Also, the synthetic data is required to be stored in audio waveform instead of features, as the different ASR systems have independent and varying feature pipelines.
Input Encoding
In this work we experimented with character embeddings as well as phoneme embeddings. The phoneme representations are based on the CMU pronunciation dictionary. We use Sequitur [43] to generate the phoneme representations for words not part of the dictionary.
Silence Preprocessing
The LibriSpeech corpus has an utterance structure that is not beneficial for TTS systems [20]. Especially longer silence parts that occur within an utterance can severely influence the model performance. In this paper we compare two approaches for silence pre-processing, namely threshold-based silence filtering and a newly introduced GMM alignmentbased silence filtering. For threshold-based filtering we use the FFMPEG silence-remove filter with a (silence-)threshold value of -40dB. For the GMM-based silence processing we train a simple GMM-HMM model starting from a linear alignment. It uses 16-dimensional MFCC features with first and second derivatives as well as energy features, as in [4]. The alignment and the mixture densities are updated over 75 iterations, followed by 10 iterations splitting and re-estimating the mixture densities. A final alignment is used to determine all silence frames. We extract timestamps for all silence regions, and remove those parts excluding a predefined amount ∆t at the silence region borders from the audio files. This means for a silence region from, e.g., 2s to 4s in an utterance we would remove the audio from 2.25s to 3.75s if ∆t = 0.5s, or all of it if ∆t = 0s. If any, we keep only silence between words, not before or after the utterance.
Speaker Modeling
For controlling the speaker characteristics of the TTS system, we compare a supervised and an unsupervised approach. For the supervised speaker embeddings we simply add a 256dimensional look-up table for all 251 speakers of the training corpus. As unsupervised method we use a reference encoder with global style tokens (GST) [44]. The reference encoder and the GST network are implemented as described in [14].
EXPERIMENTS
All our experiments are conducted by using LibriSpeech-100h as training data for the TTS and the baseline ASR systems, and the text of LibriSpeech-360h for synthesizing additional training data. We choose this scenario, as this is a very common task for semi-supervised training, and allows to compare our results with previous literature [12,14,15,16]. The best checkpoints are determined by the minimum of the negative log-likelihood score on a holdout set consisting of a subset of dev-clean and dev-other. All systems use SpecAugment [2] in training. Note that we optimized the parameters and training settings for each of the different ASR architectures presented individually in order to achieve the respective best performance on LibriSpeech-100h. The scales for language model integration were tuned on dev-clean/dev-other.
Evaluation Methods
As evaluation method for our ASR systems we use word error rate (WER). For the evaluation of our TTS systems we use two metrics as proposed in [22]: Word Deletion Rate (WDR) and Unaligned Duration Ratio (UDR). The WDR is defined as the relative amount of non-generated words by the TTS, which is the deletion rate in WER evaluation. The UDR is given by the ratio of not aligned audio segments of a length greater than a certain threshold, which was defined to be one second. For further details on these metrics refer to [22].
Stability Analysis of the TTS
For each trained TTS system, we synthesized the text of Lib-riSpeech dev-clean and test-clean and computed the UDR and WDR using existing ASR systems. For the UDR, we used a GMM-HMM ASR model and ran a forced-alignment on the synthesized data to label each frame as "speech" or "silence/noise". Following [22], we use one second as threshold for allowed non-aligned-audio. To compute the WDR, we use an attention-based LibriSpeech-960h ASR system [4], with a WDR on dev-clean and test-clean of 0.5%. First, we used the UDR to determine if using the GMM-HMM alignment to remove excessive silence helps to reduce long noise and silence sections in the synthesized audio. Table 1 shows the results with UDR and WDR for each silence preprocessing method, averaged over the remaining conditions (speaker and input encoding). By using the GMM-HMM alignment for silence pre-processing, the UDR can be reduced to an absolute zero, so there are no unaligned noise or silence parts above one second remaining in the synthesized audio. This also means that the TTS system always stops correctly at the end of an utterance. The WDR is reduced by up to 0.6%, meaning fewer words are dropped during synthesis, but is still higher than the original 0.5%. This indicates that the TTS still suffers from either early stopping or skipped words.
Synthetic data for Attention-Encoder-Decoder ASR
As AED systems use less resources in training and decoding we first tested the quality of the different TTS systems with our AED-ASR baseline. The baseline is trained for 250 [14]. For the remaining 170 sub-epochs we oversample the real data 3 times, having 3 · 100h of real data matched with ∼300h of synthetic data. With a partitioning of factor 9 this results in 83 epochs of training on LibriSpeech-100h, and about 19 epochs for the synthetic data. The results for the baseline, synthetic data from 3 different TTS systems, and the oracle performance can be found in Table 2. We see a notable jump in WER of about 2% absolute when adding the synthetic training data. The term oracle refers to using the original LibriSpeech-360h audio files instead of the synthesized one. To see the effect of the different TTS systems better, we also combined all important conditions and created synthetic data. The evaluation of the different conditions can be seen in Table 3. For each condition, we average over all possible variants for the other conditions, i.e. for the first line we took the average of the 4 experiments with threshold silence pre-processing combined with look-up or GST embedding and character or phoneme encoding. The results indicate that only the speaker embedding has a notable effect on the final ASR performance, although we observe an increased stability by using the GMM silence pre-processing method. We conclude that the ASR does not suffer from noisy or incorrect sequences, but instead they have a regularization effect on the training. For those 12 experiments, we picked 4 to also test different data ratios 1:3 and 2:3, which can be found in Table 4. Here we confirmed that the AED model with oversampling LibriSpeech-100h by factor 3 is performing best.
Internal Language Model Subtraction
Internal language model (ILM) subtraction is a method to remove the label prediction bias of ASR models that use a label prediction context such as AED systems and RNN-T systems. The first methods were presented in [25,26,27] and exten- sively investigated for AED systems in [28]. Although other variants of ILM estimation can perform better, we use the "Zero" ILM estimation approach [28] for simplicity, meaning that the acoustic context vector is set to zero for the decoder to estimate the ILM. Table 5 shows the results for the AED system when adding synthetic data to systems with and without ILM subtraction. We see that for both test sets the absolute improvement by adding synthetic data stays the same, being 2.1%/2.6% without ILM subtraction and 2.0%/2.4% with ILM subtraction for clean/other respectively. This means that the relative improvement is even larger, reaching 38% on testclean and 16% on test-other.
Synthetic Data for Hybrid ASR
As hybrid systems require a given alignment, we cannot simply add the synthetic data to the training but need to compute an alignment for the synthetic data as well. We decided to re-align both LibriSpeech-100h and the added synthetic data with the acoustic BLSTM-AM model of the baseline (see Section 2.2) and do a full re-training. The second block of Table 6 shows the baseline with the BLSTM alignment and retraining, and the same training including synthetic data from the respective TTS model. We again oversample the real data 3 times, and train for 5 full epochs in total. Although there is an improvement in the cross-entropy scores on the dev-sets during training from 0.298 to 0.275, there is no notable improvement in WER for any kind of used synthetic data, even after re-tuning the LM scales.
Synthetic Data for CTC ASR
The baseline CTC system is trained for 300 sub-epochs with a partitioning of 3, resulting in 100 training epochs. When adding the synthetic data the system was trained for 400 subepochs with a partitioning of 12. Block 3 of Table 6 shows the results for decoding. Although the CTC loss on the devset during training was noticeably lower (0.8 to 0.73) the final WER did not improve noticeably.
Synthetic Data for monotonic RNN-T ASR
The monotonic RNN-T ASR system is trained using the Viterbi-alignment generated with the respective CTC system. Both the baseline and the models including synthetic data were trained for 300 sub-epochs with a partitioning of 3 and 12 respectively. The results can be found in the last block of Table 6. It can be seen that on the "other" sets the RNN-T system which includes a prediction network is stronger than the equivalent CTC model. Nevertheless, it does not seem to improve substantially by adding synthetic data.
DISCUSSION
To put the performance of our systems into perspective, we show a full comparison of the best systems of each category in Table 6. Only the AED system improved significantly by using synthesized data from additional text, which is in line with previous publications that only show improvements for AED systems. One exception is [24] which uses an RNN-T ASR system reporting up to 12.5% improvements on LibriSpeech, but the TTS system includes additional training data which means the TTS can learn additional acoustic information. In our case, when relying only on the LibriSpeech-100h dataset as parallel data, we had no improvements with the two label context-independent models (CTC and hybrid ASR) and the context-dependent RNN-T. This indicates that the TTS system cannot produce a larger variety of audio features compared to the existing data. We observed in other experiments that AED systems did not improve when synthesizing exactly the same text that is already used for the baseline training. Similar behavior was also found in [13], and it was shown in [14,16] that training with synthetic data is complementary to using SpecAugment. Our conclusion was that the improvements for AED systems shown in many publications are due to the improved decoder. The large improvements by reducing the influence of encoder states based on synthetic data, as done with the α-factor in [16] further backed this hypothesis. Now that we see equivalent improvements when using ILM subtraction for AED-ASR and no improvements with RNN-T models, we conclude that it is not the internal language model of the decoder that benefits most from the synthetic data, but rather the attention mechanism. It was also previously shown in [14] that the effect of synthetic data from additional text and external language model fusion of the same text have independent effects. Another possibility is that RNN-T systems might not need a large label context [40,46], and thus cannot benefit from more textual data. Investigating if the label type plays a role in the effect of synthetic data can be future work.
We also compare our systems to other publications. The system from [9] is to our knowledge the best existing LibriSpeech-100h AED system without additional data, but uses nonconstrained computational resources for training, meaning the model is trained on 32 TPUs for 10 days. The system presented in [16] has the weakest baseline, but shows the largest improvements by using a TTS system with x-vector [47] speaker embeddings and the scaling of the synthetic encoder states. As seen in section 4.3, only changing the speaker embedding had a notable effect. Investigating why the speaker embedding method is important although the encoder parts of ASR systems do not seem to benefit from synthetic data can be future work, together with finding objective performance markers for the quality of synthetic data different from UDR and WDR.
CONCLUSION
In this work we presented four state-of-the-art ASR systems for LibriSpeech-100h and tried to improve the performance of each by using synthetic data from a TTS system trained on the same data. By using the alignment of a GMM-HMM system for silence removal, we were able to improve the stability of an autoregressive TTS system with respect to unaligned duration rate and word deletion rate. We found that an increase in the stability of the TTS systems is not needed to generate useful synthetic data to be used in AED-ASR training. For the first time we apply synthetic data from a TTS system to an attention-encoder-decoder, hybrid, CTC and a RNN-T system in a direct comparison, and show that only the AED system can be significantly improved by using synthetic data. We show that we can get up to 38% relative improvements by adding synthetic data to the AED-ASR system, even when using internal language model subtraction. This indicates that the benefit of adding synthetic data from additional text is mostly related to improving the robustness of the attention mechanism for sequence mapping, and not related to improving the internal language model of the decoder. Nevertheless, the improvement on AED systems is currently not sufficient to close the performance gap to a strong hybrid baseline presented in this work, which outperforms any other system in literature under the same data conditions with a word-errorrate of 3.3%/10.0% on test-clean and test-other respectively.
ACKNOWLEDGEMENTS
This work has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 694537, project "SEQCLAS"). This work was partly funded by the Google Focused Award "Pushing the Frontiers of ASR: Training Criteria and Semi-Supervised Learning". The work reflects only the authors' views and none of the funding parties is responsible for any use that may be made of the information it contains. We like to thank Mattia Di Gangi, Christoph Lüscher, Peter Vieting and Wei Zhou for additional contributions.
In the first set of experiments we trained 12 different TTS systems by adjusting 3 different conditions: • Input encoding: -Lower cased characters (char) -CMUDict-style phonemes with stress (phon) • Speaker Encoding -Trained look-up table (look-up) -Reference encoder with Global-Style-Token (GST) • Silence Pre-Processing -Threshold-based with -40dB (Threshold) -GMM-HMM alignment w. 0.0s silence (GMM 0.0s) -GMM-HMM alignment w. 0.5s silence (GMM 0.5s)
Table 1 .
1UDR and WDR measured on dev-clean and test-
clean when synthesized with TTS models using the respec-
tive preprocessed data. Values are averaged over 4 different
trained models each.
Silence
Pre-Proc.
UDR [%] WDR[%]
dev + test
∆t[s]
clean
Threshold
9.8
3.5
GMM
0.0
0.0
2.9
GMM
0.5
0.0
3.0
Table 2 .
2Results on LibriSpeech-100h with an AED model without external LM. The TTS models for data generation used a fixed look-up table and phoneme inputs.Syn.
Data
Silence
Pre-Proc.
Added
Data
WER[%]
dev
test
∆t[s]
[h]
cl. oth. cl. oth.
No
-
-
8.1 21.6 8.2 22.6
Yes
Threshold
330
5.7 19.7 6.0 20.9
GMM
0.5
345
5.6 19.8 6.1 20.5
0.0
278
5.7 19.5 5.9 20.3
Oracle
-
360
4.5 14.8 4.8 15.3
Table 3 .
3Comparison of different TTS systems. The WER score is the AED system performance after synthesizing LibriSpeech-360h and using it for training. * means averaged over all possible TTS models and respective ASR training.Silence
Pre-Proc.
Speaker
Input
Type
WDR[%] WER[%]
dev + test
test
∆t[s]
clean
cl. oth.
Threshold
*
*
3.5
6.2 20.9
GMM
0.0
*
*
2.9
6.1 20.8
GMM
0.5
*
*
3.0
6.2 20.8
*
look-up
*
3.0
6.1 20.5
*
GST
*
3.6
6.2 21.1
*
*
char
3.2
6.2 20.8
*
*
phon
3.1
6.1 20.8
Table 4 .
4Comparing the effect of different data ratios of librispeech-100h compared to the synthesized LibriSpeech-360h corpus (real : synthetic), assuming that the synthesized corpus has a duration of about 300h. The results are the average of 4 experiments with the TTS systems used as stated.Data
Ratio
Silence
Pre-Proc.
Speaker
Input
Type
WER[%]
test
cl. oth.
1:3
GMM 0.0/
0.5
GST/look-up phon
6.3 22.4
2:3
6.0 21.1
3:3
6.1 20.7
sub-epochs with a partitioning factor of 3, resulting in about
83 full epochs. We reset the learning rate to the maximum
at sub-epoch 80 for all experiments. When adding synthetic
data, we take the checkpoint of sub-epoch 80 of the baseline
as starting point to reduce the training variance for the exper-
iments
Table 5 .
5Results on LibriSpeech-100h with an AED model without LM and with external LM and internal language model subtraction using the zero method. The experiments with synthetic data are averaged over 3 runs with the respective silence pre-processing methods.External LM
ILM
subtraction
Syn.
Data
WER[%]
dev
test
cl. oth. cl. oth.
-
No
No
8.1 21.6 8.2 22.6
Transformer
5.7 15.8 6.1 16.9
Yes 3.6 13.9 4.0 14.3
Yes
No 4.6 13.7 5.3 14.8
Yes 3.0 12.0 3.3 12.4
Table 6 .
6Comparison of different systems on LibriSpeech-100h from literature and the results for each architecture described in this paper. Synthetic data describes the TTS used to generate the synthetic data, which is always based on the transcriptions of LibriSpeech-360h. Except for[4], all systems include SpecAugment.* results are averaged over multiple runs with varying silence pre-processing, see Section 4.2Architecture
Encoder Model
Label Type
LM
Synthetic Data
WER[%]
dev
test
clean other clean other
Attention
Enc.-Dec.
LAS -BLSTM [9]
16k WPM
-
-
5.3
16.5
5.5
16.9
ESPNet -Transformer [15]
5k SPM
-
-
10.3
24.0
11.2
24.9
LSTM
5.8
16.6
7.0
17.0
VAE-TTS
3.8
13.2
4.3
13.5
ESPNet -Transformer [16]
characters
LSTM
-
14.3
36.4
14.4
36.9
x-vector TTS
8.9
23.0
8.6
24.1
x-vector TTS + α
4.5
15.8
4.7
15.9
RETURNN -BLSTM [ours]
2k BPE
-
-
8.1
21.7
8.2
22.6
Transformer
+ ILM sub.
4.6
13.7
5.3
14.8
lookup-TTS *
3.0
12.0
3.3
12.4
Hybrid
RETURNN -BLSTM [4]
12k CART
4-gram
-
5.0
19.5
5.8
18.6
RETURNN -BLSTM [ours]
4.9
14.7
5.6
15.3
LSTM
3.0
9.3
3.4
10.0
lookup-TTS *
3.0
9.3
3.3
10.0
CTC
wav2letter++ -CTC [45]
characters
Transformer
-
3.3
10.8
3.8
11.3
RETURNN -BLSTM[ours] phonemes (w. EOW)
4-gram
-
5.0
15.4
5.6
16.1
LSTM
3.3
11.4
3.8
12.4
lookup-TTS *
3.2
11.6
3.6
12.5
monotonic
RNN-T
RETURNN -BLSTM[ours] phonemes (w. EOW)
4-gram
-
5.0
14.9
5.4
15.6
LSTM
3.3
10.4
3.6
11.0
lookup-TTS *
3.1
10.1
3.6
11.1
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[
"Weak and strong interaction of excitation kinks in scalar parabolic equations",
"Weak and strong interaction of excitation kinks in scalar parabolic equations"
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"J D M Rademacher ",
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| Motivated by studies of the Greenberg-Hastings cellular automata (GHCA) as a caricature of excitable systems, in this paper we study kink-antikink dynamics in the perhaps simplest PDE model of excitable media given by the scalar reaction diffusion-type θ-equations for excitable angular phase dynamics. On the one hand, we qualitatively study geometric kink positions using the comparison principle and the theory of terraces. This yields the minimal initial distance as a global lower bound, a well-defined sequence of collision data for kinks-and antikinks, and implies that periodic pure kink sequences are asymptotically equidistant. On the other hand, we study metastable dynamics of finitely many kinks using weak interaction theory for certain analytic kink positions, which admits a rigorous reduction to ODE. By blow-up type singular rescaling we show that distances become ordered in finite time, and eventually diverge. We conclude that diffusion implies a loss of information on kink distances so that the entropic complexity based on positions and collisions in the GHCA does not simply carry over to the PDE model. arXiv:2012.00309v1 [math.AP] 1 Dec 2020 | 10.1007/s10884-021-10040-2 | [
"https://arxiv.org/pdf/2012.00309v1.pdf"
]
| 227,238,853 | 2012.00309 | 0df34138b32fe910103749f6d212b26872344c61 |
Weak and strong interaction of excitation kinks in scalar parabolic equations
December 2, 2020
A Pauthier
J D M Rademacher
D Ulbrich
Weak and strong interaction of excitation kinks in scalar parabolic equations
December 2, 2020
Motivated by studies of the Greenberg-Hastings cellular automata (GHCA) as a caricature of excitable systems, in this paper we study kink-antikink dynamics in the perhaps simplest PDE model of excitable media given by the scalar reaction diffusion-type θ-equations for excitable angular phase dynamics. On the one hand, we qualitatively study geometric kink positions using the comparison principle and the theory of terraces. This yields the minimal initial distance as a global lower bound, a well-defined sequence of collision data for kinks-and antikinks, and implies that periodic pure kink sequences are asymptotically equidistant. On the other hand, we study metastable dynamics of finitely many kinks using weak interaction theory for certain analytic kink positions, which admits a rigorous reduction to ODE. By blow-up type singular rescaling we show that distances become ordered in finite time, and eventually diverge. We conclude that diffusion implies a loss of information on kink distances so that the entropic complexity based on positions and collisions in the GHCA does not simply carry over to the PDE model. arXiv:2012.00309v1 [math.AP] 1 Dec 2020
Introduction
Many spatially extended physical, chemical and biological systems form so-called excitable media, in which a supercritical perturbation from a stable equilibrium triggers an excitation that is transferred to its neighbours, followed by refractory return to the rest state.
Perhaps the simplest dynamical system that realises a caricature excitable medium is the 1D Greenberg-Hastings cellular automata (GHCA) [GH78,GHH78]. Its key features are that spatial excitation loops embedded between rest states form local pulses that travel in one direction, and a counter-propagating pair of such local pulses annihilates, leading to a pure rest state locally, cf. Fig. 1 (a). Pulse trains can be generated from local pulses by placing these at arbitrary positions; they will maintain their relative distance until a possible annihilation. In fact, the topological entropy of the GHCA results from a Devaney-chaotic closed invariant subset of the non-wandering set that consists of colliding and annihilating local pulses [DS91,KRU20]. From a dynamical systems viewpoint the non-wandering set is of prime relevance, and it turns out that for the GHCA it can be decomposed into invariant sets with different wave dynamics [KRU20].
However, the modelling of excitable media is predominantly done by parabolic partial differential equations (PDE) and systems thereof. A paradigm is the famous FitzHugh-Nagumo (FHN) equation, derived from the Hodgin-Huxley model for nerve axons [Fit61,NAY62,HH52]. A (2) of (1) (b) with f 0 = 0.05 for initial data with four pairs of pulses and kinks/antikinks, respectively; marked are the associated annihilation events. In the right panel, time is rescaled such that a single front has unit speed as in the left panel. Some details on the numerics can be found in Appendix C.
priori, for any such PDE model, the identification and description of pulse dynamics akin to GHCA is a formidable task and far from completely understood even in FHN. The PDE dynamics is in general also richer and parameter dependent, for instance self-replication of pulses has been numerically observed [CRS20,HO00], and rebound of pulses upon collision [NU00].
While the existence of arbitrary local pulse trains and their annihilation is trivial in the GHCA, already the existence proof of a single pulse in FHN is not. It is well known that FHN possesses a homoclinic travelling wave solution that is spatially asymptotic to a stable rest state and thus corresponds to a single local pulse in GHCA; by spatial reflection the direction of motion can be reversed. However, there is no meaningful notion of a local pulse since the spatial coupling by diffusion is effectively non-local through its infinite propagation speed. An analogue of pulse trains on the level of initial data is a superposition of multiple travelling wave pulses placed at a distance from each other. In the dynamics of the PDE, diffusivity immediately couples these pulses, albeit in a weak form. In the past decades, a number of results have been obtained that rigorously relate positions of these 'pulses' to an ordinary differential equation system (ODE) [Ei02,ZM09]. However, to our knowledge there are no analytical results for collisions of excitation pulses in higher order equations or systems of PDE -the closest result is the existence of an attracting invariant manifold for a sufficiently distant pair of counterpropagating pulses [SW08b,Wri09]. The situation simplifies for scalar parabolic equations; in particular, the comparison principle for scalar parabolic equations allows to study collisions. We note that also energy methods can be used [Wes20], even for the fourth order (non-excitable) Cahn-Hilliard equation [SW18].
In this paper we consider suitable scalar parabolic PDE for periodic phase dynamics as models for excitable media, and study similarities and difference between this and the local pulse dynamics of the GHCA. These so-called θ-equations for oscillator phase dynamics [RE98,BS10] are given by θ t = θ xx + f (θ), θ(t, x) ∈ S 1 = R/2πZ, x ∈ R , t > 0.
For simplicity, we specify the nonlinearity as f (θ) := cos(θ + Θ 0 ) + f 0 , where f 0 ∈ (0, 1) and Θ 0 ∈ (0, π) is chosen uniquely so that f (0) = 0, although all results are valid in general for the excitable case of [RE98].
Equation (1) possesses the stable spatially homogeneous state θ ≡ 0 and a right-moving (as well as left-moving upon reflection) travelling wave solution θ(t, x) =φ(ξ), ξ = ±x − ct with speed c > 0. The profileφ(ξ) is asymptotic to 0 ∈ S 1 for |ξ| → ∞, but with non-trivial winding number. This means that upon lifting S 1 to R, the rest state maps to the sequence 2kπ, k ∈ Z, andφ maps to heteroclinic front solutions ϕ + 2πk with ϕ(ξ) → 0 as ξ → ∞, and ϕ(ξ) → 2π as ξ → −∞.
The choice of f explicitly relates (1) to the overdamped limit of the Sine-Gordon equation, which also arises as a phase field model for certain complex Ginzburg-Landau equations with broken gauge symmetry [AK02,eqn. (91)]. Following the terminology of the Sine-Gordon equation, we refer to these fronts as kinks, and their left-moving spatial reflection as antikinks.
In the present context we view these as corresponding to a single local pulse in GHCA. We are thus concerned with the evolution of initial data built from kink-and antikink sequences, such as plotted in Fig. 1. For any given solution u(x, t) viewed in the lift to R, we geometrically define the set of positions of potential kinks and antikinks through the phase π intercepts as P (u(t)) := {ξ ∈ R|∃k ∈ Z : u(ξ, t) = (2k + 1)π},
which readily turns out to be a discrete set for t > 0. We will discuss conditions under which this set meaningfully encodes kink or antikink positions also for large t 1. A crucial ingredient is the theory of terraces.
Roughly speaking, a terrace is a superposition of finitely many fronts, each of them being a heteroclinic connection between two equilibria, such that asymptotically these fronts are separated, i.e., the distances between any two fronts diverge. To our knowledge, the terminology was introduced in [DGM14], even though the notion was already present in the seminal paper [FM77] for fronts with distinct speeds. Since then, it appeared that this notion is fundamental in the understanding of the long time behaviour of solutions of reaction-diffusion equations. Poláčik proved in [Pol20] under mild assumptions, in the context of scalar homogeneous semilinear parabolic equations, that any front-like initial data eventually leads to a terrace profile. These results were extended to localized initial data in [MP20], leading to a pair of terrace profiles going in opposite directions. Risler independently proved similar results in the more general context of gradient flows in [Ris17] for systems.
When all the considered fronts travel at asymptotically distinct speeds, the behaviour of the terrace profile, the convergence toward it (starting from front-like initial data, for example), as well as its stability are quite well understood, see [Pol20,LS15] and references therein. On the other hand, if two consecutive fronts have the same asymptotic speed, the question is much more intricate since the eventual separation is powered by weak interactions. In our context, kinks are right moving fronts, while antikinks are left moving fronts, all at the same speed. Hence, the model (1) is suited for a comparison with GHCA as it allows to study the combination of strong interactions, when a kink and an antikink collide, and weak interactions when considering consecutive kinks. However, this observation already suggests a multi-scale nature, which turns out to imply a fundamental difference for the long time dynamics of excitation pulse positions.
We informally summarize our main results. Suited to unbounded kink-antikink initial data, we prove global well-posedness for unbounded data, and infer that the set of geometric positions (2) consists of isolated points that lie on smooth curves except possibly at collisions. On the one hand, as a qualitative analysis, we rely on the comparison principle and roughly track positions which allows us to show that the minimal initial distance is a global lower bound for distances, and to abstractly identify collision times and locations. Moreover, as a model for the dynamics far from collisions, we show that under periodic boundary conditions the distances of neighboring kinks asymptotically equalise, and we numerically corroborate that this loss of information happens more broadly. On the other hand, as a quantitative analysis, we derive the ODE in the weak interaction regime following [Ei02,RLZ15] for finitely many kinks and analyse these in some detail, which extends the aforementioned qualitative terrace results. Our analysis relies on blow-up type singular rescaling and identifies the dynamics as being slaved to dynamics on a sphere at infinity, which, for instance, shows that distances become ordered in finite time. The latter is again hinting a loss of information through the dynamics: the memory of the initial distances is 'washed out'. This suggests that the chaotic dynamics of GHCA and the entropic complexity based on positional dynamics is reduced by the weak interaction in the PDE, and we do not expect a topological entropy (in a suitable sense for the PDE) -based on positions alone -which resembles that of GHCA. This paper is organized as follows. In §2 we discuss the notions of positions and the initial data. Section 3 is devoted to qualitative and quantitative aspects for bounded monotone data which are composed solely of kinks (or antikinks). In §4, we focus on the annihilation process for initial data composed of kinks and antikinks. Unbounded initial data, i.e., infinitely many kinks and antikinks, as well as the long-time complexity are considered in §5. Finally, in §6 we conclude with a discussion. In the appendices we collect some technical proofs as well as notes on the numerical implementations.
Kinks and anti-kinks and their positions
In this section we introduce initial data which has imprinted positions of kinks and anti-kinks such that the setwise definition of 'positions' P (u(t)) in (2), which is well-defined as long as u(t) is defined, turns into meaningful individual positions for all time. In order to discuss this further, we first turn to the notion of kinks and anti-kinks in more detail.
The aforementioned regime f 0 ∈ (0, 1) is termed excitable since the ODE for spatially constant data possesses a stable rest state and an unstable state which acts as a threshold for undergoing an 'excitation loop', i.e., winding once through S 1 . For f 0 > 1 the two equilibria have undergone a saddled-node bifurcation and the dynamics of this ODE is a permanent oscillation. We therefore fix an arbitrary f 0 ∈ (0, 1) throughout.
Travelling waves of (1) with velocity c solve the ODE, considered in R 2 , given by
u ξ = φ, φ ξ = −cφ − f (u),(3)
and the fundamental kinks are fronts, i.e., heteroclinic connections between the stable states 2π(k + 1), 2πk, k ∈ N. They are strictly monotone decreasing in u and their unique existence (with f 0 ∈ (0, 1)) follows, e.g., by phase plane analysis, cf. [RE98]. For k = 0, we denote the unique translate such that u(0) = π by ϕ and note that c > 0; the basic antikinks are translates of the spatial reflection ϕ(−ξ). As a scalar reaction-diffusion equation, fronts in (1), and therefore ϕ as well as ϕ(−·), are orbitally stable [Sat76,FM77]. We note that due to the periodicity in u, there do not exist heteroclinic orbits connecting 2πk and 2πk for k − k = 1.
We use the term kink and antikink more losely for monotone pieces of u(·, t) that connect even multiples of π. Concerning their positions, we refer to elements in the set P from (2) as geometric positions and will introduce a notion of analytic positions in §2.2. These types of positions generally differ, but -as will be shown -the long term dynamics leads to large distances for which the geometric and analytic positions will be exponentially close to each other. In particular, P (ϕ(t)) = {ct} is trivially a single point for all time moving with speed c.
As outlined before, we are interested in the relative motion of sequences kinks and antikinks that resemble superpositions of shifted ϕ and ϕ(−·). We start by considering discontinuous initial data built from kink or antikinks steps,
H ± (x) := 2π , ±x < 0 π , x = 0 0 x > 0.
Notably, the solution with initial data H + will converge to the above kink ϕ, while that with initial H − converges to the antikink. Next, we consider initial positions
ξ − m < ξ − m−1 < . . . < ξ − 1 < ξ + 1 < ξ + 2 < . . . < ξ + n(4)
for m kink and n antikink steps that are shifted to these positions via
H ± j (x) := H ± (x − ξ ± j+1 ).
For convenience, we smoothen H ± j in such a way that the geometric positions P (u 0 ) of the resulting u 0 coincide with those in (4). This can be realised by replacing H ± j with a convolution H ± j,ε ± j := ρ ε ± j * H ± j for a positive, symmetric and smooth mollifier ρ ε supported on [−ε, ε] and sufficiently small ε ± j depending on the neighboring initial positions, e.g., ε ± j < 1 2 min{|ξ ± j − ξ ± j−1 |, |ξ ± j − ξ ± j+1 |} for 2 ≤ j ≤ n − 1 and correspondingly for j = 1, n. Hence, we set
u 0 (x; m, n) := m−1 j=0 H − j,ε − j (x) + n−1 j=0 H + j,ε + j (x).(5)
with n, m ∈ N 0 := N ∪ {0, ∞}, i.e., possibly infinitely many kink or antikink steps. Due to the separated smoothened intervals the geometric positions of u 0 coincide with (4), i.e., P (u 0 (x; m, n)) = {ξ − 1 , . . . , ξ − m , ξ + 1 , . . . , ξ + n }.
In the following we omit the dependence on ε ± j since these do not influence the results.
Well-posedness
We consider possibly unbounded initial data, which in particular covers the case of infinitely many kinks or antikinks. In order to ensure well-posedness, one option is to consider a weight function ω : R → R, ω(x) := C −1 e −C|x| for some C > 0, and the Banach space
X ω := {v(·) ∈ C(R) : ωv ∈ L ∞ (R)} , v Xω := ωv ∞ := sup x∈R |v(x)ω(x)|.
Theorem 1. For f from (1) the initial value problem
u t = u xx + f (u), u(x, t) ∈ R, (x, t) ∈ R × R >0 u(x, 0) = u 0 (x) ∈ X ω , x ∈ R has a unique solution u ∈ C ∞ (R × R >0 , X ω ).
Proof. The proof directly follows nowadays classical techniques, and we refer to the milestone monograph [LSU68] for further details. Let us just briefly recall the main steps. Rephrasing the initial value problem as an integral equation, the local existence of a unique solution can be deduced from a standard fixed point argument. To this end, we consider the operator Φ :
C([0, T ], X ω ) → C([0, T ], X ω ) defined by Φ[u] := G t * u 0 (x) + J(x, t), J(x, t) := t 0 R G t−s (x − y)f (u(s, y)) dy ds (6) where u(x, ·) ∈ C([0, T ], X ω ) for x ∈ R and G(x, t) := G t (x) := 1 √ 4πt e − |x| 2
4t , t > 0, is the heat kernel. Moreoever, by proving the Hölder continuity of this solution, a boot-strap argument shows smoothness of the solution both in x and t.
As for ODE with bounded vector fields, the local existence of the solution can be extended to global existence (t > 0) since T is independent of the initial condition.
Geometric and analytic positions
Having established global existence, we turn to the specific notions of positions. First we note that the geometrically defined set of positions P (u(x, t; m, n)) gives locally smooth curves of positions as follows. More details for different types of initial data will be given in the subsequent sections.
Proposition 2.
For any m, n ∈ N 0 , n + m > 0, consider the global solution u(x, t) from Theorem 1 with an initial datum u 0 (x; m, n) as in (5). Then the following holds. For any t ≥ 0 the set P (u(t)) is discrete and, if n + m < ∞, consists of at most n + m elements. There is
t 1 > 0, t 1 = ∞ if nm = 0, such that for 0 ≤ t < t 1 the set P (u(t)) consist of differentiable curves ξ − m (t) < ξ − m−1 (t) < . . . < ξ − 1 (t) < ξ + 1 (t) < ξ + 2 (t) < . . . < ξ + n (t)
that coincide with the initial positions (4) at t = 0. Moreover, as long as any two such positions are defined, the number of elements in P (u(t)) between these cannot increase.
This Proposition in particular yields, at least locally in time, well-defined and regularly varying positions ξ ± j (t); we refer to ξ − j (t) as kink positions and ξ + j (t) as antikink positions.
Proof. This proposition is a consequence of the properties of the number of zeros for linear parabolic equations, see [RW74,Ang88] for a rigorous exposition and we refer to [PP18,§2.3] for an exposition of the results used next.
Let v = ∂ x u be the spatial derivative of the solution. Given u, it solves a linear parabolic equation, and at t = 0 the minimum of u yields a sign change of v at a locally unique zero.
The intervals where u is constant occur on monotone parts of u and are thus not associated to non-trivial sign changes of v.
It follows from the zero number principle that there exists T ∈ (0, ∞] such that v has a unique simple zero on (0, T ), and has constant sign on (T, ∞). Moreover, from parabolic regularity and the implicit function theorem, there exists a C 1 function t → η(t) such that v(η(t), t) = 0, for all t ∈ (0, T ). This implies that the set P (u(t)) is discrete for all t ≥ 0 and, again from the implicit function theorem, that each point lies on a differentiable curve.
Let us now turn to the case n + m < ∞. Let β ± 0 be the value of u 0 at x → ±∞. Then (see [VVV94,Theorem 4.4.2], for instance)
β ± (t) := lim x→±∞ u(x, t).
exists for all t ≥ 0, and are solutions of the initial value probleṁ
β ± = f (β ± ), β ± (0) = β ± 0
in particular, due to our choice of initial condition (5), it follows that these limits are constant steady states of the above equation. Combined with the sign properties of ∂ x u, this proves that P (t) consists of at most n + m elements.
It remains to prove that the cardinality of P (t) is non-increasing. We claim that if there exists k 0 , t 0 such that u(η(t 0 ), t 0 ) = (2k 0 − 1)π, then u(·, t) > (2k 0 − 1)π for all t > t 0 . This is a consequence of the maximum principle: let α(t) be a solution of the initial value probleṁ
α = f (α), α(t 0 ) = (2k 0 − 1)π.
Then t → α(t) is increasing and converges to 2k 0 π as t → ∞. Let w(x, t) = u(x, t) − α(t). Then, taking u as given, w solves a linear parabolic equation on (t 0 , ∞), and w(·, t 0 ) ≥ 0. It follows that w(·, t) > 0 for all t > t 0 , which concludes the claim and the proof.
In particular, for the solution u * (x, t) with a single smoothened jump kink initial data u * (x, 0) = H ε , there is a unique globally defined geometric position P (u(t)) = {ξ * (t)}. We denote its speed as c * := d dt ξ * . Recall that the single kink speed is denoted by c, and the aforementioned convergence result of u * to the kink ϕ implies c * (t) → c as t → ∞.
In order to study the evolution of distances between kinks or antikinks in more detail, it is convenient to define the following analytic positions, which however requires sufficiently large initial distances ξ i − ξ i+1 (1 ≤ j ≤ n) between neighboring kinks.
The broader task of deriving laws of motion for localized states in terms of ordinary differential equations ("laws of motion") dates back at least to the studies on metastable fronts in the Allen-Cahn equation by Carr-Pego and Fusco-Hale, who derived ODEs for the analytic positions of fronts, cf. [CP89,FH89,FM77]. This has been explored in various directions, notably to infinitely many metastable pulses in arbitrary dimension [ZM09]; we mainly follow [Ei02] and [RLZ15].
This allows to derive such ODE rigorously only for sequences of either kinks or antikinks, i.e., monotone initial data, and we therefore restrict attention to (5) with m = 0. Since in this case the overall motion of kink initial data is dominated by the drift with velocity c, we consider the deviation from this speed by introducing the comoving frame z = x − ct, which introduces the term c∂ z u on the right hand side of (1) and yields, in the covering space R,
u t = u zz + cu z + f (u).(7)
The corresponding solution u(z, t) is defined globally in t ≥ 0 by Theorem 1. We will define analytic positions η i (t) that relate to the geometric positions ξ i by ct + η i (t) ≈ ξ i (t).
For sufficiently large initial distances, the analytic positions are defined by writing u(z, t) as
u(z, t) = n i=1 ϕ i (z, t) + w(z, t), ϕ i (z, t) := ϕ(z − η i (t)),(8)
with η i (1 j n) uniquely defined through the following orthogonality condition on w as detailed in Appendix A. Let L i := ∂ 2 z + c∂ z + f (ϕ i ) denote the linearized operator of the right hand side of (7) in ϕ i , and L * i := ∂ 2 z − c∂ z + f (ϕ i ) its adjoint. The remainder term w is now supposed to be orthogonal to the adjoint eigenfunctions e
* i := e c(z−η i ) ϕ i , i.e. w, e * i = R e c(z−η i ) w(z, t)ϕ i (z, t) dz = 0, 1 i n.(9)
For initial data as in (5), w is initially nonzero, so that for the study of analytic positions it is natural -though not necessary -to replace H ± k,ε by the fronts, i.e., ϕ i as in (8). Then w(z, 0) ≡ 0 and thus remains small at least for short time. In order to control w also for larger times, we assume that the kinks are initially well-separated, i.e., |η i (0) − η i−1 (0)| 1 (2 i n).
Remark 3. For initial data v 0 as in (8), the set P (v 0 ) does not coincide with the initial positions (4), though the error is exponentially small in the minimal distance. Moreover, for nm > 0, the set P (v 0 (·; m, n)) may contain spurious points such that one generally needs to assume a minimal distance between ξ − 1 and ξ + 1 .
Remark 4. For any fixed t and i, the analytic kink position η i (t) equals the (shifted) geometric position ξ i (t)−ct if and only if w(ξ i (t), t) = (n−1)π− n j=1,j =i ϕ(ξ j (t)−ct−η j (t)), as can be seen from (8). In particular, all geometric and analytic positions coincide if and only if w vanishes simultaneously at all geometric positions. However, kinks interact eventually repulsively, i.e., their distances eventually increase (see §3 below for details) and this implies w(·, t) → 0 as t → ∞ (in L 2 or L ∞ ) so that both definitions asymptotically coincide.
Bounded monotone initial kink data
In this section we consider bounded monotone data which are composed of kinks and analyse the distances between neighboring kinks in terms of the geometric as well as analytic positions. We note that by spatial reflection the discussion equally applies to bounded sequences of antikinks. First we track geometric positions via the comparison principle, which applies for any initial distance, but only constrains the positions to lie within certain intervals, referred to as gaps, that also depend on the initial data. The analytic positions provide more specific laws of motion that apply immediately for initial data with sufficiently distant positions, or -more abstractly -from some point in time onward with a distribution of positions for which we just know the positions up to the gaps. The latter relies on the results by Poláčik and Risler, which state that front-like initial data converge to a terrace whose speeds converge and whose distances eventually diverge, albeit without a quantitative estimate. For our purposes, this can be summarized as follows.
Theorem 5. cf. [Pol20,Ris17] Let −π < u 0 < 2π(k+1) (k ∈ N) be an initial datum with lim x→−∞ u 0 (x) = 2πk, lim x→∞ u 0 (x) = 0 and corresponding solution u. Then there exist C 1 functions ξ 1 , . . . , ξ k on R satisfying
lim t→∞ ξ j (t) = 0 (j = 1, 2, . . . , k), |ξ j (t) − ξ j+1 (t)| → ∞ (j = 1, 2, . . . , k − 1)
such that the solution u(·, t) converges to the corresponding terrace:
u(·, t) − k j=1 ϕ(· − ct − ξ j (t)) ∞ −→ t→∞ 0
Remark 6. In Prop. 8 below we give a lower bound on the distances which is of course far from optimal in the asymptotics t → ∞. Naively, one might suppose that all distances monotonically increase; however, this is not true in general, as the results in §3.2 show, and numerical simulations illustrate, cf. Fig. 4
(a) and (b).
In order to describe the long-time behaviour of bounded solutions under kink-antikink annihilations, we consider limit sets. For bounded monotone solutions, this has already been donefor a much broader setup -in [Pol20] from which we take the following definition of the limit set Ω(u) := {v : u(· + x n , t n ) → v for some sequences t n → ∞ and x n ∈ R},
where u ∈ L ∞ (R × R + ) and the convergence is in L ∞ loc (R) (locally uniform convergence). Compared to the standard definition of ω-limit set, this allows to observe any finite piece of the graph of u(·, t n ). For the special situation of Theorem 5, the set is given by
Ω(u) = {ϕ(· − τ ) : τ ∈ R} ∪ {2πj : j = 0, 1, . . . , k},(10)
cf. [Pol20]. Our results add the description of ω and Ω in case m + n < ∞ and mn = 0, even though they are consequences of (10) after pairwise annihilations of kinks and antikinks.
Qualitative aspects: comparison principle
Let u 0 (x; n) := u 0 (x; n, 0) be a monotone initial datum (5) without antikinks and associated global solution u(x, t; n); recall the positions are globally defined and differentiable according to Proposition 2. Since the data has kinks only we omit the superindex + and also introduce the following notation for the speeds of positions, the nearest distances and the minimal distance (starting from a given position):
c i (t) := d dt ξ i (t), 1 i n,(11)d j (t) := ξ j (t) − ξ j+1 (t), 1 j n − 1 (12) d i (t) := min i j n−1 d j (t),(13)
as well as d min (t) := d 1 (t).
ξ 4 ξ 3 ξ 2 ξ 1 0 2π 4π 6π 8π d 0 d 0 d 0 x u 0 (x; 4, 0)
Figure 2.: Sketch of initial datum u 0 (x; 4) (black) with four antikinks at equidistant positions ξ i (0), i = 1, 2, 3, 4, such that u(ξ i (0), 0) = (2j + 1)π. The blue initial datum is given by u 0 with position τ j (0) = ξ j (0) for j = 1, 2, 3 while the red one is given byū 0 with positions ψ j (0) = ξ j+1 (0) for j = 1, 2, 3. For the purpose of illustration, the curves are plotted slightly below and above u 0 (x; 4), respectively.
Equidistant kinks. Let us first consider initial data composed of equidistant antikinks, cf. Figure 2, as it is straightforward to construct sub-and supersolutions in order to compare the speeds (11) and, consequently, to find a uniform lower bound on the distances (12).
Proposition 7. Let u(x, t; n) (n 2) be the solution with an initial datum u 0 (x; n), where d i (0) = d 0 for all 1 i n − 1 for some constant d 0 > 0. Then c 1 (t) c * (t) c n (t) for all t 0, and c 1 (0) c 2 (0)
. . . c n (0) . In particular, initially all distances d i (0) are non-decreasing in t and d i (t) d 0 for all t > 0.
Proof. We will consider solutions u(x, t; k) with initial u 0 (x; k) for the same positions ξ j , j = 1, . . . , k, but k ≤ n, and compare the speeds of kinks which we therefore denote as c k j , where we omit the argument t for readability; hence, c * = c 1 1 .
We first prove the statement for n = 2. In this case, the initial datum
u 0 (x; n) is sandwiched by H(x − ξ 1 ) u 0 (x; n) H(x − ξ 2 ) + 2π for all x ∈ R. By the comparison principle, for the associated solution u * we have u * (x − ξ + 1, t) u(x, t; n) u * (x − ξ 2 , t) + 2π for all (x, t) ∈ R × R + .
Since both H 0 and H 1 move with the same speed c * , the speeds of the kinks satisfy c 2 1 c * c 2 2 for all t 0.
For n 3, we choose sub-and supersolutions composed of n − 1 antikinks (cf. Figure 2). More precisely, let u 0 (x; n − 1) be the initial condition composed of n − 1 kinks with the same n − 1 positions, ξ i (0) for 2 i n, as u 0 (x; n). The comparison solutions arise from the initial data
u 0 (x) := u 0 (x − d 0 ; n − 1),ū 0 (x) := u 0 (x; n − 1) + 2π.
In particular, the kink positions of u 0 are ξ i (0) for 1 i n − 1 and those ofū 0 are ξ i (0) for 2 i n. We note that the proof of Prop. 7 shows that there is a hierarchy of speeds when removing kinks on the left or right. Up to equality in the bounds, kinks for equidistant data spread out, and the more kinks there are, the faster the spreading can be.
Non-equidistant kinks. For more general initial data, only the overall distance ξ n (t) − ξ 1 (t) and the smallest distance d min (t) can be controlled by our approach of sub-and supersolutions: lower bounds can be inferred for the distances only up to "gaps" rooted in the initial data. In these gaps the ordering of the associated positions and speeds cannot be further constrained by this method, cf. Fig. 3. In view of the upcoming analysis based on analytic positions, this is not surprising since distances may behave non-monotonically. Proposition 8. The solution to any initial u 0 (x; n), n 2, satisfies the following for all t 0:
ξ − 5 ξ − 4 ξ − 3 ξ − 2 ξ − 1 0 2π 4π 6π 8π 10π ξ 2 ξ 4 ξ 3 ξ 1 "Gaps" ξ 4ξ3ξ2ξ1 d 4 (0)d min (0) d 2 (0) d 1 (0) x u 0 (x; 5, 0)d dt (ξ n (t) − ξ 1 (t)) 0 , c 1 (t) c * (t) c n (t), d i (t) d min (0) ∀t > 0, 1 i n − 1.(14)
Moreover, if d min (0) = d j (0) for some 1 j n − 1 and t 0 then d j (t) is non-decreasing for all t 0. In particular, d min (t) is non-decreasing at least as long as the minimal distance is realised at the same index as initially.
Proof. The proof is analogous to that of Prop. 7. Comparing u with single kinks shifted to ξ 1 as a subsolution and to ξ n as a supersolution we immediately infer c 1 (t) c * (t) c n (t) so that d dt (ξ n (t) − ξ 1 (t)) 0. Next, we replace d 0 in the proof of of Prop. 7 by d min (0) and use again u 0 (x) := u 0 (x − d min (0); n − 1), as well as u 0 := u 0 (x; n − 1) + 2π, cf. Fig. 3. By the comparison principle this implies for all t 0 the relations c n i c n−1 i−1 for 2 i n from the supersolution, but the subsolution only yields for j such that d min (0) = d j (0) the relation c n−1 j ≤ c n j . Taken together we obtain, for all t 0, c n j c n−1 j c n j+1 , and therefore d dt d j (t) ≥ 0. In particular, d dt d min (t) 0 as long as the minimal distance is between ξ j (t) and ξ j+1 (t), but at least for t = 0.
As this use of sub-and supersolutions is limited by the described gaps, a substantial part of the next section is devoted to a deeper analysis of the behaviour of the distances in terms of the analytic positions.
t × 10 3 d i d 4 = ξ 4 − ξ 5 d 3 = ξ 3 − ξ 4 d 2 = ξ 2 − ξ 3 d 1 = ξ 1 − ξ 2 1 2 0.25 0.5 t × 10 3 d i (a) (b)
Quantitative aspects: projection scheme
The above discussion, and in particular Proposition 8, gives a first qualitative overview of interactions between stacked fronts traveling at the same asymptotic speed c. The purpose of this section is to obtain deeper insight into the relative laws of motion of the front positions, which is however valid for sufficiently large distances only. As mentioned in §2.2, we follow an approach that has developed out of the analysis of meta-stability of fronts in Allen-Cahn equation in [CP89,FH89].
Recall the notion of analytic positions from (8), (9), and denote η := (η 1 , . . . , η n ) as well as
δ j := η j − η j+1 , δ min = min{δ j , 1 ≤ j ≤ n − 1}.
Initial data (8) with w ≡ 0 form the nested n-dimensional manifolds (with δ = δ min )
K δ := n j=0 ϕ(· + z 0 − η j ) : η 1 > η 2 > · · · > η n , η i − η i+1 > δ, i = 1, . . . , n; z 0 ∈ R .
Theorem 5 shows that kinks interact eventually repulsively, which can be understood as asymptotic stability of the boundary of K 0 . In this section we will show that, for any sufficiently large δ, K δ parametrises an n-dimensional invariant manifold M, and we study the reduced dynamics on it. Moreover, M is globally attracting for initial data as in Theorem 5. In particular, any initial datum with initial data from geometric positions (5) is contained in its basin of attraction, if the initial geometric distances are sufficiently large.
This analysis relies on the following results, which can be inferred from [Ei02]. Although the latter is not intended to specifically study stacked fronts as in his paper, but rather pulses, our situation can be dealt with by exactly the same methods.
We reformulate the results that we use in our notation and towards our purposes, and refer in particular to Theorems 2.1, 2.3 and 4.1 in [Ei02] for proofs. We also provide in Appendix A some more details and a derivation of the reduced ODE system, in order to close the gap to [Ei02].
The basic observation is the following well known decomposition in such neighborhoods, which readily follows from an implicit function theorem, cf. Appendix A. Here we consider neighbourhoods of K δ defined as U ε,δ :
= K δ + v ∈ H 2 (R) : v H 1 < ε .
Lemma 9. There exist δ * , ε * > 0 such that for any v ∈ U ε * ,δ * , there exist unique η 1 , · · · η n ∈ R n and a unique function
w(z) ∈ H 2 (R) satisfying v(z) = n j=0 ϕ(z − η j ) + w(z) , R e cz w(z)ϕ(z − η j )dz = 0, for all 1 ≤ j ≤ n.(15)
We note the following direct consequence of Lemma 9 together with well-posedness.
Proposition 10. For any solution u(z, t) of (7) whose initial data lies in U ε * ,δ * we have T * := sup{t : u(z, s) ∈ U ε * ,δ * , 0 ≤ s ≤ t} > 0 and there exist unique functions η 1 (t), · · · η n (t) ∈ C 1 (0, T * ), and w(z, t) ∈ C 1 ((0, T * ) × R) such that (8) and (9) hold for all t ∈ (0, T * ).
In particular, initial conditions u 0 (x; 0, n) from (5) with d min sufficiently large lie in U ε * ,δ * so that Proposition 10 applies for those as well.
Building on the decomposition (8), the following proposition, which is a combination of results from [Ei02], cf. Appendix A, gives a quantitative description of the relative motions as a reduced ODE for the analytic positions η j . Relevant for the laws of motion are in particular the eigenvalues of the linearization of (3) in the asymptotic states (ϕ, ϕ ) = (0, 0) and (ϕ, ϕ ) = (2π, 0), which we readily determine as
λ := −c − c 2 − 4f (0) 2 < 0, µ := −c + c 2 − 4f (0) 2 > 0; (16) recall f (2π) = f (0) by periodicity of f, and note λ = −µ − c.
Proposition 11. There exist δ * ≥ δ * , ε * ≤ ε * such that U ε * ,δ * contains an n-dimensional exponentially attractive locally invariant manifold M, which is a graph over K δ * in terms of corrections w = w(η) satisfying (15). The reduced n-dimensional dynamics of (7) on M is given by the following ordinary differential equation system, which is defined for δ min ≥ δ * and where α, a R , a L > 0, and
R j (η), (1 ≤ j ≤ n) are C 1 −functions; furthermore µ + α > −λ for f 0 ≈ 0. η 1 (t) = a L e λ(η 1 −η 2 ) + R 1 (η) η j (t) = a L e λ(η j −η j+1 ) − a R e −µ(η j−1 −η j ) + R j (η), 2 ≤ j ≤ n − 1, η n (t) = −a R e −µ(η n−1 −ηn) + R n (η).(17)
The remainder terms R j (η) and the correction term w(η) satisfy, for some constant C > 0,
|R j |, w(η) H 2 (R) ≤ Ce −(µ+α)δ min .(18)
Remark 12. Remark that we do not claim that µ + α ≥ λ for all f 0 ∈ (0, 1), which means that the remainder terms R j and w are not necessarily higher order compared with the a L -terms whose exponential rate is λ.
In the remainder of this section we analyse (17) in more detail and in particular infer that, if the initial distance are large enough, then the validity constraint δ min ≥ δ * is satisfied for all t ≥ 0. This is of course consistent with Theorem 5, which moreover implies that all solutions with certain initial data eventually satisfy (8), (9) and obey (17).
Theorem 13. There is δ 1 ≥ δ * such that U ε * ,δ 1 forward invariant for (7). In particular, the decomposition (8), (9) and the ODE (17) are valid for all t ≥ 0. Moreover, for any solution u(z, t) with initial datum as in Theorem 5, there is t 1 ≥ 0 such that u(·, t 1 ) ∈ U ε * ,δ 1 .
Proof. The proof is given in §3.2.4.
The analysis of (17) is facilitated by normalizingδ i := µδ i (t/µ) and settingδ min := min{δ 1 , . . . ,δ n }, ε := min{α, c}/µ, so that (17) takes the form
δ 1 = e −δ 1 + g 1 e −(1+ε)δ min , δ j = e −δ j − e −δ j−1 + g j e −(1+ε)δ min ,(19)
where g j , 1 ≤ j ≤ n, are bounded. Note that whileδ min is in general only Lipschitz continuous due to the minimum function, the products g j e −(1+ε)δ min , j = 1, 2 are smooth. For convenience, we extend their domain beyond the definition limitδ min ≥ δ * to smooth functions on R n + with g j bounded.
Our aim is to analyze the temporal ordering and asymptotics of the distancesδ j , j = 1, 2 . . . , n.
To this end, we consider system (19) as a perturbation of the system with g j ≡ 0 for all 1 ≤ j ≤ n, which we refer to as the unperturbed system.
3.2.1. The unperturbed distance system For the unperturbed system, i.e. the system,
δ 1 = e −δ 1 , δ j = e −δ j − e −δ j−1 , 2 j n,(20)
we directly getδ 1 > 0, that is the rightmost distances is increasing. Moreoever, the distancẽ δ j (2 j n) increases if (and only if)δ j <δ j−1 , i.e., if the distance between the j-th kink and its left neighbor kink is smaller than the distance to its right neighbor. In particular, the smallest distance at time t 0 increases. This motivates the question whether the distances are eventually ordered and which asymptotics are implied by this ordering.
Theorem 14. There exists T > 0 such thatδ n (t) <δ n−1 (t) < . . . <δ 1 (t) for all t T . Moreover, lim t→∞δk (t) = ∞ for all 1 k n.
Proof. Let us first prove the divergence of the distances by induction over n. The statement is clear for k = 1 sinceδ 1 (t) = log(t + C), C > 0. Suppose the statement holds for δ k with 2 k n and, by contradiction,δ n+1 M for some M > 0. Thenδ n+1 (t) e −M − e −δn(t) > 0 for t large enough. Consequently,δ n+1 converges, hence lim t→∞δ n+1 (t) = 0 and thusδ n (t) → M as t → ∞, contradicting the induction hypothesis.
As to the eventual ordering, we first show the existence of some T 1 > 0 such thatδ 2 (t) <δ 1 (t) for all t > T 1 . To this end, assume by contradiction that for all t > 0 there exists some t 1 > t withδ 2 (t 1 ) δ 1 (t 1 ), i.e.δ 2 (t 1 ) 0. By the divergence of the distances, there exists t 2 > t 1 withδ 2 (t 2 ) > 0. Sinceδ 1 increases monotonically, this implies thatδ 2 intersects theδ 2 -nullcline {(δ 1 ,δ 2 ) :δ 1 =δ 2 } infinitely often. However, this is impossible since the set {(δ 2 ,δ 1 ) :δ 2 < δ 1 } = {(δ 2 ,δ 1 ) :δ 2 > 0} is forward invariant. In particular, ifδ 2 (t) =δ 1 (t) for some t 0, theñ δ 2 (t ) <δ 1 (t ) for all t > t. Analogously, there exists T 2 > T 1 withδ 3 (t) <δ 2 (t) for all t > T 2 . Likewise, for all remaining pairs of distances, there exists some suitable time T i > T i−1 . Setting T := T 1 concludes the proof.
In order to get a result similar to Theorem 14 for the perturbed system, we first reformulate the problem by applying a polar blow-up transformation; this enables us to apply a perturbation argument.
Polar blow-up
Substituting z j = e −δ j into (19), and setting z max := e −δ min we obtain the system
z 1 = −z 2 1 − g 1 z 1 z 1+ε max , z j = −z 2 j + z j−1 z j − g j z j z 1+ε max , 2 j n
which posssesses the non-hyperbolic equilibrium 0 = (0, 0, . . . , 0) ∈ R n . The polar blow-up transformation z = (z 1 , z 2 , . . . , z n ) T = r(t) Ψ(t) with r ∈ R and Ψ = (Ψ 1 , Ψ 2 , . . . , Ψ n ) T ∈ S n−1 yields
z 1 = −r 2 Ψ 2 1 − r 2+ε g 1 Ψ 1 Ψ 1+ε max , z j = −r 2 Ψ j (Ψ j − Ψ j−1 ) − r 2+ε g j Ψ j Ψ 1+ε max , 2 j n,(21)
where z max = rΨ max , i.e., for each t there is j such that Ψ max (t) = Ψ j (t). Note that the equilibrium 0 corresponds to r = 0. The inner product with Ψ gives
z , Ψ = r (t) = n k=1 z k Ψ k = −r 2 Σ n − O(r 2+ε )(22)
where Σ n := Ψ 3 1 − n k=2 Ψ 2 k (Ψ k−1 − Ψ k ). Using z k = r Ψ k + rΨ k together with (22) in (21) we may divide left and right hand sides by r to obtain
Ψ 1 = rΨ 1 (Σ n − Ψ 1 ) + O(r 1+ε ), Ψ j = rΨ j (Ψ j−1 − Ψ j + Σ n ) + O(r 1+ε ).
Dividing the right hand side by r results in the desingularised radial and angular equations
r = −Σ n r − O(r 1+ε ),(23)Ψ 1 = Ψ 1 (Σ n − Ψ 1 ) + O(r ε ),(24)Ψ j = Ψ j (Ψ j−1 − Ψ j + Σ n ) + O(r ε ),(25)
which, for r > 0, has the same trajectories as (22) and solutions are related by time rescaling.
Analogous to (20), we consider system (23)-(25) as a perturbation of the unperturbed polar system with g j set to zero (1 ≤ j ≤ n), i.e., zero error terms in (23)-(25), which means
r = −Σ n r,(26)Ψ j = Ψ j (Σ n − Ψ j + Ψ j−1 ), 1 j n, Ψ 0 := 0.(27)
3.2.3. Unperturbed polar system, part I: radial and angular dynamics
Since all distances are positive, we consider the part of the sphere with positive coordinates,
S + := S + (n) := x = (x 1 , x 2 , . . . , x n ) ∈ S n−1 : x i > 0 ∀i ,
on which the following holds.
Proposition 15. Σ n > 0 on S + . More specifically, Σ n > Ψ 2 j 0 /2, where the index j 0 is such that Ψ k = 0 for 1 k < j 0 and Ψ k > 0 for j 0 k n.
Note that the index j 0 exists since (Ψ 1 , . . . , Ψ n ) ∈ S n−1 does not vanish.
Proof. Rewrite Σ n as Σ n = Ψ 3 j 0 + n k=j 0 +1 Ψ 2 k (Ψ k − Ψ k−1 ) and set f k−1 (Ψ k ) := Ψ 2 k (Ψ k − Ψ k−1 ) as a function in Ψ k with parameter Ψ k−1 which has its minimum (on [0, 1]) at 2 3 Ψ k−1 , i.e.
f k−1 (Ψ k ) f k−1 2 3 Ψ k−1 = − 4 27 Ψ 3 k−1 . Hence, Σ n Ψ 3 j 0 − 4 27 n k=j 0 +1 Ψ 3 k−1 = Ψ 3 j 0 − 4 27 n−1 k=j 0 Ψ 3 k , Ψ k = 2 3 Ψ k−1 .(28)
Since Ψ k = 2 3 Ψ k−1 one gets Ψ k = 2 3 k−j 0 Ψ j 0 by recursion and consequently,
Σ n Ψ 3 j 0 − 4 27 n−1 k=j 0 2 3 3(k−j 0 ) Ψ 3 j 0 > Ψ 3 j 0 1 − 4 27 ∞ k=0 2 3 k = 15 27 Ψ 3 j 0 > 1 2 Ψ 3 j 0 Lemma 16. Ψ j > 0 for 0 < Ψ j 1 and there exist T > 0, ε > 0 such that Ψ j (t) ∈ (ε, 1 − ε) for all t T . Proof. If Ψ j = 0, we have Ψ j−1 − Ψ j + Σ n = Ψ j−1 + Σ n > 0. Choosing Ψ j > 0 small enough, we therefore get Ψ j = Ψ(Ψ j−1 − Ψ j + Σ n ) > 0.
This, together with Prop. 15, implies the second statement by contradiction.
As a consequence, the distances of the distancesδ j in the unperturbed ODE system are bounded.
Proposition 17. Solutions to (20) satisfy |δ j −δ k | > 0 andδ j −δ k = O(1) as t → ∞.
Proof. The distances are eventually ordered (cf. Thm. 14), thus |δ j −δ k | > 0 for large t > 0. To prove the boundedness, suppose thatδ j −δ k → ∞.
Then, e −(δ j −δ k ) = z j z k = Ψ j Ψ k → 0. However, due to Lemma 16, Ψ j Ψ k ∈ ε 1−ε , 1−ε ε for large t > 0.
Consequences for the perturbed polar system
We now return to the system (23)-(25). On the sphere (i.e. for r = 0), the angular equations (24) and (25) coincide with those of the unperturbed polar system (27). Likewise, near the sphere (i.e. for 0 < r 1), the radial dynamics (23) are dominated by the radial term −Σ n r, cf. (26). This allows for a perturbation argument from which one can deduce that the distances in the perturbed system diverge.
Theorem 18. There isδ 0 > 0 such for all solutions to (19) withδ min (0) >δ 0 , the statements of Theorem 14 and Proposition 17 hold true. In particular, there existsδ 1 ≥ 0 such that for all solutions to (19) withδ min (0) >δ 1 it holds thatδ min (t) ≥δ min (0) for all t ≥ 0.
Note that together with Proposition 11 this in particular implies Theorem 13.
Proof. With respect to the unperturbed polar system (26)-(27), M 0 := {r = 0} × S + is an inflowing normally hyperbolic invariant manifold (cf. [HPS77,Wig94]) since the transversal eigenvalues are strictly negative, cf. Prop. 15, and the boundary is repulsive due to Lemma 16. By robustness of such invariant manifolds, M 0 possesses a non-trivial local basin of attraction Γ for the unperturbed and perturbed polar systems (23)-(25). Theorem 14 implies that any solution to the unperturbed system with angular initial data in S + enters Γ in finite time, which therefore also holds for the perturbed system if initial distances are sufficiently large. These solutions thus converge to M 0 . The property that eventually Ψ j ∈ (ε, 1 − ε), cf. Lemma 16, is likewise structurally stable and thus remains valid for the perturbed polar system. In particular, this property means that all distancesδ j diverge.
Without loss of generality, we can choose Γ to lie in the n-dimensional local stable manifold W s of M 0 in the unperturbed system, which we write as W s (S + ) since M 0 is trivially parameterized by S + . Now, for the perturbed and unperturbed polar system, W s (S + ) is foliated by onedimensional strong stable fibers W ss (Ψ), respectively,
W s (S + ) = Ψ∈S + W ss (Ψ),(29)
which are pairwise disjoint and each intersect M 0 in their base point Ψ ∈ S + . The dynamics of the base points is given by (27) for both the perturbed and unperturbed polar systems, but the fibers differ in general.
The key point is that the flows of both (26)-(27) and (23)-(25) in Γ are slaved to the base point flow such that the perturbed flow inherits the properties of the unperturbed flow as claimed. More specifically, let Ψ(t), Ψ(0) ∈ S + be a solution to (27). Then the perturbed and unperturbed flows map their respective fibre W ss (Ψ(0)) into their respective fibre W ss (Ψ(t)); using the foliation (29) for local coordinates of the perturbed system near the sphere, the base flow (which is always the same) decouples from the transverse fibre flow, which differs between perturbed and unperturbed flow.
Since the base point flow leads to an eventually ordering of the distances for the unperturbed polar system, cf. Thm. 14, the slaving implies the same for the perturbed system (19). The remaining claims follow analogously. These imply that there isδ * such thatδ min (t) grows strictly ifδ min ≥δ * , which implies the existence of anδ 1 ≤δ * as claimed.
Unperturbed polar system, part II: local stability
Here we add more details to the dynamics of the unperturbed polar system: we prove that there is a unique equilibrium point in S + , and it is locally exponentially stable. For illustration, let us first consider the simplest cases n = 2 and n = 3.
For n = 2, the system (27), with Σ 2 = Ψ 3 1 − Ψ 1 Ψ 2 2 + Ψ 3 2 , reads
Ψ 1 = Ψ 1 (−Ψ 1 + Σ 2 ) Ψ 2 = Ψ 2 (Ψ 1 − Ψ 2 + Σ 2 ).
Let us first assume that Ψ 1 = 0 which implies Ψ 1 = Σ 2 for an equilibrium. If Ψ 2 = 0, then Ψ 2 = 2Ψ 1 and thus Ψ 1 = ± 1 5 , and yields the two equilibria E 1,2 := ± 1 5 , 2 1 5 . For Φ 2 = 0 we find the two equilibria E 3,4 := (±1, 0) and for Φ 1 = 0 the last two, E 5,6 := (0, ±1). In total, the system has S 2 := 2(2 2 − 1) = 6 equilibria.
For n = 3, the situation is already a bit more complicated. We have Σ 3 = Ψ 3 1 − Ψ 1 Ψ 2 2 + Ψ 3 2 − Ψ 2 Ψ 3 3 + Ψ 3 3 and need to distinguish the following different cases depending on the number 0 k 2 of zero coordinates, each giving a pair of equilibria: In total, the system has S 3 := 2(2 3 − 1) = 14 equilibria.
For general n ∈ N the following holds.
Proposition 19. For n ∈ N, system (27) posesses S n := 2(2 n − 1) equilibria. Specifically, there are (i) 2n equilibria which have exactly one non-zero component (which is therefore ±1) and (ii) S n − 2n − 2 equilibria with 2 j n − 1 non-zero components.
Proof. Let n ∈ N and S n denote the number of equilibria Ψ = (Ψ 1 , . . . , Ψ n ) of dimension n. First note that Ψ j = 0 implies for equilibria Ψ j = Σ n + Ψ j−1 by (27). Thus, Ψ j−1 = 0, Ψ j = 0 implies Ψ j = Σ n , and adjacent non-zero entries come as a sequence (Σ n , 2Σ n , . . . , kΣ n ), where 2 ≤ k ≤ n. Hence, an intermediate zero coordinate between two non-zero coordinates results in a triple (Σ n , 0, Σ n ).
Consequently, if Ψ 1 = 0, the S n−1 equilibria of dimension n − 1 occur with shifted index; this in particular includes the vectors with Ψ j = ±1 for some 2 ≤ j ≤ n and zero coordinates otherwise. Denote the number of remaining equilibria with Ψ 1 = 0 by M n . By the discussion above, M n is the number of possibilities to have k ∈ {0, 1, . . . , n−1} zero entries on the positions different from Ψ 1 = 0 (in particular including (±1, 0, 0, . . . , 0)). This number is given by M n = 2 n−1 k=0 n−1 k .
All together,
S n = M n + S n−1 = n j=1 M j = 2 n j=1 n−1 k=0 j − 1 k = 2 n j=1 2 j−1 = 2(2 n − 1).
By the previous lemma, E ± := (Ψ 1 , 2Ψ 1 , 3Ψ 1 , . . . , nΨ 1 ) with Ψ 1 = ± 6 n(n+1)(2n+1) are the unique equilibria with non-zero coordinates only.
Local stability of E + . In the following, we focus on E := E + since for the question of divergence of the distances in the perturbed ODE-system this is the only relevant asymptotic state. In fact, we expect that it is a global attractor on S + as can be illustrated by simulations for n = 2 and n = 3, cf. Fig. 5. However, it seems difficult to prove this rigorously in general. Instead, by linearizing (27) in E and determining the eigenvalues of the Jacobian, we show that E is locally stable on S n−1 .
Theorem 20. The equilibrium E is locally exponentially stable on S n−1 . The eigenvalues of the linearisation in E are given by − k 6 n(n+1)(2n+1), 2 k n. For the full unperturbed polar system in R n , E possesses the unstable eigendirection transverse to M 0 spanned by (1, 2, . . . , n) with corresponding eigenvalue 1 3 n(n + 1)(2n + 1).
Proof. The somewhat lengthy proof is given in Appendix B.
(a) (b) Figure 5.: Phase portraits and equilibrium E (marked in red) for (a) n = 2 and (b) n = 3, respectively. For n = 2, the phase portrait is restricted to the upper right sector and the transverse direction is neglected since it is not relevant for the angular dynamics. As these simulations suggest, E is a global attractor on S + for these cases.
Bounded initial kink-antikink data and their annihilation
Having discussed pure kink or antikink initial data, we now turn to initial data which are composed of kinks and antikinks, that is, initial data (5) with n + m < ∞ and nm = 0, cf. Fig. 6 for illustration. Here we constrain ourselves to bounded data and consider the unbounded case in §5. We aim to infer information about the process by which the respective inner pair of kink and antikink with positions ξ ± 1 (t) annihilate each other.
Annihilation process
In case m = n, the equal number of pairs of kinks and antikinks completely annihilate each other in the sense that the solution converges to the rest state 2πn given by the asymptotic state of the initial data.
Proposition 21. Let u 0 (x, n, n) be an initial datum (5) with m = n, 0 < n < ∞. Then, lim t→∞ u(·, t) = 2πn, where the convergence is uniform in x ∈ R. In particular, ω(u 0 (x; n)) = Ω(u 0 (x; n)) = {2πn}. Proof. Consider subsolutions with initial data the pure kink-or antikink sequence u 0 (x; n, 0), u 0 (x; 0, n) built from the kink-or antikink positions of u 0 (x; n, n), respectively. According to Theorem 1.1 [Pol20] the kinks and antikinks move with uniform positive, respectively negative speed. The comparison principle implies the claim.
The following proposition describes the corresponding annihilation process in some more detail by showing that the kinks-antikink pairs get annihilated successively from "bottom to top" as expected intuitively.
Proposition 22. Let u 0 (x; n, n) be an initial datum (5) with n < ∞ and corresponding solution u(x, t). Then, there are unique times 0 < t 1 < t 2 < . . . t n such that ξ − j (t j ) = ξ + i (t j ), 1 j n, i.e., the successively innermost kink and antikink collide at these times. Moreover, there are unique times t A j ∈ (t j , t j+1 ), 1 j n − 1, such that argmin x∈R {u(x, t A j )} = 2πj, i.e., at these times the successively innermost kink and antikink annihilate.
To ease notation we set t A 0 := 0.
Proof. As shown in the proof of Proposition 2, for all t > 0, the solution u(x, t) is monotonically decreasing on (−∞, η(t)) with lim x→−∞ u(x, t) = 2πn and monotonically increasing on (η(t), ∞) with lim x→∞ u(x, t) = 2πn. Together with Proposition 21, this shows that the positions ξ ± i (t), 1 i n, collide at some unique time t i > 0, respectively, i.e., u(η(t i ), t i ) = (2i−1)π and u(·, t; n) > (2i − 1)π for t > t i ; therefore ξ ± i (t) exist for [0, t i ] only. In particular, t i < t i+1 . Analogously, we identify the annihilation times t A i ∈ (t i , t i+1 ).
Next we show that during the annihilation process of the innermost kink-antikink pair, the distances between the remaining kinks (antikinks) satisfy a uniform lower bound.
Proposition 23. Let u(x, t) be the solution corresponding to an initial condition as above.
Then, d ± j (t) ≥ d ± min (0) and if j ± are indices such that d ± min (0) = d j ± (0) then d j ± (t) are nondecreasing as long as defined, i.e., t ≤ t j , respectively. In particular,
|ξ − j+1 (t) − ξ + j+1 (t)| d + min (0) + d − min (0) + |ξ − j (t) − ξ + j (t)| for all t ∈ (0, t j ].
Proof. Due to reflection symmetry, it suffices to prove the claims for the case '+'. In the proof of Proposition 8, we have constructed initial conditions u 0 andū 0 whose corresponding solutions u(x, t),ū(x, t) are sub-and supersolutions, which imply the non-decrease of the minimal distance. In the present case the same subsolutions can be used. However,ū 0 are not providing supersolutions on R, but only on (−∞, η(t)) since u(·, t) is minimal at η(t) andū(x, t) can be above u(x, t) only for x > η(t).
The claims now follow from the statement of Proposition 8.
It is tempting to suppose that the minimal distance between kinks or antikinks, respectively, that arises after annihilations can be used as a lower bound. However, it seems difficult to construct super-and subsolutions to substantiate this.
Let us now turn to the case of initial data with unequal number of kinks and antikinks, m = n; without loss of generality m < n. In this situation, the first min{m, n} innermost kink-antikink pairs annihilate and a stacked front of either kinks or antikinks remains after some finite time.
Proposition 24. Let u 0 (x; m, n) be an initial datum (5) with m < n < ∞ and mn = 0. The kinks and antikinks at ξ ± i (t), 1 i m, collide and annihilate in the sense of Proposition 22. Moreover, the limit sets are given by
ω(u 0 (x; m, n)) = {2πn}, Ω(u 0 (x; m, n)) = {2πj : m j n} ∪ {ϕ + j (· − ν) : m < j < n, ν ∈ R}.
Proof. In the course of the annihilation process of the m innermost kink-antikink pairs, u(x, t) gets uniformly close to 2πm for x ≤ η(t)). Meanwhile, the distances d + j (m ≤ j < n) of the remaining kinks are bounded from below by Proposition 23, which means the solution gets arbitrarily close to a propagating terrace. Since 2πm is a stable steady state, the solution u(x, t) converges to a propagating terrace and [Pol20, Theorem 1.1] implies the statement. Towards a more complete picture, let us briefly consider initial data with local maxima built by pairs of kinks and antikinks, cf. Fig. 7. For each such maximum with sufficiently large distance of kink and antikink, one can construct a stationary subsolution using phase plane analysis for the equation u xx + f (u) = 0; this shows that kinks and antikinks cannot annihilate at local maxima, but only between local minima as described before. In particular, the limit sets ω(u 0 ) and Ω(u 0 ) are completely determined by the numbers m, n between maxima. Unbounded initial data is most relevant for the comparison with the dynamics of the cellular automaton GHCA, such as its non-wandering set dynamics and topological entropy. However, for unbounded data several useful results cannot be directly applied, e.g., those on terraces in [Pol20].
Unbounded kink or kink-antikink data
Nevertheless, the zero number argument is still applicable (the growth condition of [Ang88, p. 80] is satisfied for u ∈ X ω ) so that we readily infer an analogon to Proposition 22 in case m = n = ∞: we can repeatedly choose initial data composed of finite kink-antikink pairs as subsolutions and obtain an infinite sequence of strictly increasing collision and annihilation times.
However, numerical simulations of (1) (lifted to u(x, t) ∈ R), cf. Figs. 8, 9, suggest that distances equilibrate asymptotically in time. Hence, initial distance information encoded far from an eventual collision is lost over time, is "washed out" -at least on the scale of the initial data. Indeed, this is consistent with our results on the dynamics of analytic positions for monotone data, and large initial distances. Therefore, as suspected from weak interaction induced by diffusion, we cannot expect entropy and dynamics for (1) with unbounded initial data are directly analogous to GHCA.
We next first corroborate the equilibration of distances for unbounded initial data by considering periodic boundary conditions and show that all solutions converge locally uniformly to equidistant staircases. Second, we discuss implications for complexity measures based on positional dynamics.
Periodic boundary conditions
Unbounded superpositions of equidistant kinks (and, analogously, antikinks) are parametrized by the uniform distance > 0, and the problem is transformed into a boundary value problem under periodic boundary conditions, up to phase rotation. More precisely, for given ∈ R + and j ∈ N, we consider the initial-boundary value problem
θ t = θ xx + f (θ), − x (30a) θ(0, x) = θ 0 (x), − x (30b) θ(− ) = θ( ) + 2πj, θ(− ) θ θ( ) (30c) θ (− ) = θ ( ) (30d)
and prove in Prop. 28 below that solutions of (30) converge to a unique equidistant staircase in terms of the ω-limit set of θ 0 .
To this end, we first consider travelling wave solutions θ(x, t) = u(x − at) =: u(z) and focus on the following boundary value problem, where = d dz .
Proposition 25. For each triple ( , j, a) ∈ R + × N × R, the boundary value problem
u + au + f (u) = 0, − < x < (31a) u(− ) = 2πj, u( ) = 0, 0 u 2πj (31b)
has a unique, monotonously decreasing solution u with ∂ x u < 0 on (− , ). Moreover, there exists a unique a( , j) such that the unique solution u of (31) corresponding to ( , j, a( , j)) additionally satisfies u (− ) = u ( ). This a( , j) is given by
a( , j) = (F (2πj) − F (0)) − (u ) 2 dx −1 ,
where F = f . In particular, lim →0 a( , j) = 0.
We remark that the case j = 1 is essentially contained in [RE98, Thm 3.5].
Proof. Sub-and super solutions are given by u ≡ 0 andū ≡ 2πj, respectively, which shows the existence of a solution u. Applying Theorem 1.4 [BN91] ("sliding method") to Ω := [− , ], together with the maximum principle, implies the monotonicity and uniqueness of this solution.
Moreover, this unique solution (i) depends continuously on a and (ii) is strictly decreasing in its dependence on a (i.e. if a 1 < a 2 , then u 2 < u 1 in Ω , where u 1 and u 2 are the solutions of (31) corresponding to ( , j, a i ), i = 1, 2, cf. Corollary 5.1 [BN92]). By Lemma 5.2 [BN92], one infers that (iii) lim a→−∞ u = 2πj and lim a→∞ u = 0, both uniformly in x. Items (i)-(iii) together imply the existence of a unique solution u and a unique a( , j) such that u (− ) = u ( ).
As to a( , j) and its asymptotics, one multiplies equation (31a) by u and integrates to get
a( , j) − (u ) 2 dx = − − u u + u f (u) dx = F (2πj) − F (0), where u (− ) = u ( ) is used. We obtain a( , j) = (F (2πj) − F (0)) − (u ) 2 dx −1 . By the Cauchy-Schwarz inequality, − (u ) 2 dx 4π 2 j 2 2
and thus a( , j) → 0 as → 0.
Remark 26. Since f is 2π-periodic, the statement clearly remains true for boundary conditions u( ) = 2πk and u(− ) = 2π(k + j), j, k ∈ N, and the solution is just u + 2πk; in particular (30c), (30d) hold. We also remark that F (2πj) = F (0) for our nonlinearity f .
In the following, we focus on the unique solution u of (31) corresponding to the triple ( , j, a( , j)). Due to its additional property u (− ) = u ( ), it is the relevant solution for the purposes of this section. As a consequence of the previous proposition, its shape is determined by single periodic solutions in the following sense; in particular all kinks in u are equidistant.
Proposition 27. Let u denote the unique solution of (31) which corresponds to the triple ( , j, a( , j)). Then u consists of j space shifted and phase rotated copies of the solutionũ of (31) corresponding to (˜ , 1, a(˜ , 1)) with˜ = /j. In particular, a( , j) = a(˜ , 1), u (± ) =ũ (±˜ ) < 0 and u has time period T =˜ /a(˜ , 1).
Proof. We split the interval [− , ] into j intervals I s := [ s+1 , s ], (0 s j − 1) of width 2˜ , where s := − 2s˜ . In particular, s > s+1 and 0 = , j = − . On each interval, we consider a space shifted and phase rotated version ofũ which together build a solution due to equal derivatives at the boundaries. More precisely, let v(x) :=ũ(x − (j − 1)˜ ) be the spatial shift ofũ to the rightmost interval I 0 and, for s ∈ {0, 1 . . . j − 1}, set
U (x) := v(x + 2s˜ ) + 2πs, x ∈ I s .
The function U is thus defined on [− , ] and solves (31) with a = a(˜ , 1) and, moreover, U (− ) = U ( ). By the uniqueness result of Prop. 25, it follows that U = u and, in particular, a( , j) = a(˜ , 1) and u (± ) =ũ (±˜ ). By the construction of U , the same proposition implies u (±˜ ) < 0 since u (x) < 0 for all x ∈ (− , ).
Having established the unique solution of problem (31) with u (− ) = u ( ), we next show that solutions of (30) converge to this solutions in the following sense; without loss of generality, we choose θ( ) = 0 and consider continuous initial conditions 0 θ 0 2πj, j ∈ N.
Proposition 28. Let u be a solution of (30) with θ( ) = 0 and initial datum 0 θ 0 2πj for some j ∈ N. Then, the ω-limit set of θ 0 consists of the orbit associated to the unique solution of (31) corresponding to ( , j, a( , j)).
Proof. This is basically a consequence of [FM89, Theorem 1] which characterises the limit set by providing a dichotomy between stationary and periodic solutions. However, in order to apply this theorem, we need to transform (30) into a problem with periodic boundary conditions.
To this end, we consider w(x, t) := θ(x, t) + πj(x− ) which transforms the problem into
w t = w xx + g(x, w),(32a)w(0, x) = w 0 (x) = θ 0 (x) + πj(x − ) ,(32b)w(− ) = w(− ), (32c) w (− ) = w ( ) (32d)
where g(x, w) := f w − πj(x− ) . In particular, there is a one-to-one relation between solutions of (30) and (32). Let u denote the unique solution of (31) associated with ( , j, a( , j)). Since
W (x, t) = u(x − a( )t) + πj(x− )
is a periodic solution solution of (32), the mentioned dichotomy implies that ω(w 0 ) consists of the periodic orbit associated with W and transforming back proves the statement.
Remark 29. The previous proposition shows that solutions of (30) converge locally uniformly to (some translate of ) the corresponding unique solution of (31), i.e., the kinks become equidistant, cf. Fig. 9. For a general analysis of convergence in one-dimensional semilinear heat equations, including other types of boundary conditions and nonlinearites, we refer to [CM89, Mat88, AF88] and references therein.
Complexity considerations
The motivation to consider kink-antikink dynamics in the θ-equations emerges from our analysis of GHCA and their (topological) complexity [KRU20]. The understanding of this complexity mainly relies on the observation that for GHCA the original and somehow abstract definitions of topological entropy [AKM65,Bow71] can be substantiated by considering simpler but equivalent definitions. In this regard, one combinatorial approach is to count possible realizations of bounded space-time windows and to determine the exponential growth rate as the size of these windows increases [DS91,KRU20]. On the other hand, the decomposition of the non-wandering set reveals that the topological complexity is completely determined by the invariant subsystem consisting of bi-infinite configurations which are composed of counter propagating pulses, (. . . , 0 k j , w R , 0 k j+1 , . . . , w R , 0 k 0 , w L , 0 k 1 , w L , . . .),
where 0 k : (0, . . . , 0) are zero blocks of length k ∈ N and w R,L represent local pulses, i.e., blocks that move to the right and left under the CA-dynamics, respectively. Thus, a more tailored way to encode the topological complexity of GHCA relies on the construction of a topological conjugacy by defining a proper homeomorphism that maps counter propagating configurations to admissible sequences of collision sequences in the form of pairs (p n , s n ) n∈N ⊂ Z×N of collision positions p n and times s n .
We stress that the consistency of these two approaches to determine the topological entropy is crucially based on the specific dynamics and topological setup of GHCA and is far from general validity.
Nevertheless, for θ-equations, tracking the geometric or analytic kink and antikink positions provides a mapping from kink-antikink initial data to collision sequences and thus allows for a comparison with those of GHCA with its encoded complexity. Hence, insight into position dynamics of kink, antikink and kink-antikink initial data is a preqrequisite for at least a first heuristic insight into the underlying complexity of the dynamics As we have shown, bounded initial data give finite sequences of this kind, but for unequal numbers of kinks and antikinks the positional dynamics remains non-trivial also after the final collision in terms of terraces. This eventually occurs on an exponentially slow time scale and distances of terraces eventually diverge, cf. Prop. 7-8, so that we do not expect a significant contribution to any position-based complexity measure. This may be corroborated further based on our lower bounds for the distances and the ODE for large initial distances, which severely constrains the positions and thus a priori reduces a position based complexity. This metastable slow dynamics already occurs before collisions and thus modifies the relation of initial positions and collision sequence. However, this is a perturbation for any fixed number of n initial kinks and antikinks, and we conjecture this can be compensated by a perturbation of initial positions.
In contrast, the above result on equilibration of distances strongly suggest that this is no longer true for unbounded kink-antikink initial data. While the dynamics of (semi-)unbounded kink or antikink data resembles the pure shift dynamics of (semi-)infinite pulse configurations of GHCA, the dynamics of unbounded kink-antikink data bears resemblance to that of infinite counter propagating pulses of GHCA -up to the aforementioned equilibriation of distances for 'late' collisions.
For a direct comparison to the complexity of GHCA in terms of positions, a more specific question is whether any given admissible sequence of collision positions and times for GHCA can also be realized by appropriate initial kink-antikink data. To be more concrete, let x = (x n ) n∈Z denote a configuration of the form (33) with pulse positions p − i (right-moving) and p + i (leftmoving), respectively, which realizes a given sequence (p n , s n ) n∈N of collision points and times. On the one hand, if we choose kink and antikink positions to be exactly the same, ξ ± i = p ± i , in general one cannot expect to observe the given collisions positions and times (cf. "washing out", Fig. 8). On the other hand, this does not rule out the possibility that a time-rescaled sequence (p n , τ s n ) n∈N , with suitable scaling factor τ , can be attained from modified initial kink and antikink positions, especially because the equilibration of distances is a phenomenon under large distances, i.e., far from collisions only. This shows that more general quantitative knowledge about the position dynamics, including less restrictive initial data, is required in order to analyze these aspects rigorously.
Discussion
Motivated by studies of the Greenberg-Hastings cellular automaton (GHCA) as a caricature of excitable systems, in this paper we have considered the θ-equations describing oscillatory phase dynamics, as the perhaps simplest PDE model of excitable media. Since the non-wandering set of GHCA in essence consists of certain excitation pulse sequences [KRU20], we have focussed on the analogue of such data in θ-equations, which consists of kinks and antikinks. Moreover, in GHCA the topological entropy can be related to kink-antikink collisions. We have therefore analyzed the dynamics of bounded and unbounded kink-antikink initial data, including pure kink and antikink data. To this end, we have defined geometric and analytic positions of kinks and antikinks, the first used for a qualitative analysis of bounded and unbounded data, the latter for quantitative results concerning bounded monotone data. For bounded initial data, the theory of terraces shows that, up to spatial reflection, the ω-limit set indeed consists essentially of finite kink-sequences that weakly interact [Pol20,Ris17].
As to the qualitative analysis, we have shown that the set of geometric positions is well-defined and consists of isolated points that lie on smooth curves up to collisions. Using the comparison principle, we have revealed that the minimal initial distance is a global lower bound for distances and that collision times and positions can be tracked abstractly. As a model for unbounded data far from collisions, we have considered monotone data with periodic boundary conditions and have shown that the initial distances asymptotically equilibrate. Consequently, information encoded in these distances is lost over time which indicates that for the PDE, contrary to GHCA, an analogous topological entropy based on positions alone cannot be expected.
As a quantitative analysis, for bounded monotone data, we have derived ODE for the analytic positions of the kinks (antikinks) within the weak interaction regime. By blow-up type singular rescaling and a perturbation argument, we have shown that the dynamics is slaved to spherical dynamics and distances become ordered in finite time, and eventually diverge; again, this incorporates a loss of information in terms of initial distances.
The combination of comparison principle and weak interaction theory has revealed that the kink-antikink collision dynamics in the PDE is a multi-scale problem, in contrast to the single scale nature of the GHCA. The fast time scale is essentially determined by the speed of an individual kink, while the slow time scale stems from 'tail' interaction that is exponentially slow in the kink (or antikink) distances.
In order to combine these approaches for unbounded data, it would be interesting to study whether the approach of [ZM09] can be used to admit infinitely many kinks (antikinks) in order to justify ODE for the positions, and whether the approach of [SW08a, BST08] -which requires different speeds of the kinks (antikinks) -can be adapted to derive motion laws for nonmonotone solutions from kink-antikink initial data. This might allow to estimate complexity measures adapted to the PDE context, and thus quantify the impact of the slow weak interaction on the fast collision dynamics.
Concerning models of excitable media from systems of PDE, such as the famous FitzHugh-Nagumo equations, there are two major issues. First, the interaction of excitation pulses no longer needs to be pure annihilation, but rebounds and even pulse replication is possible [CRS20,HO00,NU00]. Second, while weak interaction theory can be generalised to systems, the comparison principle cannot. Nevertheless, the heuristic conclusions extend to this case: weak and strong interaction yield a multi-scale problem and diffusion effects positional dynamics in a non-trivial way for infinitely many pulses, thus impacting positional complexity.
Lastly, we mention that the comparison principle has been replaced by energy methods in the Allen-Cahn and Cahn-Hilliard equations [SW18,Wes20]. However, typical PDE systems of excitable media do not seem to possess an energy structure that could be exploited.
Acknowledgements:
This work has been supported by grant RA 2788/1-1 of the German research fund (DFG).
A. Law of motion -ODE for the kink distances
This appendix is devoted to the derivation of the differential system (17). As explained in the introduction of §3.2, we follow the scheme presented in [Ei02,RLZ15] that we adapt to our situation for the convenience of the reader.
Preliminaries We consider the parabolic problem (7), i.e. already in the moving framework z = x − ct. It admits a family of standing wave profiles {ϕ(· − ξ) + 2kπ, ξ ∈ R, k ∈ Z}, where ϕ is the unique standing wave of (7) connecting 2π = ϕ(−∞) and 0 = ϕ(+∞) such that ϕ(0) = π. Analyzing (3), it is a well known fact that ϕ satisfies
ϕ(z) = 2π − a − e µz (1 + O(e γz )) z < 0, a + e λz (1 + O(e −γz )) z ≥ 0,(34)
for positive a − , a + , γ, and where λ, µ are the eigenvalues of the linearized system at the asymptotic states (ϕ, ϕ ) = (0, 0) and (ϕ, ϕ ) = (2π, 0), cf. (16). Linearizing (7) at these equilibria gives the same linear operator by periodicity of f given by L :
D(L) ⊂ X −→ X v −→ ∂ zz v + c∂ z v + f (ϕ)v.(35)
For the purpose of our analysis, one can take X = L 2 (R) and D(L) = H 2 (R). Due to the translation of the steady state equation, the operator L has a kernel: Lϕ = 0. Since ϕ has constant sign and considering the behavior of f (ϕ) as z → ±∞ it follows from Sturm-Liouville theory that 0 is the largest eigenvalue of L, it is simple, and isolated:
N (L) = N (L 2 ) = ϕ R.
It is a common result (see [Sat76] or [Roq92] for instance) that there exists a closed subspace X 1 R(L) of X such that X = X 1 ⊕ N (L). The space X 1 is the kernel of an element of the dual X * that we denote e * , with the normalization ϕ , e * = 1. It is defined through the usual inner product
ψ, e * = Λ R e cz ϕ (z)ψ(z)dz,(36)
where Λ is the normalization constant. It follows that the projection onto N (L) is given by P ψ = ψ, e * ϕ .
In our situtation, we are dealing with multiple fronts defined by their positions η 1 > η 2 > · · · > η n . With a straighforward abuse of notation, we denote
ϕ j = ϕ(· − η j ), L j = ∂ zz + c∂ z + f (ϕ j ), e * j z → e c(z−η j ) ϕ j (z).
Notice that if the η j depend on time, so do these objects.
Projection Scheme Let us now turn to the proof of Lemma 9 and define the map Φ : U ⊂ R n −→ R n (η 1 , · · · , η n ) −→ ( w, e * 1 , · · · , w, e * n )
where w = v − n j=1 ϕ j . The Jacobian
∂Φ ∂η = [ ϕ j , e * i − δ i,j w, e c(z−η i ) ϕ i ] n i,j=1
has diagonal entries close to 1 for 0 < ε, δ −1 small enough, i.e. w small and distances |η i − η j | large enough, while the off-diagonal entries are close to zero. The implicit function theorem gives the desired result.
Reduced ODE We turn out to the derivation of (17). Let us emphasize once again that we refer to [Ei02] for further details and proofs of all the underlying estimates.
To get the reduced ODE equation we use the Ansatz (8). It yields
∂ t u = − n j=1 η j ϕ j + ∂ t w = L[u],(38)
where L[v] := ∂ zz v + c∂ z v + f (v) denotes the nonlinear operator associated with our problem. Let us project (38) onto the kernel of L * j , for all j ∈ {1, . . . , n} :
− η j − i =j η i e * j , ϕ i + e * j , w t = e * j , L[u] .(39)
On the other hand, since d dt e * j , w = 0, it follows that e * j , w t = η j Λ R e c(z−η j ) w(z)ϕ j (z)dz.
The system for the positions η = (η 1 , . . . , η n ) is therefore given by Aη = β, where β j = e * j , L[u] and the matrix A = (a i,j ) is given by
a i,i = − 1 + Λ R e c(z−η j ) w(z)ϕ j (z)dz a i,j = e * j , ϕ i for i = j.
Notice that if i = j, a i,j = O(e −µδ min ) and that a i,i = −1 + O( w ), therefore in the expected asymptotics the matrix A is nearly equal to −I n . The reduced ODE for the positions becomes
− η j (t) = e * j (t), L[u(t)] + hot,(40)
where the higher order terms hot are of order w H 2 (R) + e −µδ min η i . In the following, we explain how this reduced ODE (40) leads to (17). For the sake of simplicity, during our computation we will include any other negligible terms into 'hot', without changing its name, keeping in mind the leading order we are interested in. We also focus on the (most intricate) situation 1 < j < n, the two specific cases j = 1 and j = n being similar. Let us first notice that from (8),
L[u] = n i=1 ϕ i + cϕ i + w zz + cw z + f n i=1 ϕ i + w =f n i=1 ϕ i − n i=1 f (ϕ i ) + O w W 2,∞ (R) so (40) becomes − η j (t) = e * j , f n i=1 ϕ i − n i=1 f (ϕ i ) + hot.(41)
The above bracket is an integral over R (see (36)). Let us fix M large enough, such that 0 < M < δ min , to be determined later. We split the integration over three domains
R =(−∞, η j+1 + M ] ∪ [η j+1 + M, η j−1 − M ] ∪ [η j−1 − M, ∞), so that − η j (t) = I − 1 + I 2 + I + 1 + hot.(42)
Leading order terms: integration around the front Let us first focus on I 2 , since it is the most important term. Notice that for z ∈ [η j+1 + M, η j−1 − M ] one has, from (34) and classical invariant manifold theory that
ϕ i (z) = a + e λ(z−η j ) + hot, for i > j 2π − ϕ i (z) = a − e µ(z−η i ) + hot, for i < j,(43)
and that their derivatives satisfy similar estimates. Then, using the 2π−periodicity of f, and linearizing at ϕ j and 0, one has (see also (51) and (52)):
1 Λ I 2 = η j−1 −M η j+1 +M e * j (z) f n i=1 ϕ i (z) − n i=1 f (ϕ i (z)) dz = η j−1 −M η j+1 +M e * j f ϕ j + i<j (ϕ i − 2π) + i>j ϕ i − f (ϕ j ) − i<j f (ϕ i − 2π) − i>j f (ϕ i ) dz = η j−1 −M η j+1 +M e * j (z) f (ϕ j (z)) − f (0) i<j (ϕ i (z) − 2π) + i>j ϕ i (z) dz + hot(44)
where the higher order terms in (44) are of order hot = O (δ j + δ j−1 )e −2µM as can be seen by (43). We split the integral in (44) between the left interacting fronts (i > j) and the right interacting fronts (i < j). Once again, the leading order terms are determined by the closest fronts, ϕ j+1 and ϕ j−1 respectively. It follows that
1 Λ I 2 = η j−1 −M η j+1 +M e * j (z) f (ϕ j (z)) − f (0) (ϕ j−1 (z) − 2π) dz + η j−1 −M η j+1 +M e * j (z) f (ϕ j (z)) − f (0) (ϕ j+1 (z)) dz + hot I 2 =I R 2 + I L 2 + hot.
From (43) and the definition of e * j , we infer that
I L 2 =e λ(η j −η j+1 ) Λa + R e * (z) f (ϕ(z) − f (0)) e λz dz + hot = − a L e λ(η j −η j+1 ) + hot,(45)I R 2 = − e µ(η j −η j−1 ) Λa − R e * (z) f (ϕ(z)) − f (0) e µz dz + hot =a R e µ(η j −η j−1 ) + hot,(46)
where the coefficients a L , a R are precisely those given in (17). To complete our derivation of (17) it remains to determine the sign of these coefficients. We present the computation for a L , the case a R being similar. It uses estimates (34) for ϕ and its derivatives up to order 2, as well as the observation that ϕ f (ϕ) = −ϕ − cϕ . It follows that
a L = − Λa + R e cz ϕ (z) f (ϕ(z) − f (0)) e λz dz = − Λa + R e −µz −ϕ (z) − cϕ (z) − f (0)ϕ (z) dz (47) =Λa + 2µ 2 a − + cµa − + µ 2 + cµ + f (0) R e −µz ϕ (z)dz(48)
with three successive integration by parts from (47) to (48). Finally, notice that µ 2 +cµ+f (0) = 0, and one gets
a L = Λa + a − µ (2µ + c) > 0.(49)
A quite similar computation gives
a R = Λa + a − λ (2λ + c) > 0 (50)
Negligible terms: integration away from the front It remains to prove that the integral I 2 is actually the one that drives the dynamics, and to discuss which value of M would be appropriate.We focus on I − 1 for the computations, the situation being similar for I + 1 . Le us first notice that linearizing at any front, we obtain that for all z ∈ R,
f n i=1 ϕ i (z) − n i=1 f (ϕ i (z)) = O e − µ 2 δ min ,(51)
while, linearizing around at ϕ j+1 , we have that for all z ∈ [η j+1 , η j+1 + M ],
f n i=1 ϕ i (z) − n i=1 f (ϕ i (z)) = O e µ(z−η j ) + e λ(z−η j−2 ) .(52)
Then, once again we split the integration domain into two.
1 Λ I − 1 = η j+1 +M −∞ e * j (z) f n i=1 ϕ i (z) − n i=1 f (ϕ i (z)) dz = η j+1 −∞ e * j f n i=1 ϕ i − n i=1 f (ϕ i ) dz + η j+1 +M η j+1 e * j f n i=1 ϕ i − n i=1 f (ϕ i ) dz = J 1 + J 2 .
Combining (51), (34) and the definition of e * j , we get that
J 1 = η j+1 −∞ e c(z−η j ) ϕ (z − η j )f n i=1 ϕ i (z) − n i=1 f (ϕ i (z))dz |J 1 | ≤ C η j+1 −∞ e c(z−η j ) e µ(z−η j ) dz e − µ 2 δ min ≤ Ce (λ−µ/2)δ min(53)
for some positive constant C. On the other hand, using (52) and (34), one has that for some positve constant C, possibly distinct from the above one,
|J 2 | ≤ C η j+1 +M η j+1 e −λ(z−η j ) e µ(z−η j ) + e λ(z−η j+2 ) dz ≤ C e (−2µ−c)(δ j −M ) + M e λ(δ j +δ j−1 ) ≤ C e (−2µ−c)(δ min −M ) + M e 2λδ min .(54)
Combining (53) and (54), we get, for some positive constant C,
I − 1 ≤ C e (λ−µ/2)δ min + e (−2µ−c)(δ min −M ) + M e 2λδ min .(55)
Similar computations give the following estimates, for some positive constant C,
I + 1 ≤ C e − 3 2 µδ min + e (λ−µ)(δ min −M ) + M e −2µδ min .(56)
It remains to choose M and determine α > 0 such that both I − 1 , I + 1 , and the higher order terms in (44) are of order e −(µ+α)δ min , as long as δ min is large enough. Let us fix α > 0 small enough, and ε = α µ : we fix M = 1 2 + ε δ min , and the estimates are as required.
B. Proof of Thm. 20 (local stability of E)
We write Σ n = n k=1 Ψ 3 k − n−1 k=1 Ψ k Ψ 2 k+1 so (27) becomes Ψ j = Ψ 1 Ψ 3 1 − Ψ 1 − Ψ 1 Ψ 2 2 + n k=2 Ψ 3 k − n−1 k=2 Ψ k Ψ 2 k+1 , j = 1 Ψ j Ψ 3 j − Ψ j + Ψ j−1 − Ψ j Ψ 2 j+1 − Ψ j−1 Ψ 2 j + n k=1, k =j Ψ 3 k − n−1 k=1, k / ∈{j−1,j} Ψ k Ψ 2 k+1 , 2 j n − 1 Ψ n Ψ 3 n − Ψ n + Ψ n−1 − Ψ n−1 Ψ 2 n + n−1 k=1 Ψ 3 k − n−2 k=1 Ψ k Ψ 2 k+1 , j = n
In the following, χ A denotes the characteristic function which is either 1 if condition A is satisfied or 0 otherwise. For instance, for given i ∈ N, χ {i=2} = 1 if i = 2 and 0 otherwise. The Jacobian J := ∂f j ∂Ψ i 1 i,j n is then given by
∂f 1 ∂Ψ i = −2Ψ 1 + 4Ψ 3 1 − 2Ψ 1 Ψ 2 2 + n k=2 Ψ 3 k − n−1 k=2 Ψ k Ψ 2 k+1 , i = 1 3Ψ 1 Ψ 2 i − 2χ {i=2} Ψ 2 1 Ψ 2 − Ψ 1 Ψ 2 i+1 − 2χ {i>2} Ψ 1 Ψ i−1 Ψ i , 2 i n − 1 3Ψ 1 Ψ 2 n − 2Ψ 1 Ψ n−1 Ψ n , i = n
and, for 2 j n − 1 and 1 i n,
∂f j ∂Ψ i = Ψ j−1 − 2Ψ j + 4Ψ 3 j − 2Ψ j Ψ 2 j+1 − 3Ψ j−1 Ψ 2 j + n k=1, k =j Ψ 3 k − n−1 k=1, k / ∈{j−1,j} Ψ k Ψ 2 k+1 , i = j Ψ j + 3Ψ 2 j−1 Ψ j − Ψ 3 j − 2χ {j>2} Ψ j−2 Ψ j−1 Ψ j , i = j − 1 3Ψ j Ψ 2 j+1 − 2Ψ 2 j Ψ j+1 − Ψ j Ψ 2 j+2 , i = j + 1 < n 3Ψ n−1 Ψ 2 n − 2Ψ 2 n−1 Ψ n , i = j + 1 = n 3Ψ j Ψ 2 i − Ψ j Ψ 2 i+1 − 2χ {i>1} Ψ j Ψ i−1 Ψ i , 1 i j − 2 3Ψ j Ψ 2 i − 2Ψ j Ψ i−1 Ψ i − χ {i<n} Ψ j Ψ 2 i+1 , j + 2 i n
Finally, for j = n and 1 i n, ∂f n ∂Ψ i = 4Ψ 3 n − 2Ψ n + Ψ n−1 − 3Ψ n−1 Ψ 2 n + n−1 k=1 Ψ 3 k − n−2 k=1 Ψ k Ψ 2 k+1 , i = n Ψ n − Ψ 3 n + 3Ψ n Ψ 2 n−1 − 2Ψ n−2 Ψ n−1 Ψ n ,
i = n − 1 3Ψ n Ψ 2 i − Ψ n Ψ 2 i+1 − 2χ {i>1} Ψ n Ψ i−1 Ψ i , else
It remains to determine the eigenvalues of J(E). To this end, since n is fixed throughout, we set b := n(n + 1)(2n + 1) = 2n 3 + 3n 2 + n and β = b 6 , α := 1/β. Then α = 6 n(n+1)(2n+1) so that E = α (1, 2, . . . , n) . After some straightforward computations, the first and last row of J(E) are given by
∂f j ∂Ψ i (E) = α 3 ·
−j(1 + β(n)), i = j j(β(n) − 1), i = j − 1 −j, i = j + 1 < n n 3 + n 2 − 2n, i = j + 1 = n −j, 1 i j − 2 jn(n + 2), j + 2 i = n −j, j + 2 i < n
Let C := { c 1 , c 2 , . . . , c n } := α −3 J(E), where c i are the column vectors. First note that the radial direction from E, v 1 := (1, 2, 3, . . . , n) , is an eigenvector of C with eigenvalue λ 1 := 2β > 0. However, this unstable direction is transverse to the invariant sphere and thus irrelevant here. Also by invariance, all remaining eigenvectors are orthogonal to v 1 .
A basis of the orthogonal complement (span( v 1 )) ⊥ is given by V := { v 2 , v 3 , . . . , v n }, v j := (j, 0, . . . , 0, −1, 0, . . . , 0) , j = 2, . . . , n − 1,
where the entry −1 in v j is on the j-th position. In order to reduce the matrix C onto the tangential dynamics, we orthogonalize the vectors v j , j = 1, 2, . . . , n by Gram-Schmidt and normalize afterwards. This results in the column vectors of the matrix W given by n(n + 1)(2n + 1) < 0.
In order to prove this, we first note that with d k := ( w k · c 1 , w k · c 2 , . . . , w k · c n ) we have b k,k = d k · w k so it remains to determine d k for each k ∈ {2, 3, . . . , n}.
For 2 k n − 1 and 1 j n − 1,
w k · c j = −q k k−1 i=1 i 2 + ka k q k , k < j −q k k−1 i=1 i 2 + ka k q k (1 + β), k = j −q k j−1 i=1 i 2 − j 2 q k (1 + β) − ka k q k (β − 1), k = j + 1 −q k 1 i k−1, i / ∈{j,j+1}
i 2 − j 2 q k (1 + β) + ka k q k + (j + 1) 2 q k (β − 1), k j + 2 For 2 k n − 1, we have w k · c n = q k n(n + 2) k−1 i=1 i 2 − ka k ; finally, for j < n, w n · c j = −q n 1 i n−1, i / ∈{j,j+1} i 2 + j(1 + β) − (j + 1) 2 (β − 1) + a n n(n 2 + 2n − β) and w n · c n = q n n(n + 2) n−1 i=1 i 2 − a n n(n 2 + 2n − β) .
This leads to the form b k,k = A 1 + A 2 + A 3 with (after straightforward computations) i 2 + (k − 1) 2 (1 + β) + ka k (β − 1) = 72 2k + 1 − 36 k + 1 − 3 2k − 1 − 3 β,
A 3 := a k q 2 k k−1 i=1 i 2 − ka 2 k q 2 k (1 + β) = 3 2k + 1 + 3 − k − 6 k + 1 β,
and thus b k,k = −kβ. We conclude that the eigenvalues of C are λ 1,2 = ±2β, λ k = −kβ, 3 k n.
and thus E is locally exponentially stable on S n−1 .
C. Implementation of the simulations
In this section, we briefly describe the numerical methods and software used in the illustrating figures. The software packages we use are JCASim [FW00] for simulating cellular automata, pde2path [HUR14] for PDE simulation, continuation and bifurcation package, and Mathematica.
The space-time plot of the Greenberg-Hastings cellular automaton ( Fig. 1 (a)) has been created with JCASim via an adaption of the method transition, where we specify the transition from one generation of cell states to the next according to the Greenberg-Hastings rule for e = 2 excited and r = 4 refractory states.
The phase portraits on the spheres (Fig. 5) have been created with Mathematica using the standard routines StreamPlot and TransformedField to change from Cartesian to spherical coordinates.
For the simulations of the PDE (1) plotted in 1 (b), Fig. 4, 8 and 9 , we have used pde2path with f 0 = 0.05 or f 0 = 0.2, respectively. As discretization of the interval (-lx,lx) by mesh width 2*lx/dsc we have chosen lx=10 and dsc=500 which corresponds to a mesh width of h = 0.04. Initial data p.u(1:p.nu) have been step functions with equilibrium levels at eq+2*pi*j for j = 1, 2, . . . , k, k ∈ N.
For the time stepping the pde2path routines tints or tintsfreeze have been used since most simulations go over a rather long time period in order to catch exponentially small interaction effects between kinks (antikinks). tintsfreeze removes the translational symmetry so that the required spatial interval need not be excessively long. In particular, since annihilations of kinks and antikins on the one side and interactions of kinks (antikinks) on the other side happen at different time scales (the annihilations being much faster), it is nearly impossible to observe both aspects simultaneously (even when choosing very large distances or a large space interval). We have circumvented this problem in Fig. 8 by starting with pure kink data "sending in" antikinks from the left space boundary repeatedly after time has allowed for observable exponentially small effects between the kinks that are left after each annihilation. This has been done by calling tintsfreeze multiple times and using the perturbed result as initial condition for the next call.
The detection of the kink (antikink) positions has been done via a function that stores the (geometrically defined) positions for each time step. The differences between these positions are the basis for the distance plots in Fig. 4, 8 and 9. For better illustration we have smoothened the distance plots by averaging the data and interpolating by splines using the MATLAB routine interp1. Finally, for general information about pde2path and, in particular, the simulation with periodic boundary conditions (Fig. 9), we refer to [DRUW14] and the pde2path webpage, respectively.
Figure 1 .
1: Space-time plots of pulse positions for GHCA (a) and positions
Since u andū are translates of each other, their positions ξ i ,ξ i , 1 i n − 1 are equal up to shift by d 0 and have the same speeds c n−1 i , for all t 0. By construction, u 0 u 0 ū 0 and thus the corresponding solutions satisfy u u(·; n) ū for all (x, t) ∈ R×R + by the comparison principle. We can therefore compare the speeds c n j with the speeds c for i = 1, . . . , n − 1. By iterating this construction of sub-and supersolutions,
Figure 3 .
3: Sketch of gaps (hatched): ξ i (0) < ξ i (0) for i = 1, 2, 4. The associated speeds need not be ordered. In particular, the distances may be increasing or decreasing.
Figure 4 .
4: Simulation of kink distances (a) (with zoom (b)), positions (c) and speeds (d), starting from an initial condition with five kinks and f 0 = 0.2. As can be seen from (a) and (b), the distances need not be monotone functions, but are eventually ordered, cf.§3.2. Moreover, the speeds of the kinks converge to the single front speed (d).
) k = 2 : E 9,10 = (±1, 0, 0), E 11,12 = (0, ±1, 0), E 13,14 = (0, 0, ±1)
Figure 6 .
6: Sketch of initial datum (5) with m = n = 4.
Figure 7 .
7: Sketch of initial datum with local maximum and subsolution (red). Kink and antikink which built the local maximum cannot collide.
Figure 8 .
8: Simulations to illustrate the loss of information (f 0 = 0.2). (a) Information encoded in the initial kink distances is locally "washed out" at t = 250000, independent of three 'fast' annihilations due to incoming antikinks as plotted in (b). See Appendix C for details on the implementation.
Figure 9 .
9: Simulation of (a) kink distances and (b) corresponding solutions done with pde2path by freezing under periodic boundary conditions and with f 0 = 0.2, cf. Appendix C. The initial datum converges to an equidistant state.
w 1 := q 1 v 1
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[
"LOCAL HEURISTICS AND AN EXACT FORMULA FOR ABELIAN SURFACES OVER FINITE FIELDS",
"LOCAL HEURISTICS AND AN EXACT FORMULA FOR ABELIAN SURFACES OVER FINITE FIELDS"
]
| [
"ANDJeffrey D Achter ",
"Cassandra Williams "
]
| []
| []
| Consider a quartic q-Weil polynomial f . Motivated by equidistribution considerations we define, for each prime ℓ, a local factor which measures the relative frequency with which f mod ℓ occurs as the characteristic polynomial of a symplectic similitude over F ℓ . For a certain class of polynomials, we show that the resulting infinite product calculates the number of principally polarized abelian surfaces over F q with Weil polynomial f . | 10.4153/cmb-2015-050-8 | [
"https://arxiv.org/pdf/1403.3037v3.pdf"
]
| 54,574,257 | 1403.3037 | 71f97d15e1f82ad7b83ad37cf88274e8a254854a |
LOCAL HEURISTICS AND AN EXACT FORMULA FOR ABELIAN SURFACES OVER FINITE FIELDS
12 Mar 2014
ANDJeffrey D Achter
Cassandra Williams
LOCAL HEURISTICS AND AN EXACT FORMULA FOR ABELIAN SURFACES OVER FINITE FIELDS
12 Mar 2014
Consider a quartic q-Weil polynomial f . Motivated by equidistribution considerations we define, for each prime ℓ, a local factor which measures the relative frequency with which f mod ℓ occurs as the characteristic polynomial of a symplectic similitude over F ℓ . For a certain class of polynomials, we show that the resulting infinite product calculates the number of principally polarized abelian surfaces over F q with Weil polynomial f .
INTRODUCTION
Consider abelian varieties over a finite field. To each such X/F q one may associate a characteristic polynomial of Frobenius, f X/F q (T) ∈ Z[T]; and two abelian varieties X and Y are isogenous if and only if f X/F q (T) = f Y/F q (T). In this way, isogeny classes of abelian varieties over F q are parametrized by suitable q-Weil polynomials f (T).
Conversely, given such a polynomial f , it is of intrinsic interest to calculate how many abelian varieties are in the corresponding isogeny class. In fact, a polarized variant of this problem seems even more natural. Let A g be the moduli space of principally polarized abelian varieties of dimension g, and let
A g (F q ; f ) = {(X, λ) ∈ A g (F q ) : f X/F q (T) = f (T)}.
Armed with an (overly optimistic) equidistribution philosophy, one might attempt to estimate #A g (F q ; f ) in the following fashion. To the extent possible, Frobenius elements of abelian varieties are equidistributed in GSp 2g (Z/ℓ). By somehow multiplying together, over all ℓ, the frequency with which f (T) mod ℓ occurs as the characteristic polynomial of a symplectic similitude, one might try to apprehend #A g (F q ; f ).
As written, this strategy is nonsense; for given g and ℓ, mod ℓ Frobenius elements are equidistributed only if q ≫ g ℓ. Nonetheless, such congruence considerations apparently control the sizes of isogeny classes.
Our main result is as follows. For f in a certain class of simple, ordinary q-Weil polynomials of degree 4 (see Section 2), we define for each prime ℓ a quantity ν ℓ ( f ) (see (4.1)) which measures its relative frequency as the characteristic polynomial of an element GSp 4 (Z/ℓ). After defining a Sato-Tate term ν ∞ ( f ), we show that
(1.1) ν ∞ ( f ) ∏ ℓ ν ℓ ( f ) = #A 2 (F q ; f )
(where a principally polarized abelian surface is given mass inversely proportional to the size of its automorphism group).
The present work is inspired by work of Gekeler [5], who derived a version of (1.1) for elliptic curves over a finite prime field. Our perspective was influenced by Katz's analysis [8] of Gekeler's product formula.
JDA was partially supported by a grant from the Simons Foundation (204164). 1
ABELIAN VARIETIES AND WEIL POLYNOMIALS
Let X/F q be an abelian variety of dimension g over a finite field with q = p e elements, and let f X/F q (T) ∈ Z[T] be the characteristic polynomial of its Frobenius endomorphism (acting on, say, any of the Tate modules T ℓ X with ℓ = p). Then f X/F q (T) is a q-Weil polynomial, i.e., the complex roots α 1 , · · · , α 2g of f X/F q (T) may be ordered so that α j α g+j = q for 1 ≤ j ≤ g, and in fact α j = √ q for each j.
Now assume that f (T) is a q-Weil polynomial of degree 4; such a polynomial corresponds to a (possibly empty) isogeny class I f of abelian surfaces over F q . In the sequel, we will assume that f is:
(W.1) ordinary its middle coefficient is relatively prime to p; (W.2) principally polarizable there exists a principally polarized abelian surface with characteristic polynomial f ; (W.3) Galois the polynomial f (T) is irreducible over Q, and K f := Q(T)/ f (T) is Galois and unramified at p;
(W.4) maximal let ̟ f be a (complex) root of f (T), with complex conjugate ̟ f . Then O f := Z[̟ f , ̟ f ],∈ Gal(K f /Q), since K f is a CM field.
As in the description of condition W.4, we will often denote the action of ι on an element α ∈ K f by α = ι(α). If R is any ring, then ι acts on O K f ⊗ R via the first component.
Example 2.1. The polynomial f (T) = T 4 + 29T 3 + 331T 2 + 1769T + 3721 is a 61-Weil polynomial which is ordinary, principally polarizable, Galois and maximal. In fact, O f is the ring of integers in Q(ζ 5 ), and f is the characteristic polynomial of Frobenius of the Jacobian of the curve with affine equation
y 2 = x 5 − 2.
Define the conductor of f , cond( f ), as the index of
Z[̟ f ] ∼ = Z[T]/ f (T) in O f . If f (T) = T 4 − aT 3 + bT 2 − aTq + q 2 , let f + (T) = T 2 − aT + (b − 2q); then f + (T) is the minimal poly- nomial of ̟ f + ̟ f , and K + f := Q[T]/ f + (T) is the maximal totally real subfield of K f . Denote the discriminants of f (T) and f + (T) by ∆ f and ∆ f + , respectively. Similarly, let ∆ O represent the discriminant of an order O; notice that ∆ Z[̟ f ] = ∆ f and ∆ O K + = ∆ f + . Lemma 2.2. The index of Z[̟ f ] in O f is q.
Proof. Using the above definition of f and Propositions 9.4 and 9.5 of [7],
∆ O f = ∆ 2 f + · N K f /Q (̟ f − ̟ f ) = (a 2 − 4b + 8q) 2 (b 2 + 4bq + 4q 2 − 4a 2 q). The discriminant of Z[̟ f ] is given by ∆ Z[̟ f ] = ∆ f = q 2 (a 2 − 4b + 8q) 2 (b 2 + 4bq + 4q 2 − 4a 2 q) = q 2 ∆ O f and ∆ f = [O f : Z[̟ f ]] 2 ∆ O f .
Then the desired index is q.
Corollary 2.3. If ℓ = p, then O K f ⊗ Z (ℓ) ∼ = Z (ℓ) [T]/ f (T).
Similarly, Z[T]/ f + (T) is maximal:
Lemma 2.4. The order Z[T]/ f + (T) is the maximal order O K + f . Proof. Condition W.4 implies that O f ∩ K + f = O K f ∩ K + f = O K + f . Certainly Z[T]/ f + (T) = Z[̟ f + ̟ f ] ⊆ O f ∩ K + f . Consider a ∈ O f ; then a = a 0 + a 1 ̟ f + a 2 ̟ f + a 3 ̟ f ̟ f for some integers a i . We have ̟ f ̟ f = q as f is a q-Weil polynomial, and a ∈ K + f if and only if a 1 = a 2 . Then a ∈ O f ∩ K + f has the form a = (a 0 + a 3 q) + a 1 (̟ f + ̟ f ) and O f ∩ K + f ⊆ Z[̟ f + ̟ f ]. Thus, Z[̟ f + ̟ f ] = O K + f .
CONJUGACY CLASSES IN SYMPLECTIC GROUPS
If X/F q is a principally polarized abelian surface, then the four-dimensional F ℓ -vector space X ℓ := X[ℓ](F q ) is naturally equipped with a symplectic form. We collect some notation concerning symplectic (similitude) groups.
3.1. Symplectic groups. Let V be a vector space of dimension 2g over a field k, equipped with a perfect, skew-symmetric form ·, · . The symplectic similitude group of V is the group of automorphisms which preserve this form up to a multiple. Concretely,
GSp(V, ·, · ) = {γ ∈ GL(V) : ∃m(γ) ∈ k × : ∀u, v ∈ V, γu, γv = m(γ) u, v }.
The group of automorphisms of the symplectic space is the symplectic group, Sp(V, ·, · ), and these groups sit in an exact sequence
1 ✲ Sp(V, ·, · ) ✲ GSp(V, ·, · ) mult ✲ k × ✲ 1 γ ✲ m(γ). For m ∈ k × , we let GSp(V, ·, · ) (m) = mult −1 (m).
Call a decomposition V = W 1 ⊕ W 2 symplectic if, for each i, ·, · | W i is a perfect pairing; and isotropic if, for some i, ·, · | W i = 0.
In fact, any symplectic space V of dimension 2g is isomorphic to k ⊕2g , equipped with the pairing described by the 2g × 2g matrix
J = 0 1 g −1 g 0 ;
the associated similitude and symplectic groups are GSp 2g (k) and Sp 2g (k), respectively.
Shapes of conjugacy classes.
In a general linear group, semisimple conjugacy classes are parametrized by characteristic polynomials; arbitrary conjugacy classes are determined by their characteristic polynomial and additional partition data. The classification of conjugacy classes in GSp 2g (k) is more intricate for two reasons. First, for elements with repeated eigenvalues, the presence of the symplectic form places nontrivial restrictions on allowable partition data. Second, elements of GSp 2g (k) which are conjugate in GL 2g (k) need not be conjugate in the symplectic similitude group; certain GL 2g -conjugacy classes decompose into classes indexed by k × /(k × ) 2 .
(For details of this decomposition, see, for example, [4] or [10]. Alternatively, compare our results to those of [1] or [9].)
In the sequel, we will only need the case where g = 2 and k = F ℓ is a finite field. Let C(γ) denote the conjugacy class of γ. We distinguish conjugacy classes by the factorization pattern of their characteristic polynomials f γ (T) into irreducible polynomials over F ℓ , and then refine this with additional combinatorial data, if necessary. The resulting collection of data associated to C(γ) will be called the shape of γ, or of its conjugacy class.
Our enumeration of conjugacy classes in GSp 4 (F ℓ ) is purposefully incomplete; we only include those which arise in our subsequent study of abelian surfaces. First, we only consider those classes for which all irreducible factors of the characteristic polynomial have the same degree. (Briefly call such a class "relevant".) Second, we only list those conjugacy classes corresponding to regular, or cyclic, elements. In general, an element of an algebraic group γ ∈ G(k) is called regular if the dimension of its centralizer, dim Z G (γ), is minimal, i.e., equal to the rank of G. In the case of G = GSp 4 , it is equivalent to insist that γ be cyclic in the standard representation, i.e., that there
exists v ∈ V such that {γ i v : i ≥ 0} spans V.
Note that an element is cyclic if and only if its minimal polynomial and characteristic polynomial coincide. As usual, a semisimple element is regular if and only if its eigenvalues are distinct.
Let f γ (T) = ∏ j g j (T) e j be the factorization of f γ (T) into powers of distinct, irreducible monic polynomials of equal degree. To this factorization of f γ (T) there is an associated factorization V ∼ = ⊕W j , where γ| W j has characteristic polynomial g j (T) e j . For each j, either ·, · | W j is zero or it is perfect; call these factorizations isotropic and symplectic, respectively.
Case 1: Regular semisimple elements
A regular semisimple conjugacy class is one for which the elements have a squarefree characteristic polynomial. We classify such conjugacy classes by the factorization of f γ (T) (over F ℓ ) and by m(γ); let a i ∈ F ℓ be distinct and g 1 = g 2 . Then Table 3.1 is a complete classification of relevant regular semisimple conjugacy class shapes.
Class shape f γ (T) m(γ) Split ∏ 4 j=1 (T − a j ) m = a 1 a 3 = a 2 a 4 DQ-S g 1 (T)g 2 (T) m = g j (0) (symplectic) DQ-I g 1 (T)g 2 (T) m = g j (0), m 2 = g 1 (0)g 2 (0) (isotropic) Quartic g(T) m 2 = g(0)
Case 2: Non-semisimple elements
If f γ (T) is not squarefree, then γ is cyclic (if and) only if the associated partition data is maximal.
In fact, such a conjugacy class is determined by a signed partition, and Table 3.2 completes our list of relevant cyclic conjugacy class shapes. (As above, the a i ∈ F ℓ are distinct.)
Class shape f γ (T) m(γ) Partition QRL (T − a) 4 m = a 2 [4] DRL-S (T − a) 2 (T + a) 2 m = a 2 (symplectic) {[2], [2]} ± DRL-I (T − a 1 ) 2 (T − a 2 ) 2 m = a 1 a 2 (isotropic) [2] RQ-1 [g(T)] 2 m = g(0) [2] RQ-2 (T 2 − m) 2 m = ✷ [2]±
(For the conjugacy class shape DRL-I, the partition [2] corresponds to the factor (T − a 1 )(T − a 2 ) in f γ (T), and thus to a subspace with characteristic polynomial (T − a 1 )(T − a 2 ).)
Note that, for a fixed characteristic polynomial f of shape DRL-S or RQ-2, the set of cyclic elements with characteristic polynomial f forms two conjugacy classes. For example, for a nonsquare x ∈ F ℓ ,
γ 1 = a 1 −a 1 a −a and γ 2 = a 1 −a x a −a
are both elements of shape DRL-S. (Verification of this fact is discussed in the proof of Lemma 3.2.) The matrix
Z = z 1 z 3 z 2 z 4 z 1 z 2 x
conjugates γ 1 to γ 2 over GL 4 (F ℓ ), but is an element of GSp 4 (F ℓ ) if and only if z 2 1 = z 2 2 x. Since x is nonsquare, γ 1 and γ 2 are not conjugate in GSp 4 (F ℓ ), although they are conjugate in GSp 4 (F ℓ 2 ). (A similar argument shows that we also have two classes of shape RQ-2.)
Centralizer orders.
We determine the size of each of the conjugacy classes C(γ) listed in Tables 3.1 and 3.2 by computing the order of the centralizer of the representative γ. Let Z GSp 4 (F ℓ ) (γ) denote the centralizer of γ in GSp 4 (F ℓ ).
A representative of a regular semisimple conjugacy class is an element of a unique maximal torus of GSp 4 (F ℓ ), and the centralizer of such a γ is that maximal torus [3]. We use the structure of these maximal tori to compute the sizes of the centralizers of the regular semisimple class shapes.
#Z GSp 4 (F ℓ ) (γ) = (ℓ − 1) 3 if C(γ) is Split, (ℓ + 1) 2 (ℓ − 1) if C(γ) is DQ-S, (ℓ + 1)(ℓ − 1) 2 if C(γ) is DQ-I, (ℓ 2 + 1)(ℓ − 1) if C(γ) is Quartic.
Proof. In each case, we determine the size of the appropriate torus. For example, if C(γ) is of shape Quartic, the polynomial f γ (T) has roots t, t ℓ , t ℓ 2 , and t ℓ 3 in F × ℓ 4 in one orbit under the action of Galois. Two pairs of roots have product m(γ)
∈ F × ℓ since γ ∈ GSp 4 (F ℓ ). The element tt ℓ cannot lie in F × ℓ when t ∈ F ℓ 4 F ℓ 2 , thus tt ℓ 2 = m(γ). The map t → tt ℓ 2 is the norm map of F ℓ 4 over F ℓ 2 . There are ℓ 4 −1 ℓ 2 −1 · (ℓ − 1) = (ℓ 2 + 1)(ℓ − 1) elements of F × ℓ 4 whose F ℓ 2 -norm lies in F × ℓ ,
which is the size of the torus and thus the centralizer. The other centralizer orders are computed analogously.
Determining the centralizer orders of the non-semisimple class shapes requires more effort. Table 3
Lemma 3.2. Suppose C(γ) has one of the conjugacy class shapes listed in
.2. Then
#Z GSp 4 (F ℓ ) (γ) = ℓ 2 (ℓ − 1) if C(γ) is QRL, 2ℓ 2 (ℓ − 1) if C(γ) is DRL-S, ℓ(ℓ − 1) 2 if C(γ) is DRL-I, ℓ(ℓ 2 − 1) if C(γ) is RQ-1, 2ℓ 2 (ℓ − 1) if C(γ) is RQ-2.
Proof. For each conjugacy class shape, find an explicit cyclic representative γ ∈ GSp 4 (F ℓ ) such that f γ and m(γ) are as given in Table 3.2. Then find a generic member C of the centralizer of γ and use it to find the size of Z GSp 4 (F ℓ ) (γ). (For the DRL-S and RQ-2 shapes, two distinct non-conjugate representatives are needed for the + and − classes.)
As an example, reconsider our previous example where C(γ) is DRL-S. It is easy to verify that the representatives γ 1 and γ 2 given earlier are cyclic elements of GSp 4 (F ℓ ) with characteristic polynomial (T − a) 2 (T + a) 2 and m(γ) = a 2 . The matrix
C = c 1 c 3 c 2 c 4 c 1 c 2
centralizes both γ 1 and γ 2 with the conditions that c 1 ∈ F × ℓ , c 2 = ±c 1 , and c 3 , c 4 ∈ F ℓ . Then each class has a centralizer of order 2ℓ 2 (ℓ − 1).
The rest of the computations are similar and are omitted here.
LOCAL FACTORS FOR f
Given f , we will define terms ν ℓ ( f ) for each finite prime of ℓ, as well as an archimedean term ν ∞ ( f ). For finite primes ℓ = p, we let ν ℓ ( f ) be the probability that a random element of GSp 4 (F ℓ ) (q) has characteristic polynomial f , and compare this probability to the average such chance. The definitions of ν p ( f ) and ν ∞ ( f ) are more intricate, but guided by a similar philosophy. 4.1. ν ℓ ( f ). First, suppose ℓ = p is a finite rational prime. The Frobenius endomorphism of a principally polarized abelian variety X/F q , thought of as an automorphism of X ℓ , is an element of GSp(X ℓ ) (q) ∼ = GSp 4 (F ℓ ) (q) . There are ℓ 2 polynomials which occur as characteristic polynomials of elements of GSp 4 (F ℓ ) (q) . Consequently, at least for ℓ unramified in K f , we measure the departure of the frequency of f from the average by
(4.1) ν ℓ ( f ) = #{γ ∈ GSp 4 (F ℓ ) (q) : f γ ≡ f mod ℓ} # GSp 4 (F ℓ ) (q) /ℓ 2 .
(An extension of this definition to all ℓ = p is given below in (5.1).)
4.2.
ν p ( f ). By way of motivation, suppose that X/F q is an ordinary abelian surface, with characteristic polynomial of Frobenius In particular (X[p] tor ) * (F q ) ∼ = (Z/p) 2 , and the action of Frobenius on this Galois module (again) has characteristic polynomial g X/F q (T).
f X/F q (T) = T 4 − a X T 3 + b X T 2 − qa X T + q 2 . Since X
Finally, recall that the Frobenius operator must preserve the canonical decomposition of X[p] into itsétale and toric parts.
Because of these considerations, we set
ν p ( f ) = #{γ ∈ GSp 4 (F p ) (b 2 ) : f γ ≡ (T 2 − aT + b) 2 mod p and γ semisimple} # GSp 4 (F p ) (b 2 ) /p 2 .
4.3. ν p (∞). It remains to define an archimedean term; our choice comes from the Sato-Tate measure, which (conjecturally) explains the distribution of Frobenius elements of abelian surfaces.
Recall that semisimple conjugacy classes in the compact group USp 4 are parametrized by ("Frobenius angles") 0 ≤ θ 1 ≤ θ 2 ≤ π. The Sato-Tate measure on the space of Frobenius angles is simply the pushforward of Haar measure. Explicitly, the Weyl integration formula [11, p218,7.8B] shows that this measure is
µ ST (θ 1 , θ 2 ) = 16 π 2 (cos(θ 2 ) − cos(θ 1 )) 2 sin 2 (θ 1 ) sin 2 (θ 2 ) dθ 1 dθ 2 .
Once q is fixed, a pair of angles {θ 1 , θ 2 } gives rise to a q-Weil polynomial
∏ j=1,2 (T − √ q exp(iθ j ))(T − √ q exp(−iθ j ));
the induced measure on the space of q-Weil polynomials is
µ ST (a, b) = 1 4q 3 π 2 (a 2 − 4b + 8q) 2 (b 2 + 4bq + 4q 2 − 4a 2 q) da db.
Note that, since there are approximately q dim A 2 = q 3 principally polarized abelian surfaces over F q , abelian varieties, q 3 µ ST (a, b) is a sort of archimedean prediction for #A 2 (F q ; f ). We set
(4.2) ν ∞ ( f ) = 1 cond( f ) 4π 2 ∆ f ∆ f + .
THE SHAPE OF FROBENIUS
Fix a q-Weil polynomial satisfying conditions W.1-W.4. Suppose X/F q is a principally polarized abelian variety such that O f ⊆ End(X); we choose the polarization so that the Rosati involution on End(X) induces complex conjugation on O f . On ℓ-torsion, the principal polarization induces a symplectic pairing on X ℓ ; complex conjugation on O f ⊗ Z/ℓ is adjoint with respect to this pairing; and we obtain ρ ℓ (̟ f ) ∈ GSp(X ℓ ). Our goal in the present section is to relate the shape of ρ ℓ (̟ f ) (in the sense of Section 3.2) to the structure of f (T) mod ℓ.
Of course, all of this can be formulated without recourse to abelian varieties. Let κ(ℓ) = O K ⊗ Z/ℓ; it is a four-dimensional vector space over F ℓ . Choose a symplectic pairing ·, · on κ(ℓ) for which complex conjugation on O K ⊗ Z/ℓ is the adjoint with respect to ·, · . (If ℓ ∤ ∆ K , one may explicitly construct such a pairing as follows. Choose α ∈ O K relatively prime to ℓ such that α = −α. Then the reduction modulo ℓ of the pairing
O K × O K ✲ Z (x, y) ✲ tr K/Q (αxy) is a suitable form.
) Such a form is canonically defined up to scaling, and in particular its group of symplectic similitudes is independent of the choice of form.
Then ̟ f acts on κ(ℓ). Let γ ℓ be the image of ̟ f in GSp(κ(ℓ)); our goal is to use the splitting behavior of f (T) mod ℓ to compute the cyclic shape of γ ℓ , i.e., the shape of any cyclic element whose semisimplification is conjugate to γ ℓ .
In fact, we define
(5.1) ν ℓ ( f ) = #{γ ∈ GSp 4 (F ℓ ) : γ is cyclic, with semisimplification γ ℓ } # GSp 4 (F ℓ ) (q) /ℓ 2 .
Lemma 5.1. If ℓ ∤ p∆ K , then definitions (4.1) and (5.1) coincide.
Proof. If ℓ ∤ p∆ K , then ℓ ∤ ∆ f . Therefore, f (T) mod ℓ has distinct roots, and γ ℓ is regular semisimple. The classification in Table 3.1 shows that if f (T) mod ℓ is either irreducible or a product of linear factors, then any element with characteristic polynomial f (T) mod ℓ is actually conjugate to γ ℓ . If f (T) mod ℓ is a product of distinct irreducible quadratic polynomials, then the possible shapes of γ ℓ are distinguished by their multiplier; but one knows that the multiplier of γ ℓ is q. The claim now follows once one recalls that a regular semisimple element is cyclic.
Note that, tautologically, the characteristic polynomial of γ ℓ is exactly the reduction of f (T). If, for instance, f (T) mod ℓ is irreducible, then a moment's reflection (or a glance at Tables 3.1 and 3.2) reveals that γ ℓ is Quartic.
However, it sometimes happens (e.g., with DQ-S and DQ-I) that the factorization pattern of f alone does not determine the shape of γ.
To ease notation slightly, we will write K for K f and, given assumption W.
4, write O K for O f . Since κ(ℓ) = O K /ℓ ∼ = F ℓ [T]/ f (T) (Corollary 2.3)
, the factorization of f (T) mod ℓ is precisely determined by the splitting of ℓ in O K . We have
κ(ℓ) ∼ = λ|ℓ κ(ℓ) λ ,
where λ ranges over all primes of K which lie over ℓ. (In fact, the dimension of κ(ℓ) λ over the residue field O K /λ is e(λ/ℓ), the ramification index of λ.)
For the sequel, it is worth singling out the following immediate observation:
Lemma 5.2.
The symplectic pairing induces a perfect duality between κ(ℓ) λ and κ(ℓ) λ .
Proof. We have chosen ·, · such that the involution induced by complex conjugation is the adjoint with respect to ·, · .
Consider a finite Galois extension L/Q with Gal(L/Q) = G. If ℓ is a rational prime and λ is a prime of O L lying over ℓ, the decomposition group and inertia group of λ are, respectively,
D(λ/ℓ) = {σ ∈ G : σ(λ) = λ} I(λ/ℓ) = {σ ∈ G : ∀β ∈ O L : σ(β) ≡ β mod λ}.
Then I(λ/ℓ) is normal in D(λ/ℓ). In fact, we will only use these notions for the abelian extension K/Q, and thus the inertia and decomposition groups depend only on ℓ, and not on the choice of λ. Hence we write I(ℓ) and D(ℓ) for I(λ/ℓ) and D(λ/ℓ).
Let f (T) ≡ ∏ 1≤j≤r g j (T) e j mod ℓ be the factorization of f (T) mod ℓ into irreducible monic poly-
nomials. Since O K /ℓ ∼ = F ℓ [T]/ f (T)
, there are r primes, λ 1 , · · · , λ r of K lying over ℓ; O K /λ i has degree deg g i over F ℓ ; and the ramification index of λ i is e i . Note that the quantities deg g i and e i are independent of i as K/Q is Galois. (This is why we restricted to relevant conjugacy classes in Section 3.2.)
Finally, if ℓ ∤ ∆ f then there exists an element of Gal(K/Q) which induces the canonical generator of Gal(κ(λ)/F ℓ ). Let Frob K (ℓ) ∈ Gal(K/Q) be this element, called the Frobenius endomorphism of λ over ℓ.
K cyclic.
Suppose that Gal(K/Q) is cyclic, with generator σ. Note that complex conjugation is given by ι = σ 2 . We classify the splitting behavior of rational primes ℓ in K by enumerating the possibilities for D(ℓ) and I(ℓ). Table 5.1.
Lemma 5.3. Suppose f satisfies assumptions W.1-W.4 with cyclic Galois group generated by σ. Let ℓ = p be a rational prime. The cyclic shape of γ ℓ is determined by the decomposition and inertia groups D(ℓ) and I(ℓ) as in
Thus, for instance, Lemma 5.3 asserts that if D(ℓ) = σ 2 and I(ℓ) = 1 , then γ ℓ has cyclic shape DQ-S. Note that if γ ℓ has cyclic shape RQ-2, then there are two conjugacy classes with cyclic shape γ ℓ . Otherwise, the cyclic shape of γ ℓ determines a unique conjugacy class.
Proof. In Table 5 Of the remaining cases, we first consider those in which D(ℓ) = σ 2 . Let λ be one of the two primes of K lying over ℓ. Lemma 5.2 shows that κ(ℓ) λ is symplectic if and only if complex conjugation stabilizes λ, i.e., if and only if ι = σ 2 ∈ D(ℓ). Since this happens in the two cases under consideration, the induced decomposition is symplectic and the cyclic shape of γ ℓ is the S variant.
Finally, we analyze the situation in which I(ℓ) = σ 2 ⊂ D(ℓ) = σ ; we must decide whether the cyclic shape is RQ-1 or RQ-2. Let λ be the prime of K lying over ℓ. Consider the Frobenius element ̟ f as an element of O K . Then f (T) = ∏ 0≤j≤3 (T − σ j (̟ f )). The ramification hypothesis implies that σ 2 (̟ f ) ≡ ̟ f mod λ, and we have the factorization
f (T) ≡ g(T) 2 mod λ where g(T) ≡ (T − ̟ f )(T − σ(̟ f )) mod λ.
By comparing constant terms, we find that
(̟ f σ(̟ f )) 2 ≡ q 2 mod λ
and in particular
̟ f σ(̟ f ) ≡ ±q mod λ.
However, if we had ̟ f σ(̟ f ) ≡ q mod λ, then we would know that σ(̟ f ) ≡ σ 2 (̟ f ) mod λ, which contradicts the hypothesis that σ ∈ I(ℓ).
Therefore, the constant term of the irreducible factor g(T) is −q mod λ, and the cyclic shape of γ ℓ is RQ-2.
D(ℓ) I(ℓ) Frob K (ℓ) (e, f , r) Class shape {1} {1} 1 (1,1,4) Split σ 2 {1} σ 2 (1,2,2) DQ-S σ 2 σ 2 - (2,1,2) DRL-S σ {1} σ or σ 3 (1,4,1) Quartic σ σ 2 - (2,2,1) RQ-2 σ σ - (4,1,1) QRL 5.2. K biquadratic. Suppose instead that K = Split( f ) is biquadratic.
Then K is the compositum of quadratic imaginary fields K 1 and K 2 , and we have Gal(K/Q) ∼ = Gal(K 1 /Q) ⊕ Gal(K 2 /Q) ∼ = Z/2 ⊕ Z/2; let τ i generate Gal(K/K i ). Then complex conjugation is given by ι = τ 1 τ 2 ; its fixed field is the real quadratic subfield K + . We again classify the splitting behavior of primes ℓ by considering pairs D(ℓ) and I(ℓ). Table 5.2.
Lemma 5.4. Suppose f satisfies conditions W.1-W.4 with K = K f biquadratic. Let ℓ = p be a rational prime. The cyclic shape of γ ℓ is determined by the decomposition and inertia groups D(ℓ) and I(ℓ) as in
As before, the cyclic shape of γ ℓ determines a unique conjugacy class except when that shape is either DRL-S or RQ-2. In these cases, there are two conjugacy classes with the cyclic shape given by γ ℓ .
Proof. Proceed as in the proof of Lemma 5.3. Note that, by Lemma 5.2, κ(ℓ) λ is isotropic if and only if complex conjugation acts nontrivially on λ, i.e., if and only if ι = τ 1 τ 2 / ∈ D(ℓ). In these cases the induced decomposition is isotropic and the cyclic shape of γ ℓ is the I variant. All ambiguous cases where r > 1 can be identified by the action of the pairing on the induced decomposition. Now (without loss of generality) suppose that I(ℓ) = τ 1 ⊂ D(ℓ) = τ 1 , τ 2 , and let λ be the prime lying over ℓ. As in Lemma 5.3, we recall that
f (T) ≡ (T − ̟ f )(T − τ 1 (̟ f ))(T − τ 2 (̟ f ))(T − τ 1 τ 2 (̟ f )) mod λ.
The assumption on ramification implies that ̟ f ≡ τ 1 (̟ f ) mod λ, and thus that
τ 2 (̟ f ) ≡ τ 1 τ 2 (̟ f ) mod λ. Therefore, f (T) factors as f (T) ≡ ((T − ̟ f )(T − τ 2 (̟ f ))) 2 mod λ.
Moreover, ̟ f τ 2 (̟ f ) ≡ ̟ f τ 1 τ 2 (̟ f ) ≡ q mod λ. Therefore, f (T) ≡ g(T) 2 mod λ where g(0) = q, and the cyclic shape of γ ℓ is RQ-1.
The remaining cases follow in an analogous fashion.
D(ℓ) I(ℓ) Frob K (ℓ) (e, f , r) Class shape {1} {1} 1 (1,1,4) Split τ i {1} τ i (1,2,2) DQ-I τ i τ i - (2,1,2) DRL-I τ 1 τ 2 {1} τ 1 τ 2 (1,2,2) DQ-S τ 1 τ 2 τ 1 τ 2 - (2,1,2) DRL-S τ 1 , τ 2 τ i - (2,2,1) RQ-1 τ 1 , τ 2 τ 1 τ 2 - (2,2,1) RQ-2
6. LOCAL TERMS FOR K Let Gal(K/Q) * be the character group of the Galois group of K. For χ ∈ Gal(K/Q) * , let K χ be the subfield of K fixed by ker χ. For a rational prime ℓ, let χ(ℓ) = χ(Fr K χ (ℓ)) if ℓ is unramified in K χ , and let χ(ℓ) = 0 otherwise.
Since K + is a subextension of K, Gal(K + /Q) * is naturally a subgroup of Gal(K/Q) * , and we define
ν ℓ (K) = ∏ χ∈S(K) 1 1 − χ(ℓ)/ℓ (6.1) where S(K) = Gal(K/Q) * Gal(K + /Q) * .D(ℓ) I(ℓ) {χ(ℓ), χ(ℓ)} Class shape {1} {1} {1, 1} Split σ 2 {1} {−1, −1} DQ-S σ 2 σ 2 {0, 0} DRL-S σ {1} {−i, i} Quartic σ σ 2 {0, 0} RQ-2 σ σ {0, 0} QRL 6.2. K biquadratic.
As in Section 5.2, let Gal(K/Q) = τ 1 , τ 2 . Denote the quadratic imaginary subfields of K by K 1 and K 2 and let Gal(K/K i ) = τ i . For i ∈ {1, 2}, define the character
φ i (τ j ) := −1 i = j, 1 i = j.
Then S(K) = {φ 1 , φ 2 }. Lemma 6.2. Let ℓ be a rational prime. The multiset of values {φ 1 (ℓ), φ 2 (ℓ)} is determined by the decomposition and inertia groups D(ℓ) and I(ℓ) as in Table 6.2.
D(ℓ) I(ℓ) {φ 1 (ℓ), φ 2 (ℓ)} Class shape {1} {1} {1, 1} Split τ i {1} {−1, 1} DQ-I τ i τ i {0, 1} DRL-I τ 1 τ 2 {1} {−1, −1} DQ-S τ 1 τ 2 τ 1 τ 2 {0, 0} DRL-S τ 1 , τ 2 τ i {0, −1} RQ-1 τ 1 , τ 2 τ 1 τ 2 {0, 0} RQ-2 6.3. Matching.
In this section, we show that each of the local factors naïvely assigned to f matches a factor intrinsic to the splitting field K = K f .
Proposition 6.3. If ℓ = p, then ν ℓ ( f ) = ν ℓ (K).
Proof. Let γ ℓ be as in Section 5. We first assume that the cyclic shape of γ ℓ determines a unique conjugacy class in GSp 4 (F ℓ ), and then indicate what must be changed to accommodate the remaining cases.
Thus, let γ be any cyclic element whose semisimplification is γ ℓ , and assume the shape of γ is neither DRL-S nor RQ-2. By Lemmas 6.1 and 6.2, the set of character values
{χ(ℓ) : χ ∈ S(K)}
depends only on the shape of γ; and tautologically, the size of the conjugacy class C(γ) only depends on the shape of γ, too.
On one hand, by (5.1) we have
ν ℓ ( f ) = #C(γ) # GSp 4 (F ℓ ) (q) /ℓ 2 = # GSp 4 (F ℓ )/#Z(γ) # GSp 4 (F ℓ ) (q) /ℓ 2 = ℓ 2 (ℓ − 1) #Z (γ) .
Lemmas 3.1 and 3.2 supply column 2 of Table 6.3, and applying this simple calculation provides column 3.
On the other hand, recall that (6.1) gives
ν ℓ (K) = ∏ χ∈S(K) 1 1 − χ(ℓ)/ℓ .
Lemmas 6.1 and 6.2 provide column 4 of Table 6.3, and we compute column 5 using (6.1).
If γ ℓ has cyclic shape of type DRL-S or RQ-2, then there are two cyclic conjugacy classes with semisimplification γ ℓ . For a representative γ of each class, #C(γ)/(# GSp 4 (F ℓ ) (q) /ℓ 2 ) = 1 2 , and thus ν ℓ ( f ) = 1 2 + 1 2 . As columns 3 and 5 are equal, the theorem is proven.
TABLE 6.3. ν ℓ ( f ) and ν ℓ (K) Class shape #Z (γ) ν ℓ ( f ) {χ(ℓ) : χ ∈ S(K)} ν ℓ (K) Split (ℓ − 1) 3 ℓ 2 (ℓ−1) 2 {1, 1} ℓ ℓ−1 · ℓ ℓ−1 DQ-S (ℓ + 1) 2 (ℓ − 1) ℓ 2 (ℓ+1) 2 {−1, −1} ℓ ℓ+1 · ℓ ℓ+1 DQ-I (ℓ + 1)(ℓ − 1) 2 ℓ 2 ℓ 2 −1 {−1, 1} ℓ ℓ+1 · ℓ ℓ−1 Quartic (ℓ 2 + 1)(ℓ − 1) ℓ 2 ℓ 2 +1 {−i, i} ℓ ℓ−i · ℓ ℓ+i QRL ℓ 2 (ℓ − 1) 1 {0, 0} 1 · 1 DRL-S 2ℓ 2 (ℓ − 1) 1 {0, 0} 1 · 1 DRL-I ℓ(ℓ − 1) 2 ℓ ℓ−1 {0, 1} ℓ ℓ−1 · 1 RQ-1 ℓ(ℓ 2 − 1) ℓ ℓ+1 {−1, 0} ℓ ℓ+1 · 1 RQ-2 2ℓ 2 (ℓ − 1) 1 {0, 0} 1 · 1
Similarly:
Lemma 6.4. We have ν p ( f ) = ν p (K).
Proof. Since we have assumed p unramified in K (W.3), g(T) := T 2 − aT + b is not a square. For convenience, we recall the definition
ν p ( f ) = #{γ ∈ GSp 4 (F p ) (b 2 ) : f γ (T) ≡ g(T) 2 mod p and γ semisimple} # GSp 4 (F p ) (b 2 ) /p 2 .
First suppose that K/Q is cyclic. Then p splits completely in K (e.g., [6, Table 3]), and g(T) factors (in F p ). The set of semisimple elements with characteristic polynomial g(T) 2 has the same cardinality as a conjugacy class of type Split. From (the first line of) Table 6.3, we see that ν p ( f ) = ν p (K). Now instead suppose that K/Q is biquadratic. Then either p splits completely in K, or p splits in exactly one of the K i ([6, Table 4]). The former case has already been addressed. For the latter case, the set of semisimple elements with characteristic polynomial g(T) 2 has the same cardinality as a conjugacy class of type DQ-I. Again we conclude from Table 6.3 that ν p ( f ) = ν p (K).
Finally, we compute: Lemma 6.5. We have
ν ∞ ( f ) = 1 4π 2 ∆ K ∆ K + .
Proof. From Lemma 2.2 we have cond( f ) = q and ∆ f = q 2 ∆ K . Additionally, Lemma 2.4 implies that ∆ f + = ∆ K + . Then (4.2) gives the result.
MAIN RESULT
In the following, we will have several occasions to consider conditionally convergent infinite products. For a sequence of numbers {a ℓ } indexed by finite primes, let
(7.1) ∏ ℓ a ℓ = lim X→∞ ∏ ℓ<X a ℓ .
With this convention, we have (∏ ℓ a ℓ ) · (∏ ℓ b ℓ ) = ∏ ℓ (a ℓ b ℓ ).
For a number field L, let h(L), ω L and R L denote, respectively, the class number, number of roots of unity, and regulator of L.
Theorem 7.1. Let f be a degree 4 q-Weil polynomial which is ordinary, principally polarizable, Galois and maximal. Let K f be the splitting field of f , and let K + f be its maximal totally real subfield. Then
(7.2) ν ∞ ( f ) ∏ ℓ ν ℓ ( f ) = 1 ω K h(K f ) h(K + f ) .
Proof. We write K and K + for K f and K + f . By the analytic class number formula, the ratio of class numbers on the right-hand side of (7.2) is Finally, O × K and O × K + agree up to torsion so R K = 2R K + , and K + is a real field so ω K + = 2. Consequently,
h(K) h(K + ) = lim s→1 (s − 1)ζ K (s) (s − 1)ζ K + (s) |∆ K |2 2 ω K R K + |∆ K + |(2π) 2 ω K + R K .h(K) h(K + ) = ω K 1 4π 2 |∆ K | |∆ K + | ∏ ℓ ν ℓ (K) = ω K ν ∞ ( f ) ∏ ℓ ν ℓ ( f )
by Proposition 6.3 and Lemmas 6.4 and 6.5.
In fact, (7.2) has a natural interpretation in terms of abelian varieties. Proof. If (X, λ) ∈ A 2 (F q ; f ), then # Aut(X, λ) = ω K . By [2], h(K)/h(K + ) is the (unweighted) size of A 2 (F q ; f ). Now invoke Theorem 7.1.
In fact, unpublished work of Howe shows that in much greater generality, h(K)/h(K + ) computes the size of a suitable isogeny class. Thus, we expect (7.3) to also hold when K f is biquadratic.
Lemma 3. 1 .
1Let C(γ) have one of the conjugacy class shapes listed in
.1, we have enumerated all possibilities for pairs of subgroups I(ℓ) ⊆ D(ℓ) ⊆ Gal(K/Q). For each such pair, in the prime factorization of f (T) mod ℓ, there are r = # Gal(K/Q)/#D(ℓ) distinct irreducible factors. Each has degree f = #D(ℓ)/#I(ℓ) and multiplicity e = #I(ℓ) . When I(ℓ) ⊆ D(ℓ) is either {1} ⊆ {1}, {1} ⊂ σ , or σ ⊆ σ , this factorization pattern already determines the cyclic shape of γ ℓ .
. K cyclic. As in Section 5.1, let Gal(K/Q) = σ . Let χ be a faithful character of Gal(K/Q), so that χ := χ • ι is the other faithful character of Gal(K/Q) and S(K) = {χ, χ}. Lemma 6.1. Let ℓ be a rational prime. The multiset of values {χ(ℓ), χ(ℓ)} is determined by the decomposition and inertia groups D(ℓ) and I(ℓ) as inTable 6.1.Proof. This follows from the definition of {χ, χ} and the calculation of Frobenius elements in Lemma 5.3. In particular, {χ(ℓ), χ(ℓ)} depends only on the order of D(ℓ) and I(ℓ), and not on their canonical generators; and is also independent of the choice of generator of Gal(K/Q) * .
For a finite abelian extension L/Q, we havelim s→1 (s − 1)ζ L (s) = ∏ χ∈Gal(L/Q) * id L(1, χ),where the product is over nontrivial characters of Gal(L/Q) and, as in (7.1), we interpret L(1, χ) as the conditionally convergent product L(1, χ) K and K + are abelian, as they are Galois over Q of degrees 4 and 2, respectively. By definition (6.1), for each ℓ,∏ χ∈Gal(K/Q) * id 1 1 − χ(ℓ)/ℓ ∏ χ∈Gal(K + /Q) * id 1 1 − χ(ℓ)/ℓ = ν ℓ (K).
Corollary 7. 2 .
2For f as in Theorem 7.1, suppose further that Gal(K f /Q) is cyclic. Then(7.3) ν ∞ ( f ) ∏ ℓ ν ℓ ( f ) = #A 2 (F q ; f ),the number of isomorphism classes of principally polarized abelian surfaces over F q with characteristic polynomial of Frobenius f , weighted by (inverse) size of automorphism group.
a priori an order in K f , is actually the maximal order O K f . Conditions W.1 and W.3 imply that any abelian surface in A 2 (F q ; f ) is ordinary and simple. Condition W.2 is explicitly characterized, in terms of the coefficients of f , in [7, Thm. 1.3]. Condition W.4 is indeed an extra hypothesis, which we hope to relax in a future work. The isomorphism class of O f , as an abstract order, is independent of the choice of ̟ f . Note that Gal(K f /Q) is abelian, since [K f : Q] = 4; and there is an intrinsically defined complex conjugation ι
TABLE 3 .
31. Regular semisimple conjugacy class shapes
TABLE 3 .
32. Non-semisimple cyclic conjugacy class shapes
Table 3 .1. Then
3
Now, X[p] tor is connected, and specifically X[p] tor (F q ) is a single point; but its Cartier dual isétale.is or-
dinary (and F q is perfect), there is a canonical decomposition X[p] ∼ = X[p] et ⊕ X[p] tor of the
p-torsion group scheme intoétale and toric components. Note that X p := X[p](F q ) is actu-
ally X[p] et (F q ) ∼ = (Z/p) 2 . The F q -rational structure of X[p] et is captured by the action of the
q-power Frobenius on X p . In fact, ̟ X/F q acts invertibly on X p , with characteristic polynomial
g X/F q (T) := T 2 − a X T + b X mod p.
TABLE 5 .
51. Prime factorizations and conjugacy class shapes for K/Q cyclic
TABLE 5 .
52. Prime factorizations and conjugacy class shapes for K/Q biquadratic
TABLE 6 .
61. Values of imaginary characters on Frobenius elements for K/Q cyclic.
TABLE 6 .
62. Values of imaginary characters on Frobenius elements for K/Q biquadratic.
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The characters of the finite conformal symplectic group, CSp(4, q). K.-I Shinoda, Comm. Algebra. 1013K.-i. Shinoda. The characters of the finite conformal symplectic group, CSp(4, q). Comm. Algebra, 10(13):1369-1419, 1982.
On the conjugacy classes in the unitary, symplectic and orthogonal groups. G E Wall, J. Austral. Math. Soc. 3G. E. Wall. On the conjugacy classes in the unitary, symplectic and orthogonal groups. J. Austral. Math. Soc., 3:1-62, 1963.
H , The classical groups. Princeton Landmarks in Mathematics. Princeton, NJ; Princeton PaperbacksPrinceton University PressTheir invariants and representations, Fifteenth printingH. Weyl. The classical groups. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1997. Their invariants and representations, Fifteenth printing, Princeton Paperbacks.
. Colorado, Fort University, Collins, COLORADO STATE UNIVERSITY, FORT COLLINS, CO 80523-1874
E-mail address: [email protected]. E-mail address: [email protected] URL: http://www.math.colostate.edu/~achter
. James Madison University, Harrisonburg, JAMES MADISON UNIVERSITY, HARRISONBURG, VA 22807
| []
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[
"Introduction to the Renormalization Group with Applications to Non-Relativistic Quantum Electron Gases",
"Introduction to the Renormalization Group with Applications to Non-Relativistic Quantum Electron Gases"
]
| [
"Cime Lectures ",
"Vincent Cetraro ",
"Rivasseau "
]
| []
| []
| We review the rigorous work on many Fermions models which lead to the first constructions of interacting Fermi liquids in two dimensions, and allowed to prove that there are different scaling regimes in two dimensions, depending on the shape of the Fermi surface. We also review progress on the three dimensional case.We start with a pedagogical introduction on quantum field theory and perturbative renormalization. Emphasis is then put on using renormalization around the Fermi surface in a constructive way, in which all orders of perturbation theory are summed rigorously. arXiv:1102.5117v1 [math-ph] | 10.1007/978-3-642-29511-9_1 | [
"https://arxiv.org/pdf/1102.5117v1.pdf"
]
| 55,774,786 | 1102.5117 | 3f322748283c4e9033f179a072c816f7d2aff892 |
Introduction to the Renormalization Group with Applications to Non-Relativistic Quantum Electron Gases
January 13, 2013
Cime Lectures
Vincent Cetraro
Rivasseau
Introduction to the Renormalization Group with Applications to Non-Relativistic Quantum Electron Gases
January 13, 2013
We review the rigorous work on many Fermions models which lead to the first constructions of interacting Fermi liquids in two dimensions, and allowed to prove that there are different scaling regimes in two dimensions, depending on the shape of the Fermi surface. We also review progress on the three dimensional case.We start with a pedagogical introduction on quantum field theory and perturbative renormalization. Emphasis is then put on using renormalization around the Fermi surface in a constructive way, in which all orders of perturbation theory are summed rigorously. arXiv:1102.5117v1 [math-ph]
1 Introduction to QFT and Renormalization
Gaussian Measures
A finite dimensional centered normalized Gaussian measure dµ C is defined through its covariance. Consider a finite dimensional space R N and a symmetric positive definite N by N matrix A. The inverse of the matrix A is also a definite positive symmetric N by N matrix C = A −1 called the covariance associated to A. The corresponding centered normalized Gaussian measure is
dµ C = (2π) −N/2 √ det A e − 1 2 t XAX d N X,(1.1)
so that dµ C = 1.
To understand dµ C it is better to know C than A since the moments or correlation functions of a Gaussian measure can be expressed simply as sums of monomials in C. In fact formula (1.1) perfectly makes sense if C is non invertible, and even for C = 0; but the corresponding measure has no density with respect to the Lebesgue measure in this case (for C = 0 dµ C is just Dirac's δ function at the origin). The reader familiar with eg ordinary linear PDE's knows that the essential point is to invert the matrix or the operator, hence to know the "Green's function". But quadratic forms have linear equations as their variational solutions, so both problems are linked.
Sine any function can be approximated by polynomials, probability measures are characterized by their moments, that is by the integrals they return for each polynomial of the integration variables.
The corresponding theorem which computes the moments of a Gaussian measure in terms of the covariance is fundamental in QFT and known there under the name of Wick's theorem. It expresses the result as the sum over all possible pairings or the variables of a product of covariances between the paired variables:
X i1 ...X in dµ C = G ∈G C i b( ) ,i e( ) ,(1.2)
where G runs over all Wick contractions or pairings of the labels 1, ..., n. Each pair is pictured as a line joining two labels b( ) and e( ) (which we call arbitrarily the "beginning" and "end" of the line).
The theory of Gaussian measures and Wick's theorem extends to infinite dimensional spaces, in which the covariance C may become a positive kernel C(x, y) in a distribution space. We recall that such a kernel is an operator acting on functions through C.f = C(x, y)f (y) dy. The identity operator is represented by the Dirac kernel C(x, y) = δ(x − y). But if C is positive definite with some regularity it can be also considered the covariance of a truly well defined Gaussian measure in some infinite dimensional space of distributions, through an extension of Bochner's theorem known as Minlos Theorem [1].
Functional integrals
In QFT, like in grand-canonical statistical mechanics, particle number is not conserved. Cross sections in scattering experiments contain the physical information of the theory 1 . They are the matrix elements of the diffusion matrix S. Under suitable conditions they are expressed in terms of the Green functions G N of the theory through so-called "reduction formulae".
Green functions are time ordered vacuum expectation values of the field φ, which is operator valued and acts on the Fock space:
G N (z 1 , ..., z N ) =< ψ 0 , T [φ(z 1 )...φ(z N )]ψ 0 > . (1.3)
Here ψ 0 is the vacuum state and the T -product orders φ(z 1 )...φ(z N ) according to increasing times. Consider a Lagrangian field theory, and split the total Lagrangian as the sum of a free plus an interacting piece, L = L 0 + L int . The Gell-Mann-Low 1 Correlation functions play this fundamental role in statistical mechanics formula expresses the Green functions as vacuum expectation values of a similar product of free fields with an e i Lint insertion:
G N (z 1 , ..., z N ) = < ψ 0 , T φ(z 1 )...φ(z N )e i dxLint(φ(x)) ψ 0 > < ψ 0 , T (e i dxLint(φ(x)) )ψ 0 > . (1.4) In the functional integral formalism proposed by Feynman [2], the Gell-Mann-Low formula is replaced by a functional integral in terms of an (ill-defined) "integral over histories" which is formally the product of Lebesgue measures over all space time. The corresponding formula is the Feynman-Kac formula:
G N (z 1 , ..., z N ) == j φ(z j )e i L(φ(x))dx Dφ e i L(φ(x))dx Dφ .
(1.5)
The integrand in (1.5) contains now the full Lagrangian L = L 0 +L int instead of the interacting one. This is interesting to expose symmetries of the theory which may not be separate symmetries of the free and interacting Lagrangians, for instance gauge symmetries. Perturbation theory and the Feynman rules can still be derived as explained in the next subsection. But (1.5) is also well adapted to constrained quantization and to the study of non-perturbative effects.
For general references on QFT, see [3,4,5].
Statistical Mechanics and Thermodynamic Quantities
There is a deep analogy between the Feynman-Kac formula and the formula which expresses correlation functions in classical statistical mechanics. The partition function of a statistical mechanics grand canonical ensemble described by a Hamiltionian H at temperature T and chemical potential µ is Z Λ = T r(e −β(H−µN ) ), (1.6) where β = 1/kT , and the trace may be either a classical integration in phasespace or in the quantum case a trace on the relevant Hilbert space (Fock space). The main problem is to compute the logarithm of the partition function. Indeed thermodynamic quantities such as eg the mean energy of the system (1.9) or the heat capacity (at fixed volume) 10) follow from that computation.
< H > T,µ = T r(He −β(H−µN ) ) Z Λ = − ∂ log Z λ ∂β ,(1.C V = ∂ < H > T,µ ∂T µ = − 1 β 2 ∂ < H > T,µ ∂β µ (1.
In fact all the detailed information on the equilibrium states is encoded in the list of their correlation functions, which are derivatives of the logarithm of the partition function with respect to appropriate sources. For instance for a lattice Ising model the partition function is where x labels the discrete sites of the lattice. The sum is over configurations {σ x = ±1} which associate a "spin" with value +1 or -1 to each such site and L(σ) contains usually nearest neighbor interactions and possibly a magnetic field h: L(σ) =
Z Λ =
x,y nearest neighbors Jσ x σ y +
x hσ x .
(1.13)
By analytically continuing (1.5) to imaginary time, or Euclidean space, it is possible to complete the analogy with (1.12), hence to establish a firm contact between Euclidean QFT and statistical mechanics [6,7,8].
Schwinger Functions
This idea also allows to give much better meaning to the path integral, at least for a free Bosonic field. Indeed the free Euclidean measure can be defined easily as a Gaussian measure, because in Euclidean space L 0 is a quadratic form of positive type 2 .
The Green functions continued to Euclidean points are called the Schwinger functions of the model, and are given by the Euclidean Feynman-Kac formula:
S N (z 1 , ..., z N ) = Z −1 N j=1 φ(z j )e − L(φ(x))dx Dφ,
(1.14)
Z = e − L(φ(x))dx Dφ.
(1. 15) The simplest interacting field theory is the theory of a one component scalar bosonic field φ with quartic interaction λφ 4 (φ 3 , which is simpler, is unstable). In R d it is called the φ 4 d model. For d = 2, 3 this model is superrenormalizable and has been built non perturbatively by constructive field theory (see [1,9]). In these dimensions the model is unambiguously related to its perturbation series [10,11] through Borel summability [12]. For d = 4 the model is just renormalizable, and provides the simplest pedagogical introduction to perturbative renormalization theory. But because of the Landau ghost or triviality problem explained in subsection 1.11, the model presumably does not exist as a true interacting theory at the non perturbative level (see [9] for a discussion of this subtle issue).
Formally the Schwinger functions of φ 4 d are the moments of the measure:
dν = 1 Z e − λ 4! φ 4 −(m 2 /2) φ 2 −(a/2) (∂µφ∂ µ φ) Dφ,(1.16)
where • λ is the coupling constant, usually assumed positive or complex with positive real part; remark the arbitrary but convenient 1/4! factor to take into account the symmetry of permutation of all fields at a local vertex.
• m is the mass, which fixes an energy scale for the theory;
• a is the wave function constant. It can be set to 1 by a rescaling of the field.
• Z is a normalization factor which makes (1.16) a probability measure;
• Dφ is a formal (mathematically ill-defined) product where Z 0 is again the normalization factor which makes (1.17) a probability measure.
More precisely if we consider the translation invariant propagator C(x, y) ≡ C(x − y) (with slight abuse of notation), whose Fourier transform is
C(p) = 1 (2π) d 1 p 2 + m 2 ,(1.18)
we can use Minlos theorem and the general theory of Gaussian processes to define dµ(φ) as the centered Gaussian measure on the Schwartz space of tempered distributions S (R d ) whose covariance is C. A Gaussian measure is uniquely defined by its moments, or the integral of polynomials of fields. Explicitly this integral is zero for a monomial of odd degree, and for even n = 2p it is equal to
φ(x 1 )...φ(x n )dµ(φ) = W ∈W C(x b( ) , x e( ) )
, (1.19) where the sum runs over all the 2p!! = (2p − 1)(2p − 3)...5.3.1 Wick pairings W of the 2p arguments into the p disjoint pairs = (b( ), e( )). Note that since for d ≥ 2, C(p) is not integrable, C(x, y) must be understood as a distribution. It is therefore convenient to also use regularized kernels, for instance
C κ (p) = 1 (2π) d e −κ(p 2 +m 2 ) p 2 + m 2 = ∞ κ e −α(p 2 +m 2 ) dα (1.20)
whose Fourier transform C κ (x, y) is a smooth function and not a distribution:
C κ (x, y) = ∞ κ e −αm 2 −(x−y) 2 /4α dα α D/2 . (1.21) α −D/2 e −(x−y) 2 /4α
is the heat kernel. Therefore this α-representation has also an interpretation in terms of Brownian motion:
C κ (x, y) = ∞ κ dα exp(−m 2 α) P (x, y; α) (1.22)
where P (x, y; α) = (4πα) −d/2 exp(−|x − y| 2 /4α) is the Gaussian probability distribution of a Brownian path going from x to y in time α.
Such a regulator κ is called an ultraviolet cutoff, and we have (in the distribution sense) lim κ→0 C κ (x, y) = C(x, y). Remark that due to the non zero m 2 mass term, the kernel C κ (x, y) decays exponentially at large |x − y| with rate m. For some constant K and d > 2 we have:
|C κ (x, y)| ≤ Kκ 1−d/2 e −m|x−y| .
(1.23)
It is a standard useful construction to build from the Schwinger functions the connected Schwinger functions, given by: (1.24) where the sum is performed over all distinct partitions of {1, ..., N } into k subsets P 1 , ..., P k , P i being made of p i elements called j 1 , ..., j pi . For instance in the φ 4 theory, where all odd Schwinger functions vanish due to the unbroken φ → −φ symmetry, the connected 4-point function is simply:
C N (z 1 , ..., z N ) = P1∪...∪P k ={1,...,N }; Pi∩Pj =0 (−1) k+1 k i=1 S pi (z j1 , ..., z jp i ),C 4 (z 1 , ..., z 4 ) = S 4 (z 1 , ..., z 4 ) − S 2 (z 1 , z 2 )S 2 (z 3 , z 4 ) (1.25) −S 2 (z 1 , z 3 )S 2 (z 2 , z 4 ) − S 2 (z 1 , z 4 )S 2 (z 2 , z 3 ).
Feynman Graphs
The full interacting measure may now be defined as the multiplication of the Gaussian measure dµ(φ) by the interaction factor:
dν = 1 Z e − λ 4! φ 4 (x)dx dµ(φ) (1.26)
and the Schwinger functions are the normalized moments of this measure:
S N (z 1 , ..., z N ) = φ(z 1 )...φ(z N )dν(φ). (1.27)
Expanding the exponential as a power series in the coupling constant λ, one obtains a formal expansion for the Schwinger functions:
S N (z 1 , ..., z N ) = 1 Z ∞ n=0 (−λ) n n! φ 4 (x)dx 4! n φ(z 1 )...φ(z N )dµ(φ). (1.28)
It is now possible to perform explicitly the functional integral of the corresponding polynomial. The result is at any order n a sum over (4n + N − 1)!! Wick contractions schemes W, i.e. over all the ways of pairing together 4n + N fields into 2n + N/2 pairs. The weight or amplitude of such a scheme W is the spatial integral over x 1 , ..., x n of the integrand ∈W C(x i b( ) , x ie( ) ) times the factor 1 n! ( −λ 4! ) n . Such amplitudes are functions (in fact distributions) of the external positions z 1 , ..., z N . They may diverge either because they are integrals over all of R 4 (no volume cutoff) or because the integrand is typically unbounded due to the singularities in the propagator C at coinciding points.
Labeling the n dummy integration variables in (1.28) as x 1 , ..., x n , we draw a line for each contraction of two fields. Each position x 1 , ..., x n is then associated to a four-legged vertex and each external source z i to a one-legged vertex, as shown in Figure 1.
It is convenient to draw these Wick contractions and to regroup all contractions which give rise to the same drawing or graph. There are some subtleties about labels. Example 1.1 For the normalization at order 1 we have 4 fields, hence 3 Wick contractions, which all give the same graph. For the 2 point function at order 1 we have 6 fields, and 15 Wick contractions which fall into 2 categories with weight 3 and 12.
We have additional observations • The great advantage of Feynman graphs is that they form a combinatoric species in the sense of Joyal [13] whose logarithm can be computed as the species of connected graphs. As we already remarked, the computation of this logarithm is the key physical problem.
• However Feynman graphs proliferate, that is their generating functional n an n! λ n has zero radius of convergence in λ. At the heart of any constructive strategy [1,9,14,15,16,17,18], lies the replacement of the proliferating species of Feynman graphs by a better one [19], typically the species of forests. The corresponding connected species is the species of trees, which does not proliferate. Indeed by Cayley's theorem there are only n n−2 labeled trees on n vertices. This is why constructive expansions converge while ordinary perturbative expansions dont. Constructive theory ultimately may be considered just as repacking Feynman graphs in some clever way according to underlying forests [20]. See also the discussion in subsection 1.16 and below.
• The computation factorizes nicely into the connected components of the graphs. These components may or may not have external arguments. In the expansion for the normalized functions the vacuum components (i.e. those without external arguments) factor out and disappear. Only graphs whose connected components all contain external arguments remain.
• If we further search for elementary bricks of the expansion, we can consider the connected Schwinger functions like (1.25). In the expansion of these functions only the graphs with a single connected component containing all external arguments survive.
Feynman Rules
The "Feynman rules" summarize how to compute the amplitude associated to a Feynman graph with its correct combinatoric factor. We always use the following notations for a graph G:
• n(G) or simply n is the number of internal vertices of G, or the order of the graph.
• l(G) or l is the number of internal lines of G, i.e. lines hooked at both ends to an internal vertex of G.
• N (G) or N is the number of external vertices of G; it corresponds to the order of the Schwinger function one is looking at. When N = 0 the graph is a vacuum graph, otherwise it is called an N -point graph.
• c(G) or c is the number of connected components of G,
• L(G) or L is the number of independent loops of G.
For a regular φ 4 graph, i.e. a graph which has no line hooked at both ends to external vertices, we have the relations:
l(G) = 2n(G) − N (G)/2, (1.29) L(G) = l(G) − n(G) + c(G) = n(G) + 1 − N (G)/2. (1.30)
where in the last equality we assume connectedness of G, hence c(G) = 1.
A subgraph F of a graph G is a subset of internal lines of G, together with the corresponding attached vertices. Lines in the subset defining F are the internal lines of F , and their number is simply l(F ), as before. Similarly all the vertices of G hooked to at least one of these internal lines of F are called the internal vertices of F and considered to be in F ; their number by definition is n(F ). Finally a good convention is to call external half-line of F every half-line of G which is not in F but which is hooked to a vertex of F ; it is then the number of such external half-lines which we call N (F ). With these conventions one has for φ 4 subgraphs the same relation (1.29) as for regular φ 4 graphs.
To compute the amplitude associated to a φ 4 graph, we have to add the contributions of the corresponding contraction schemes. This is summarized by the "Feynman rules":
• To each line with end vertices at positions x and y , associate a propagator C(x , y ).
• To each internal vertex, associate (−λ)/4!.
• Count all the contraction schemes giving this diagram. The number should be of the form (4!) n n!/S(G) where S(G) is an integer called the symmetry factor of the diagram. The 4! represents the permutation of the fields hooked to an internal vertex.
• Multiply all these factors, divide by n! and sum over the position of all internal vertices.
The formula for the bare amplitude of a graph is therefore, as a distribution in z 1 , ....z N :
A G (z 1 , ..., z N ) ≡ n i=1 dx i ∈G C(x i b( ) , x i e( ) ).
(1.31) This is the "direct" or "x-space" representation of a Feynman integral. As stated above, this integral suffers of possible divergences. But the corresponding quantity with both volume cutoff Λ and ultraviolet cutoff κ, namely
A κ G,Λ (z 1 , ..., z N ) ≡ Λ n n i=1 dx i ∈G C κ (x i b( ) , x i e( ) ), (1.32)
is well defined. The integrand is indeed bounded and the integration domain Λ is assumed compact.
The unnormalized Schwinger functions are therefore formally given by the sum over all graphs with the right number of external lines of the corresponding Feynman amplitudes:
ZS N = φ 4 graphs G with N (G)=N (−λ) n(G) S(G) A G . (1.33)
Z itself, the normalization, is given by the sum of all vacuum amplitudes:
Z = φ 4 graphs G with N (G)=0 (−λ) n(G) S(G) A G . (1.34)
We already remarked that the species of Feynman graphs proliferate at large orders. More precisely the total number of φ 4 Feynman graphs at order n with N external arguments is (4n + N )!!. Taking into account Stirling's formula and the symmetry factor 1/n! from the exponential we expect perturbation theory at large order to behave as K n n! for some constant K. Indeed at order n the amplitude of a Feynman graph is a 4n-dimensional integral. It is reasonable to expect that in average it should behave as c n for some constant c. But this means that one should expect zero radius of convergence for the series (1.33). This is not too surprising. Even the one-dimensional integral
F (g) = +∞ −∞ e −x 2 /2−λx 4 /4! dx (1.35)
is well-defined only for λ ≥ 0. We cannot hope infinite dimensional functional integrals of the same kind to behave better than this one dimensional integral. In mathematically precise terms, F is not analytic near λ = 0, but only Borel summable. Borel summability [12] is therefore the best we can hope for the φ 4 theory, and we mentioned that it has indeed been established for the φ 4 theory in dimensions 2 and 3 [10,11]. From translation invariance, we do not expect A κ G,Λ to have a limit as Λ → ∞ if there are vacuum subgraphs in G. But obviously an amplitude factorizes as the product of the amplitudes of its connected components.
With simple combinatoric verification at the level of contraction schemes we can factorize the sum over all vacuum graphs in the expansion of unnormalized Schwinger functions, hence get for the normalized functions a formula analog to (1.33):
S N = φ 4 graphs G with N (G)=N G without any vacuum subgraph (−λ) n(G) S(G) A G . (1.36)
Now in (1.36) it is possible to pass to the thermodynamic limit (in the sense of formal power series) because using the exponential decrease of the propagator, each individual graph has a limit at fixed external arguments. There is of course no need to divide by the volume for that because each connected component in (1.36) is tied to at least one external source, and they provide the necessary breaking of translation invariance. Finally one can find the perturbative expansions for the connected Schwinger functions and the vertex functions. As expected, the connected Schwinger functions are given by sums over connected amplitudes:
C N = φ 4 connected graphs G with N (G)=N (−λ) n(G) S(G) A G (1.37)
and the vertex functions are the sums of the amputated amplitudes for proper graphs, also called one-particle-irreducible. They are the graphs which remain connected even after removal of any given internal line. The amputated amplitudes are defined in momentum space by omitting the Fourier transform of the propagators of the external lines. It is therefore convenient to write these amplitudes in the so-called momentum representation:
Γ N (z 1 , ..., z N ) = φ 4 proper graphs G with N (G)=N (−λ) n(G) S(G) A T G (z 1 , ..., z N ), (1.38) A T G (z 1 , ..., z N ) ≡ 1 (2π) dN/2 dp 1 ...dp N e i pizi A G (p 1 , ..., p N ), (1.39) A G (p 1 , ..., p N ) = internal line of G d d p p 2 + m 2 v∈G δ( (v, ) p ). (1.40)
Remark in (1.40) the δ functions which ensure momentum conservation at each internal vertex v; the sum inside is over both internal and external momenta; each internal line is oriented in an arbitrary way (from b( ) to e( )) and each external line is oriented towards the inside of the graph. The incidence matrix (v, ) captures in a nice way the information on the internal lines 3 . It is 1 if the line arrives at v, -1 if it starts from v and 0 otherwise. Remark also that there is an overall momentum conservation rule δ(p 1 + ... + p N ) hidden in (1.40). The drawback of the momentum representation lies in the necessity for practical computations to eliminate the δ functions by a "momentum routing" prescription, and there is no canonical choice for that. Although this is rarely explicitly explained in the quantum field theory literature, such a choice of a momentum routing is equivalent to the choice of a particular spanning tree of the graph.
Scale Analysis and Renormalization
In order to analyze the ultraviolet or short distance limit according to the renormalization group method [22], we can cut the propagator C into slices C i so that C = ∞ i=0 C i . This can be done conveniently within the parametric representation, since α in this representation roughly corresponds to 1/p 2 . So we can define the propagator within a slice as
C 0 = ∞ 1 e −m 2 α− |x−y| 2 4α dα α d/2 , C i = M −2(i−1) M −2i e −m 2 α− |x−y| 2 4α dα α d/2 for i ≥ 1.
(1.41) where M is a fixed number, for instance 10, or 2, or e. We can intuitively imagine C i as the piece of the field oscillating with Fourier momenta essentially of size M i . In fact it is easy to prove the bound (for d > 2)
|C i (x, y)| ≤ K.M (d−2)i e −M i |x−y| (1.42)
where K is some constant. Now the full propagator with ultraviolet cutoff M ρ , ρ being a large integer, may be viewed as a sum of slices:
C ≤ρ = ρ i=0 C i .
(1.43)
Then the basic renormalization group step is made of two main operations:
• A functional integration
• The computation of a logarithm Indeed decomposing a covariance in a Gaussian process corresponds to a decomposition of the field into independent Gaussian random variables φ i , each distributed with a measure dµ i of covariance C i . Let us introduce
Φ i = i j=0 φ j .
(1.44) This is the "low-momentum" field for all frequencies lower than i. The RG idea is that starting from scale ρ and performing ρ − i steps, one arrives at an effective action for the remaining field Φ i . Then, writing Φ i = φ i + Φ i−1 , one splits the field into a "fluctuation" field φ i and a "background" field Φ i−1 . The first step, functional integration, is performed solely on the fluctuation field, so it computes
Z i−1 (Φ i−1 ) = dµ i (φ i )e −Si(φi+Φi−1) . (1.45)
Then the second step rewrites this quantity as the exponential of an effective action, hence simply computes
S i−1 (Φ i−1 ) = − log[Z i−1 (Φ i−1 )] (1.46)
Now Z i−1 = e −Si−1 and one can iterate! The flow from the initial bare action S = S ρ for the full field to an effective renormalized action S 0 for the last "slowly Figure 2: A high energy subgraph S seen from lower energies looks quasi-local.
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varying" component φ 0 of the field is similar to the flow of a dynamical system. Its evolution is decomposed into a sequence of discrete steps from S i to S i−1 . This renormalization group strategy can be best understood on the system of Feynman graphs which represent the perturbative expansion of the theory. The first step, functional integration over fluctuation fields, means that we have to consider subgraphs with all their internal lines in higher slices than any of their external lines. The second step, taking the logarithm, means that we have to consider only connected such subgraphs. We call such connected subgraphs quasi-local. Renormalizability is then a non trivial result that combines locality and power counting for these quasi-local subgraphs.
Locality, Power Counting
Locality simply means that quasi-local subgraphs S look local when seen through their external lines. Indeed since they are connected and since their internal lines have scale say ≥ i, all the internal vertices are roughly at distance M −i . But the external lines have scales ≤ i − 1, which only distinguish details larger than M −(i−1) . Therefore they cannot distinguish the internal vertices of S one from the other. Hence quasi-local subgraphs look like "fat dots" when seen through their external lines, see Figure 2. Obviously this locality principle is completely independent of dimension.
Power counting is a rough estimate which compares the size of a fat dot such as S in Figure 2 with N external legs to the coupling constant that would be in front of an exactly local φ N (x)dx interaction term if it were in the Lagrangian. To simplify we now assume that the internal scales are all equal to i, the external scales are O(1), and we do not care about constants and so on, but only about the dependence in i as i gets large. We must first save one internal position such as the barycenter of the fat dot or the position of a particular internal vertex to represent the dx integration in φ N (x)dx. Then we must integrate over the positions of all internal vertices of the subgraph save that one. This brings about a weight M −di(n−1) , because since S is connected we can use the decay of the internal lines to evaluate these n − 1 integrals. Finally we should not forget the prefactor M (D−2)li coming from (1.42), for the l internal lines. Multiplying these two factors and using relation (1.29)-(1.30) we obtain that the "coupling constant" or factor in front of the fat dot is of order
M −di(n−1)+2i(2n−N/2) = M ω(G) , if we define the superficial degree of divergence of a φ 4
d connected graph as:
ω(G) = (d − 4)n(G) + d − d − 2 2 N (G). (1.47)
So power counting, in contrast with locality, depends on the space-time dimension.
Let us return to the concrete example of Figure 2. A 4-point subgraph made of three vertices and four internal lines at a high slice i index. If we suppose the four external dashed lines have much lower index, say of order unity, the subgraph looks almost local, like a fat dot at this unit scale. We have to save one vertex integration for the position of the fat dot. Hence the coupling constant of this fat dot is made of two vertex integrations and the four weights of the internal lines (in order not to forget these internal line factors we kept internal lines apparent as four tadpoles attached to the fat dot in the right of Figure 2). In dimension 4 this total weight turns out to be independent of the scale.
Renormalization, Effective Constants
At lower scales propagators can branch either through the initial bare coupling or through any such fat dot in all possible ways because of the combinatorial rules of functional integration. Hence they feel effectively a new coupling which is the sum of the bare coupling plus all the fat dot corrections coming from higher scales. To compute these new couplings only graphs with ω(G) ≥ 0, which are called primitively divergent, really matter because their weight does not decrease as the gap i increases.
-If d = 2, we find ω(G) = 2 − 2n, so the only primitively divergent graphs have n = 1, and N = 0 or N = 2. The only divergence is due to the "tadpole" loop d 2 p (p 2 +m 2 ) which is logarithmically divergent.
-If d = 3, we find ω(G) = 3 − n − N/2, so the only primitively divergent graphs have n ≤ 3, N = 0, or n ≤ 2 and N = 2. Such a theory with only a finite number of "primitively divergent" subgraphs is called superrenormalizable.
-
If d = 4, ω(G) = 4 − N .
Every two point graph is quadratically divergent and every four point graph is logarithmically divergent. This is in agreement with the superficial degree of these graphs being respectively 2 and 0. The couplings that do not decay with i all correspond to terms that were already present in the Lagrangian, namely φ 4 , φ 2 and (∇φ).(∇φ) 4 . Hence the structure of the Lagrangian resists under change of scale, although the values of the coefficients can change. The theory is called just renormalizable.
-Finally for d > 4 we have infinitely many primitively divergent graphs with arbitrarily large number of external legs, and the theory is called nonrenormalizable, because fat dots with N larger than 4 are important and they correspond to new couplings generated by the renormalization group which are not present in the initial bare Lagrangian.
To summarize:
• Locality means that quasi-local subgraphs look local when seen through their external lines. It holds in any dimension.
• Power counting gives the rough size of the new couplings associated to these subgraphs as a function of their number N of external legs, of their order n and of the dimension of space time d.
The BPHZ Theorem
The BPHZ theorem is both a brilliant historic piece of mathematical physics which gives precise mathematical meaning to the notion of renormalizability, using the mathematics of formal power series, but it is also ultimately a dead end and a bad way to understand and express renormalization. Let us try to explain both statements. For the massive Euclidean φ 4 4 theory we could for instance state the following normalization conditions on the connected functions in momentum space at zero momenta:
C 4 (0, 0, 0, 0) = −λ ren , (1.48) C 2 (p 2 = 0) = 1 m 2 ren , (1.49) d dp 2 C 2 | p 2 =0 = − a ren m 4 ren .
(1.50)
Usually one puts a ren = 1 by rescaling the field φ.
Using the inversion theorem on formal power series for any fixed ultraviolet cutoff κ it is possible to reexpress any formal power series in λ bare with bare propagators 1/(a bare p 2 + m 2 bare ) for any Schwinger functions as a formal power series in λ ren with renormalized propagators 1/(a ren p 2 + m 2 ren ). The BPHZ theorem then states that that formal perturbative formal power series has finite coefficients order by order when the ultraviolet cutoff κ is lifted. The first proof by Hepp [25] relied on the inductive Bogoliubov's recursion scheme [?]. Then a completely explicit expression for the coefficients of the renormalized series was written by Zimmermann and many followers [27]. The coefficients of that renormalized series can ne written as sums of renormalized Feynman amplitudes. They are similar to Feynman integrals but with additional subtractions indexed by Zimmermann's forests. Returning to an inductive rather than explicit scheme, Polchinski remarked that it is possible to also deduce the BPHZ theorem from a renormalization group equation and inductive bounds which does not decompose each order of perturbation theory into Feynman graphs [28]. This method was clarified and applied by C. Kopper and coworkers, see [?].
The solution of the difficult "overlapping" divergence problem through Bogoliubov's or Polchinski's recursions and Zimmermann's forests becomes particularly clear in the parametric representation using Hepp's sectors. A Hepp sector is simply a complete ordering of the α parameters for all the lines of the graph. In each sector there is a different classification of forests into packets so that each packet gives a finite integral [30,31,32].
But from the physical point of view we cannot conceal the fact that purely perturbative renormalization theory is not very satisfying. At least two facts hint at a better theory which lies behind:
• The forest formula seems unnecessarily complicated, with too many terms.
For instance in any given Hepp sector only one particular packet of forests is really necessary to make the renormalized amplitude finite, the one which corresponds to the quasi-local divergent subgraphs of that sector. The other packets seem useless, a little bit like "junk DNA". They are there just because they are necessary for other sectors. This does not look optimal.
• The theory makes renormalized amplitudes finite, but at tremendous cost! The size of some of these renormalized amplitudes becomes unreasonably large as the size of the graph increases. This phenomenon is called the "renormalon problem". For instance it is easy to check that the renormalized amplitude (at 0 external momenta) of the graphs P n with 6 external legs and n + 2 internal vertices in Figure 3 becomes as large as c n n! when n → ∞. Indeed at large q the renormalized amplitude A R G2 in Figure 5 grows like log |q|. Therefore the chain of n such graphs in Figure 3 behaves as [log |q|] n , and the total amplitude of P n behaves as
[log |q|] n d 4 q [q 2 + m 2 ] 3 n→∞ c n n! (1.51)
So after renormalization some families of graphs acquire so large values that they cannot be resummed! Physically this is just as bad as if infinities were still there.
These two hints are in fact linked. As their name indicates, renormalons are due to renormalization. Families of completely convergent graphs such as the graphs Q n of Figure 4, are bounded by c n , and produce no renormalons.
Studying more carefully renormalization in the α parametric representation one can check that renormalons are solely due to the forests packets that we compared to "junk DNA". Renormalons are due to subtractions that are not Figure 3: A family of graphs P n producing a renormalon. Figure 4: A family of convergent graphs Q n , that do not produce any renormalon.
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necessary to ensure convergence, just like the strange log |q| growth of A R G0 at large q is solely due to the counterterm in the region where this counterterm is not necessary to make the amplitude finite.
We can therefore conclude that subtractions are not organized in an optimal way by the Bogoliubov recursion. What is wrong from a physical point of view in the BPHZ theorem is to use the size of the graph as the relevant parameter to organize Bogoliubov's induction. It is rather the size of the line momenta that should be used to better organize the renormalization subtractions.
This leads to the point of view advocated in [9]: neither the bare nor the renormalized series are optimal. Perturbation should be organized as a power series in an infinite set of effective expansions, which are related through the RG flow equation. In the end exactly the same contributions are resummed than in the bare or in the renormalized series, but they are regrouped in a much better way.
The Landau ghost and Asymptotic Freedom
In the case of φ 4 4 only the flow of the coupling constants really matters, because the flow of m and of a for different reasons are not very important in the ultraviolet limit:
-the flow of m is governed at leading order by the tadpole. The bare mass m 2 i corresponding to a finite positive physical mass m 2 ren is negative and grows as λM 2i with the slice index i. But since p 2 in the i-th slice is also of order M 2i but without the λ, as long as the coupling λ remains small it remains much larger than m 2 i . Hence the mass term plays no significant role in the higher slices. It was remarked in [9] that because there are no overlapping problem associated to 1PI two point subgraphs, there is in fact no inconvenience to use the full renormalized m ren all the way from the bare to renormalized scales, with subtractions on 1PI two point subgraphs independent of their scale. -the flow of a is also not very important. Indeed it really starts at two loops because the tadpole is exactly local. So this flow is in fact bounded, and generates no renormalons. In fact as again remarked in [9] for theories of the φ 4 4 type one might as well use the bare value a bare all the way from bare to renormalized scales and perform no second Taylor subtraction on any 1PI two point subgraphs.
But the physics of φ 4 4 in the ultraviolet limit really depends of the flow of λ. By a simple second order computation there are only 2 connected graphs with n = 2 and N = 4 pictured in Figure 5. They govern at leading order the flow of the coupling constant.
In the commutative φ 4 4 theory the graph G 1 does not contribute to the coupling constant flow. This can be seen in many ways, for instance after mass renormalization the graph G 1 vanishes exactly because it contains a tadpole which is not quasi-local but exactly local. One can also remark that the graph is one particle reducible. In ordinary translation-invariant, hence momentumconserving theories, one-particle-reducible quasi-local graphs never contribute significantly to RG flows. Indeed they become very small when the gap i between internal and external scales grows. This is because by momentum conservation the momentum of any one-particle-reducible line has to be the sum of a finite set of external momenta on one of its sides. But a finite sum of small momenta remains small and this clashes directly with the fact that being internal its momentum should grow as the gap i grows. Remark that this is no longer be true in non commutative vulcanized φ 4 4 , because that theory is not translation invariant, and that's why it will ultimately escape the Landau ghost curse.
So in φ 4 4 the flow is intimately linked to the sign of the graph G 2 of Figure 5. More precisely, we find that at second order the relation between λ i and λ i−1 is
λ i−1 λ i − βλ 2 i (1.52)
(remember the minus sign in the exponential of the action), where β is a constant, namely the asymptotic value of j,j / inf(j,j )=i d 4 yC j (x, y)C j (x, y) when i → ∞. Clearly this constant is positive. So for the normal stable φ 4 4 theory, the relation (1.52) inverts into
λ i λ i−1 + βλ 2 i−1 ,(1.53)
so that fixing the renormalized coupling seems to lead at finite i to a large, diverging bare coupling, incompatible with perturbation theory. This is the Landau ghost problem, which affects both the φ 4 4 theory and electrodynamics. Equivalently if one keeps λ i finite as i gets large, λ 0 = λ ren tends to zero and the final effective theory is "trivial" which means it is a free theory without interaction, in contradiction with the physical observation e.g. of a coupling constant of about 1/137 in electrodynamics.
But in non-Abelian gauge theories an extra minus sign is created by the algebra of the Lie brackets. This surprising discovery has deep consequences. The flow relation becomes approximately
λ i λ i−1 − βλ i λ i−1 ,(1.54)
with β > 0, or, dividing by λ i λ i−1 ,
1/λ i 1/λ i−1 + β, (1.55) with solution λ i λ0 1+λ0βi
. A more precise computation to third order in fact leads to
λ i λ 0 1 + λ 0 (βi + γ log i + O(1))
.
( 1.56) Such a theory is called asymptotically free (in the ultraviolet limit) because the effective coupling tends to 0 with the cutoff for a finite fixed small renormalized coupling. Physically the interaction is turned off at small distances. This theory is in agreement with scattering experiments which see a collection of almost free particles (quarks and gluons) inside the hadrons at very high energy. This was the main initial argument to adopt quantum chromodynamics, a non-Abelian gauge theory with SU (3) gauge group, as the theory of strong interactions [?].
Remark that in such asymptotically free theories which form the backbone of today's standard model, the running coupling constants remain bounded between far ultraviolet "bare" scales and the lower energy scale where renormalized couplings are measured. Ironically the point of view on early renormalization theory as a trick to hide the ultraviolet divergences of QFT into infinite unobservable bare parameters could not turn out to be more wrong than in the standard model. Indeed the bare coupling constants tend to 0 with the ultraviolet cutoff, and what can be farther from infinity than 0?
Recently it has been shown to all orders of perturbation theory that there should be no Landau ghost but an asymptotically safe fixed point for the similar RG flow of the non-commutative Grosse-Wulkenhaar φ 4 4 model [34,35,36,37]. Therefore this model is a kind of "Ising model" for just renormalizable QFT, that is a simple model in which one can presumably fully mathematically control at last the phenomenon of renormalization in ll its aspects.
Grassmann representations of determinants and Pfaffians
Independent Grassmann variables χ 1 , ..., χ n satisfy complete anticommutation relations
χ i χ j = −χ j χ i ∀i, j (1.57)
so that any function of these variables is a polynomial with highest degree one in each variable. The rules of Grassmann integration are defined by linearity and
dχ i = 0, χ i dχ i = 1.
plus the rule that all dχ symbols also anticommute between themselves and with all χ variables.
The main important facts are
• Any function of Grassmann variables is a polynomial with highest degree one in each variable.
• Pfaffians and determinants can be nicely written as Grassmann integrals;
The determinant of any n by n matrix can indeed be expressed as a Grassmann Gaussian integral over 2n independent Grassmann variables which it is convenient to name asψ 1 , . . . ,ψ n , ψ 1 , . . . , ψ n , although the bars have nothing yet at this stage to do with complex conjugation. The formula is
det M = dψ i dψ i e − ijψ iMij ψj . (1.58)
Remember that for ordinary commuting variables and a positive n by n Hermitian matrix M
1 π n +∞ −∞ i dφ i dφ i e − ijφ iMij φj = 1 det M . (1.59)
In short Grassmann Gaussian measures are simpler than ordinary Gaussian measures for two main reasons:
• Grassmann Gaussian measures are associated to any matrix M , there is no positivity requirement for M like for ordinary Gaussian measures.
• Their normalization directly computes the determinant of M , not the inverse (square-root of) the determinant of M . This is essential in many areas where factoring out this determinant is desirable; it explains in particular the success of Grassmann and supersymmetric functional integrals in the study of disordered systems.
The stubborn reader which remembers the square-root formula in (1.1) and would like to understand the corresponding "real version" of (1.58) is rewarded by the beautiful theory of Pfaffians. Clearly commuting Gaussian real integrals involve symmetric matrices, but Grassmann Gaussian with only n "real" integrals must involve n by n antisymmetric matrices.
The Pfaffian Pf(A) of an antisymmetric matrix A is defined by
det A = [Pf(A)] 2 .
(1. 60) and is known to be polynomila in the coefficients of A. This fact is recovered easily by writing it as
Pf(A) = dχ 1 ...dχ n e − i<j χiAij χj = dχ 1 ...dχ n e − 1 2 i,j χiAij χj . (1.61) Indeed we have det A = i dψ i dψ i e − ijψ iAij ψj . (1.62)
Performing the change of variables (which a posteriori justifies the complex notation)ψ
i = 1 √ 2 (χ i − iω i ), ψ i = 1 √ 2 (χ i + iω i ), (1.63)
whose Jacobian is i −n , the new variables χ and ω are again independent Grassmann variables. Now a short computation using
A ij = −A ji gives det A = i −n i dχ i dω i e − i<j χiAij χj − i<j ωiAij ωj = i dχ i e − i<j χiAij χj i dω i e − i<j ωiAij ωj ,(1.64)
where we used that n = 2p has to be even and that a factor (−1) p is generated when changing i dχ i dω i into i dχ i i dω i . Equation There are also normalized Grassmann Gaussian integrals which may be expressed formally as For a much more detailed introduction to the rules of Grassmann calculus in QFT, we refer to [41].
dµ M = dψ i dψ i e − ijψ iM −1 ij ψj dψ i dψ i e − ijψ
Trees, forests and the parametric representation
Classical evolution can be expanded perturbatively into sums indexed by trees whether in quantum field theory the loops of Feynman graphs are essential.
The hidden trees of the classical system inside QFT can be revealed only under scale analysis, since they do not correspond to ordinary spanning trees of the graphs, but to the abstract inclusion relations of short range effects (high energy quasi local subgraphs) inside larger ones. This point of view has been progressively formalized over the years from Bogoliubov to Zimmermann to the most recent formalization by D. Kreimer and A. Connes in terms of Hopf algebras.
But ordinary spanning trees of a connected graph also enter in a fascinating way in the computation of its amplitude. Since the heat kernel is quadratic it is possible to explicitly compute all spatial integrations in a Feynman amplitude. One obtains the so-called parametric representation. The result is expressed in terms of topological or so-called "Symanzik" polynomials [23,24].
The amplitude of an amputated graph G with external momenta p is, up to a normalization, in space-time dimension D:
A G (p) =δ( p) ∞ 0 e −V G (p,α)/U G (α) U G (α) D/2 l (e −m 2 α l dα l ) .
(1.68)
The first and second Symanzik polynomials U G and V G are
U G = T l ∈T α l , (1.69a) V G = T2 l ∈T2 α l ( i∈E(T2) p i ) 2 , (1.69b)
where the first sum is over spanning trees T of G and the second sum is over two trees T 2 , i.e. forests separating the graph in exactly two connected components E(T 2 ) and F (T 2 ); the corresponding Euclidean invariant ( i∈E(T2) p i ) 2 is, by momentum conservation, also equal to ( i∈F (T2) p i ) 2 . The proof of relations (1.69a-1.69b) is a special case of the Tree matrix Theorem, which we now explain following [42] Theorem 1.3 (Tree Matrix Theorem) Let A be an n by n matrix such that
n i=1 A ij = 0 ∀j .
(1.70)
Obviously det A = 0. But let A 11 be the matrix A with line 1 and column 1 deleted. Then This theorem has both a positivity and a democracy aspect: all trees contribute with positive, equal weights to the determinant. Proof of Theorem 1.3: We use Grassmann variables to write the determinant of a matrix with one line and one raw deleted as a Grassmann integral with two corresponding sources:
det A 11 = T ∈T A i ,j ,(1.det A 11 = (dψdψ) (ψ 1 ψ 1 )e −ψAψ (1.72)
The trick is to use (1.70) to write
ψAψ = n i,j=1 (ψ i − ψ j )A ij ψ j (1.73)
and to obtain
det A 11 = dψdψ (ψ 1 ψ 1 ) exp − n i,j=1 A ij (ψ i − ψ j )ψ j (1.74) = dψdψ (ψ 1 ψ 1 ) n i,j=1 1 − A ij (ψ i − ψ j )ψ j (1.75)
by the Pauli exclusion principle. We now expand to get
det A 11 = G =(i,j)∈G (−A ij ) Ω G (1.76)
where G is any subset of [n] × [n], and we used the notation
Ω G def = dψdψ (ψ 1 ψ 1 ) (i,j)∈G (ψ i − ψ j )ψ j (1.77)
The tree matrix theorem then follows from the following Lemma 1.4 Ω G = 0 unless the graph G is a tree directed away from 1 in which case Ω G = 1.
Proof of the lemma : Trivially, if (i, i) belongs to G, then the integrand of Ω G contains a factor ψ i − ψ i = 0 and therefore Ω G vanishes. But the crucial observation is that if there is a loop in G then again Ω G = 0. This is because then the integrand of Ω F ,R contains the factor
ψ τ (k) − ψ τ (1) = (ψ τ (k) − ψ τ (k−1) ) + · · · + (ψ τ (2) − ψ τ (1) ) (1.78)
Now, upon inserting this telescoping expansion of the factor ψ τ (k) − ψ τ (1) into the integrand of Ω F ,R , the latter breaks into a sum of (k − 1) products. For each of these products, there exists an α ∈ Z Z/kZ Z such that the factor (ψ τ (α) − ψ τ (α−1) ) appears twice : once with the + sign from the telescopic expansion of (ψ τ (k) − ψ τ (1) ), and once more with a + (resp. −) sign if (τ (α), τ (α − 1)) (resp. (τ (α − 1), τ (α))) belongs to F. Again, the Pauli exclusion principle entails that Ω G = 0. Now every connected component of G must contain 1, otherwise there is no way to saturate the dψ 1 integration.
This means that G has to be a directed tree on {1, ...n}. It remains only to see that G has to be directed away from 1, which is not too difficult.
Relations (1.69a-1.69b) follow rather easily from the tree matrix theorem and the direct representation of Feynman amplitudes (1.80).
In [21] a deeper proof of these relations is given. It relies on the more canonical phase-space parametric representation, which we briefly describe now. Let us limit ourselves to "semi-regular" graphs, which have no "tadpoles" that it no line starting and ending at the same vertex. These graphs (once their lines have been oriented in an arbitrary way) are nicely characterized by their incidence matrix, which is a regular l(G) by n(G)
matrix v with v = −1 if line exits vertex v v = +1 if line enters vertex v v = 0 otherwise (1.79)
There are also external momenta p f , f = 1, · · · , N , which we could also also orient through a matrix f v . The momentum parametric representation then writes
A T G (p 1 , ..., p N ) = δ( f,v f v p f ) l(G) =1 dα d d k e −α (k 2 +m 2 ) n(G)−1 v=1 δ( f v p f + v k ).
But there is no canonical way to solve for the delta functions, something known in physics as the procedure of momentum attribution. So it is better to rewrite these amplitudes in the phase-space parametric representation
A T G (p 1 , ..., p N ) = l(G) =1 dα e −α m 2 d d k V −1 v=1 d d x v e −α k 2 +2i(p f vf xv+k v xv) ,
Integrating over momenta leads to the direct space parametric representation:
A T G (p 1 , ..., p N ) = l(G) =1 dα e −α m 2 α d/2 n(G)−1 v=1 d d x v e 2ip f vf xv−xv·x v v v /α .
In [21] it is shown how the representation (1.80) together with the quasi-Pfaffian representation of Lemma 1.2 leads to deletion-contraction relations for the Symanzik polynomials which allow to compute (1.69a-1.69b) from the theory of the universal Tutte polynomial.
BKAR Forest Formula
Since we want to implement Renormalization Group in a non-perturbative or constructive way, we need tools to compute connected functions in a nonperturbative way, with the right scaling properties for the convergence radius of the expansion. For instance in the just renormalizable case, we need a convergence radius in the coupling constant which is uniform in the scale index.
The main such tool is a canonical forest formula [38,39] which allows to package a perturbative expansion in terms of trees rather than Feynman graphs. The advantage was already mentioned several times: the species of trees does not proliferate [19,20] at large orders, in contrast with the species of Feynman graphs.
Consider n points; the set of pairs P n of such points which has n(n − 1)/2 elements = (i, j) for 1 ≤ i < j ≤ n, and a smooth function f of n(n − 1)/2 variables x , ∈ P n . Noting ∂ for ∂ ∂x , the standard forest formula is
f (1, . . . , 1) = F ∈F 1 0 dw [ ∈F ∂ ]f [x F ({w })] (1.80) where
• the sum over F is over forests over the n vertices, including the empty one
• x F ({w }) is the infimum of the w for in the unique path from i to j in F, where = (i, j). If there is no such path, x F ({w }) = 0 by definition.
• The symmetric n by n matrix X F ({w}) defined by X F ii = 1 and X F ij = x F ij ({w }) for 1 ≤ i < j ≤ n is positive. This formula can be viewed as a tool to associate to any pair made of a graph G and a spanning forest F ⊂ G a unique rational number or weight w(G, F ) between 0 and 1, called the relative weight of T in G. These weights are barycentric or percentage factors, ie for any G The numbers w(G, F ) are multiplicative over disjoint unions 5 . Hence it is enough to give the formula for (G, F ) only when G is connected and F = T is a spanning tree in it 6 . The definition of these weights is
Definition 1 w(G, T ) = ∈T 1 0 ∈T dw ∈T x T ({w}) (1.82)
where x T ({w}) is again the infimum over the w parameters over the lines forming the unique path in T joining the ends of .
Consider the expansion in terms of Feynman amplitudes of a connected quantity S. The most naive way to reorder Feynman perturbation theory according to trees rather than graphs is to insert for each graph the relation (1.81)
S = G A G = G T ⊂G w(G, T )A G (1.83)
and exchange the order of the sums over S and T . Hence it writes
S = T A T , A T = G⊃T w(G, T )A G . (1.84)
This rearranges the Feynman expansion according to trees, but each tree has the same number of vertices as the initial graph. Hence it reshuffles the various terms of a given, fixed order of perturbation theory. Remark that if the initial graphs have say degree 4 at each vertex, only trees with degree less than or equal to 4 occur in the rearranged tree expansion.
For Fermionic theories this is typically sufficient and one has for small enough coupling
T |A T | < ∞ (1.85)
because Fermionic graphs essentially mostly compensate each other at a fixed order by Pauli's principle; mathematically this is because these graphs form a determinant and the size of a determinant is much less than what its permutation expansion usually suggests. This is well known [46,62,47]. But this repacking fails completely for Bosonic theories, because the only compensations there occur between graphs of different orders. Hence if we perform this naive reshuffling, eg on the φ 4 0 theory we would still have
T |A T | = ∞. (1.86)
Recently a new expansion called the Loop Vertex Expansion has been found [16] which overcomes this difficulty by exchanging the role of vertices and propagators before applying the forest formula. It can also be seen as a combination of the forest formula with the so-called intermediate field representation, which expands into essentially square roots of a stable Bosonic interaction. We refer the reader to [16,17,19,40,20] but wont review this expansion here, since from now on we are mostly going to deal with Fermions.
Gram and Hadamard Bounds
These two bounds on a determinant are often confused! The Gram bound applies to a matrix A = a ij whose entries are scalar product. This means we suppose that there exists some Hilbert space H and 2n vectors f i , i = 1, · · · , n, g j , j = 1, · · · , n with a ij =< f i , g j > H .
(1.87)
The Gram bound states
| det A| ≤ n i=1 f i H n j=1 g j H .
(1.88)
Of course any matrix A can always be written of the Gram type, eg with H = R n , f i = (a i,k ) and g j = δ j,k , k = 1, · · · , n, or conversely. Hence there are two corresponding asymmetric Hadamard bounds, one for rows and one for columns:
| det A| ≤ n i=1 n j=1 a 2 ij , (1.89) | det A| ≤ n j=1 n i=1 a 2 ij (1.90)
and also a symmetric Hadamard bound involving the supremum of the matrix elements:
| det A| ≤ n n/2 sup i,j |a ij | n . (1.91)
Remark that the symmetric Hadamard bound means that a determinant of a large matrix is always much smaller than what its permutation expansion plus naive bounds would suggest, which is the "stupid bound"
| det A| ≤ n! sup i,j |a ij | n . (1.92)
This difference in constructive theory is essential. Indeed for Fermionic theories with bounded propagators and a quartic interaction, the matrix A at n-th order of perturbation is a 2n × 2n matrix, with propagators as matrix elements, and there is a 1/n! symmetry factor. Hence the bound (1.92) would lead to believe that the radius of convergence of the partition function is 0, like in the Bosonic case. But the Hadamard bound (1.91) proves that it is at least finite. Moreover usually it is possible to write the propagators as scalar products in L 2 (R) d of functions which also have bounded L 2 norms 7 . In that case the Gram bound (1.88) shows that the partition function is in fact an entire function, as it shows no factorial dependence at all as n → ∞!
Single Scale Constructive Theory for a Toy Model
Consider a just-renormalizable QFT theory. The key problem is to compute connected quantities with an expansion which converges for a small coupling constant, with a propagator limited to a single renormalization group scale, uniformly in the slice index. In the simple Fermionic case, this can be done through applying first the BKAR formula then checking convergence through a Gram bound.
To discuss the type of Fermionic models met in condensed matter it is appropriate to consider first a toy Fermionic d-dimensional QFT model. It is made of a single an infrared slice with N colors. Suppose the propagator is diagonal in color space and satisfies the bound
|C j,ab (x, y)| ≤ δ ab M −dj/2 √ N e −M −j |x−y| . (1.93)
We say that the interaction is of the vector type (or Gross-Neveu type) if it is of the form
V = λ d d x N a=1ψ a (x)ψ a (x) N b=1ψ b (x)ψ b (x) (1.94)
where λ is the coupling constant. We claim that Lemma 1.5 The perturbation theory for the connected functions of this single slice model has a radius of convergence in λ which is uniform in j and N .
To prove this lemma, expands the partition function Z(Λ) through the forest formula, and take the logarithm to obtain a tree formula for the pressure
p = lim Λ→∞ 1 |Λ| log Z(Λ). (1.95)
This is completely straightforward, the only difficulty being notational. Using the notations of [47] p = lim The sum over the a i 's and b i 's are over the colors of the fields and antifields of the vertices obtained by expanding the interaction and of the form:
ψ ai (x i )ψ ai (x i )ψ bi (x i )ψ bi (x i ) (1.97) with 1 ≤ i ≤ n.
The sum over T is over all trees which connect together the n vertices at x 1 , . . . , x n . The sum over Ω is over the compatible ways of realizing the bonds l = {i, j} ∈ T as contractions of a ψ andψ between the vertices i and j (compatible means that we do not contract twice the same field or antifield). (T , Ω) is a sign which is not important for the bound (see [47] for its explicit computation). For any l ∈ T , i(l) ∈ {1, . . . , n} labels the vertex where the field, contracted by the procedure Ω concerning the link l, was chosen. Likewise j(l) is the label for the vertex containing the contracted antifield. δ l is 1 if the colors (among a 1 , . . . , a n , b 1 , . . . , b n ) of the field and antifield contracted by l are the same and else is 0. Finally the matrix (C αβ ) α,β of the remaining "loop lines" is defined in the following manner. The row indices α label the 2n fields produced by the n vertices, so that α = (i, σ) with 1 ≤ i ≤ n and σ takes two values 1 or 2 to indicate whether the field is the second or the fourth factor in (1.97) respectively.
The column indices β label in the same way the 2n antifields, so that β = (j, τ ) with 1 ≤ j ≤ n and τ = 1 or 2 according to whether the antifield is the first or the third factor in (1.97) respectively. The α's and β's are ordered lexicographically. We denote by c(i, σ) the color of the field labeled by (i, σ) that is a i , if σ = 1, and b i if σ = 2. We introduce the similar notationc(j, τ ) for the color of an antifield. Now
C (i,σ)(j,τ ) = w T ,BK ij (w)C(x i , x j )δ(c(i, σ),c(j, τ )).
(1.98)
Finally each time a field (i, σ) is contracted by Ω the corresponding row is deleted from the 2n × 2n matrix (C αβ ). Likewise, for any contracted antifield the corresponding column is erased. A and B denote respectively the set of remaining rows and the set of remaining columns. The minor determinant featuring as det remaining in (1.97) is now det(b αβ ) α∈A,β∈B which is (n + 1) × (n + 1). Indeed for each of the n − 1 links of T , a row and a column have been erased. Suppose we have written C j (x k , y m ) < f j,k , g j,m > L 2 (1.99) (this is realized through f j,k = f j (x k , ·) and g j,m = g j (·, y m ) iff j (p).ĝ j (p) = C j (p)). and that the power counting is conserved by taking square-roots, namely that the L 2 norms of g j and f j scale as M −dj/4 N 1/4 . Then applying the Gram inequality will lead to the proof that the radius of convergence of the pressure is uniform in j and N for this toy model. Indeed the following lemma shows that the presence of the weakening factors w does not change the outcome of the Gram bound.
Lemma Let A = (a αβ ) α,β be a Gram matrix: a αβ =< f α , g β > for some inner product < ., . >. Suppose each of the indices α and β is of the form (i, σ) where the first index i, 1 ≤ i ≤ n, has the same range as the indices of the positive matrix (w F ,BK ij (w)) ij , and σ runs through some other index set Σ. Let C = (C αβ ) α,β be the matrix with entries C (i,σ)(j,τ ) = w F ,BK ij (w). < f (i,σ) , g (j,τ ) > and let (C αβ ) α∈A,β∈B be some square matrix extracted from C, then for any w we have the Gram inequality:
| det(C αβ ) α∈A,β∈B | ≤ α∈A ||f α || β∈B ||g β || (1.101)
Proof Indeed we can take the symmetric square root v of the positive matrix w F ,BK so that w F ,BK ij = n k=1 v ik v kj . Let us denote the components of the vectors f and g, in an orthonormal basis for the scalar product < ., . > with q elements, by f m (i,σ) and g m (j,τ ) , 1 ≤ m ≤ q. (Indeed even if the initial Hilbert space is infinite dimensional, the problem is obviously restricted to the finite dimensional subspace generated by the finite set of vectors f and g). We then define the tensorized vectors F (i,σ) and G (j,τ ) with components F km
(i,σ) = v ik f m (i,σ)
and G km (j,τ ) = v jk g m (j,τ ) where 1 ≤ k ≤ n and 1 ≤ m ≤ q. Now considering the tensor scalar product < ., . > T we have
< F (i,σ) , G (j,τ ) > T = n k=1 q m=1 v ik v jk f m (i,σ) g m (j,τ ) = b (i,σ)(j,τ ) . (1.102)
By Gram's inequality using the < ., . > T scalar product we get
| det(C αβ ) α∈A,β∈B )| ≤ α∈A ||f α || T β∈B ||g β || T (1.103) but ||F (i,σ) || 2 T = n k=1 q m=1 (F km (i,σ) ) 2 = n k=1 q m=1 v 2 ik (f m (i,σ) ) 2 = w ii q m=1 (f m (i,σ) ) 2 = ||f (i,σ) || 2 , (1.104) since w ii = 1 for any i, 1 ≤ i ≤ n.
Let us apply the Gram bound and this last Lemma to bound the determinant in (1.97).
• There is a factor M −dj/2 per line, or M −dj/4 per field ie entry of the loop determinant. This gives a factor M −dj per vertex.
• There is a factor M +dj per vertex spatial integration (minus one)
Hence the λ radius of convergence is uniform in j.
• There is a factor N −1/2 per line, or N −1/4 per field ie entry of the loop determinant. This gives a factor N −1 per vertex
• There is a factor N per vertex (plus an extra one)
This leads to a bound in N.M 2j [cλ] n for the n-th order of the pressure, hence to a radius of convergence at least 1/c. As expected, this is a bound uniform in the slice index j. Hence the λ radius of convergence is uniform in N .
The last item, namely the factor N n+1 for the colors sums is the only one not obvious to prove. Indeed don't know all the graph, but only a spanning tree. We need to organize the sum over the colors from the leaves to the root of the tree. In this way the pay a factor N at each leaf to know the color index which does not go towards the root, then prune the leaf and iterate. The last vertex (the root) is the only special one as it costs two N factors.
Let us remark that to treat the corresponding toy model in the Bosonic case the standard constructive method would be to perform a cluster expansion with respect to a lattice of cubes, then a Mayer expansion which further removed the remaining hardcore constraints with respect to the cubes [9]. Both expansions needed to use the forest formula. This was simplified by the invention of the Loop vertex expansion, in which cubes, cluster and Mayer expansions are no longer needed. In addition the Loop vertex model leads to uniform bounds also for matrix toy models, a result which cannot be obtained up to now with other methods [16].
Interacting Fermions in Two Dimensions
Introduction
One of the main achievements in renormalization theory has been the extension of the renormalization group of Wilson (which analyzes long-range behavior governed by simple scaling around the point singularity p = 0 in momentum space) to long-range behavior governed by extended singularities [51,52,53]. This very natural and general idea is susceptible of many applications in various domains, including condensed matter (reviewed here, in which the extended singularity is the Fermi surface) but also other ones such as diffusion in Minkowski space (in which the extended singularity is the mass shell). In this section we will discuss interacting Fermions models such as those describing the conduction electrons in a metal.
The key features which differentiate electrons in condensed matter from Euclidean field theory, and make the subject in a way mathematically richer, is that space-time rotation invariance is broken, and that particle density is finite. This finite density of particles creates the Fermi sea: particles fill states up to an energy level called the Fermi surface.
The field theory formalism is the best tool to isolate fundamental issues such as the existence of non-perturbative effects. In this formalism the usual Hamiltonian point of view with operators creating electrons or holes is superseded by the more synthetic point of view of anticommuting Fermionic fields with two spin indices and arguments in d + 1 dimensional space-time,. Beware however of the QFT-convention to always call dimension the dimension of space-time, whether from now on we have to stick to the usual condensed matter conven-tion which is to always call dimension the dimension of space only. So one dimensional interacting Fermions correspond at zero temperature to a two dimensional QFT, two dimensional Fermions correspond at zero temperature to a three dimensional QFT and so on.
After the discovery of high temperature superconductivity, a key question emerged. Do interacting Fermions in 2 dimensions (above their low-temperature phase) resemble more three dimensional Fermions, i.e. the Fermi liquid, or one dimensional Fermions, i.e. the Luttinger liquid? The short answer to this controversial question is that it was solved rigorously by mathematical physics and that the answer depends on the shape of the Fermi surface. Interacting Fermions with a round Fermi surface behave more like three dimensional Fermi liquids, whether interacting Fermions with the square Fermi surface of the Hubbard model at half-filling behave more like a one-dimensional Luttinger liquid.
This statement has been now proved in full mathematical rigor, beyond perturbation theory, in the series of works [54,55,56,57,58,59,60].
The existence of usual 2D interacting Fermi liquids was established in [54,55] using the mathematically precise criterion of Salmhofer [48] in the case of a temperature infra-red regulator. Using a magnetic field regulator that breaks parity invariance of the Fermi surface it was also established in the initial sense of a discontinuity at the Fermi surface in the series of papers [49].
The Models:
J 2 , J 3 , H 2 ...
We consider a gas of Fermions in thermal equilibrium at temperature T , with coupling constant λ. The free propagator for this model iŝ
C ab (k) = δ ab 1 ik 0 − e(k) (2.1)
with k being the d-dimensional momentum, e(k) = (k) − µ, (k) being the kinetic energy and µ the chemical potential. The a, b ∈ {↑, ↓} index is for spin hence can take two values (remember spin is treated non-relativistically). At finite temperature, since Fermionic fields have to satisfy antiperiodic boundary conditions, the component k 0 in (2.1) can take only discrete values (called the Matsubara frequencies) : so the integral over k 0 is really a discrete sum over n.
These Matsubara frequencies are:
k 0 = 2n + 1 β π , n ∈ Z (2.2)
where β = (kT ) −1 . For any n we have k 0 = 0, so that the denominator in C(k) can never be 0. This is why the temperature provides a natural infrared cut-off. We can think of k 0 as the Fourier dual to an imaginary Euclidean-time continuous variable taking values in a circle, with length proportional to inverse temperature β. When T → 0 + , (which means β → +∞), k 0 becomes a continuous variable, the corresponding discrete sum becomes an integral, and the corresponding propagator C 0 (x) becomes singular on the Fermi surface defined by k 0 = 0 and e(k) = 0.
This Fermi surface depends on the kinetic energy (k) of the model. For rotation invariant models, (k) = k 2 /2m where m is some effective or "dressed" electron mass. In this case the energy is invariant under spatial rotations and the Fermi surface is simply a circle in two dimensions and a sphere in three dimensions, with radius √ 2mµ. This jellium isotropic propagator is realistic in the limit of weak electron densities. We call this propagator the jellium propagator. We always consider this model, the most natural one in the continuum, together with an ultraviolet cutoff (which it is natural to also take rotation invariant) 8 .
Another model considered extensively is the half-filled 2d Hubbard model, nicknamed H 2 . In this model the position variable x lives on the lattice Z 2 , and (k) = cos k 1 + cos k 2 so that at µ = 0 the Fermi surface is a square of side size √ 2π, joining the points (π, 0), (0, π) in the first Brillouin zone. This propagator is called the Hubbard propagator.
Interaction, Locality
The physical interaction between conduction electrons in a solid could be very complicated; the naive Coulomb interaction is in fact subject to heavy screening, the main effective interaction being due to lattice phonons exchange and other effects. But we are interested in long-range physics, Hence we should use a quasi-local action which decays It is a bit counterintuitive but in fact perfectly reasonable for a mathematical idealization to use in fact a fully local interaction. This should capture all essential mathematical difficulties of the corresponding renormalization group. But there is a unique exactly local such interaction, namely
S V = λ V d d+1 a∈{↑,↓}ψ a (x)ψ a (x) 2 , (2.3)
where V := [−β, β[×V and V is an auxiliary volume cutoff in two dimensional space, that will be sent to infinity in the thermodynamic limit. Indeed any local polynomial of higher degree is zero since Fermionic fields anticommute. Remark it is of the same form than (??), with spin playing the role of color. Hence from the mathematical point of view, in contrast with the propagator, the interesting condensed matter interaction is essentially unique.
The models with jellium propagator and such an interaction (2.3) are respectively nicknamed J 2 and J 3 in dimensions 2 and 3. The model with Hubbard propagator and interaction (2.3) is nicknamed H 2 .
It is possible to interpolate continuously between H 2 and J 2 by varying the filling factor of the Hubbard model. Lattice models with next-nearest neighbor hopping are also interesting, as they are really the ones used to model the high T c superconducting phase in cuprates, but we shall not consider them here for simplicity.
The basic new feature which changes dramatically the power counting of the theory is that the singularity of the jellium propagator is of codimension 2 in the d + 1 dimensional space-time. Instead of changing with dimension, like in ordinary field theory, perturbative power counting is now independent of the dimension, and is the one of a just renormalizable theory. Indeed in a graph with 4 external legs, there are n vertices, 2n − 2 internal lines and L = n − 1 independent loops. Each independent loop momentum gives rise to two transverse variables, for instance k 0 and |k| in the jellium case, and to d − 1 inessential bounded angular variables. Hence the 2L = 2(n − 1) dimensions of integration for the loop momenta exactly balance the 2n − 2 singularities of the internal propagators, as is the case in a just renormalizable theory.
In one spatial dimension, hence two space-time dimensions, the Fermi surface reduces to two points, and there is also no proper BCS theory since there is no continuous symmetry breaking in two dimensions (by the "Mermin-Wagner theorem"). Nevertheless the many Fermion system in one spatial dimension gives rise to an interesting non-trivial behavior, called the Luttinger liquid [43].
The marvel is that although the renormalization group now pinches a non trivial extended singularity, the locality principle still works. The leading part of the four point quasi-local graphs is still local, like the interaction of the initial theory. This is very surprising since quasi-local graphs have internal lines which simply carry excitations farther from the Fermi surface than their external ones. Momenta close to the Fermi surface, when moved to a barycenter of the graph, should react through a non trivial phase factor. But the miracle is that the only divergent part of the main one-loop contribution, the "bubble graph" comes when the combination of the two external legs at each end carries approximately zero total momentum. This is because only then can the two inner bubble propagators both range over the full Fermi sphere. This special configuration, with proper spin and arrows to ensure no oscillations occur, is really the Cooper pair! Then these two external legs with approximately zero total momentum can be moved together like a single low momentum leg in an ordinary Wilsonian renormalization. This is in essence why renormalization still works in condensed matter and beyond the two point function renormalization (Fermi radius renormalization) only changes the value of the coupling constant for the local interaction (2.3). Of course if contributions beyond one loop are taken into account, the story becomes more complicated and the renormalization group flow can in fact involve infinitely many coupling constants [64].
A still more surprising case where locality works in a new form, called Moyality, is the case of non commutative field theory on Moyal space [34]. There again the divergent subgraphs are exactly the only ones that can be renormalized through counterterms of the initial form of the Lagrangian [50]. This lead us to hope that another still generalized form of locality might hold in quantum gravity.
A Brief Review of Rigorous Results
What did the programs of rigorous mathematical study of interacting Fermi systems accomplish until now? Recall that in dimension 1 there is neither superconductivity nor extended Fermi surface, and Fermion systems have been proved to exhibit Luttinger liquid behavior [43]. The initial goal of the studies in two or three dimensions was to understand the low temperature phase of these systems, and in particular to build a rigorous constructive BCS theory of superconductivity. The mechanism for the formation of Cooper pairs and the main technical tool to use (namely the corresponding 1/N expansion, where N is the number of sectors which proliferate near the Fermi surface at low temperatures) have been identified [62]. But the goal of building a completely rigorous BCS theory ab initio remains elusive because of the technicalities involved with the constructive control of continuous symmetry breaking. So the initial goal was replaced by a more modest one, still important in view of the controversies over the nature of two dimensional Fermi liquids [61], namely the rigorous control of what occurs before pair formation.
As is well known, sufficiently high magnetic field or temperature are the two different ways to break the Cooper pairs and prevent superconductivity. Accordingly two approaches were devised for the construction of "Fermi liquids. One is based on the use of non-parity invariant Fermi surfaces to prevent pair formation. These surfaces occur physically when generic magnetic fields are applied to two dimensional Fermi systems. In the large series of papers [49], the construction of two dimensional Fermi liquids for a wide class of non-parity invariant Fermi surfaces has been completed in great detail by Feldman, Knörrer and Trubowitz. These papers establish Fermi liquid behavior in the traditional sense of physics textbooks, namely as a jump of the density of states at the Fermi surface at zero temperature, but they do not apply to the simplest Fermi surfaces, such as circles or squares, which are parity invariant.
The other approach is based on Salmhofer's criterion, in which temperature is the cutoff which prevents pair formation. The corresponding program studies whether given models satisfy Salmhofer's criterion or not. The study of each model has been divided into two main steps of roughly equal difficulty, the control of convergent contributions and the renormalization of the two point functions. In dimension two the corresponding analysis has been completed for J 2 , a Fermi liquid in the sense of Salmhofer, and for H 2 which is not, and is a Luttinger liquid with logarithmic corrections, according to [54,55,56,57,58].
Similar results similar have been also obtained for more general convex curves not necessarily rotation invariant such as those of the Hubbard model at low filling, where the Fermi surface becomes more and more circular, including an improved treatment of the four point functions leading to better constants [59,60]. Therefore as the filling factor of the Hubbard model is moved from halffilling to low filling, we conclude that there must be a crossover from Luttinger liquid behavior to Fermi liquid behavior. This sheds light on the controversy [61] over the Luttinger or Fermi nature of two-dimensional many-Fermion systems above their critical temperature.
Multiscale Analysis, Angular Sectors
For any two-dimensional model built until now in the constructive sense, the strategy is the same. It is based on some kind of multiscale expansion, which keeps a large fraction of the theory in unexpanded determinants. The global bound on these determinant (using determinant inequalities such as Gram inequality) is much better than if the determinant was expanded into Feynman graphs which would then be bounded one by one, and the bounds summed. The bound obtained in this way would simply diverge at large order (i.e. not prove any analyticity at all in the coupling constant) simply because there are too many Feynman graphs at large order. But the divergence of a bound does not mean the divergence of the true quantity if the bound is bad. Constructive analysis, which keeps loops unexpanded is the correct way to obtain better bounds, which do prove that the true series in fact does not diverge, i.e. has a finite convergence radius in the coupling constant. This radius however shrink when the temperature goes to 0, and a good constructive analysis should establish the correct shrinking rate, which is logarithmic. This is where multiscale rather than single scale constructive analysis becomes necessary.
The basic idea of the multiscale analysis is to slice the propagator according to the size of its denominator so that the slice with index j corresponds to
|ik 0 + e(k)| M −j , where M is some fixed constant.
This multiscale analysis is supplemented within each scale by an angular "sector analysis. The number of sectors should be kept as small as possible, so each sector should be as large as possible in the directions tangent to the Fermi surface in three dimensions, or to the Fermi curve in two dimensions. What limits however the size of these sectors is the curvature of the surface, so that stationary phase method could still relate the spatial decay of a propagator within a sector to its dual size in momentum space. In the case of a circle, the number of sectors at distance M −j of the singularity grows therefore at least like M j/2 , hence like a power of T . However for the half-filled Hubbard model, since the curvature is "concentrated at the corners the number of sectors grows only like | log T |. In one dimension there are really only two sectors since the Fermi singularity is made of two points. A logarithm is closer to a constant than to a power; this observation is the main reason for which the half-filled Hubbard model is closer to the one-dimensional Luttinger liquid than to the three dimensional Fermi liquid.
Momentum conservation rules for sectors which meet at a given vertex in general are needed to fix the correct power counting of the subgraphs of the model. In the Hubbard case at half filling, these rules are needed only to fix the correct logarithmic power counting, since the growth of sectors near the singularity is only logarithmic. In both cases the net effect in two dimensions of these conservation rules is to roughly identify two pairs of conserved sectors at any vertex, so that in each slice the model resembles an N -component vector model, where N is the number of sectors in the slice.
The multiscale renormalization group analysis of the model then consists essentially in selecting, for any graph, a tree which is a subtree in each of the quasi-local connected components of the graph accord-ing to the momentum slicing. These connected components are those for which all internal lines are farther from the Fermi surface than all external lines. The selection of this tree can be performed in a constructive manner, keeping the remaining loop fields in a determinant. The combinatoric difficulty related to the fact that a graph contains many trees is tackled by the forest formula.
Once the scale analysis has been performed, a partial expansion of the loop determinant can detect all the dangerous two and four point functions which require renormalization. A key point is that this expansion can be done without destroying the Gram bound, and the corresponding sum is not too big (this means its cardinal remains bounded by K n (where K is a constant)) because in typical graphs there are not many two and four point subgraphs.
One and Two Particle Irreducible Expansions
Salmhofer's criterion is stated for the self-energy, i.e. the sum of all one-particle irreducible graphs for the two point function. Its study requires the correct renormalization of these contributions. Since angular sectors in a graph may vary from one propagator to the next in a graph, and since different sectors have different decays in different directions, we are in a delicate situation. In order to prove that renormalization indeed does the good that it is supposed to do, one cannot simply rely on the connectedness of these self-energy graphs, but one must use their particle irreducibility explicitly. So the proof requires a constructive particle irreducible analysis of the self-energy. The following the-orem summarizes the results of [54,55]:
Theorem 2.1
The radius of convergence of the jellium two-dimensional model perturbative series for any thermodynamic function is at least c/| log T |, where T is the temperature and c some numerical constant. As T and λ jointly tend to 0 in this domain, the self-energy and its first two momentum derivatives remain uniformly bounded so that the model is a Fermi liquid in the sense of Salmhofer.
In the case of the jellium model J 2 , this analysis can be performed at the level of one-particle irreducible graphs [55]. The half-filled Hubbard model, however, is more difficult. Although there is no real divergence of the self-energy (the associated counterterm is zero thanks to the particle hole symmetry of the model at half-filling) one really needs a two-particle and one-vertex irreducible constructive analysis to establish the necessary constructive bounds on the selfenergy and its derivatives [57]. For parity reasons, the self-energy graphs of the model are in fact not only one-particle irreducible but also two particle and one vertex irreducible, so that this analysis is possible.
This analysis leads to the explicit construction of three line-disjoint paths for every self-energy contribution, in a way compatible with constructive bounds. On top of that analysis, another one which is scale-dependent is performed: after reduction of some maximal subsets provided by the scale analysis, two vertexdisjoint paths are selected in every self-energy contribution. This construction allows to improve the power counting for two point subgraphs, exploiting the particle-hole symmetry of the theory at half-filling, and leads to the desired analyticity result.
Finally an upper bound for the self energy second derivative is combined with a lower bound for the explicit leading self energy Feynman graph [58]. This completes the proof that the Hubbard model violates Salmhofer's criterion, hence is not a Fermi liquid, in contrast with the jellium two dimensional model. More precisely the following theorem summarizes the results of [56,57,58] Theorem 2.2 The radius of convergence of the Hubbard model perturbative series at half-filling is at least c/| log T | , where T is the temperature and c some numerical constant. As T and λ jointly tend to 0 in this domain, the self-energy of the model does not display the properties of a Fermi liquid in the sense of Salmhofer, since the second derivative is not uniformly bounded.
We would like now to enter into more technical detail, without drowning the reader. Hence we shall limit ourselves here to the non-perturbative analysis of connected functions in a single RG scale, which is the core mathematical problem. We compare the various models J 2 , J 3 and H 2 to the toy model of section 1.16, and explain why sector analysis plus momentum conservation is suited to analyze the two-dimensional models but fails in three dimensions.
2D Jellium Model: Why Sectors Work
We claim that the J 2 model in a slice is roughly similar to the Toy Model, with dimension d = 3, provided the momentum slice is divided into angular subpieces called sectors, which play the role of colors.
The naive estimate on the slice propagator is (using integration by parts)
|C j (x, y)| ≤ M −j e −[M −j |x−y|] 1/2 (2.4)
(using Gevrey cutoffs f j to get fractional exponential decay). The prefactor M −j corresponds to the volume of integration of the slice, M −2j , divided by the slice estimate of the denominator, M −j . This is much worse than the factor M −3j/2 that would be needed. But the situation improves if we cut the Fermi slice into smaller pieces (called sectors). Suppose we divide the j-th slice into M j sectors, each of size roughly M −j in all three directions.
A sector propagator C j,a has now prefactor M −2j corresponding to the volume of integration of the sector M −3j divided by the slice estimate of the denominator, M −j . Using integration by parts and Gevrey cutoffs f ja for fractional power decay we get without too much effort the bound
|C j,ab (x, y)| ≤ δ ab M −2j e −[M −j |x−y|] 1/2 . (2.5) But since N = M j M −2j = M −3j/2 √ N (2.6)
so that this bound is identical to that of the toy model. It remains just to explain why the interaction of the model is approximatley of the vector type. This is because of the momentum conservation rule at every vertex.
In two dimensions a rhombus (i.e; a closed quadrilateral whose four sides have equal lengths) is a parallelogram. Hence an approximate rhombus should be an approximate parallelogram.
Momentum conservation δ(p 1 + p 2 + p 3 + p 4 ) at each vertex follows from translation invariance of J 2 . Hence p 1 , p 2 , p 3 , p 4 form a quadrilateral. For j large we have |p k | √ 2M µ, hence the quadrilateral is an approximate rhombus. Hence the four sectors to which p 1 , p 2 , p 3 and p 4 should be roughly equal two by two (parallelogram condition).
It means that the interaction is roughly of the color (or Gross-Neveu) type with respect to these angular sectors:
aψ a ψ a bψ b ψ b (2.7)
In fact this "rhombus rule" is not fully correct for almost degenerate rhombuses. The correct statement is
Lemma 2.3 Fix m ∈ Z 3 . The number of 4-tuples {S 1 , · · · S 4 } of sectors for which there exist k i ∈ R 2 , i = 1, · · · , 4 satisfying k i ∈ S i , |k i − k i | ≤ const M −j , i = 1, · · · , 4
(2.8)
and
|k 1 + · · · + k 4 | ≤ const (1 + |m|) M −j (2.9)
is bounded by const(1 + |m|) 2 M 2j {1 + j} .
(2.10)
The 1 + j factor is special to dimension 2 and is the source of painful technical complications which were developed by Feldman, Magnen, Trubowitz and myself.
The solution uses in fact M j/2 anisotropic angular sectors, which are longer in the tangential direction (of length M −j/2 ). The corresponding propagators still have dual spatial decay because the sectors are still aprroximately flat.
Ultimately the conclusion is unchanged: the radius of convergence of J 2 in a slice is independent of the slice index j.
Why Sectors Fail in d = 3
In 3D sectors and Gram's bound fail by a full power per vertex! There is indeed no rhombus rule in d = 3. A closed quadrilateral with equal sides is not a parallelogram because it can be non-planar; hence it is obtained by rotating half of a planar parallelogram around the diagonal by an arbitrary twisting angle. Therefore the jellium model interaction is not of the vector type. More precisely the analog of Lemma 2.3 is, with similar notations Lemma 2.4 The number of 4-tuples {S 1 , · · · S 4 } of sectors for which there exist k i ∈ R 3 , i = 1, · · · , 4 satisfying
k i ∈ S i , |k i − k i | ≤ const M −j , i = 1, · · · , 4
(2.11)
and
|k 1 + · · · + k 4 | ≤ const (1 + |m|) M −j (2.12)
is bounded by
const(1 + |m|) 3 M 5j . (2.13)
Remark the absence of the log factor that was present in d = 2. But for d = 3 we find M 5j 4-tuples which correspond to the choice of two sectors (M 2j × M 2j ) and of one angular twist M j . The power counting corresponds to M −3j per sector propagator. Two propagators pay for one vertex integration (M 4j ) and one sector choice (M 2j ) but there is nothing to pay for the angular twist. Going to anisotropic sectors is possible but there remains still in this case a M j /2 twist factor. After many years of effort, we concluded that the sector method and Gram bound apparently cannot be improved to do better. Hence although J 3 is expected to be a Fermi liquid in the sense of Salmhofer, new methods have to be developed to treat it constructively [66,67,68].
The Hadamard method in x-space
The idea of using Hadamard's inequality in x-space to overcome the constructive power counting problem of Fermions in three dimensions took Jacques Magnen and myself four years of continuous hard work, with dozens and dozens of various failed trials, from 1991 to 1995 [66]. Another four or five years took place to fine-tune this idea for the multidimensional case with M. Disertori [67]. Another ten years have passed to find a momentum-conserving version, which should at last allow for the proof of Salmhofer's criterion in the three dimensional jellium model [68]. Let us describe the main idea in an informal way. We remember that the naive estimate (2.4) is far from sufficient for correct power counting. But we also know from perturbative power counting in momentum space that the theory should be just renormalizable. Therefore the j-slice propagator C j should behave more as the one of an infrared φ 4 4 theory, hence should be bounded by
KM −2j e −[M −j |x−y|] 1/2 . (2.14)
In fact this is not totally correct, because it can be shown that the j-th slice propagator at almost coinciding points, hence at |x − y| 0 is not bounded by M −2j . Still (2.14) is correct at typical distances for the J 3 propagator in the j-th slice. We know from eg (2.4) that these typical space-time distances should be |x − y| M j .
Indeed integrating over angles on the Fermi sphere leads to an additional 1/|x − y| decay, because π 0 sin θdθdφe i cos θ|x−y| = sin |x − y|/|x − y| (2.15) Hence for typical distances |x − y| M j the propagator indeed obeys the improved estimate
|C j (x, y) |x−y| M j | ≤ KM −2j e −[M −j |x−y|] 1/2 . (2.16)
The problem is that this bound is wrong at small distances. However at small distances there is a bonus, namely the corresponding integration volumes are smaller. Another related problem is that if we use the Gram bound (1.88) to bound a determinant with all the matrix elements C j (x p − y q ) corresponding to large distances |x p − y q |, we still loose the improvement (2.16), because the L 2 norms in (1.88) will correspond again to propagators at coinciding points!
The solution is to use the Hadamard bounds (1.89)-(1.90) because they conserve the decay of the typical propagators.
Remember however, as seen conveniently in (1.91) that Hadamard bounds consume the 1/n! symmetry factor for n vertices. Hence we can no longer use the explicit tree formula and the method of [46,47].
But we can still use the standard old-fashioned cluster expansion between cubes. Summarizing:
• Just renormalizable power counting is recovered for the main part of the theory if we use Hadamard's bound rather than Gram's bound.
• The factor n! is lost in the Hadamard bound at order n; this forces us to rely on the on-canonical tool of cluster expansion between cubes Still, this solves the constructive problem only for the main part of the propagator, the one at typical distances |x − y| M j . However at smaller distances there is a bonus, namely the volume factors for spatial integration are also smaller.
It turns out that the problem of smaller than typical distances can be solved with an auxiliary superrenormalizable decomposition
C j = j k=0 C jk (2.17)
of the propagator. Roughly speaking C j,k corresponds to |x−y| M k , It means that even the single slice theory in d ≥ 3 is a non-trivial theory that contains a rather non-trivial renormalization, like the one of φ 4 3 . This renormalization can be analyzed by means of the auxiliary scales. The solution is in fact more complicated that what we sketch here, and has to take into account the anisotropy between the space and the imaginary time variables [66,67].
The use of non-canonical lattices of cubes in this method is the signal that we have probably still not found the optimal constructive treatment of J 3 . Recently we found a better decomposition that should allow to check Salmhofer's criterion for J3 [68]. However in its present stage it will still use a non-canonical cluster expansion between cubes. It would be interesting to find a solution such as the loop vertex expansion that solves the constructive problem in a truly canonical way.
2D Hubbard Model
The Hubbard model lives on the square lattice Z Z 2 , so that the three dimensional vector x = (x 0 , x) is such that x = (n 1 , n 2 ) ∈ Z Z 2 . From now on we write v 1 and v 2 for the two components of a vector v along the two axis of the lattice.
At half-filling and finite temperature T , the Fourier transform of the propagator of the Hubbard model is:
C ab (k) = δ ab 1 ik 0 − e(k)
, e(k) = cos k 1 + cos k 2 ,
(2.18)
where a, b ∈ {↑, ↓} are the spin indices. The vector k lives on the two-dimensional torus IR 2 /(2πZ Z) 2 . Hence the real space propagator is
C ab (x) = 1 (2π) 2 β k0 π −π dk 1 π −π dk 2 e ikxĈ ab (k) . (2.19)
Recall that |k 0 | ≥ π/β = 0 hence the denominator in C(k) again can never be 0 at non zero temperature. This is why the temperature provides a natural infrared cut-off. When T → 0 (which means β → ∞) k 0 becomes a continuous variable, the discrete sum becomes an integral, and the corresponding propagator C 0 (x) becomes singular on the Fermi surface defined by k 0 = 0 and e(k) = 0. This Fermi surface is a square of side size √ 2π (in the first Brillouin zone) joining the corners (±π, 0), (0, ±π). We call this square the Fermi square, its corners and faces are called the Fermi faces and corners. Considering the periodic boundary conditions, there are really four Fermi faces, but only two Fermi corners.
In the following to simplify notations we will write:
d 3 k ≡ 1 β k0 d 2 k , d 3 x ≡ 1 2 β −β dx 0 x∈Z Z 2 .
(2.20)
In determining the spatial decay we recall that by anti-periodicity
C(x) = f (x 0 , x) := m∈Z Z (−1) m C 0 x 0 + m T , x . (2.21)
where C 0 is the propagator at T = 0. Indeed the function f is anti-periodic and its Fourier transform is the right one.
The interaction of the Hubbard model is again (2.3):
S V = λ V d 3 x ( aψ ψ) 2 (x) ,(2.22)
where V := [−β, β] × V and V is an auxiliary finite volume cutoff in two dimensional space that will be sent later to infinity.
Scale Analysis
The theory has a natural lattice spatial cutoff. To implement the renormalization group analysis, we introduce as usually a compact support function u(r) ∈ C ∞ 0 (R) (it is convenient to choose it to be Gevrey of order α < 1 so as to ensure fractional exponential decrease in the dual space) which satisfies: u(r) = 0 for |r| > 2 ; u(r) = 1 for |r| < 1 .
(2.23)
With this function, given a constant M ≥ 2, we can construct a partition of unity
1 = ∞ i=0 u i (r) ∀r = 0 ; u 0 (r) = 1 − u(r) ; u i (r) = u(M 2(i−1) r) − u(M 2i r) for i ≥ 1 . (2.24)
The propagator is then divided into slices according to this partition
C(k) = ∞ i=0 C i (k) (2.25) where C i (k) = C(k)u i [k 2 0 + e 2 (k)] .
(2.26) (indeed k 2 0 + e 2 (k) ≥ T 2 > 0). In a slice of index i the cutoffs ensure that the size of k 2 0 + e 2 (k) is roughly M −2i . More precisely in the slice i we must have
M −2i ≤ k 2 0 + e 2 (k) ≤ 2M 2 M −2i . (2.27)
The corresponding domain is a three dimensional volume whose section through the k 0 = 0 plane is the shaded region pictured in Figure 6.
Remark that at finite temperature, the propagator C i vanishes for i ≥ i max (T ) where M imax(T ) 1/T (more precisely i max (T ) = E(log M √ 2
πT / log M ), where E is the integer part), so there is only a finite number of steps in the renormalization group analysis.
Let us state first a simple result, for a theory whose propagator is only C i , hence corresponding to a generic step of the renormalization group:
R i ≥ c/i . (2.28)
The rest of this section is devoted to the definitions and properties of H 2 sectors, their scaled decay and momentum conservation rules. As discussed already this result is a first step towards the rigorous proof [57,58] that H 2 is not a Fermi liquid in the sense of Salmhofer.
Sectors
This section is extracted from [56].
The "angular" analysis is completely different from the jellium case. We remark first that in a slice, k 2 0 + e 2 (k) is of order M −2i , but this does not fix the size of e 2 (k) itself, which can be of order M −2j for some j ≥ i. In order for sectors defined in momentum space to correspond to propagators with dual decay in direct space, it is essential that their length in the tangential direction is not too big, otherwise the curvature is too strong for stationary phase methods to apply. This was discussed first in [62]. This leads us to study the curve (cos k 1 + cos k 2 ) 2 = M −2j for arbitrary j ≥ i. We can by symmetry restrict ourselves to the region 0 ≤ k 1 ≤ π/2, k 2 > 0. It is then easy to compute the curvature radius of that curve, which is R = (sin 2 k 1 + sin 2 k 2 ) 3/2 | cos k 1 sin 2 k 2 + cos k 2 sin 2 k 1 | .
(2.29)
We can also compute the distance d(k 1 ) to the critical curve cos k 1 + cos k 2 = 0, and the width w(k 1 ) of the band M −j ≤ | cos k 1 + cos k 2 | ≤ √ 2M.M −j . We can then easily check that
d(k 1 ) w(k 1 ) M −j M −j/2 + k 1 , (2.30) R(k 1 ) k 3 1 + M −3j/2 M −j ,(2.31)
where f g means that on the range 0 ≤ k 1 ≤ π/2 we have inequalities cf ≤ g ≤ df for some constants c and d.
Defining the anisotropic length
l(k 1 ) = w(k 1 )R(k 1 ) M −j/2 + k 1 ,(2.32)
the condition in [62] is that the sector length should not be bigger than that anisotropic length. This leads to the idea that k 1 or an equivalent quantity should be sliced according to a geometric progression from 1 to M −j/2 to form the angular sectors in this model. For symmetry reasons it is convenient to introduce a new orthogonal but not normal basis in momentum space (e + , e − ), defined by e + = (1/2)(π, π) and e − = (1/2)(−π, π). Indeed if we call (k + , k − ) the coordinates of a momentum k in this basis, the Fermi surface is given by the simple equations k + = ±1 or k − = ±1. This immediately follows from the identity cos k 1 + cos k 2 = 2 cos(πk + /2) cos(πk − /2) .
(2.33) (Note however that the periodic b.c. are more complicated in that new basis). Instead of slicing e(k) and k 1 , it is then more symmetric to slice directly cos(πk + /2) and cos(πk − /2). We remark that using (2.27) in order for C i,σ not to be 0, we need to have s + + s − ≥ i − 2. We define the "depth" l(σ) of a sector to be l = s + + s − − i + 2.
To get a better intuitive picture of the sectors, we remark that they can be classified into different categories:
• the sectors (0,i) and (i,0) are called the middle-face sectors
• the sectors (s,i) and (i,s) with 0 < s < i are called the face sectors • the sector (i,i) is called the corner sector
• the sectors (s,s) with (i − 2)/2 ≤ s < i are called the diagonal sectors
• the others are the general sectors Finally the general or diagonal sectors of depth 0 for which s + + s − = i − 2 are called border sectors.
If we consider the projection onto the (k + , k − ) plane, taking into account the periodic b.c. of the Brillouin zone, the general and diagonal sectors have 8 connected components, the face sectors have 4 connected components, the middle face sectors and the corner sector have 2 connected components. In the three dimensional space-time, if we neglect the discretization of the Matsubara frequencies, these numbers would double except for the border sectors. Proof This is essentially Fourier analysis and integration by parts. If x = (n 1 , n 2 ) ∈ Z Z 2 , we define (x + , x − ) = (π/2)(n 1 +n 2 , n 2 −n 1 ). The vector (x + , x − ) then belongs to (π/2)Z Z 2 but with the additional condition that x + and x − have the same parity. Defining, for X ∈ [(π/2)Z Z ] 2 D i,σ (X) = (1/2) 1 8β we note that C i,σ (X) = D i,σ (X) for X satisfying the parity condition.
Scaled decay
(Remember the Jacobian π 2 2 from dk 1 dk 2 to dk + dk − , and the initial domain of integration that is doubled.)
The volume of integration trivially gives a factor M −i for the k 0 sum and factors M −s+ and M −s− for the k + and k − integration (see (2.44) below). The integrand is trivially bounded by M i on the integration domain, and this explains the prefactor cM −i−l in (2.37).
We then apply standard integration by parts techniques to formulate the decay. From e.g. Lemma 10 in [54] we know that to obtain the scaled decay of Lemma 2.5 we have only to check the usual derivative bounds in Fourier space: where n = n 0 + n + + n − , and the derivative ∂ ∂k0 really means the natural finite difference operator (1/2πT )(f (k 0 + 2πT ) − f (k 0 )) acting on the discrete Matsubara frequencies. The norm is the ordinary sup norm.
But from (2.39),
D i,σ (k) = 1 16β u i [k 2
0 + 4 cos 2 (πk + /2) cos 2 (πk − /2)] ik 0 − 2 cos(πk + /2) cos(πk − /2) v s+ [cos 2 (πk + /2)] v s− [cos 2 (πk − /2)] (2.41) and the derivatives are bounded easily using the standard rules for derivation, product and composition of Gevrey functions, or by hand, using the support properties of the v s+ and v s− fonctions. For instance a derivative ∂ ∂k+ can act on the v s+ [cos 2 (πk + /2)] factor, in which case it is easily directly bounded by cM s+ for some constant c. When it acts on u i [k 2 0 + 4 cos 2 (πk + /2) cos 2 (πk − /2)] it is easily bounded by c.M 2i−s+−2s− hence by c.M s+ , using the relation s + + s − ≥ i − 2. When it acts on the denominator [ik 0 − 2 cos(πk + /2) cos(πk − /2)] −1 . it is bounded by c.M i−s− , hence again by c.M s+ , using the relation s + + s − ≥ i − 2. Finally when it acts on a cos(πk + /2) created by previous derivations, it costs directly c.M s+ . The factorial factor (n!) 1/α in (2.40) comes naturally from deriving the cutoffs, which are Gevrey functions of order α; deriving other factors give smaller factorials (with power 1 instead of 1/α).
Finally a last remark: to obtain the Lemma for the last slice, i = i max (T ), one has to take into account the fact that x 0 lies in a compact circle, so that there is really no long-distance decay to prove.
Support Properties
If C i,σ (k) = 0, the momentum k must obey the following bounds:
|k 0 | ≤ √ 2M M −i (2.42)
M −1 ≤ | cos(πk ± /2)| ≤ 1 for s ± = 0 , M −s±−1 ≤ | cos(πk ± /2)| ≤ √ 2M −s± for 1 ≤ s ± ≤ i − 1 , | cos(πk ± /2)| ≤ √ 2M −i for s ± = i .
(2.43)
In the support of our slice in the first Brillouin zone we have |k + | < 2 and |k − | < 2 (this is not essential but the inequalities are strict because i ≥ 1). It is convenient to associate to any such component k ± a kind of "fractional part" called q ± defined by q ± = k ± − 1 if k ± ≥ 0 and q ± = k ± + 1 if k ± < 0, so that 0 ≤ |q ± | ≤ 1. Then the bounds translate into
2/πM ≤ |q ± | ≤ 1 for s ± = 0 , 2M −s± /πM ≤ |q ± | ≤ √ 2M −s± for 1 ≤ s ± ≤ i − 1 , |q ± | ≤ √ 2M −i for s ± = i .
(2.44)
Momentum conservation rules at a vertex
Let us consider that the four momenta k 1 , k 2 , k 3 , k 4 , arriving at a given vertex v belong to the support of the four sectors σ 1 , σ 2 , σ 3 , σ 4 , in slices i 1 , i 2 , i 3 , i 4 . In Fourier space the vertex (2.22) implies constraints on the momenta. Each spatial component of the sum of the four momenta must be an integer multiple of 2π in the initial basis, and the sum of the four Matsubara frequencies must also be zero. In our tilted basis (e + , e − ), this translates into the conditions: k 1,0 + k 2,0 + k 3,0 + k 4,0 = 0 , (2.45) k 1,+ + k 2,+ + k 3,+ + k 4,+ = 2n + , (2.46)
k 1,− + k 2,− + k 3,− + k 4,− = 2n − ,(2.47)
where n + and n − must have identical parity. We want to rewrite the two last equations in terms of the fractional parts q 1 , q 2 , q 3 and q 4 .
Since an even sum of integers which are ±1 is even, we find that (2.46) and (2.47) imply q 1,+ + q 2,+ + q 3,+ + q 4,+ = 2m + , (2.48) q 1,− + q 2,− + q 3,− + q 4,− = 2m − , (2.49) with m + and m − integers. Let us prove now that except in very special cases, these integers must be 0. Since |q j,± | ≤ 1, |m ± | ≤ 2. But |q j,± | = 1 is possible only for s j,± = 0. Therefore |m ± | = 2 implies s j,± = 0 ∀j. Now suppose e.g. |m + | = 1. Then s j,+ is 0 for at least two values of j. Indeed for s j,± = 0 we have |q j,± | ≤ √ 2M −1 , and assuming 3 √ 2M −1 < 1, equation (2.48) could not hold.
We have therefore proved Lemma 2.6 m + = 0 unless s j,+ is 0 for at least two values of j, and m − = 0 unless s j,− is 0 for at least two values of j.
Let us analyze in more detail equations (2.48) and (2.49) for |m + | = |m − | = 0. Consider e.g. (2.48). By a relabeling we can assume without loss of generality that s 1,+ ≤ s 2,+ ≤ s 3,+ ≤ s 4,+ Then either s 1,+ = i 1 or s 1,+ < i 1 if M > 3π/ √ 2, which we assume from now on. The conclusion is: Lemma 2.7 If m ± = 0, either the smallest index s 1,± coincides with its scale i 1 , or the two smallest indices among s j,± differ by at most one unit. Now we can summarize the content of both Lemmas in a slightly weaker but simpler lemma:
Lemma 2.8 A) (single slice case)
The two smallest indices among s j,+ for j = 1, 2, 3, 4 differ by at most one unit, and the two smallest indices among s j,− for j = 1, 2, 3, 4 differ by at most one unit.
B) Multislice case
The two smallest indices among s j,+ for j = 1, 2, 3, 4 differ by at most one unit or the smallest one, say s 1,+ must coincide with its scale i 1 , which must then be strictly smaller than the three other scales i 2 , i 3 and i 4 . Exactly the same statement holds independently for the minus direction.
This lemma allows to check that again the single scale analysis works and leads to a radius of convergence independent of the slice [56].
Multiscale Analysis
The half-filling point is very convenient since the particle-hole exact symmetry at this point ensures that there is no flow for the Fermi surface itself.
Using the sector decomposition and the momentum conservation we find that power counting of the 2D Hubbard model is essentially similar to one dimensional case with logarithmic corrections [57]. Expanding the two point function to second order we can even find lower bounds which prove that this model is not a Fermi liquid in the sense of Salmhofer [58].
Acknowledgments
every point of R d .The Gaussian part of the measure isdµ(φ) = 1 Z 0 e −(m 2 /2) φ 2 −(a/2) (∂µφ∂ µ φ) Dφ.(1.17)
Figure 1 :
1A possible contraction scheme with n = N = 4.
•
Renormalizability (in the ultraviolet regime) holds if the structure of the Lagrangian resists under change of scale, although the values of the coefficients or coupling constants may change. For φ 4 it occurs if d ≤ 4, with d = 4 the most interesting case.
Figure 5 :
5The φ 4 connected graphs with n = 2, N = 4.
( 1 .
164) shows why det A is a perfect square and proves(1.61).A useful Lemma is:Lemma 1.2 The determinant of a matrix D + A where D is diagonal and A antisymmetric has a "quasi-Pfaffian" representation det(D + A) = i dχ i dω i e − i χiDiiωi− i<j χiAij χj + i<j ωiAij ωj . (1.65) Proof The proof consists in performing the change of variables (1.63) and canceling carefully the i factors.
are characterized by their two point function or covarianceψ i ψ j dµ M = M ij .(1.67)plus the Grassmann-Wick rule that n-point functions are expressed as sum over Wick contractions with signs.
IR nd dx 1 ...dx n δ(x 1 = 0) l∈T C(x i(l) , x j(l) )δ l × det remaining (C αβ ) α∈A,β∈B
Figure 6 :
6A single slice of the renormalization group Theorem 1 The Schwinger functions of the theory with propagator C i and interaction (2.22) are analytic in λ in a disk of radius R i which is at least c/i for a suitable constant c:
(r) = 1 − u(M 2 r) v s = u s+1 for1 ≤ s ≤ i − 1 v i (r) = u(M 2i r) i,σ (k) = C i (k)v s+ [cos 2 (πk + /2)] v s− [cos 2 πk − /2)] .(2.36)
Lemma 2. 5
5Using Gevrey cutoffs of degree α < 1, the propagator C i,σ obeys the scaled decay|C i,σ | ≤ c.M −i−l e −[di,σ(x,y)] α (2.37) where d i,σ (x, y) = {M −i |x 0 − y 0 | + M −s+ |x + − y + | + M −s− |x − − y − |} .(2.38)
2 (πk + /2) cos 2 (πk − /2)] ik 0 − 2 cos(πk + /2) cos(πk − /2) v s+ [cos 2 (πk + /2)] v s− [cos 2 (πk − /2)](2.39)
σ ≤ A.B n M in0 M s+n+ M s−n− (n!) 1/α (2.40)
I thank J. Magnen, M. Disertori, M. Smerlak and L. Gouba for contributing various aspects of this work.
, in which case combining equations (2.48) and (2.44) we must have:3
√
2M −s2,+ ≥ 2M −s1,+ /πM ,
(2.50)
which means
s 2,+ ≤ s 1,+ + 1 +
log(3π/
√
2)
log M
.
(2.51)
This implies
|s 2,+ − s 1,+ | ≤ 1
(2.52)
However the functional space that supports this measure is not in general a space of smooth functions, but rather of distributions. This was already true for functional integrals such as those of Brownian motion, which are supported by continuous but not differentiable paths. Therefore "functional integrals" in quantum field theory should more appropriately be called "distributional integrals".
Strictly speaking this is true only for semi-regular graphs, i.e. graphs without tadpoles, i.e. without lines which start and end at the same vertex, see[21].
Because the graphs with N = 2 are quadratically divergent we must Taylor expand the quasi local fat dots until we get convergent effects. Using parity and rotational symmetry, this generates only a logarithmically divergent (∇φ).(∇φ) term beyond the quadratically divergent φ 2 . Furthermore this term starts only at n = 2 or two loops, because the first tadpole graph at N = 2, n = 1 is exactly local.
And also over vertex joints of graphs, just as in the universality theorem for the Tutte polynomial6 It is enough in fact to compute such weights for 1-particle irreducible and 1-vertexirreducible graphs, then multiply them in the appropriate way for the general case.
This is usually easily done by taking some kind of "square roots" in momentum space.
The question of whether and how to remove that ultraviolet cutoff has been discussed extensively in the literature, but we consider it as unphysical for a non-relativistic model of condensed matter, which is certainly an effective theory at best.
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| We consider a quantum spin Hall system in a two-terminal setup, with an extended tunneling contact connecting upper and lower edges. We analyze the effects of this geometry on the backscattering current as a function of voltage, temperature, and strength of the electron interactions. We find that this configuration may be useful to confirm the helical nature of the edge states and to extract their propagation velocity. By comparing with the usual quantum point contact geometry, we observe that the power-law behaviors predicted for the backscattering current and the linear conductance are recovered for low enough energies, while different power-laws also emerge at higher energies. | 10.1103/physrevb.85.195138 | [
"https://arxiv.org/pdf/1203.4486v2.pdf"
]
| 55,861,174 | 1203.4486 | 4f8de1332aef55385265d160f0ef85c06fdcad17 |
Tunneling between helical edge states through extended contacts
7 Jun 2012
G Dolcetto
Dipartimento di Fisica
Università di Genova
Via Dodecaneso 3316146GenovaItaly
CNR-SPIN
Via Dodecaneso 3316146GenovaItaly
INFN
Via Dodecaneso 3316146GenovaItaly
S Barbarino
Dipartimento di Fisica
Università di Genova
Via Dodecaneso 3316146GenovaItaly
D Ferraro
Dipartimento di Fisica
Università di Genova
Via Dodecaneso 3316146GenovaItaly
CNR-SPIN
Via Dodecaneso 3316146GenovaItaly
INFN
Via Dodecaneso 3316146GenovaItaly
N Magnoli
Dipartimento di Fisica
Università di Genova
Via Dodecaneso 3316146GenovaItaly
INFN
Via Dodecaneso 3316146GenovaItaly
M Sassetti
Dipartimento di Fisica
Università di Genova
Via Dodecaneso 3316146GenovaItaly
CNR-SPIN
Via Dodecaneso 3316146GenovaItaly
Tunneling between helical edge states through extended contacts
7 Jun 2012(Dated: May 5, 2014)APS/123-QEDnumbers: 7323-b7110Pm7343-f
We consider a quantum spin Hall system in a two-terminal setup, with an extended tunneling contact connecting upper and lower edges. We analyze the effects of this geometry on the backscattering current as a function of voltage, temperature, and strength of the electron interactions. We find that this configuration may be useful to confirm the helical nature of the edge states and to extract their propagation velocity. By comparing with the usual quantum point contact geometry, we observe that the power-law behaviors predicted for the backscattering current and the linear conductance are recovered for low enough energies, while different power-laws also emerge at higher energies.
I. INTRODUCTION
Since the discovery of the Quantum Hall Effect (QHE) 1 , the condensed matter community devoted great efforts in finding other topological states of matter in which fundamental physical properties are insensitive to smooth changes in material parameters and can be modified only through quantum phase transitions. In recent years, a new class of these peculiar systems have been experimentally observed: the topological insulators 2,3 . Their main characteristics are the presence of a gap in the bulk, analogous to the one of the ordinary band insulators, and gapless edge states protected by timereversal symmetry. In two spatial dimensions they are the first realization of the Quantum Spin Hall Effect (QSHE), theoretically predicted in graphene with spinorbit interaction 4,5 , in strained semiconductors 6 and in Mercury-Telluride quantum wells 7,8 . The edge states of the QSHE are helical 9 , namely their electrons have spin direction and momentum locked each other. In presence of intra-edge interactions they can be described in terms of a helical Luttinger liquid 9 . The experimental measurement of non-local transport in multi-terminal setups, according with the prediction of the Landauer-Buttiker theory 10 , represented an important test of the existence of helical edge states 11 . The fast technical developments in this field will make shortly possible to realize interesting experimental geometries, like the Quantum Point Contact (QPC) [12][13][14] that already revealed extremely useful to extract information on the edge properties in the fractional QHE [15][16][17][18][19][20] . Various theoretical proposals have investigated this geometry focusing on both the two-terminal 21 and four-terminal [22][23][24] setups.
Possible interference experiments 25,26 , as well as quantum pumps 27 , involving two point contacts have also been considered. The possibility offered by the Mercury-Telluride quantum wells to realize a QPC by means of electrostatic gates or, more realistically, by etching the sample in the desired shape makes possible to have a great control on the geometry and allows to study the evolution of the transport properties as a function of the constriction geometrical parameters. An analysis of the effects of extended contacts 28-31 on the transport properties have been already addressed for the QHE showing deviations from the standard power-law behavior of the current as a function of the voltage at zero temperature. Finite temperature effects were also considered for composite fractional QH systems 29 , demonstrating that extended contacts may provide information about the neutral mode propagation velocity along the edge, provided that it is very small with respect to the one of the charged mode. In this paper we propose to investigate the extended contact geometry for the helical edge states of the QSHE by properly taking into account the role played by interactions. We will evaluate the backscattering current as a function of voltage and temperature. We will demonstrate that all the deviations with respect to the pointlike case can be included in a modulating function. We will demonstrate that, at low enough temperatures, a peak appears in the differential conductance, which provides evidence of the helical nature of the edge states and gives information about the propagation velocity of the edge modes. At low energies the backscattering current and the linear conductance are described by the same power-law behaviors predicted for the QPC geometry. Even more interestingly, power-laws are recovered also at higher energies, but with different exponents. The paper is divided as follows. In Sec. II we recall the main results of the helical Luttinger liquid description of edge states of a QSH system. In Sec. III we analyze the extended contact geometry introducing the modulating function both in the non-interacting and in the interacting case. Sec. IV contains the main results on transport properties. Sec. V is devoted to the conclusions.
II. MODEL
We consider a QSH insulator with one Kramers doublet of helical edge states in the two terminal configuration (see Fig. 1). On the upper edge (1) one has right-moving spin up and left-moving spin down electrons, on the lower edge (2) the opposite. The corresponding free Hamiltonians are 21,22 ( = 1)
H 1(2) = −iv F dx ψ † R,↑(↓) ∂ x ψ R,↑(↓) − ψ † L,↓(↑) ∂ x ψ L,↓(↑)
(1) where ψ R,↑ (ψ L,↑ ) annihilates right (left)-moving electron with spin up, and analogous for the spin down, and v F is the Fermi velocity, estimated 8,32 about 5 · 10 5 m/s. For sake of simplicity we assume infinite edges, even if a more realistic description based on finite length edges coupled to non-interacting leads can also be considered 23,24 . This so called g(x) model [33][34][35] reveals crucial in order to recover the proper quantization of the conductance of one dimensional channels and leads to finite length corrections to physical quantities, that however are not crucial in the considered setup 23 . Concerning interactions, we consider terms which preserve time-reversal symmetry near the Fermi surface for a single Kramers doublet of helical edge states 36 . They are a subset of all possible contributions analyzed by the so called g-hology 37,38 represented by the dispersive
H d = g 2⊥ dx ψ † R,↑ ψ R,↑ ψ † L,↓ ψ L,↓ + ψ † L,↑ ψ L,↑ ψ † R,↓ ψ R,↓(2)
and the forward scattering
H f = g 4 2 α=R,L;σ=↑,↓ dxψ † α,σ ψ α,σ ψ † α,σ ψ α,σ .(3)
Note that possible Umklapp terms, which are important only at certain commensurate fillings 9 , are here neglected.
The bosonized procedure of the Luttinger liquid allows to write the electronic field operator in the form 37
ψ R/L,σ (x) = F R/L,σ √ 2πa e ±ikF x e −i √ 2πϕ R/L,σ (x) ,(4)
with ϕ R/L,σ (x) a bosonic field (σ =↑, ↓), F R/L,σ the Klein factor, necessary to give the proper commutation relation between electrons belonging to different edges, a a finite length cut-off and k F the Fermi momentum. The bosonic field ϕ R/L,σ (x) is related to the electron density
through ρ R/L,σ (x) = ∓ 1 √ 2π ∂ x ϕ R/L,σ (x).
According to the standard bosonization procedure 37,38 the interaction terms in Eqs.
ϕ 1(2) (x) = 1 √ 2 ϕ L,↑(↓) (x) − ϕ R,↓(↑) (x) ,(5)
with their canonical conjugates
θ 1(2) (x) = 1 √ 2 ϕ L,↑(↓) (x) + ϕ R,↓(↑) (x) ,(6)H = v 2 i=1,2 dx 1 K (∂ x ϕ i ) 2 + K (∂ x θ i ) 2 . (7)
Here, K = 2πvF +g 4 −g 2⊥
2πvF +g 4 +g 2⊥ is the interaction parameter
and v = v F 1 + g 4 2πvF 2 − g 2⊥ 2πvF 2
the renormalized velocity. For Coulomb repulsion g 4 = g 2⊥ and therefore v = v F /K. In the following we will assume this condition, despite other possible interactions can be straightforwardly taken into account.
III. EXTENDED CONTACT
In presence of an external voltage V , right (left) -moving electrons feel a chemical potential µ L (µ R ), with µ L − µ R = eV . Spatial separation prevents electron tunneling between edges leading to conductance quantization 8 G = e 2 π . In order to study tunneling effects the system is pinched by means of a gate voltage 21 or, more realistically, by etching the sample 39 creating a tunneling region 13 . Previous theoretical works have studied this configuration 12,[21][22][23]25 , both in two-terminal and in four-terminal setups, assuming a point-like tunneling. In what follows, we will generalize this assumption, taking into account the possibility of tunneling events occurring in an extended region (see Fig. 1). Our aim is to investigate the effects induced by a long contact on the backscattering current. The backscattering Hamiltonian connecting the two helical edge states is represented by (8) with Λ x,y the tunneling amplitude in which a left-moving electron is destroyed in y on one edge and recreated as a right-moving electron in x on the other one. A reasonable choice for Λ x,y is to assume a separable form 30
H B = dx dy σ=↑,↓ Λ x,y ψ † R,σ (x)ψ L,σ (y) + h.c.,Λ x,y = Λ 0 f l (|x + y|) f c (|x − y|) .(9)
The function f l , indicated as lateral contribution, specifies the average location of the tunneling events [28][29][30] while f c , dubbed crossed, allows to take into account non perfectly vertical events 30 . This assumption is reasonable for smooth tunneling junctions. Both functions are maximal around zero and decrease by increasing their arguments. With this requirement, the longer the tunneling path the smaller the corresponding local amplitude. Note that Eq. (8) describes spin-conserving tunneling processes only, since spin-flipping tunneling terms give no contribution in our two-terminal setup 23,25 .
Furthermore we neglect tunneling of either charged (∼ cos 1 √ π (ϕ 1 + ϕ 2 ) ) or spinful (∼ cos 1 √ π (θ 1 − θ 2 ) ) particle pairs, although for strong enough electron interactions they could compete with single-particle tunneling processes 12,24 . Note that all these processes are irrelevant 12 , in the RG sense, for 0.5 < K < 2. We limit our analysis to repulsive interaction 0.5 < K < 1, and we treat the tunneling current as a small perturbation. The tunneling Hamiltonian in Eq. (8) induces no net charge transfer between the two edges, but leads to a net spin tunneling current. The corresponding spin current operator is
(∼ cos 1 2 √ π (ϕ 1 + ϕ 2 ) cos 1 2 √ π (θ 1 − θ 2 ) )I S = − i 2 σ=↑,↓ dx dy Λ x,y ψ † R,σ (x)ψ L,σ (y) + h.c.,(10)
according to the requirement of absence of spin flipping and multiple-particles contributions. In linear response approximation in the tunneling hamiltonian, the stationary expectation value of the spin current in Eq. (10) can be written in terms of the tunneling rates Γ L,σ→R,σ and Γ R,σ→L,σ as
I S = 1 2 σ=↑,↓ [Γ L,σ→R,σ − Γ R,σ→L,σ ] .(11)
Note that the functional dependence of rates and other physical quantities from bias and temperature is understood for notational convenience.
One can easily realize that this spin tunneling current is responsible for a reduction of the net current flowing from one lead to the other 21,23 , i.e. I = e 2 π V − I BS , with I BS the backscattering current, related to I S by
I BS = 2e I S .(12)
We can thus measure the spin tunneling current by measuring the ordinary backscattering current 21 . By taking into account the spin independence of the tunneling rates and by considering the detailed balance relation Γ R,σ→L,σ = e −βeV Γ L,σ→R,σ , (β = 1/k B T the inverse temperature) one has
I BS = 2e 1 − e −βeV Γ L,↑→R,↑ .(13)
According to Eq. (13), we can consider only the tunneling rate Γ ≡ Γ L,↑→R,↑ given by
Γ = dx dy dx ′ dy ′ Λ x,y Λ * x ′ ,y ′ × dt e ieV t G > L (y ′ − y, t)G < R (x ′ − x, t),(14)
with They do not depend on spin and can be written in terms of the chiral ones W ± (x, t)
G > R/L (x, t) = e ∓ikF x 2πa e W R/L (x,t) (15) G < R/L (x, t) = e ±ikF x 2πa e W R/L (x,t)(16)W R (x, t) = c (+) K W + (x, t) + c (−) K W − (x, t)(18)W L (x, t) = c (−) K W + (x, t) + c (+) K W − (x, t),(19)
with
W ± (x, t) = W t ∓ x v(20)
and
W(t) = ln Γ 1 + 1 βωc − i t β 2 Γ 2 1 + 1 βωc (1 + iω c t) .(21)
Here, Γ(x) is the Euler Gamma function, c
(±) K = 1 4 √ K ± 1 √ K 2
are the interaction dependent tunneling coefficients and ω c = v/a the energy bandwidth. By replacing the above expressions into Eq. (14) one obtains
Γ K = dx dy dx ′ dy ′ Λ x,y Λ * x ′ ,y ′ (2πa) 2 e ikF (y ′ −y+x ′ −x) dt e ieV t e c (+) K W(t− x ′ −x v )+c (−) K W(t+ x ′ −x v )+c (−) K W(t− y ′ −y v )+c (+) K W(t+ y ′ −y v ) ,(22)
where we explicitly indicate the dependence on the interaction parameter K.
In what follows we will first analyze the non-interacting case, which can be thought as a superposition of two independent integer QH systems subjected to opposite magnetic fields. Later we will address the case of interacting helical edge states.
A. Non-interacting helical edge states
In the non-interacting case (K = 1), one has c
Γ 1 = d x d y Λ x,y Λ * x ′ ,y ′ (2πa) 2 e ikF (y ′ −y+x ′ −x) × dt e ieV t e W(t− x ′ −x v )+W(t+ y ′ −y v )(23)
where we introduced the short hand notation d x ≡ dx · dx ′ , d y ≡ dy · dy ′ . In terms of the new variables 30
τ = t − y−y ′ −x+x ′ 2v and z = y−y ′ +x−x ′ 2 one has Γ 1 = d x d y Λ x,y Λ * x ′ ,y ′ (2πa) 2 e i[k+(x ′ −x)+k−(y ′ −y)] × dτ e ieV τ e [W(τ− z v )+W(τ + z v )] ,(24)
with k ± = k F ± eV /2v. This can be further expressed as
Γ 1 = d x d y Λ x,y Λ * x ′ ,y ′ (2πa) 2 e i[k+(x ′ −x)+k−(y ′ −y)]F 1 (z, eV ) (25) wherẽ F g (z, ω) = dτ e iωτ P g τ − z v P g τ + z v(26)
and P g (t) = e gW(t) (cf. Eq. (21)). The separability assumption in Eq. (9) allows to factorize the tunneling amplitude as
Γ 1 = 4 |Λ 0 | 2 (2πa) 2 d y cos eV v (y ′ − y) f c (|2y|)f c (|2y ′ |) × d x cos [2k F (x ′ − x)] f l (|2x|)f l (|2x ′ |)F 1 (x ′ − x, eV ).(27)
To better characterize the effects of the extended contact geometry it is useful to represent Γ 1 in terms of the point contact rate Γ (point) 1
as
Γ 1 = λ 1 × Γ (point) 1 .(28)
This can be done regardless of the form of the tunneling amplitude but, as we will see, the separability assumption of Eq. (9) allows to give a closed form for the modulating function. From Eq. (13) and Eq. (28) follows that
I BS = λ 1 × I (point) BS .(29)
For any interaction K, the point-like current is given by 21
I (point) BS = 2e(1 − e −βeV ) |Λ 0 | 2 (2πa) 2P 2dK (eV ) (30) with d K ≡ c (+) K + c (−) K = 1 2 K + 1 K so that d K = 1 in the non-interacting case. The functioñ P g (ω) = dt e iωt P g (t)(31)
has the following form 29 for energies lower than the bandwidth ω c
P g (E) = 2π Γ(g)ωc E ωc g−1 θ(E) (T = 0) 2π βωc g−1 e βE 2 ωc B g 2 − i βE 2π , g 2 + i βE 2π (T = 0)(32)
with θ(x) the Heaviside step function and B [x, y] the Euler Beta function. The modulating function λ 1 in Eq. (28) represents the influence of the extended region and is given by
λ 1 = 4 d y cos eV v (y ′ − y) f c (|2y|)f c (|2y ′ |) × d x cos [2k F (x ′ − x)] f l (|2x|)f l (|2x ′ |) ×F 1 (x ′ − x, eV ) P 2 (eV ) .(33)
It can be written as a product of crossed and lateral con-
tribution λ 1 = λ c 1 λ l 1 , with λ c 1 = 2 d y cos eV v (y ′ − y) f c (|2y|)f c (|2y ′ |) (34) λ l 1 = 2 d x cos [2k F (x ′ − x)] f l (|2x|)f l (|2x ′ |) ×F 1 (x ′ − x, eV ) P 2 (eV ) .(35)
Notice that, while λ c 1 depends on the crossed contribution f c only, λ l 1 contains also the electronic Green's functions throughF 1 .
In order to perform an analysis of the extended contact, we consider a separable gaussian form 29,30
Λ x,y = Λ 0 2πξ c ξ l e − (x−y) 2 4ξ 2 c e − (x+y) 2 4ξ 2 l .(36)
The parameter ξ l is related to the extension of the contact, while ξ c allows to take into account non perfectly vertical events. In this sense a realistic assumption for modeling an extended contact is ξ c ≪ ξ l . Note that in the limits ξ c,l → 0 we recover the point-like tunneling amplitude Λ x,y → Λ 0 δ(x)δ(y), so that I BS → I
λ c 1 = e − 1 2 ( ξceV v ) 2 (37) λ l 1 = 1 √ 2π dxe − x 2 2 cos (2k F ξ l x)F 1 (ξ l x, eV ) P 2 (eV ) .(38)
By exploiting the convolution properties
F g (z, ω) = 1 2π dE e i 2z v EP g ω 2 + E P g ω 2 − E ,(39)
the tunneling amplitude can be written in the form
λ 1 = e − 1 2 (ξc eV v ) 2 −2(kF ξ l ) 2 dE 2π e −2(ξ l E v ) 2 cosh 4k F ξ 2 l E v ×P 1 eV 2 + E P 1 eV 2 − E P 2 (eV ) .(40)
This result is valid also at finite temperature and extends what done in Ref. 30 for the QHE at T = 0. Note that the crossed contribution to the modulating function comes into play only at high bias voltage. For an extended contact with length ∼ (0.1 ÷ 1)µm, one has ξ l ∼ (0.1 ÷ 1)µm and ξ c ≪ ξ l , e.g. ξ c ∼ 10 nm. With this assumption the crossed contribution is crucial only for relatively high bias 0.1 V, not considered here. This fact allows to choose λ c 1 ≈ 1 and to focus only on the lateral contribution which, as we will see in the following, shows strong modifications with respect to the point-like case also at low bias.
B. Interacting helical edge states
Starting from the general expression in Eq. (22) and proceeding as in the previous section, one can express the interacting modulating function as (K = 1)
λ K = dE 1 dE 2 dE 3 (2π) 3 e − 1 2 [ ξc v (eV −2E2−2E3)] 2 − 1 2 ξ l v (eV −2E1−2E2−2kF v) 2Pc (+) K (E 1 )P c (−) K (E 2 )P c (−) K (E 3 )P c (+) K (eV − i=1,2,3 E i ) P 2dK (eV ) .(41)
Due to the natural constraints imposed by the functional form ofP (E) in Eq. (32) it is possible to neglect the crossed contribution, present in the first gaussian term, as far as eV, k B T ≪ v/ξ c . Under this condition and noting that
∞ −∞ dE 2πP g1 (E)P g2 (ω − E) =P g1+g2 (ω),(42)
Eq. (41) becomes
λ K = e −2α 2 l dE 2π e −2 Kα l E ǫ F 2 cosh 4Kα 2 l E ǫ F ×P dK eV 2 + E P dK eV 2 − E P 2dK (eV ) .(43)
Here, we introduced the Fermi energy ǫ F = k F v F and the dimensionless parameter α l = k F ξ l . The modulating function thus depends on the length of the contact ξ l and on the Fermi momentum only through their product. By inserting Eq. (32) in Eq. (43) one has
λ K = Γ(2d K )e −2α 2 l 8π 2 Γ 2 (d K ) dxe − 1 2 (Kα l k B T ǫ F x) 2 cosh 2Kα 2 l k B T ǫ F x × B [γ +,+ (x), γ +,− (x)] B [γ −,+ (x), γ −,− (x)](44)
with (η, η ′ = ±)
γ η,η ′ (x) = d K 2 + η i 4π eV k B T + η ′ x .(45)
To conclude we observe that also in the interacting case the backscattering current can be written as
I BS (V, T ) = λ K (V, T ) × I (point) BS (V, T )(46)
with I
(point) BS
(V, T ) given in Eq. (29) and where we explicitly reintroduced the dependence on bias and temperature. Note that for α l = 0, Eq. (44) reduces to λ K = 1, and the point-like tunneling case is recovered.
IV. RESULTS
Since the modulating function depends on bias and temperature, it will influence the behavior of transport properties with respect to the point-like tunneling case. It is then useful to investigate it in details. Fig. 2 shows λ K as a function of voltages (a) or temperatures (b). Fig. 2(a) presents a maximum at V ≈V ≡ 2ǫ F /eK, becoming more and more pronounced by increasing α l , that is the length of the contact. In the limit α l → 0 it is washed out and λ K (V, T ) → 1. As already noted for QHE 29 , this maximum is determined by the two phases that control tunneling, one set by the Fermi momentum (2k F x) and the other by the voltage drop (eV t). The peak occurs when the two phases are equal:
eV /ǫ F k B T /ǫ F λ K (V, T ) λ K (V, T ) (a) (b)eV = 2k F x/t = 2k F v = 2ǫ F /K.
A maximum is present also in Fig. 2(b), but it originates from a dephasing mechanism, induced by finite temperature, similar to what was found in interferometric geometries with two or several QPCs, both in QH 40 and in QSH systems 26 , where the dephasing was depending on the distance among the QPCs. The extended contact geometry can be seen indeed as an infinite series of QPCs with different tunneling amplitudes, with infinitesimal distance dx between them, and the backscattering current is now given by integrating over the contact region. For all interaction strengths 0.5 < K < 1 we find the maximum at a positionT of the order of ǫ F /k B , vanishing as α l → 0, reproducing in this case the point-like regime with λ K (V, T ) → 1.
Note that for vanishing bias and temperature the modulating function is exponentially suppressed by the length of the contact, namely λ K (V = 0, T = 0) = e −2α 2 l . We can also study the asymptotic behavior of λ K at low bias or low temperatures. Introducing the energy scales eV α l = ǫ F /(Kα l ) and k B T α l = ǫ F /(Kα l ) one finds
λ K (V, T ≪ eV /k B ) ∼ constant V ≪ V α l V −1 V −V ≫ V α l(47)
and contrast to the point-like case, reminiscent of the form of λ K (see Fig. 2). More quantitatively, focusing on a given length, we can study the dependence on interactions. Fig. 4 shows the differential conductance as a function of bias, varying the electron interaction. The conductance shows a peak at V ≈V , which depends on the velocity of the excitations (V = 2k F v/e). Thanks to this behavior, we argue that an extended contact geometry could be fruitful to extract information about the velocity of the excitation modes along the edges, by experimentally measuring the peak of the conductance, varying the Fermi energy and the bias voltage 8,11 . Furthermore, it must be stressed that in presence of an ordinary Luttinger liquid we should expect two different peaks, as a consequence of the spincharge separation, which leads to two different propagation velocities, one for the charge modes and one for the spin modes 37,38,41,42 . The single-peak structure of Fig. 4, instead, provides evidence of the close connection between spin and charge typical of the helical edge states of QSHE, where these degrees of freedom are locked each other and propagate with the same velocity. We remark that, as expected, the peak in the differential conductance is reduced by increasing temperature and finally washed out for temperatures 2πk B T ∼ eV , eV /ǫ F G(V, T )/G 0
λ K (V ≪ k B T /e, T ) ∼ constant T ≪ T α l T −1 T −T ≫ T α l .(48eV /ǫ F k B T /ǫ F G(V, T )/G 0 G(T )/G 0 (a) (b)I (point) BS (V, T ≪ eV /k B ) ∼ V 2dK −1 (49) G (point) (T ) ∼ T 2dK−2 .(50)
Despite these trends are here no longer valid, they still survive at bias or temperatures lower enough, namely for V ≪ V α l or T ≪ T α l respectively, as shown in Fig. 6. Interestingly, by increasing energies, new power-law behaviors are recovered, however, with different exponents
k B T /ǫ F eV /ǫ F G(T )/G 0 I BS (V, T ) /I 0 (a) (b)I BS (V, T ≪ eV /k B ) ∼ V 2dK −2 (V −V ≫ V α l )(51)
and
G(T ) ∼ T 2dK−3 (T −T ≫ T α l ).(52)
This is a consequence of the asymptotic behavior of the modulating function (cf. Eqs. (47)-(48)). It is worth noting that the effective visibility of these high energy power-laws crucially depends on the Fermi energy ǫ F of the system, that can be easily tuned experimentally by means of an external gate 8 , and the natural cut-off energy ω c of the theory. The latter can be reasonably identified as the energy at which additional bulk effects have to be taken into account, thus the presented helical Luttinger liquid picture holds for energies lower than ω c .
V. CONCLUSIONS
We proposed a model for an extended tunneling through contact region in QSH system. We demonstrated that it is possible to take into account the extended nature of the contact through a modulating function, which renormalizes the transport properties of the point-like case. We showed that, due to the extended nature of the contact and for low enough temperatures, the differential conductance shows a pronounced peak that can be used to extract information about the propagation velocity of the excitations along the edge. The presence of a unique peak is a signature of the helical nature of the edge states in QSHE.
We analyzed the backscattering current in the low temperature regime and the linear conductance, showing that the power-law behaviors predicted in the point-like case survive at progressively lower energies by increasing the length of the contact. Remarkably enough, new powerlaws emerge also at higher energies, but with different exponents.
(2)-(3) are quadratic in the electron density. Introducing the helical edge basis on the upper and lower edge38
Figure 1 .
1(Color online) Extended contact geometry for a quantum spin Hall system with one Kramers doublet of helical edge states. The full (dashed) lines represent helical edge states carrying electrons with spin up (down). Right (left)moving electrons are in equilibrium with the left (right) contact at chemical potential µL (µR). The black arrow represents a possible spin-conserving electron tunneling event through the extended region.
the greater and lesser electron Green's functions associated to the right (R) and left (L) movers. The corresponding bosonic Green's functions are W R/L (x, t) = 2π ϕ R/L,σ (x, t)ϕ R/L,σ (0, 0) −2π ϕ R/L,σ (0, 0)ϕ R/L,σ (0, 0) . (17)
Figure 2 .
2(Color online) Modulating function as a function of (a) bias V (in units of ǫF /e) at low temperature (kBT = 10 −2 ǫF ) and (b) temperature T (in units of ǫF /kB) at low bias (eV = 10 −2 ǫF ), for different lengths of the contact: α l = 1 (long dashed red), 2 (dashed green), 5 (short dashed blue). Note that the behavior at low temperature in (a) is indistinguishable from the T = 0 case. This comment holds as well for panel (b) between low V and V = 0. Other parameters: K = 0.75.
) Fig. 3
)3shows the differential conductance G(V, T ) = d I BS (V, T ) /dV as a function of bias (a) and the linear conductance G(T ) = G(V = 0, T ) as a function of temperature (b). They both show a peaked structure, in
Figure 3 . 2 F
32(Color online) (a) Differential conductance as a function of bias V (in units of ǫF /e) at low temperature (kBT = 10 −2 ǫF ) and (b) linear conductance as a function of the temperature T (in units of ǫF /kB), for different lengths of the contact: α l = 1 (long dashed red), 2 (dashed green), 5 (short dashed blue). Units of the conductance: G0 = 2e 2 ǫ |Λ 0 | 2 (2πa) 2 (kF a) 2d K . Other parameters: K = 0.75.
Figure 4 .Figure 5 .
45(Color online) Differential conductance as a function of bias V (in units of ǫF /e) for different interaction strengths: K = 1 (long dashed red), 0.75 (dashed green), 0.5 (short dashed blue). Note that the conductance is plotted in unity of G0 as in Fig. 3, which depends on K and thus not allow for a direct comparison on the size between the different curves. Other parameters: α l = 5; kBT = 10 −2 ǫF . (Color online) Differential conductance as a function of bias V (in units of ǫF /e) for different temperatures (in units of ǫF /kB): T = 0.1 (long dashed red), T = 0.5 (dashed green), T = 2 (short dashed blue). Units of G0 as in Fig. 3. Other parameters: α l = 5; K = 0.75.as shown inFig. 5. Information about velocity are not the only ones that can be extracted by means of this setup. Theoretical works concerning point-like tunneling predict power-law behaviors for current21,24
Figure 6 .
6(Color online) Log-Log plot (a) of the backscattering current (in units of I0 ≡ (ǫF /e)G0) as a function of the bias voltage V (in units of ǫF /e) at low temperature (kBT = 10 −2 ǫF ) (long dashed red curve) and (b) of the linear conductance (in units of G0) as a function of temperature (in units of ǫF /kB) (long dashed red curve). Other parameters: α l = 1, K = 0.75. Straight lines represent the asymptotic power-law behavior with exponent (a) 2dK − 1 = 13/12 (dashed green line) and 2dK − 2 = 1/12 (short dashed blue line) and (b) 2dK − 2 = 1/12 (dashed green line) and 2dK − 3 = −11/12 (short dashed blue line).
ACKNOWLEDGEMENTSWe thank A. Braggio, M. Carrega, and T. Martin for useful discussions. The support of CNR STM
ITN-2008-234970 NANOCTM and CNR-SPIN via Seed Project PGESE001 is acknowledged. Eu-Fp7 Via Grant Program, No, program, EU-FP7 via Grant No. ITN-2008- 234970 NANOCTM and CNR-SPIN via Seed Project PGESE001 is acknowledged.
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| []
|
[
"Distributed SMC-PHD Fusion for Partial, Arithmetic Average Consensus",
"Distributed SMC-PHD Fusion for Partial, Arithmetic Average Consensus"
]
| [
"Tiancheng Li "
]
| []
| []
| We propose an average consensus approach for distributed SMC-PHD (sequential Monte Carlo-probability hypothesis density) fusion, in which local filters extract Gaussian mixtures (GMs) from their respective particle posteriors, share them (iteratively) with their neighbors and finally use the disseminated GM to update the particle weight. The resulting particle distribution is the arithmetic average of the disseminated GM-posteriors. There are two distinguishable features of our approach compared to exiting approaches. First, a computationally efficient particles-to-GM (P2GM) conversion scheme is developed based on the unique structure of the SMC-PHD updater in which the particle weight can be exactly decomposed with regard to the measurements and misdetection. Only significant components of higher weight are utilized for parameterization and so the disseminated information is only a part of that of local posteriors. The consensus, conditioned on partial information dissemination over the network, is called "partial consensus". Second, importance sampling (IS) is employed to re-weight the local particles for integrating the received GM information, without changing the states of the particles. By this, the local prior PHD and likelihood calculation can be carried out in parallel to the dissemination & fusion procedure.To assess the effectiveness of the proposed P2GM parameterization approach and IS approach, two relevant yet new distributed SMC-PHD fusion protocols are introduced for comparison. One uses the same P2GM conversion and GM dissemination schemes as our approach but local particles are regenerated from the disseminated GMs at each filtering iteration -in place of the IS approach. This performs similar to our IS approach (as expected) but prevents any parallelization as addressed above. The other is disseminating the particles between neighborsin place of the P2GM conversion. This avoids parameterization but is communicatively costly. This protocol, essentially seeking complete (posterior) consensus, however, does not perform better than the GM-dissemination based partial consensus. Different to these arithmetic average consensus approaches, the state-of-theart exponential mixture density approach that seeks geometric average consensus is also realized for comparison. | null | [
"https://arxiv.org/pdf/1712.06128v1.pdf"
]
| 34,081,586 | 1712.06128 | 59af635aa9a5c72854d87f55461312e5c62022e6 |
Distributed SMC-PHD Fusion for Partial, Arithmetic Average Consensus
17 Dec 2017
Tiancheng Li
Distributed SMC-PHD Fusion for Partial, Arithmetic Average Consensus
17 Dec 20171Index Terms-Distributed trackingaverage consensusPHD filterparticle filterGaussian mixturepartial consensusarith- metic averagegeometric average
We propose an average consensus approach for distributed SMC-PHD (sequential Monte Carlo-probability hypothesis density) fusion, in which local filters extract Gaussian mixtures (GMs) from their respective particle posteriors, share them (iteratively) with their neighbors and finally use the disseminated GM to update the particle weight. The resulting particle distribution is the arithmetic average of the disseminated GM-posteriors. There are two distinguishable features of our approach compared to exiting approaches. First, a computationally efficient particles-to-GM (P2GM) conversion scheme is developed based on the unique structure of the SMC-PHD updater in which the particle weight can be exactly decomposed with regard to the measurements and misdetection. Only significant components of higher weight are utilized for parameterization and so the disseminated information is only a part of that of local posteriors. The consensus, conditioned on partial information dissemination over the network, is called "partial consensus". Second, importance sampling (IS) is employed to re-weight the local particles for integrating the received GM information, without changing the states of the particles. By this, the local prior PHD and likelihood calculation can be carried out in parallel to the dissemination & fusion procedure.To assess the effectiveness of the proposed P2GM parameterization approach and IS approach, two relevant yet new distributed SMC-PHD fusion protocols are introduced for comparison. One uses the same P2GM conversion and GM dissemination schemes as our approach but local particles are regenerated from the disseminated GMs at each filtering iteration -in place of the IS approach. This performs similar to our IS approach (as expected) but prevents any parallelization as addressed above. The other is disseminating the particles between neighborsin place of the P2GM conversion. This avoids parameterization but is communicatively costly. This protocol, essentially seeking complete (posterior) consensus, however, does not perform better than the GM-dissemination based partial consensus. Different to these arithmetic average consensus approaches, the state-of-theart exponential mixture density approach that seeks geometric average consensus is also realized for comparison.
I. INTRODUCTION
D ISTRIBUTED target tracking (DTT) based on wireless sensor networks (WSNs) has received considerable research interest in the last decade. It basically involves a number of spatially distributed, low-powered, interconnected sensors that are equipped with a signal processing unit, allowing them to carry out sensing, calculation and communication with the neighbors without a fusion center [1], [2]. These sensors T. Li is with the School of Sciences, University of Salamanca, 37007 Salamanca, Spain, E-mail: [email protected]; [email protected]. This work is in part supported by the Marie Skłodowska-Curie Individual Fellowship (H2020-MSCA-IF-2015) under Grant no. 709267 cooperatively track the targets based on their local measurement and the information disseminated from the others, which facilitates better estimation accuracy and overcomes many deficiencies that an isolated sensor suffers from such as false and missing measurement and limited fields of view. Typically, it is expected that local nodes reach a single "consensus" [3], [4] conditioned on the common information they share after sufficient peer-to-peer (P2P) communication.
To deal with the nonlinearity and non-normal non-Gaussian uncertainty that are involved in the statistical models regarding the targets, the scenario and/or the sensors, the sequential Monte Carlo (SMC) approach provides one of the most vital tools for realizing sequential Bayesian inference (SBI), which is also known as the particle filter (PF); see some recent advances [5]. Realizing the PF on the WSN leads to a quite universal distributed tracking framework, which has invited many specific implementation protocols; see literature reviews offered in [6]- [8] primarily regarding a single target.
For multitarget tracking (MTT) in the presence of false and missing data, measurement-to-target association is typically needed which entails either computationally intensive calculation or ad-hoc strategy (such as gating) design [9], [10]. To overcome this difficulty, random finite set (RFS) has emerged as a powerful and versatile tool. In particular, instead of propagating the full multitarget density which has been considered computationally intractable, the PHD (probability hypothesis density) filter propagates the first order statistical moment of the multitarget RFS [11] and avoids measurementto-target association. Consequently, many RFS-models based PFs have been developed [12], which have become a new driving force for the flourishing of both the PF algorithm and the RFS filtering family.
Nonetheless, exact implementation of the multisensor PHD filter involves summing over all partitions of the measurements from different sensors which is intractable in computation (if not impossible) and typically, one has to resort to simplifying approximation [13], [14]. Alternatively, immense interest has been seen recently for extending the theory of "average consensus" to DTT, in which the item being estimated may be the arithmetic average [2] (AA, akin to the linear opinion pool [3], [4]) or the geometric average [15] (GA, akin to the logarithmic opinion pool [16]) of the initial values. AA and GA differ in measuring the distance for calculating the "average". In the former, it is the Euclidean distance while in the latter it is the Kullback-Leibler divergence [17], [18].
Notably, the GA fusion coincides with the covariance intersection approach [17], [19]- [21], a type of Chernoff fusion, which was originally developed for addressing unknown infor-mation correlation among sensors and for avoiding information double-counting. This approach has been widely applied for distributed PHD fusion, based on either the Gaussian mixture (GM) implementation [22]- [25] or the PF [26], [27]. Literally, the GA is also referred to as the Kullback-Leibler average (KLA) [22], [28], exponential/geometric mixture density (EMD/GMD) [17], [27] or generalized covariance intersection (GCI) [29]- [31]).
However, the GA rarely admits closed-form solution for a mixture distribution such as GM, not to say an arbitrary particle distribution. The only existing GA-based SMC-PHD fusion resorts to clustering and converting the particle distribution to continuous distribution approximations [27], which disseminates both particles and the continuous functions, suffering from very intensive communication. Moreover, the GA fusion suffers from several deficiencies such as: 1) Delay in detecting new appearing targets [23]; 2) Failure to handle closely distributed targets and/or low SNR background [32]; 3) Prone to mis-detection [24], [33] or incorrect data [34]. For GM-PHD average consensus, we have demonstrated that the apparently simple AA offers a promising alternative to the GA, yielding higher filtering accuracy, better reliability in cluttered environment with mis-detection, and lower communication and fusion-computation (C&F) cost [35]. Clearly, the AA is universal and unlimited to the GM filter. As we will show in this paper, it also applies to the RFS-PFs, for which the C&F challenges are:
• Information dissemination: it is practically preventable to disseminate the particle set, but instead, efficient parameterization such as particle-to-Gaussian/GM conversion to compromise between approximation accuracy and communication cost is on great demand. • Information fusion: the immediate challenge to the parametric posterior dissemination is how to efficiently integrate/incorporate the parametric information into the particle distribution for improving upon its fitness to the global optimum, which is deemed to be an "average" of the interested initial posteriors. • Real time networking: In the most favorable situation, the C&F should be carried out in parallel with local filter calculations to avoid any time delay to the filter, which is referred to as real time networking. However, in almost all existing DTT systems, the filtering calculation (at one stage or another) depends on the shared information gained by the C&F which is exactly how the filters benefit from networking. As a result, local filtering calculation and neighbor-wise communication are performed interactively in time. This may not be allowed in reality. These challenges motivate our work. On the one hand, for posterior parameterization, we investigate the unique structure of the PHD filter whose posterior can be decomposed with regard to measurements and misdetection. In consistent with the notion of "partial consensus" [35], only significant components (in the format of few parameters) are disseminated among sensors while the insignificant components are not involved in C&F. The idea of sharing only a part of the information over the network has appeared earlier in the centralized network with a fusion center [36], [37] and in the diffusion network [38], [39], in which the benefit is primarily limited to saving communication/memory cost. However, we demonstrate (besides [35]) that, the partial consensus does not only save communication/memory but also, more importantly, improve the estimation accuracy.
On the other hand, we propose an efficient, fully distributed, means to update weights of local particles (without changing their states) according to the disseminated GMs based on importance sampling (IS), leading to an exact weighted AA of the disseminated GM-posteriors.
The resulting framework enjoys two novel features:
• Local filtering calculation is allowed to be carried out partially in parallel to the C&F, for reducing the network dissemination delay. • The framework can cooperate seamlessly with the distributed GM-PHD filter [35] for a hybrid sensor network consisting of both GM-PHD and SMC-PHD filters, without any special algorithm design for either. The remainder of this paper is organized as follows. Primary notations and assumptions are listed next immediately. The basics of the SMC-PHD filter and the particle weight decomposition are given in Section II. The proposed distributed SMC-PHD fusion approach is detailed in Section III. Simulations are given in Section IV. We conclude in Section V.
A. Notation and Assumption
The sensor network is represented by a directed graph G = (V, E) with the set of sensors V = {1, 2, · · · , N } and the set of edges E ⊆ V × V. In the directed graph, any edge is denoted by an ordered pair of sensors (a, b) ∈ E, which means node b is directly reachable from node a, where a is called the in-neighbor of b while b is the out-neighbor of a. For any b ∈ V, denote N b := {a ∈ V|(a, b) ∈ E, a = b}, which is the set of all the in-neighbors of node b excluding node b itself. Undirected graph is a special type of directed graph where for any (a, b) ∈ E, we must have (b, a) ∈ E.
The collections of target states and measurements at time k can be represented as finite sets X k = {x k,1 , · · · , x k,N k } and Z k = {z k,1 , · · · , z k,M k }, where N k and M k are the number of targets and of measurements, respectively. The cardinality (number of elements) of a finite set I is denoted by |I|. Therefore, we have N k = |X k | and M k = |Z k |. A Gaussian probability density function (PDF) of a random variable x with mean m and covariance P is denoted by G(x; m, P) and the Kronecker delta function is denoted as δ y (x).
At each time k, a random Poisson number of targets appear according to the new-born intensity function γ k (x). We do not particularly consider target spawn. Each target is assumed to evolve and generate measurements independently of others. More specifically, a target with state x k−1 may either disappear with probability 1 − p S,k (x k−1 ), or continue to exist at time k with survival probability p S,k (x k−1 ) and move to a new state with a transition probability density
f k|k−1 (x k |x k−1 ).
A target with state x k ∈ X k is either miss-detected with probability 1−p D,k (x k ) or detected with probability p D,k (x k ) and generates an measurement z k ∈ Z k with likelihood g k (z k |x k ).
One target can generate no more than one measurement at each scan. In addition, the number of clutter points at time k is subject to a random Poisson distribution κ k (z) and is independent of the real measurement of targets.
II. DECOMPOSITION OF PARTICLE PHD A. RFS and PHD
A RFS variable X is a random variable that takes values as unordered finite sets and is uniquely specified by its cardinality distribution ρ(n) Pr[|X| = n] and a family of symmetric joint distributions p n (x 1 , x 2 , · · · , x n ) that characterize the distribution of its elements over the state space, conditioned on the set cardinality n. The PDF f (X) of a RFS variable X is given as f ({x 1 , x 2 , · · · , x n }) = n!ρ(n)p n (x 1 , x 2 , · · · , x n ).
The PHD D S (x) of a multitarget RFS variable X with the PDF f (X) in a measurable region S ⊆ R d is given as:
D S (x) = S δ X (x)f (X)δX ,(1)
where δ X (x) y∈X δ y (x) and the RFS integral reads:
S f (X)δX f (∅) + ∞ n=1 S n f ({x1,x2,··· ,xn}) n! dx 1 dx 2 · · · dx n .
B. PHD Filtering Recursion
Denote by D k|k−1 (x) and D k|k (x) the PHD of the prior and posterior point processes X k |Z 1:k−1 and X k |Z 1:k , respectively. Omitting the conditioning on the measurements for convenience, the PHD prediction-updating recursion can be given as follows [11]:
· · · → D k−1|k−1 (x) → D k|k−1 (x) → D k|k (x) → · · · (2)
To be more specific, 1) Time update step (to calculate the prior PHD):
D k|k−1 (x) =γ k (x)+ p S,k (x ′ )f k|k−1 (x|x ′ )D k−1|k−1 (x ′ )dx ′ .(3)
2) Measurement update step (to calculate the posterior PHD):
D k|k (x) = 1 − p D,k (x) D k|k−1 (x)+ z∈Z k p D,k (x)g k (z|x)D k|k−1 (x) κ k (z) + p D,k (x)g k (z|x)D k|k−1 (x)dx .
(4) There have been many implementations of the PHD filter based on different types of PFs since the considered standard implementation [40], including auxiliary PF [41], marginalized PF [42] and box PF [43]. Without loss of generality, we adopt the standard implementation [40] (except the estimate extraction part for which we will employ computationally much faster approaches, to be addressed in Section III-E) without giving its detail here.
C. Particle (posterior) Weight Decomposition
The representation of the posterior PHD D k|k (x) by using J k particles with state x (j) k and nonnegative weight w (j) k|k , j = 1, 2, · · · , J k , can be written as [40]
D k|k (x) ≈ J k j=1 w (j) k|k δ x (j) k (x) ,(5)
where (cf. (4)) w (j)
k|k = 1 − p D,k x (j) k w (j) k|k−1 + z k ∈Z k p D,k x (j) k g k z k |x (j) k w (j) k|k−1 κ k (z k ) + J k j=1 p D,k x (j) k g k z k |x (j) k w (j) k|k−1 ,(6)
and w (j) k|k−1 is the prediction weight of particle j (either evolved from time k − 1 or new born at time k; see [40] for detail) and admits
D k|k−1 (x) ≈ J k j=1 w (j) k|k−1 δ x (j) k (x) .
Obviously, (6) can be decomposed with regard to the measurements
w (j) k|k (z k ) 1 − p D,k x (j) k w (j) k|k−1 if z k = z 0 , p D,k x (j) k g k z k |x (j) k w (j) k|k−1 κ k (z k )+ J k j=1 p D,k x (j) k g k z k |x (j) k w (j) k|k−1 if z k ∈ Z k .(7)
where the pseudo-measurement z 0 is introduced to represent the misdetection. w (j) k|k (z k ) implies how much each z k contributes to the weight of particle j. Straightforwardly, we have
w (j) k|k = z k ∈{z0}∪Z k w (j) k|k (z k ) .(8)
Furthermore, we define the sum of weight components of all particles with regard to measurement z k as
W k (z k ) J k j=1 w (j) k|k (z k ) ,(9)
which indicates the probability that the underlying measurement is from a real target (z k ∈ Z k ) or that misdetection
occurs (z k = z 0 ). Obviously, ∀z k ∈ Z k , W k (z k ) ∈ [0, 1] [44]
and, the weight sum admits
z k ∈{z0}∪Z k W k (z k ) = J k j=1 w (j) k|k W k .(10)
D. Multitarget RFS Cardinality Estimation
The expectation of the total number of targets N k conditioned on the PHD is given by its integral in the entire state space, which is approximated by the weight sum W k of all particles as in (5); see the detailed derivation given in Appendix A. That is,
E[N k |D k|k (x)] ≈ W k .(11)
For estimate extraction, a common approach to estimate the number of targets is rounding the total weight sum, i.e.,
N k = [W k ] ,(12)
where the operator [·] rounds the content to the nearest integer.
III. OUR PROPOSAL
We consider now a sensor network where all sensors synchronously observe the scenario, affected with independent noises and clutter. The local sensor performs particle prediction, updating and resampling exactly as in the centralized case, except that an additional C&F scheme is carried out once the posterior PHD is achieved at each filtering iteration, which uses the information disseminated from the other sensors to "re-weight" the underlying particle set for consensus.
In the sequel, we shall only concentrate on the C&F part, which consists of three key components:
• Extract a GM from each local particle set and disseminate them in neighborhood, perhaps in multiple iterations and with mixture reduction applied; see Section III-A. • Update the weight of local particles to integrate the posterior information carried in the disseminated GM that is comprised of components both received from the other sensors and generalized locally; see Section III-C. • Seek cardinality AA consensus in neighborhood in parallel to the above schemes of GM dissemination and particle-GM fusion; see Section III-D.
A. Particle to GM Conversion
We propose to extract GMs from the a posteriori particle distribution at each sensor based on the weight decomposition as in (7) as that each measurement corresponds to one Gaussian component (GC). By this, as many as M k + 1 GCs can be obtained which, however, could still be too communicatively costly. Arguably, the GM should contain sufficient information of the potential targets subject to the communication limitation. Following the partial consensus principle [35], only the significant GC, namely corresponding to high W k (z), should be disseminated for consensus, while the insignificant GC should be less likely involved.
The number of significant measurements can be determined either by the estimated number of targetsN k as in (12) or as that of the measurements corresponding to W k (z k ) larger than a threshold T c (usually, T c ∈ [0.1, 0.5]). The former is referred to as the Rank rule while the latter is the Threshold rule (considered as the default in our approach), akin to the notions used in [45] and [35]. Either way, we denote the selected measurements by a subset Z k,T ⊆ Z k .
For each z k ∈ Z k,T , a GC G x;m k (z k ),P k (z k ) weighted by W k (z k ) can be extracted from the weight component-based
particle set (x (j) k , w (j) k|k (z k )) J k j=1 , i.e., cf.(5) W k (z k )G x;m k (z k ),P k (z k ) ≈ J k j=1 w (j) k|k (z k )δ x (j) k (x),(13)
where W k (z k ) is already given in (9) while the mean and the covariance of the formed GC are given as follows, respectively,m
k (z k ) = J k j=1 w (j) k|k (z k )x (j) k ,(14)P k (z k ) = J k j=1 w (j) k|k (z k ) x (j) k −m k (z k ) x (j) k −m k (z k ) T .
(15) It is necessary to note that, such a Gaussian approximate can only become accurate when the prior PHD is Gaussiandistributed and the likelihood function is Gaussian. Otherwise, the parameterization is no more than approximation.
Various particles-to-GC/GM converting approaches have been developed for distributed particle filtering [6]- [8], [27], [46]- [48], mostly based on either ad-hoc strategy or sophisticated learning algorithm. Thanks to the unique weightdecomposition property of the SMC-PHD updater, our approach is computationally very efficient and reliable.
We may define the "partial PHD" D k,T (x) as a congregation of all the significant components of the PHD, namely the part to be extracted for parameterization, i.e., cf.(5)
D k,T (x) J k j=1 w (j) k,T δ x (j) k (x) ,(16)
where w
(j) k,T = z∈Z k,T w (j) k|k (z k ) ≤ w (j)
k|k denotes all the significant part of the weight of particle j. In contrast, the remaining components w
(j) k|k − w (j)
k,T are considered insignificant and will not be involved in the C&F procedure.
Substituting (13) into (16) gives an explicit GM approximation of the partial PHD, i.e.,
D k,T (x) ≈ z∈Z k,T W k (z k )G x;m k (z k ),P k (z k ) . (17)
Remark 1 When multiple communication iterations are performed, the local sensor will have an iteration-increasing GM size unless mixture reduction is applied [7]. To save the communication, GM merging [35] may be performed at all or some iterations, e.g., when the size exceeds a predefined upper threshold, which, however, may lead to information double-counting in addition to merging error. For example, if a GC sent from sensor a to sensor b is merged with another GC at sensor b, the resulting fused GC will be sent back to sensor a in the next communication iteration. In this case, appropriate (sensor-oriented) fusion weights shall be designed for fast consensus convergence.
B. Weighted, Arithmetic Average of Partial PHDs
In the following formulation, we will use subscripts a and b ∈ N a to distinguish between two neighboring sensors. In the proposed protocol, the local partial PHD D a,k,T (x) at sensor a will be linearly averaged with the received GM/partial PHD from the neighbors D b,k,T (x), ∀b ∈ N a , i.e.,
D a,k,T (x) = b∈{a}∪Na ω b→a D b,k,T (x) ,(18)
where the fusion weights ω b→a ≥ 0, b∈{a}∪Na ω b→a = 1 indicating that the fusion result is an "average".
As the key of our approach, we will use the arithmetically averaged partial PHD to replace the local PHD, by means of re-weighting the local particles. To this end, we apply the the Metropolis weights [49], [50] which are given as
ω b→a = 1 1+max (|Na|,|N b |) if b ∈ N a , b = a , 1 − l∈Na ω l→a if b = a .(19)
To note, the purpose of Metropolis weights here is the same to that in the original proposal [49], [50], which is for fast AA convergence and has no explicit connection to issues such as "dividing the common information" or "coping with the correlation" between sensors [51]. Therefore, we will not address on those issues (for which a recent review [52] is available) but instead we concentrate our distributed fusion goal on "average consensus".
Remark 2 The rationale for calculating the AA of PHDs is based on the essential property of the PHD that the integral of PHD in any region gives the expected number of targets in that region -cf. (1). This renders the AA calculation as in (18) a meaningful interpretation. Also, there is an important assumption behind the PHD filter: both misdetection and clutter are random and are independent of the real measurement of targets. So, it is unlikely for the same target to be missed in detection, or to say a target does not form a significant GC, in the majority of all sensors, or false alarms coincidentally occur in the same area in the majority of all sensors [35]. Using the principle of "majority rule", the AA can compensate for the false/missing data of a single sensor. Indeed, the AA is provably immune to either false or missing data problem. Exactly because of this, it is reasonable to abandon the insignificant GCs without worrying about misdetection or false alarms, namely partial AA consensus. Doing so does not only save communication but also tend to ameliorate the local signal-to-noise ratio (SNR) [35], [51].
C. Particle Updating w.r.t. GMs based on IS
Denote by (x (j) a,k , µ a ) J a,k
j=1 the uniformly weighted particles yielded by resampling after the PHD updating at sensor a at time k. Given the local weight sum W a,k , we have µ a = W a,k J a,k . As long as the resampling scheme adopted is unbiased [53] and J a,k is large enough, the new particle set still admits an appropriate approximation of the posterior PHD, namely
D a,k (x) ≈ J a,k j=1 µ a δ x (j) a,k (x) .(20)
Arguably, these particles by assigning appropriate weights can approximate any PHD that has the same support space as D a,k (x). In particular, for
D b,k,T (x), b ∈ {a} ∪ N a , we have D b,k,T (x) ≈ J a,k j=1 w b x (j) a,k δ x (j) a,k (x) .(21)
where
w b x (j)
a,k is the new weight assigned to particle x (j) a,k . Substituting (21)
into (18) yields D a,k,T (x) ≈ J a,k j=1w (j) a,k δ x (j) a,k (x),(22)wherew (j) a,k = b∈{a}∪Na ω b→a w b x (j) a,k .(23)
To
determine w b x (j)
a,k to fulfill (21), we employ the classic IS approach. The idea of IS is to choose a proposal distribution q(x) in place of the target probability distribution p(x). The support of q(x) is assumed to cover that of p(x). Rewrite a general integration problem as
S f (x)p(x)dx = S f (x) p(x) q(x) q(x)dx,(24)
where f (x) is an integrable function in a measurable space S. The IS [54, Chapter 3.3] is to use a number, to say J, of independent samples drawn from q(x) to obtain a weighted sum to approximate (24):
f p = 1 J J i=1 w(x (i) )f (x (i) ),(25)
where the importance weights/ratios are
w(x (i) ) = p(x (i) ) q(x (i) ) .(26)
If both q(x) and p(x) are discrete, i.e., the random variable x can only take on discrete values from a set X , p(x (i) ) and q(x (i) ) are actually known as the probability mass function (PMF) pmf x (y) of the discrete random variable x, which is defined as
pmf x (y) = Pr[x = y] if y ∈ X , undefined if y / ∈ X .(27)
Here, we extend the PMF definition from the discrete "probabilities" to the "intensity/PHD" so that the value is not limited to be in the scope of [0, 1], termed as intensity mass function (IMF), which reads
imf x (y) = D[x = y] if y ∈ X , undefined if y / ∈ X .(28)
where D[x = y] is in sharp the weight of the particle x = y, which can be larger than 1. Now, the uniformly weighted particles given by resampling as shown in (20) are just samples randomly drawn from the proposal D a,k (x) and the discrete set X in (28) is just the particle state set x (j) a,k J a,k j=1 . To get the desired PHD distribution D b,k,T (x) as shown in (21), it is as easy as weighting these particles by
w b x (j) a,k = imf b,k,T x (j) a,k imf a,k x (j) a,k .(29)
where
imf b,k,T x (j)
a,k is to evaluate the IMF at state x (j) a,k w.r.t. the GM-PHD disseminated from sensor b ∈ {a} ∪ N a as in (14), (15) and (9), which is given as
imf b,k,T x (j) a,k = z k ∈Z b,k,T W b,k (z k )G x (j) a,k ;m b (z k ),P b (z k ) ,(30)
and
imf a,k x (j) a,k = D a,k [x = x (j) a,k ],(31)
which is nothing else but just the weight of the "mother" particle from which particle x (j) a,k is resampled, denoted as w [j] a,k|k . Therefore, to avoid repeated computing, we need to store the weights of the particles prior to resampling.
Substituting (31) and (30) into (29) yields
w b x (j) a,k ∝ z k ∈Z b,k,T W b,k (z k )G x (j) a,k ;m b (z k ),P b (z k ) w [j] a,k|k ,(32)
which is subject to the cardinality consistence (for which, in fact, a separate cardinality AA consensus scheme will be applied in parallel as addressed next), i.e.,
J a,k j=1 w b x (j) a,k = z k ∈Z b,k,T W b,k (z k ).(33)
Remark 3 A key for the success of the IS approach is that the support of the proposal q(x) covers that of the target distribution p(x), both distributions better of the similar shape. This is true in our case as that the posteriors obtained at local sensors corresponding to the same multitarget RFS are approximately identical in general.
D. Cardinality AA Consensus
In parallel to the above C&F procedure, the standard AA consensus [49], [50] is also applied to update the local weight sum at each communication iteration, namely cardinality AA consensus or simply cardinality consensus (CC), as follows:
W a,k = b∈{a}∪Na ω b→a W b,k ,(34)
where the local weight sum
W b,k = J b,k j=1 w (j)
b,k . The resulting new weight sum will be used for re-scaling the weightw (j) a,k of each particle given in (23), i.e., the afterconsensus (AC) weight of particle j is
w (j),AC a,k =w (j) a,k J a,k j=1w (j) a,kW a,k .(35)
E. Estimate Extraction
Estimates of the targets' states can be extracted in two means. One is carried out prior to the C&F and is purely based on the local particle-posterior, for which the usual estimate extraction procedures proposed for the centralized SMC-PHD filter such as multi-EAP [45] or other computing fast measurement-driven approaches e.g., [55], [56] which actually extract the means of the formed GC as in (14) of selected measurements corresponding to significant weights W a,k (z k ), which is taken as the default way.
The advantage of this means is that, the estimate extraction does not need to "wait" for the C&F procedure and is therefore able to be carried out timely. The disadvantage is that, the latest information from the other sensors is not used, although the neighbor information has been used in the previous filtering iterations by C&F and is reflected in the prior. We refer to this as "real-time/before-consensus (BC) estimation".
The other means to extract estimates is carried out with regard to the disseminated GCs, taking into account the latest information from the other sensors. This is referred to as "delayed/AC estimation" as it can only be performed after the C&F. There are two typical ways to do so. One is merging closely-distributed GCs and extracting the mean(s) of the GC with larger weights. The other way is clustering the weighted GCs, and extracting the centroid of each significant cluster, which is taken as the default way in our approach. Either way, the number of estimates can be determined by the consensus on the cardinality which is rounding (34). To note, if the weight sum of one cluster or the weight of a single GC is closer to another integer n ≥ 2 rather than 1 (indicating multiple targets in that cluster), n estimates should be extracted from that cluster or that GC.
There is still space for optimizing these estimate extraction algorithms on the basis of either the particle set or the disseminated GM. For example, for multisensor data clustering, it is useful to set constraints on the size of cluster [57], to avoid false alarm (e.g., a cluster of too small size) and to deal with overlapped clusters (e.g., a cluster of over large size because of closely-distributed targets). By this, the clustering scheme may automatically determine the number of estimates. Extensions in this regard are however beyond the focus of this work.
F. Parallelization of Local Filtering and C&F
In summary, a complete filtering iteration of the proposed distributed SMC-PHD filter is illustrated in Algorithm 1. We have an important note on the parallel processing of local filtering calculation and the C&F procedure.
Remark 4 The proposed IS which preserves the state of local particles renders filtering-C&F parallelization possible: in parallel to the network C&F at time k, some of local filtering calculations required for the filter iteration k + 1 can be executed including 1)
Step 1 of Algorithm 1: calculating the prior PHD as in (3), including generating new-born particles and propagating particles inherent from time k, and 2) Step 2-1 of Algorithm 1: calculating the likelihoods g k+1 z a,k+1 |x (j) a,k+1|k and the detection probabilities
p D,k+1 x (j)
a,k+1|k for all particles j = 1, 2, · · · , J a,k , and the clutter intensities κ k+1 (z a,k+1 ) regarding all measurements z k ∈ Z a,k . The parallelization is feasible because all of these calculations do not need the knowledge of the particle weights until
Step 2-2 where the particle weights are needed and so the calculations thereafter can only be performed after (35).
G. Hybrid GM-and SMC-PHD Sensor Network
The proposed distributed SMC-PHD filtering protocol is naturally incorporable to the distributed GM-PHD filter [35]. That is, some sensors operate SMC-PHD filters while the others operate GM-PHD filters, both disseminating and receiving GMs. We demonstrate this hybrid filter network in our simulation in Section IV.
Algorithm 1 Distributed SMC-PHD Iteration at sensor a based on GM for dissemination and IS for fusion Input:
• All statistical models as required, the measurement RFS Z a,k received at time k, and the posterior particle set
ξ a,k−1 (x (j) a,k−1 , w (j) a,k−1 ) J a,k−1 j=1
for time k − 1. Output:
• State-estimates of the targets and a new particle set ξ a,k (x
(j) a,k , w (j) a,k ) J a,k j=1
. Procedure:
Step 1 Existing particle propagation and new particle generation:
• Update the state of particles x (j) a,k−1 to x (j)
a,k , j = 1, · · · , J a,k−1 according to the state transition model φ a,k|k−1 (x|x ′ ), preserving the weight w
(j) a,k|k−1 = w (j) a,k−1 . • Add new particles (x (j) a,k , w (j) a,k|k−1 ) J ′ a,k
j=J a,k−1 +1 according to the new-born target intensity model γ a,k (x).
Step 2 Particle weight updating:
• Re-weight all particles as in (6), yielding a new particle
set ξ ′ a,k (x (j) a,k , w (j) a,k ) J ′ a,k
j=1 , consisting of two steps: 1) Calculate the likelihood g k z a,k |x (j) a,k and detection probability p D,k x (j) a,k for each particle j = 1, 2, · · · , J ′ a,k , and the clutter intensity κ k (z a,k ) w.r.t. all measurements z a,k ∈ Z a,k . 2) Calculate the finally updated weight as in (6), taking into account the prediction weight w (j) a,k|k−1 .
Step 3 Real-time/BC estimate extraction:
• Extract totally [W a,k ] state-estimates as in (14) and (15) for z k ∈ Z a,k of higher weights W a,k (z k ) as in (9). • Determine Z a,k,T ⊂ Z a,k satisfying that ∀z k ∈ Z a,k,T : W a,k (z k ) ≥ T c -Threshold rule.
Step 4 Resampling:
• Resample from ξ ′ a,k to get a uniformly weighted new particle set ξ a,k (x (j) a,k , µ a ) J a,k j=1 as in (20). Store the initial weight w [j] a,k of the mother particle from which particle j is sampled.
Step 5 Partial consensus via IS:
• Extract a GC for each z k ∈ Z a,k,T as in (14) and (15). • Disseminate the GCs and W a,k to the neighbors and receive theirs; this step may be carried out for multiple iterations and if necessary (e.g., when the number of GCs exceeds a specific threshold), mixture reduction may be performed at some iterations. Target1: k [5,37] Target2: k [7,35] Target3: k [16,60] Target4: k [18,18] Target5: k [20,60] Target6: k [48,60] Linear sensors Nonlinear sensors Each target has a time-constant survival probability p S (x k ) = 0.98 and the survival target follows a nearly constant velocity motion as given
• Calculate w b x (j) a,k J a,kx k = 1 ∆ 0 0 0 1 0 0 0 0 1 ∆ 0 0 0 1 x k−1 + ∆ 2 /2 0 ∆ 0 0 ∆ 2 /2 0 ∆ u k ,(36)
where x k = [p x,k ,ṗ x,k , p y,k ,ṗ y,k ] T with the position [p x,k , p y,k ] T and the velocity [ṗ x,k ,ṗ y,k ] T , the sampling interval ∆ = 1s, and the state transition noise u k ∼ G(0 2 , 25I 2 ).
We deploy two different types of sensors, with either linear measurement models or nonlinear measurement models, as marked in Fig.1. The linear position measurement model is given as follows
z k = 1 0 0 0 0 0 1 0 x k + v k,1 v k,2 ,(37)
with v k,1 and v k,2 as mutually independent zero-mean Gaussian noise with the same standard deviation of 10.
The FOV (field of view) of each nonlinear sensor is a disc of radius 3000m centralized with the sensor's position [s n,x , s n,y ] T , which fully covers the scenario. The range and bearing measurement is given by
z k =
(p x,k − s n,x ) 2 + (p y,k − s n,y ) 2 arctan (p y,k − s n,y )/(p x,k − s n,x )
+ v k ,(38)
where
v k ∼ N (; 0, R k ), with R k = diag [σ 2 r , σ 2 θ ] T , σ r = 10m, σ θ = π/90 rad/s.
The linear sensors have the same and constant target detect probability p D (x k ) = 0.95 while the nonlinear sensors have p D (x k ) = 0.95N ([|p x,k − s n,x |, |p y,k − s n,y |] T ; 0, 6000 2 I 2 )/N (0; 0, 6000 2 I 2 ). Clutter is uniformly distributed over each sensor's FOV with an average rate of 10 points per scan, which indicates clutter intensity κ k = 10/2000 2 for the linear sensors and κ k = 10/3000/2π for the nonlinear sensors.
Two scenarios have been considered. First, the sensor network consists of only SMC-PHD filters, namely a pure SMC-PHD filter network. Second, each linear sensor operates a GM-PHD filter while each nonlinear sensor operates a SMC-PHD filter, namely a hybrid network consisting of both GM-PHD and SMC-PHD filters. In the next subsection, different C&F schemes are designed for comparison with our approach.
For mixture reduction regarding to the GM: GCs with a weight lower than 10 −4 will be truncated, any two GCs closer than Mahalanobis-distance τ = 4 will be merged and the maximum number of significant GCs to be transmitted and owned by a sensor is 100. The GC is identified as a significant GC if its weight is larger than T c = 0.45 and will be transmitted among neighbors.
The optimal sub-pattern assignment (OSPA) metric [58] is used to evaluate the estimation accuracy of the filter, with cut-off parameter c = 1000 and order parameter p = 2. Furthermore, we define OSPAs over all filtering steps. To evaluate the communication cost, we record a GC that consists of a weight parameter (1 tuple), a 4-dimensional vector mean (4 tuples), and a 4×4 -dimension symmetric matrix covariance (10 tuples) as 15 tuples and the scalevalued cardinality parameter as 1 tuple. In addition, each weighted particle takes 5 tuples (4 for the state vector and 1 for the weight). For the SMC-PHD filter, N p = 200 particles are assigned for each expected target during the resampling scheme to adjust the number of particles in time series. It is worth noting that strategies such as roughening [59] that is to add a small zero-mean random variable to the state of each resampled particle [5], is useful to increase the diversity of particles after resampling for the PF and is adopted in our implementations.
Each simulation was performed 100 runs with independently generated measurement series, each run consisting of 100 filtering iterations. Different numbers t of neighbor/P2P communication iterations from 0 (without applying any communication between sensors) to 10 or 5 are applied to all consensus schemes.
A. Comparison approachs 1) GM-EMD-IS:
The state-of-the-art distributed SMC-PHD fusion given in [27] is based on EMD, which consists of two essential parts: 1) convert the particle distribution to continuous distribution approximations and 2) construct the multitarget EMD. In the former, the work [27] proposed a clustering approach for kernel learning which according to our experience is computationally intensive and unstable. Instead, in our implementation, we use the proposed P2GM strategies (with a much lower threshold T c = 0.1 for selecting a sufficiently large number of GCs for accurate approximation) for generating the continuous kernel distribution (namely GM). In the latter, to construct the multitarget EMD, the IS approach can be applied as is done in our approach to update the fused particles. But there are two key differences:
• In our approach, there is no particle communication between sensors. In the EMD approach [27], the particles are the union of the particle sets that are sampled from neighbor sensors. Because of this, it needs to disseminate both particles and the corresponding kernel/GM function parameters at each communication iteration; • In our approach the target density is the AA of disseminated GMs while for EMD it is the GA of local PHDs. More specifically, denoting the union of uniformly weighted particle sets from neighbor sensors as x (j) a,k , µ a J a,k j=1 (we will present a particle resampling dissemination approach later, which is used here as well), the EMD-IS approach determines the weight of each particle as follows (cf. (23)):
w (j) a,k ∝ b∈{a}∪Na w b x (j) a,k ω b→a ,(39)
where
w b x (j)
a,k is coherent with (32). Also, different to our cardinality AA consensus (34), the cardinality consensus is implemented in the GA sense, i.e.
W a,k = b∈{a}∪Na W b,k ω b→a .(40)
However, we did not employ the sophisticated fusion weight strategy proposed in [27] but still applied the Metropolis weights, which have been widely used for the GCI based GM-PHD fusion since [22]. Therefore, the resulting EMD fusion approach is indeed a modified implementation of that given in [27] and is referred to as "GM-EMD-IS".
Remark 5 The logarithmic fusion is arguably more compelling than the linear fusion for dealing with multi-sensor likelihood fusion as it is external Bayesian [18]. However, as addressed in this paper, the items to be fused are the unknown-cross-correlated PHDs (which is more meaningful to calculate the AA, cf. Remark 2) and the sum of fusion weights is presumed being unity, Bayesian optimality is impossible. Instead, the problem of false and missing data is more prominent in distributed MTT. This together the consideration on communication and computation form the essential reason we advocate AA rather than GA for "average consensus".
2) Particle Resampling Dissemination (PRD): In contrast to the GA that rarely admits closed-form solution for mixture distributions, an exact solution for AA of PHDs can be easily given by disseminating particles. That is, different to the partial consensus as in (18) and (45), complete consensus is sought by linearly averaging the local PHD D a,k (x) at sensor a with that from the neighbors D b,k (x), ∀b ∈ N a , i.e.,
D a,k (x) = b∈{a}∪Na ω b→a D b,k (x) .(41)
However, disseminating all particles requires tremendous communication and will unnecessarily lead to an increasing number of particles at local sensors. To maintain a stable number of particles and to reduce the communication cost, the number J b→a,k of transmitting particles from sensor b ∈ N a to sensor a can be determined as
J b→a,k = [N p W b→a,k ],(42)
where N p = 200 as specified and W b→a,k = ω b→a W b,k , relying on the particle weight sum of the particle-sending sensor b and fusion weights to the particle-receiving sensor a; for ω b→a , we apply the Metropolis weights again. These particles are obtained by unbiased (re)sampling retaining the identical representation of the intial particle posterior [60]. Each resampled particle is weighted as
w (j) b→a,k = W b→a,k J b→a,k ≈ 1 N p .(43)
It is straightforward to have the weight sum of all particles received at sensors (cf. (34)
) b∈{a}∪Na J b→a,k j=1 w (j) b→a,k = b∈{a}∪Na W b→a,k =W a,k(44)
We detail this protocol in Appendix B, with reference to Algorithm 1. As shown, the resampling scheme plays a key role for "selecting" the particles for dissemination, which strives to gain a trade-off between communication cost and approximation accuracy. We refer to this protocol as PRD. Particularly different to the other protocols, no GM is involved. However, for estimate output, we still apply the similar estimate extraction solution as in (14), (15) and (9) from the particle distribution. Then, the component of weight greater than T e = 0.5 will be identified as a target estimate.
3) GM-Re-Sampling (RS): In place of the IS approach, new particles can be re-generated from the disseminated GMs via a standard sampling algorithm, namely recovering particles from GM after the C&F procedure. We refer to this approach as GM-RS, which is realized for the specific purpose to evaluate the effectiveness of the proposed IS approach.
Whether RS or IS, the GM can be merged to control the mixture size or simply remain unchanged (to avoid any fusion error or information double-counting) during their dissemination at each communication iteration, as addressed in Remark 1. In the latter, it is commonly referred to as distributed flooding [7] while in the former we use the conservative GM merging (CGMM) scheme [35]. Overall, on disseminating the GMs, there are different combinations of strategies: either RS or IS and either flooding or CGMM.
4) CC-AA:
The cardinality consensus based on AA (CC-AA) approach only disseminates the weight sum information between sensors to re-scale the weight of particles.
5) No consensus:
We also realize a centralized protocol in which all local filters do not share any information with each other referred to as "no consensus".
As addressed in Section III-E, all these distributed SMC-PHD filtering protocols, except the non-consensus protocol, can extract estimates at two different stages: the BC manner and the AC manner, as addressed in Step 3 and Step 6 in Algorithm 1, respectively. In the former, we simply extract the mean of the extracted GMs as in (14) as the estimates while the number of estimates is determined by (12), which resembles the approaches given in [55], [56]. In the latter, we apply the well-known k−means clustering algorithm to the disseminated GCs for estimate extraction, where the number of clusters is specified by rounding (34) and the estimates are given as the center of the formed clusters.
B. WSN of Pure SMC-PHD Filters
In this case, each sensor runs a SMC-PHD filter individually, using either the linear measurement model/data or the nonlinear measurement model/data.
When t = 5, the Network OSPA, the online estimated number of targets, and the computing time of different consensus protocols for each filtering step are given in the upper row of Fig. 2, separately. For different numbers of communication iterations from t = 0 to t = 10, the time-averaged network OSPA and network communication of different consensus protocols are given in the bottom row of Fig. 2, separately. We have the following key findings with regard to the filtering accuracy, communication and computation cost, respectively: 1) On filtering accuracy:
• All C&F protocols improve the filtering accuracy by reducing the filter OSPA as compared to the protocol applying no consensus. The OSPA reduction basically increases with that of the number of the P2P communication iterations, till to a convergent/consensual level. • Except CC-AA for which both BC and AC outputs are the same, all AC estimates are significantly better than the BC estimates indicating that the C&F is immediately beneficial for improving the local filter accuracy at each filtering step. • The GM flooding protocols that avoids information double-counting outperform the CGMM protocol. • As expected, the CC-AA which is the least communicatively and computationally costly, benefits the filter the least among all distributed C&F schemes. • Roughly speaking, the IS approaches perform similar to the RS approaches in each corresponding category when applying CGMM/Flooding or in BC/AC manners. More precisely, the RS approaches outperform the IS approaches in the BC estimation case (whether Flooding or CGMM is applied), performs very similar in the CGMM-AC case and inferior in the GM flooding-AC case, as compared with the IS approaches.
• Surprisingly, all GM based C&F approaches including CGMM/GM-flooding combined with RS/IS, and EMD, outperform the PRD approach, in both BC and AC manners. The result is intuitively surprising because, despite the communication cost, the particles set contains more complete information of the posterior PHD than the GM extracted from particles. However, more does not alway mean better. The partial consensus by which the sensors share only the significant components of the GM rather than all, is supposed to be help reduce the affection of No. Tuples the false alarm, as has been illustrated in [35]. This leads to an unique advantage that the PRD approach or the complete consensus does not have. • On the AC estimation, the GM-EMD-IS approach performs slightly better than the CGMM-(RS/IS) approaches, inferior to the GM flooding-(RS/IS) approach, while on the BC estimation, the GM-EMD-IS approach performs favorably which is better than all the others (especially for large t).
2) On communication:
• Both the GM-EMD-IS approach and the PRD protocol, which disseminate a large number of weighted particles are unsurprisingly the most communicatively costly. • Except t = 1, the flooding is slightly more communicatively costly than the CGMM which applies mixture reduction during communication. • The CC-AA approach is computationally ignorable.
3) On computation:
• The EMD approach is the most computationally costly while the CC-AA is the least and is ignorable. • The CGMM approaches are more computationally costly than the flooding approaches. • The RS approach is even more computationally costly than the IS approach. • The PRD approach and the GM flooding-IS approach are similar in computational cost and are very efficient. With particular regard to the proposed P2GM based IS approach, we have the following conclusions • The flooding-communication approach achieves the best accuracy benefit among all; in fact, it reaches very close to the maximal accuracy gain that the algorithm converges to by only 1 or 2 communication iterations; • When CGMM is applied during the C&F, the communication cost will be reduced somewhat (depending on how the merging threshold is set) while the computational cost is increased and the accuracy benefit is reduced. • The IS approach performs similar to the RS approach in accuracy and communication cost, but computes faster. The RS approach prevents any parallelization of the filtering calculation and the C&F. • The GA-based EMD approach is significantly more communicatively and computationally intensive than AA based approaches while it only shows insignificant superiority in improving the BC estimation accuracy.
C. Hybrid WSN of SMC-PHD and GM-PHD filters
In this case, we study a hybrid sensor network: each linear sensor operates a GM-PHD filter [35] while each nonlinear sensor operates a SMC-PHD filter. However, the PRD, GM-RS and GM-EMD-IS approaches are based to the PF and do not apply to the GM-PHD filter and therefore, will not be realized here. To integrate the disseminated GM into the local posterior, rather than the IS approach for the SMC-PHD filter (where the local posterior is represented by particles), straightforward GM union is applied for the GM-PHD filter (where the local posterior is a GM). To note, in the distributed GM-PHD filter [35], the fused AA PHD at sensor a is given by linearly averaging the initial complete posterior PHD D a,k (x) (a GM) with the received partial PHD (also a GM, which only represents a part of the corresponding posterior) from the neighbors D b,k,T (x), ∀b ∈ N a , i.e., (cf. (18)) D a,k (x) = ω a→a D a,k (x) + b∈Na ω b→a D b,k,T (x) , (45) which is slightly different to the AA implemented for the SMC-PHD filter in (18) as here, the local sensor contributes the whole PHD D a,k (x) rather than only the partial PHD.
To show the simulation result, similar contents (up to t = 5) given in Fig. 3 correspond to those in Fig. 2, respectively. The results are highly consistent, confirming the effectiveness of our approach for the hybrid filter network. For example,
• All C&F approachs converge with the increase of the number of P2P communication iterations; the proposed IS-AC approach demonstrates again fast convergence; • The AC estimation is more accurate than the BC estimation in all approachs; • The CC-AA is ignorable in either computation or communication; • The CGMM scheme saves communication but costs more computation as compared to the GM flooding approach. However, different to what shown in the last simulation, the GM flooding-IS approach performs very similar with the CGMM-IS approach (except t = 1) in the sense of OSPA reduction. We conjecture that this is because the CGMM does not cause approximation error for the GM-PHD filter as significantly as it did to the SMC-PHD filter based on P2GM. Therefore, appropriate merging does not have to sacrifice the filter accuracy.
V. CONCLUSION We present a "partial, arithmetic average consensus" approach to distributed SMC-PHD fusion. Our approach is composited of two major parts. One part is regarding particlesto-GM conversion, which constructs a GM from the particle set at each local sensor for parameterized information dissemination. The GM represents only the significant part of the particle posterior rather than the complete, for a trade-off between approximation accuracy and communication cost. The disseminated GMs are linearly/arithmetically averaged over the network for consensus. The other part is an importance sampling approach for re-weighting the local particles according to the disseminated GM without changing their states. This allows parallel implementation of the local calculation for the prior PHD and likelihood, and the network communication, combating or even avoiding time delay to the filter. The effectiveness and reliability of our approach have been demonstrated in both regards: the particles-to-GM conversion and the importance sampling.
The proposed distributed SMC-PHD filter can be seamlessly cooperated with the distributed GM-PHD filter, leading to a promising, hybrid sensor framework in which each sensor, using whether linear or nonlinear measurement model, may operate a PHD filter implemented by means of either GM or SMC according to its realistic needs or conditions.
APPENDIX A CARDINALITY ESTIMATION CONDITIONED ON PHD
Denoting the PDF of the multi-target RFS variable X k = x 1 , x 2 , . . . , x n as f (X k ), we have ρ(n) = |X k |=n f (X k )δX k (46) From (5) and R d δ
x (j) k (x)dx = 1, we have R d D k (x)dx ≈ W k(47)
Substituting (1) and (46) to (47) yields
R d D k (x)dx = R d R d δ X k (x)f (X k )δX k dx = R d R d δ X k (x)dxf (X k )δX k = R d nf (X k )δX k = n≥0 nρ(n)(48)
Estimation (48) is known as expected a posteriori (EAP) estimate of the number of targets at time k, i.e.,
N EAP k = R d D k (x)dx ≈ W k .(49)
APPENDIX B PRD BASED DISTRIBUTED SMC-PHD FILTER Algorithm 2 Distributed SMC-PHD Iteration at sensor a based on particle resampling and dissemination Input and Output are the same to Algorithm 1.
Procedure:
Step 1-Step 3 are the same as that of Algorithm 1.
Step 4 Resampling: • Calculate the local particle weight sum W a,k . • Resample from ξ ′ a,k to get |N a | + 1 new particle sets ξ a→b,k x (j) a,k , µ a J a→b,k j=1 for b ∈ {a} ∪ N a as in (42) and (43).
Step 5 Partial consensus via particle dissemination:
• Disseminate the weighted particles ξ a→b,k together with parameter W a→b,k , and the local BC estimates yielded in Step 3 to sensor b ∈ N a ; simultaneously, receive their transmissions. • CalculateW a,k as in (34) and use it to replace W a,k . • The resulting particle set ξ a,k is given by ξ a,k = ∪ b∈{a}∪Na ξ b→a,k
Step 6: Iteration, if necessary:
• Steps 4 and 5 may be carried out for multiple iterations (set ξ ′ a,k ← ξ a,k before redo resampling).
Step 7 Delayed/AC estimate extraction:
• Similar to Step 6 of Algorithm 1, but differently the items to be clustered are the BC estimates obtained in Step 3.
as in(23), and calculateW a,k as in (34); • Calculate w (j),AC a,k J a,k j=1 as in(35) as the final AC weight of each particle.Step 6 Delayed/AC estimate extraction:• Apply the k−means clustering on the gathered GCs and extract the centroid of each cluster as estimates, with the number of estimates given by [W a,k ].
Fig. 1 .
1Tracking scenario: target trajectories (starting at '△' and ending at ' ') and a sensor network consisting of both linear and nonlinear sensors. IV. SIMULATIONS The simulations are set up in a scenario over the planar region [−1000, 1000]m × [−1000, 1000]m which is monitored fully by a connected undirected sensor network. The trajectories of totally 6 targets are given in Fig. 1 with the starting and ending times of each trajectory noted. The target birth process follows a Poisson RFS with intensity function γ k (x) = 3 i=1 λ i N (.; m i , Q r ), with Poisson rates λ 1 = λ 2 = λ 3 = 0.05 and the Gaussian parameters m 1 = [0, 0, 950, −30] T , m 2 = [−100, 10, −800, 30] T , m 3 = [−800, 20, −500, 0] T , and Q r = diag([100, 25, 100, 25] T ), where diag(a) represents a diagonal matrix with diagonal a.
•
Network OSPA: the average of OSPAs obtained by all sensors in the network at each sampling step; • Time-average Network OSPA: the average of the Network
Fig. 2 .
2Average performance of 10 SMC-PHD filters when different number of P2P communication iterations are performed between them. Their connection is shown in Fig.1. The upper row: Network OSPA (shown in the left), online estimated number of targets (middle) and computing time (right) of different consensus protocols for each filtering step, respectively, when 5 P2P communication iterations are applied between neighbors. The bottom row: Time-averaged network OSPA (shown in the left) and network communication cost (middle) for different numbers of communication iterations. CC-AA: Cardinality consensus based on arithmetic average; CGMM: Conservative Gaussian mixture merging [35], PRD: Particle resampling dissemination; RS: Re-sampling; IS: Importance sampling; BC: Before consensus; AC: After consensus; EMD: Exponential mixture density.
Fig. 3 .
3Average performance of 5 SMC-PHD filters and 5 GM-PHD filters when different number of communication iterations are performed between them. The upper row: Network OSPA (shown in the left), online estimated number of targets (middle) and computing time (right) of different consensus protocols for each filtering step, respectively, when 5 P2P communication iterations are applied. The bottom row: Time-averaged network OSPA (shown in the left) and network communication cost (middle) for different numbers of communication iterations. CC-AA: Cardinality consensus based on arithmetic average; CGMM: Conservative Gaussian mixture merging [35], IS: Importance sampling; BC: Before consensus; AC: After consensus.
ACKNOWLEDGMENT This draft of this work has been discussed with Prof. Franz Hlawatsch who directed the author's secondment at the Institute of Telecommunications, Vienna University of Technology from July to September 2017. The author would like to acknowledge his comments and suggestion. This work has also been presented in part at the ATR laboratory in NUDT, Changsha China on Dec. 12 2017, which was invited by Dr. Hongqi Fan.
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2013 and his second Doctoral degree in Mechatronics Engineering from Northwestern Polytechnical University (China), in 2015. He has been a Research Associate with the BISITE Research Group, University of Salamanca (Spain) since June 2014 and presently, he holds a Marie Skodowska-Curie Individual Fellowship (H2020-MSCA-IF-2015) with the Host. Harbin Engineering University (China) ; his first PhD in Electrical and Electronic Engineering from London South Bank University (United Kingdom ; University of Salamanca and the Partner Vienna University of Technology (AustriaTiancheng Li received his Bachelors degree in Mechanical and Electrical EngineeringTiancheng Li received his Bachelors degree in Mechanical and Electrical Engineering, with a minor degree in Business Administration, from Harbin Engineering University (China), in 2008, his first PhD in Electrical and Electronic Engineering from London South Bank University (United Kingdom), in 2013 and his second Doctoral degree in Mecha- tronics Engineering from Northwestern Polytech- nical University (China), in 2015. He has been a Research Associate with the BISITE Research Group, University of Salamanca (Spain) since June 2014 and presently, he holds a Marie Skodowska-Curie Individual Fellowship (H2020-MSCA-IF-2015) with the Host University of Salamanca and the Partner Vienna University of Technology (Austria).
His research interests lie in the general area of statistical signal processing and distributed information fusion, with particular emphasis on novel Markovfree solutions for realistic multisensor multiobject detection, tracking and forecasting by means of sensor data clustering. He has been on the Editorial Board of two peer-reviewing journals: FITEE. 17Advances in Distributed Computing and Artificial Intelligence Journal) as well as on the organizing and/or technical program committee of several international symposiums and conferences including FUSION. fitting and miningHe has been on the Editorial Board of two peer-reviewing journals: FITEE (2017-, Frontiers of Information Technology & Electronic Engineering) and ADCAIJ (2016-, Advances in Distributed Computing and Artificial Intel- ligence Journal) as well as on the organizing and/or technical program committee of several international symposiums and conferences including FUSION 2014-17, ACM-SAC 2015-17 and DCAI 2015-18. His research interests lie in the general area of statistical signal processing and distributed information fusion, with particular emphasis on novel Markov- free solutions for realistic multisensor multiobject detection, tracking and forecasting by means of sensor data clustering, fitting and mining.
| []
|
[]
| [
"Willi-Hans Steeb ",
"Yorick Hardy [email protected] \nDepartment of Mathematical Sciences\nUniversity of South Africa\nPretoriaSouth Africa\n",
"\nInternational School for Scientific Computing\nUniversity of Johannesburg\n2006Auckland ParkSouth Africa\n"
]
| [
"Department of Mathematical Sciences\nUniversity of South Africa\nPretoriaSouth Africa",
"International School for Scientific Computing\nUniversity of Johannesburg\n2006Auckland ParkSouth Africa"
]
| []
| Pauli spin matrices, Pauli group, commutators, anti-commutators and the Kronecker product are studied. Applications to eigenvalue problems, exponential functions of such matrices, spin Hamilton operators, mutually unbiased bases, Fermi operators and Bose operators are provided. | null | [
"https://arxiv.org/pdf/1405.5749v2.pdf"
]
| 119,331,620 | 1405.5749 | 79c35176d42450082830170ef371069a96503d40 |
2 Jun 2014
Willi-Hans Steeb
Yorick Hardy [email protected]
Department of Mathematical Sciences
University of South Africa
PretoriaSouth Africa
International School for Scientific Computing
University of Johannesburg
2006Auckland ParkSouth Africa
2 Jun 2014Spin Operators, Pauli Group, Commutators, Anti-Commutators, Kronecker product and Applications
Pauli spin matrices, Pauli group, commutators, anti-commutators and the Kronecker product are studied. Applications to eigenvalue problems, exponential functions of such matrices, spin Hamilton operators, mutually unbiased bases, Fermi operators and Bose operators are provided.
Introduction
We investigate the commutators and anticommutators of Pauli spin matrices and their Kronecker product. Let
σ 1 = 0 1 1 0 , σ 2 = 0 −i i 0 , σ 3 = 1 0 0 −1
be the Pauli spin matrices. We include σ 0 = I 2 , where I 2 is the 2 × 2 unit matrix. The Pauli spin matrices are unitary and hermitian with eigenvalues +1 and −1.
Then the spin matrices are given by s 1 = 1 2 σ 1 , s 2 = 1 2 σ 2 , s 3 = 1 2 σ 3 with eigenvalues +1/2 and −1/2. For the Pauli spin matrices we find that σ 1 σ 2 = iσ 3 , σ 2 σ 3 = iσ 1 , σ 3 σ 1 = iσ 2 and the commutators are given by [σ 1 , σ 2 ] = 2iσ 3 , [σ 2 , σ 3 ] = 2iσ 1 , [σ 3 , σ 1 ] = 2iσ 2 .
The matrices iσ 1 , iσ 2 , iσ 3 form a basis of the simple Lie algebra su (2). The anticommutators of the Pauli spin matrices vanish, i.e.
[σ 1 , σ 2 ] + = 0 2 , [σ 2 , σ 3 ] + = 0 2 , [σ 3 , σ 1 ] + = 0 2 where 0 2 is the 2 ×2 zero matrix. Here we study the commutators and anticommutators of the Kronecker product ⊗ of the Pauli spin matrices [1]. Thus we study the commutator and anticommutator of the 2 n × 2 n unitary matrices of the form
(−i) j 0 n t=1 σ jt
where j 0 ∈ {0, 1, 2, 3} and j t ∈ {0, 1, 2, 3}. These matrices are elements of the Pauli group [2], [3]. Furthermore the square of the matrices n t=1 σ jt is the 2 n × 2 n unit matrix. Then
Π 1 = 1 2 I 2 n + n t=1 σ jt , Π 2 = 1 2 I 2 n − n t=1
σ jt are 2 n × 2 n projection matrices. Whether the commutator or anticommutator of such matrices vanishes is helpful for the eigenvalue problem of such matrices and for the calculation of the exponential function of such matrices. Other applications discussed concern spin-Hamilton operators and the projection to sub-Hilbert space and mutually unbiased bases. Finally application with Fermi operators are described.
Kronecker Product of Pauli Spin Matrices
Let us first give some examples where the commutator or the anticommutator vanishes. Consider first the 4 × 4 matrices
σ 12 := σ 1 ⊗ σ 2 , σ 23 := σ 2 ⊗ σ 3 , σ 31 := σ 3 ⊗ σ 1 .
Then we find that the commutators vanish, i.e.
= −2σ 3 ⊗ σ 1 = −2σ 31 [σ 23 , σ 31 ] + = 2i 0 0 0 1 0 0 −1 0 0 1 0 0 −1 0 0 0 = −2σ 1 ⊗ σ 2 = −2σ 12 [σ 31 , σ 12 ] + = 2i 0 0 1 0 0 0 0 −1 −1 0 0 0 0 1 0 0 = −2σ 2 ⊗ σ 3 = −2σ 23 .
Consider now the three hermitian and unitary 8 × 8 matrices
σ 123 := σ 1 ⊗ σ 2 ⊗ σ 3 , σ 312 := σ 3 ⊗ σ 1 ⊗ σ 2 , σ 231 := σ 2 ⊗ σ 3 ⊗ σ 1 .
The commutators are non-zero and we find the skew-hermitian invertible 8 × 8 matrices
[σ 123 , σ 312 ] = 2 0 2 0 2 σ 1 0 2 0 2 0 2 0 2 −σ 1 −σ 1 0 2 0 2 0 2 0 2 σ 1 0 2 0 2 = 2iσ 2 ⊗ σ 3 ⊗ σ 1 = 2iσ 231 [σ 312 , σ 231 ] = 2 0 2 0 2 0 2 σ 3 0 2 0 2 −σ 3 0 2 0 2 σ 3 0 2 0 2 −σ 3 0 2 0 2 0 2 = 2iσ 1 ⊗ σ 2 ⊗ σ 3 = 2iσ 123 [σ 231 , σ 123 ] = 2i 0 2 σ 2 0 2 0 2 σ 2 0 2 0 2 0 2 0 2 0 2 0 2 −σ 2 0 2 0 2 −σ 2 0 2 = 2iσ 3 ⊗ σ 1 ⊗ σ 2 = 2iσ 312 .
However the anti-commutators vanish, i.e.
[σ 123 , σ 312 ]
+ = 0 8 , [σ 312 , σ 231 ] + = 0 8 , [σ 231 , σ 123 ] + = 0 8 .
As mentioned above for the Pauli spin matrices σ 1 , σ 2 , σ 3 we find that the anticommutators vanish and the commutators are given by [σ 1 ,
σ 2 ] = iσ 3 , [σ 2 , σ 3 ] = iσ 1 , [σ 3 , σ 1 ] = iσ 2 .
Consider now the three unitary and hermitian matrices
σ 11 := σ 1 ⊗ σ 1 , σ 22 := σ 2 ⊗ σ 2 , σ 33 := σ 3 ⊗ σ 3 .
Then the commutators vanish, i.e. [σ 11 ,
σ 22 ] + = −2σ 3 ⊗ σ 3 = −2σ 33 [σ 22 , σ 33 ] + = −2σ 1 ⊗ σ 1 = −2σ 11 [σ 33 , σ 11 ] + = −2σ 2 ⊗ σ 2 = −2σ 22 .
Now consider the 8 × 8 hermitian and unitary matrices
σ 111 := σ 1 ⊗ σ 1 ⊗ σ 1 , σ 222 := σ 2 ⊗ σ 2 ⊗ σ 2 , σ 333 := σ 3 ⊗ σ 3 ⊗ σ 3 .
Here the anticommutators vanish, i.e.
[σ 111 , σ 222 ] = −2iσ 3 ⊗ σ 3 ⊗ σ 3 = −2iσ 333 [σ 222 , σ 333 ] = −2iσ 1 ⊗ σ 1 ⊗ σ 1 = −2iσ 111 [σ 333 , σ 111 ] = −2iσ 2 ⊗ σ 2 ⊗ σ 2 = −2iσ 222 .
Consider now the general case of the three unitary and hermitian matrices
σ 11...1 , σ 22...2 , σ 33...3
with n Kronecker products. If n is odd the three matrices form a basis of a simple Lie algebra. For n odd the anti-commutators vanish. If n is even the commutators vanish and the anti-commutators can be expressed as Kronecker products of σ 1 , σ 2 and σ 3 .
Another useful case is that
[σ 1 ⊗ σ 2 , σ 3 ⊗ σ 3 ] = [σ 2 ⊗ σ 1 , σ 3 ⊗ σ 3 ] = 0 4 .
This implies that
[σ 0 ⊗ σ 0 ⊗ σ 1 ⊗ σ 0 ⊗ · · · σ 0 ⊗ σ 2 ⊗ σ 0 ⊗ · · · ⊗ σ 0 , σ 3 ⊗ σ 3 ⊗ · · · ⊗ σ 3 ] = 0 2 n
with σ 1 and σ 2 at the j'th and k'th position (j = k) with j, k = 1, 2, . . . , n.
From the commutators given above we can also infer that the Hamilton operator
H = J 12 4 n j=1 (σ 1,j σ 1,j+1 + σ 2,j σ 2,j+1 ) + J 3 4 n j=1 (σ 3,j σ 3,j+1 )
commutes with σ 3 ⊗ σ 3 ⊗ · · · ⊗ σ 3 for both both open end boundary conditions and periodic boundary conditions, where σ α,j = σ 0 ⊗ · · · σ 0 ⊗ σ α ⊗ σ 0 ⊗ · · · ⊗ σ 0 with σ α (α = 1, 2, 3) at the j-th position.
Pauli Group, Commutator and Anticommutator
The n-qubit Pauli group is defined by
P n := { I 2 , σ 1 , σ 2 , σ 3 } ⊗n ⊗ { ±1, ±i }
where σ 1 , σ 2 , σ 3 are the 2 × 2 Pauli matrices and σ 0 ≡ I 2 is the 2 × 2 identity matrix. The dimension of the Hilbert space under consideration is dim H = 2 n . Thus each element of the Pauli group P n is (up to an overall phase ±1, ±i) a Kronecker product of Pauli matrices and 2 × 2 identity matrices acting on n qubits. The order of the Pauli group is 2 2n+2 . Thus for n = 1 we have the order 16.
Let j 0 , . . . , j n , k 0 , . . . , k n ∈ {0, 1, 2, 3} and
A = (−i) j 0 n t=1 σ jt , B = (−i) k 0 n t=1 σ kt .
Using the fact that σ j σ k = δ j,k I 2 + i
A = σ 1 ⊗ σ 1 ⊗ σ 1 , B = σ 3 ⊗ σ 3 ⊗ σ 3 with [A, B] + = 0 8 . Thus since +1 is a eigenvalue of A we find that −1 is an eigenvalue of A with the eigenvector (σ 3 ⊗ σ 3 ⊗ σ 3 )v where Av = v.
One of the main calculations in quantum theory in the Hilbert space C n is to find
e A Be −A ,λ j I dim(V j )
.
From e A Be −A = B we obtain Λ = Ue A Be −A U * = (Ue A U * )Λ(Ue A U * ) −1 .
Since the eigenvalues λ j are distinct we have There is also a lesser known expansion using the anti-commutator ( [4], [5])
e A = U * m j=1 P j U where each P j is an invertible dim(V j ) × dim(V j ) matrix. When A = (−i) j 0 n t=1 σ jt , B = (−i) k 0e A Be A = B + [A, B] + + 1 2! [A,Let A = σ 111 = σ 1 ⊗ σ 1 ⊗ σ 1 , B = σ 222 = σ 2 ⊗ σ 2 ⊗ σ 2 .
Then [A, B] + = 0 8 and A 2 = I 8 so that
e A Be −A = e 2A B = (cosh(2)I 8 + sinh(2)σ 1 ⊗ σ 1 ⊗ σ 1 )(σ 2 ⊗ σ 2 ⊗ σ 2 ).
Another application is for spin-Hamilton operators and projection matrices. Let A be an hermitian d × d matrix with A 2 = I d . Then
Π + = 1 2 (I d + A), Π − = 1 2 (I d − A)
are projection matrices which can be used to decompose the Hilbert space C d into invariant sub Hilbert spaces. Consider for example the spin-Hamilton operatorŝ
H = 2 j=1 (σ j ⊗ σ j ⊗ I 2 + I 2 ⊗ σ j ⊗ σ j ) andK = 2 j=1 (σ j ⊗ σ j ⊗ I 2 + I 2 ⊗ σ j ⊗ σ j + σ j ⊗ I 2 ⊗ σ j ).
Then bothĤ andK commute with the operator σ 3 ⊗ σ 3 ⊗ σ 3 which is an element of the Pauli group with (σ 3 ⊗ σ 3 ⊗ σ 3 ) 2 = I 8 . Thus we have projection matrices
Π + = 1 2 (I 8 + σ 3 ⊗ σ 3 ⊗ σ 3 ), Π − = 1 2 (I 8 − σ 3 ⊗ σ 3 ⊗ σ 3 )
which decomposes the Hilbert space C 8 into two four-dimensional sub Hilbert spaces. Then the eigenvalue problem can be solved in these sub Hilbert spaces. | e j , f k | = 1 √ d .
For the Pauli spin matrices σ 3 , σ 1 , σ 2 the normalized eigenvectors
B 3 = 1 0 , 0 1 B 1 = 1 √ 2 1 1 , 1 √ 2 1 −1 B 2 = 1 √ 2 1 i , 1 √ 2 1 −i
each form an orthonormal basis in C 2 . Furthermore this is a set of mutually unbiased bases. Consider now σ 3 ⊗ σ 3 , σ 1 ⊗ σ 1 , σ 2 ⊗ σ 2 . Then
B 3 = 1 0 ⊗ 1 0 , 1 0 ⊗ 0 1 , 0 1 ⊗ 1 0 , 0 1 ⊗ 0 1 B 1 = 1 2 1 1 ⊗ 1 1 , 1 2 1 1 ⊗ 1 −1 , 1 2 1 −1 ⊗ 1 1 , 1 2 1 −1 ⊗ 1 −1 B 2 = 1 2 1 i ⊗ 1 i , 1 2 1 i ⊗ 1 −i , 1 2 1 −i ⊗ 1 i , 1 2 1 −i ⊗ 1 −i
[σ 12
12, σ 23 ] = 0 4 , [σ 23 , σ 31 ] = 0 4 , [σ 31 , σ 12 ] = 0 4 where 0 4 is the 4 × 4 zero matrix. For the anticommutators we find the hermitian invertible matrices which can expressed as Kronecker products of the Pauli spin matrices [σ 12 , σ 23 ]
[σ 11 , σ 22 ] = 0 4 , [σ 22 , σ 33 ] = 0 4 , [σ 33 , σ 11 ] = 0 4 and the anticommutators can be written as Kronecker products, i.e.
[σ 111 , σ 222 ] + = 0 8 , [σ 222 , σ 333 ] + = 0 8 , [σ 333 , σ 111 ] + = 0 8 and the commutators can be written as Kronecker products, i.e.
ǫ
, using ǫ jt,kt,lt = −ǫ kt,jt,lt , AB = (−i) jt,kt,lt σ lt .Now, noting thatδ jt,kt I 2 − i 3 lt=1 ǫ jt,kt,lt σ lt = (−1) 1−δ j t ,k t δ jt,kt I 2 + i 3 lt=1 ǫ jt,kt,lt σ lt ,we find the following expressions for the commutator and anticommutator of A and B[A, B] = (−i) j 0 +k 0 δ j t ,k t = (−1) n .Consequently [A, B] = 0 if and only if the number of coincidences j t = k t is even if n is even, and odd when n is odd. Similarly [A, B] + = 0 if and only if the number of coincidences j t = k t is even if n is odd, and odd when n is even.4 Applications First we look at the eigenvalue problem. Let A, B be two nonzero n × n matrices. Let Av = λv (v = 0) be the eigenvalue equation. If [A, B] = 0 n , then A(Bv) = λ(Bv). Consequently if Bv = 0, then Bv is an eigenvector of the matrix A. Now let us assume that the anti-commutator vanishes, i.e [A, B] + = 0 n . Then we obtain A(Bv) = −λ(Bv). Thus if Bv = 0, then Bv is an eigenvector of A corresponding to the eigenvalue −λ. An application is given in section 2 with
where A, B are n × n matrices. This is utilized in the solution of the Heisenberg equation of motion. Now it is well-known that e A Be −A = B + [A, If the commutator [A, B] vanishes, we find e A Be −A = B. Suppose B is normal with spectral decomposition B = m j=1 λ j Π j where λ j are the m distinct eigenvalues of B and Π j are the projections onto the corresponding eigenspaces V j . There exists a unitary n × n matrix U such that Λ := UBU * = m j=1
(
with A, B = I 2 n ) we find m = 2 and dim(V 1 ) = dim(V 2 ) = 2 n−1 so that A = V [I 2 n−1 ⊕ (−I 2 n−1 )]V * and B = U[I 2 n−1 ⊕ (−I 2 n−1 )]U * for some unitary U and V . Thus e A = V [eI ⊕ (1/e)I]V * = U * [P 1 ⊕ P 2 ]U. It follows that e A Be −A = B if (UV )[eI ⊕ (1/e)I](UV ) * is a direct sum of two 2 n−1 × 2 n−1 matrices.
If the anti-commutator of A and B vanishes we obtain e A Be −A = Be −2A and e A Be −A = e 2A B.[A, B] + ] + +
1
3!
[A, [A, [A, B] + ] + ] + + · · ·
It follows that
e A Be −A = (B + [A, B] + +
1
2!
[A, [A, B] + ] + +
1
3!
[A, [A, [A, B] + ] + ] + + · · ·)e −2A
e A Be −A = e 2A (B − [A, B] + +
1
2!
[A, [A, B] + ] + −
1
3!
[A, [A, [A, B] + ] + ] + + · · ·)
Two orthogonal bases in the Hilbert spaceC d A = { e 1 , . . . , e d } , B = { f 1 , . . . , f d } are called unbiased if for every 1 ≤ j, k ≤ d ([6],[7],[8])
AcknowledgmentThe authors are supported by the National Research Foundation (NRF), South Africa. This work is based upon research supported by the National Research Foundation. Any opinion, findings and conclusions or recommendations expressed in this material are those of the author(s) and therefore the NRF do not accept any liability in regard thereto.
Matrix Calculus and Kronecker Product: A Practical Approach to Linear and Multilinear Algebra. W.-H Steeb, Y Hardy, World ScientificSingapore2nd editionW.-H. Steeb and Y. Hardy, Matrix Calculus and Kronecker Product: A Practi- cal Approach to Linear and Multilinear Algebra, 2nd edition, World Scientific, Singapore (2011)
M A Nielsen, I L Chuang, Quantum Computing and Quantum Information. CambridgeCambridge University PressM. A. Nielsen and I. L. Chuang, Quantum Computing and Quantum Infor- mation, Cambridge University Press, Cambridge (2000)
W.-H Steeb, Y Hardy, Quantum Mechanics using Computer Algebra. SingaporeWorld Scientific2nd editionW.-H. Steeb and Y. Hardy, Quantum Mechanics using Computer Algebra, 2nd edition, World Scientific, Singapore (2010)
Anticommutator analogue of the Baker-Hausdorff lemma. I Mendaš, P Milutinović, J. Phys. A: Math. Gen. 22I. Mendaš and P. Milutinović, "Anticommutator analogue of the Baker- Hausdorff lemma", J. Phys. A: Math. Gen. 22, L687-L689 (1989)
W.-H Steeb, I Tanski, Y Hardy, Groups, Lie Groups and Lie Algebras with Applications. SingaporeWorld ScientificW.-H. Steeb, I. Tanski and Y. Hardy, Groups, Lie Groups and Lie Algebras with Applications, World Scientific, Singapore (2012)
A Fourier analytic approach to the problem of mutually unbiased bases. M Matolcsi, arXiv:1009.2407v1M. Matolcsi, "A Fourier analytic approach to the problem of mutually unbi- ased bases", arXiv:1009.2407v1
Geometrical view of the Mean King Problem. M Revzen, arXiv:1205.5406v1M. Revzen, "Geometrical view of the Mean King Problem", arXiv:1205.5406v1
A geometrical relation between symmetric operators and mutually unbiased operators. A Kalev, arXiv:1305.6044v1A. Kalev, "A geometrical relation between symmetric operators and mutually unbiased operators", arXiv:1305.6044v1
| []
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[]
| [
"Tatyana Ivanova s:[email protected] \nDepartment of Mathematical logic and its applications\nFaculty of Mathematics and Informatics\nSofia University\n\n",
"Dimiter Vakarelov \nDepartment of Mathematical logic and its applications\nFaculty of Mathematics and Informatics\nSofia University\n\n"
]
| [
"Department of Mathematical logic and its applications\nFaculty of Mathematics and Informatics\nSofia University\n",
"Department of Mathematical logic and its applications\nFaculty of Mathematics and Informatics\nSofia University\n"
]
| []
| The notion of contact algebra is one of the main tools in the region based theory of space. It is an extension of Boolean algebra with an additional relation C called contact. The elements of the Boolean algebra are considered as formal representations of spatial regions as analogs of physical bodies and Boolean operations are considered as operations for constructing new regions from given ones and also to define some mereological relations between regions as part-of, overlap and underlap. The contact relation is one of the basic mereotopological relations between regions expressing some topological nature. It is used also to define some other important mereotopological relations like non-tangential inclusion, dual contact, external contact and others. Most of these definitions are given by means of the operation of Boolean complementation. There are, however, some problems related to the motivation of the operation of Boolean complementation. In order to avoid these problems we propose a generalization of the notion of contact algebra by dropping the operation of complement and replacing the Boolean part of the definition by distributive lattice. First steps in this direction were made in[8,9]presenting the notion of distributive contact lattice based on contact relation as the only mereotopological relation. In this paper we consider as nondefinable primitives the relations of contact, nontangential inclusion and dual contact, extending considerably the language of distributive contact lattices. Part I of the paper is devoted to a suitable axiomatization of the new language called extended distributive contact lattice (EDC-lattice) by means of universal first-order axioms true in all contact algebras. EDClattices may be considered also as an algebraic tool for certain subarea of mereotopology, called in this paper distributive mereotopology. The main result of Part I of the paper is a representation theorem, stating that each EDC-lattice can be isomorphically embedded into a contact algebra, showing in this way that the presented axiomatization preserves the meaning of mereotopological relations without considering Boolean complementation. Part II of the paper is devoted to topological representation theory of EDC-lattices, transferring into the distributive case important results from the topological representation theory of contact algebras. It is shown that under minor additional assumptions on distributive lattices as extensionality of the definable relations of overlap or underlap one can preserve the good topological interpretations of regions as regular closed or regular open sets in topological space. | 10.1007/s10472-016-9499-5 | [
"https://arxiv.org/pdf/1901.10442v1.pdf"
]
| 17,126,852 | 1901.10442 | 8004750c6fb1e2caa1e4bd78c0dce199dca01102 |
29 Jan 2019
Tatyana Ivanova s:[email protected]
Department of Mathematical logic and its applications
Faculty of Mathematics and Informatics
Sofia University
Dimiter Vakarelov
Department of Mathematical logic and its applications
Faculty of Mathematics and Informatics
Sofia University
29 Jan 2019DISTRIBUTIVE MEREOTOPOLOGY: Extended Distributive Contact Latticesmereotopologydistributive mereotopologycontact algebrasdistributive contact algebrasextended distributive contact algebrastopological representations
The notion of contact algebra is one of the main tools in the region based theory of space. It is an extension of Boolean algebra with an additional relation C called contact. The elements of the Boolean algebra are considered as formal representations of spatial regions as analogs of physical bodies and Boolean operations are considered as operations for constructing new regions from given ones and also to define some mereological relations between regions as part-of, overlap and underlap. The contact relation is one of the basic mereotopological relations between regions expressing some topological nature. It is used also to define some other important mereotopological relations like non-tangential inclusion, dual contact, external contact and others. Most of these definitions are given by means of the operation of Boolean complementation. There are, however, some problems related to the motivation of the operation of Boolean complementation. In order to avoid these problems we propose a generalization of the notion of contact algebra by dropping the operation of complement and replacing the Boolean part of the definition by distributive lattice. First steps in this direction were made in[8,9]presenting the notion of distributive contact lattice based on contact relation as the only mereotopological relation. In this paper we consider as nondefinable primitives the relations of contact, nontangential inclusion and dual contact, extending considerably the language of distributive contact lattices. Part I of the paper is devoted to a suitable axiomatization of the new language called extended distributive contact lattice (EDC-lattice) by means of universal first-order axioms true in all contact algebras. EDClattices may be considered also as an algebraic tool for certain subarea of mereotopology, called in this paper distributive mereotopology. The main result of Part I of the paper is a representation theorem, stating that each EDC-lattice can be isomorphically embedded into a contact algebra, showing in this way that the presented axiomatization preserves the meaning of mereotopological relations without considering Boolean complementation. Part II of the paper is devoted to topological representation theory of EDC-lattices, transferring into the distributive case important results from the topological representation theory of contact algebras. It is shown that under minor additional assumptions on distributive lattices as extensionality of the definable relations of overlap or underlap one can preserve the good topological interpretations of regions as regular closed or regular open sets in topological space.
Introduction
In this paper we continue the research line started in the publications [8,9], devoted to certain non-classical approach to the region-based theory of space (RBTS), which roots goes back mainly to Whitehead [30]. In contrast to the classical Euclidean approach, in which the notion of point is taken as one of the basic primitive notions in geometry and geometric figures are considered as sets of points, RBTS adopts as primitives the more realistic spatial notion of region (as an abstraction of spatial or physical body), together with some basic relations and operations on regions. Some of these relations come from mereology (see [20]): e.g., part-of (x ≤ y), overlap (xOy), its dual underlap (x Oy), and some others definable in terms of these. RBTS extends classical mereology by considering some new relations among regions which are topological in nature, such as contact (xCy), nontangential part-of (x ≪ y), dual contact (x Cy), and some others definable by means of the contact and part-of relations. This is one of the reasons that the extension of mereology with these new relations is commonly called mereotopology. There is no clear difference in the literature between RBTS and mereotopology, and by some authors RBTS is related rather to the so called mereogeometry, while mereotopology is considered only as a kind of point-free topology, considering mainly topological properties of things. In this paper we consider all these names almost as synonyms representing collections of various point-free theories of space. According to Whitehead the point-free approach to space should not disregard points at all -on the contrary, they are suitable high level abstractions which, as such, should not be put on the base of the theory, but have to be definable by means of the other primitive notions of the theory. The Whitehead's criticism is based on the fact that points, as well as the other primitive notions in Euclidean geometry like lines and planes, do not have separate existence in reality, while for instance, spatial bodies as cubes, prisms, pyramids, balls, etc are things having analogs in reality. In this sense the point-free approach to space can be considered as certain equivalent re-formulation of the classical point-based approach by means of more realistic primitive notions.
Survey papers about RBTS (and mereotopology) are [24,4,14] (see also the handbook [1] and [3] for some logics of space). Let us mention that in a sense RBTS had been reinvented in computer science, because of its more simple way of representing qualitative spatial information and in fact it initiated a special field in Knowledge Representation (KR) called Qualitative Spatial Representation and Reasoning (QSRR). One of the most popular systems in QSRR is the Region Connection Calculus (RCC) introduced in [18]. Note that RCC influenced various investigations in the field both of theoretical and applied nature. Survey papers about applications of RBTS and mereotopology in various applied areas are, for instance, [5] and the book [16].
Let us note that one of the main algebraic tools in mereotopology is the notion of contact algebra, which appears in the literature under different names and formulations as extensions of Boolean algebra with some mereotopological relations [27,21,25,26,4,10,6,7]. The simplest system, called just contact algebra was introduced in [6] as an extension of Boolean algebra B = (B, 0, 1, ., +, * ) with a binary relation C called contact and satisfying several simple axioms: The elements of the Boolean algebra are called regions and the Boolean operations can be considered as some constructions of new regions by means of given ones. In this definition Boolean algebra stands for the mereological component, while the contact relation C stands for the mereotopological component of the system. For instance the mereological relations overlap O, underlap (dual overlap) O and part-of ≤ have the following definitions: aOb ↔ def a.b = 0, a Ob ↔ def a + b = 1 and ≤ is just the lattice ordering. The unite element 1 is the region containing as its parts all regions, and the zero region 0 symbolize the non-existing region and can be used to define the ontological predicate of existence: a exists ↔ def a = 0. According to these definitions the axiom (C1) says that if a and b are in a contact then they exist, and axiom (C5) says that overlapping regions are in a contact.
(
By means of the contact relation one can define other mereotopological relations: dual contact a Cb ↔ def a * Cb * , non-tangential part-of a ≪ b ↔ def aCb * , and some others.
Intuitively if we consider regions as certain sets of points, then contact aCb means that a and b share a common point, part-of a ≤ b means that all points of a are points of b, overlap aOb means that a and b share an existing region (just a.b = 0 is a part both of a and of b), underlap a Ob means that there exists a non-universal region containing both a and b (just a + b = 1 contains both a and b).
Let us note that standard model of Boolean algebra is the algebra of subsets of a given universe, so in such a model regions are pure sets and the mereological relations between regions are just the Boolean relations between sets. In this model one can not distinguish boundary and internal points of a given region and hence it can not express all kinds of contact, for instance, the so called external contact in which the contacting regions share only a boundary point (external contact is definable by the formula aCb ∧ aOb). For this reason standard point models of contact algebras are of topological nature and consist of the Boolean algebras of regular closed sets in a given topological space and the contact between two such sets means that they have a common point. Another topological model of contact algebra is the Boolean algebra of regular open sets of a topological space, but in this model contact is not so intuitive and is definable by the formula: aCb ↔ def Cl(a) ∩ Cl(b) = ∅, where Cl(a) is the topological closure operation. Let us mention that the topological representation theory of contact algebras can be treated just as a realization of the Whitehead's idea of defining points and of recreation the point-based structure of the corresponding kind of space within a point-free system (see, for instance, the surveys [24,4]).
One of the motivations to put Boolean algebra on the base of the notion of contact algebra is based on the remark given by Tarski (see for this [20], page 25) that one of the most popular mereological systems, namely the system of Lesniewski, can be identified with the complete Boolean algebra with zero deleted. If we are not interested in infinite unions and intersections then we can accept just Boolean algebra (with zero considered as non-existing region, as mentioned above). In the papers [8,9] a generalization of the notion of contact algebra is presented just by replacing the Boolean algebra by means of a (bounded) distributive lattice and obtaining in this way the notion of distributive contact lattice. Some motivations for this generalization are the following. First, that Boolean algebra is a bounded distributive lattice and that the axioms of the contact relation do not use the operation of Boolean complementation * and have the same formulation in the language of bounded distributive lattice. Second, that the same can be said for the basic mereological relations partof, overlap and underlap -they have definitions in the language of distributive lattice without the operation of Boolean complement. Third, that the representation theory for distributive lattices is quite similar to the corresponding theory of Boolean algebras and we wanted to see if this can help us in transferring the topological representation theory of contact algebras to the more general theory of distributive contact lattices, keeping the topological meaning of regions as regular closed sets. And finally, one philosophical motivation: the meaning of the Boolean complementation a * is not well motivated: if the region a represents a physical body, then what kind of body represents a * ? In the point-based models this is "the rest out of a" from the "whole space", the latter identified with the sum of all observed regions, the unit region 1. However, if we extend the area of our observation we will obtain another unit, and then a * will be changed. But it is natural to assume that physical bodies should not depend on the area of observation in which they are included. As a result of this generalization, one can see that the paper [9] generalizes almost all from the topological representation theory of contact algebras developed for instance in [10,6] and even more; on the distributive case one can see some deep features which can not be observed in the Boolean case. For instance in the Boolean case mereological relations have some hidden properties which in the distributive case are not always fulfilled and have to be postulated explicitly (this is the so called extensionality property for the underlap and overlap relations). However, the obtained generalization in [8,9] has some open problems. The mereotopological relations of non-tangential part-of and dual contact in contact algebras have definitions by means of the operation of complementation. However these relations have a meaning in topological representation of contact algebras which does not depend on the operation of complementation on regular closed sets. Namely, if a and b are regular closed subsets of a topological space X, then a ≪ b iff a ⊆ Int(b) and a Cb iff Int(a) ∪ Int(b) = X, where Int is the topological operation of interior of a set. Thus, it will be interesting to add these relations as primitives to the language of distributive contact lattices and to axiomatize them by means of a set of universal first-order axioms and then to extend the topological representation theory from [9]. This is one of the main open problems in [9] which positive solution is subject of the present paper. One of the motivations for this extension of the language of distributive contact lattice is that in this way we obtain a system with full duality: contact C is dual to the dual contact C and non-tangential part-of ≪ is dual to it converse ≫ and this symmetry makes possible to obtain proofs by duality. The obtained new algebraic mereotopological system is named Extended Distributive Contact Lattice, EDC-lattice for short. We will consider in the paper the topological representation theory of some axiomatic extensions of EDC-lattices with new axioms yielding representations in better topological spaces, generalizing in this way the existing representation theory for contact algebras. Since all these investigations form a special subfield of mereotopology based on distributive lattices, we introduce for this subfield a special name -distributive mereotopology, which is included in the title of the present paper. Having in mind this terminology, then the subarea of mereotopology based on Boolean algebras should be named Boolean mereotopology. Similar special names for other subfields of mereotopology depending on the corresponding mereological parts also can be suggested: for instance the mereotopology considered in [15,28,29] is based on some non-distributive lattices -hence non-distributive mereotopology, and the mereotopological structures considered, for instance, in [17,13] are pure relational and without any algebraic lattice-structure in the set of regions -hence relational mereotopology.
The paper is divided in two parts. Part I is devoted to the axiomatization of the three mereotopological relations of contact C, dual contact C and nontangential part-of ≪ taken as primitives on the base of distributive lattice by means of universal first-order axioms, which remain true in contact algebras. The main result of this part is the abstract notion of Extended Distributive Contact Lattice (EDC-lattice) and an embedding theorem of EDC-latices into contact algebras, showing in this way that the meaning of the contact, dual contact and non-tangential part-of relations is preserved in the language of EDC-lattices. The method is based on a certain generalization of the Stone representation theory of distributive lattices [22,2]. As a consequence of the embedding theorem one can consider EDC-lattice also as the universal fragment of contact algbera based on the signature of distributive lattice and mereotopological relations of contact C, dual contact C and non-tangential inclusion ≪. Relations of EDC-lattices with other mereotopological systems are also considered: EDC-lattices are relational mereotopological systems in the sense of [17], and the well known RCC-8 system of mereotopological relations is definable in the language of EDC-lattices.
Part II of the paper is devoted to the topological representation theory of EDC-lattices and some of their axiomatic extensions yielding representations in T 1 and T 2 spaces. Special attention is given to dual dense and dense representations (defined in Section 5.1) in contact algebras of regular closed and regular open subsets of topological spaces. The method is an extension of the representation theory of distributive contact lattices [9] and adaptation of some constructions from the representation theory of contact algebras [6,7]. In the concluding Section we discuss some open problems and future plans with applications in qualitative spatial representation and reasoning. Choosing the right axioms 2.1 Contact algebras, distributive contact lattices and extended distributive contact lattices
As it was mention in the Introduction, contact algebra is a Boolean algebra B = (B, ≤, 0, 1, ·, +, * , C) with an additional binary relation C called contact, and satisfying the following axioms: Let us note that on the base of (C4) we have (C3') (a + b)Cc implies aCc or bCc.
Remark 2.1 Observe that the above axioms are universal first-order conditions on the language of Boolean algebra with the C-relation and not containing the Boolean complementation * . This fact says that the axioms of C will be true in any distributive sublattice of B.
The Remark 2.1 was one of the formal motivations for the definition of distributive contact lattice introduced in [8,9]: the definition is obtained just by replacing the underlying Boolean algebra by a bounded distributive lattice (D, ≤, 0, 1, +, ·) and taking for the relation C the same axioms. This makes possible to consider the main standard models of contact algebras, namely the algebras of regular closed or regular open sets of a topological space, also as the main models for distributive contact lattices, just by ignoring the Boolean complementation * in this models. This was guaranteed by Theorem 7 from [9] stating that every distributive contact lattice can be isomorphically embedded into a contact algebra, which fact indicates also that the choice of the set of axioms for distributive contact lattice is sufficient for proving this theorem. Since our main goal in the present paper is to obtain a definition of distributive contact lattice extended with relations of dual contact C and nontangential part-of ≪, we will follow here the above strategy, namely to choose universal firs-order statements for the relations C, C, ≪ as additional axioms which are true in arbitrary contact algebras and which guarantee the embedding into a contact algebra. The obtained algebraic system will be called extended distributive contact lattice. The next definition is a result of several preliminary experiments for fulfilling the above program.
Definition 2.2 Extended distributive contact lattice. Let D = (D, ≤ , 0, 1, +, ·, C, C, ≪) be a bounded distributive lattice with three additional relations C, C, ≪, called respectively contact, dual contact and nontangential part-of. The obtained system, denoted shortly by D = (D, C, C, ≪), is called extended distributive contact lattice ( EDC-lattice, for short) if it satisfies the axioms listed below. Notations: if R is one of the relations ≤, C, C, ≪, then its complement is denoted by R. We denote by ≥ the converse relation of ≤ and similarly ≫ denotes the converse relation of ≪.
Axioms for C alone: The axioms (C1)-(C5) mentioned above.
Axioms for C alone: Axioms for ≪ alone: For the language of EDCL we can introduce the following principle of duality: dual pairs (0, 1), (·, +), (≤, ≥), (C, C), (≪ , ≫). By means of these pairs for each statement (definition) A of the language we can define in an obvious way its dual A. Then by a routine verification one can see that for each axiom Ax from the list of axioms of EDCL its dual Ax is also true. On the base of this observation the proofs of dual statements will be omitted. Note, for instance, that each axiom from the first group (axioms for C alone) is dually equivalent to the corresponding axiom from the second group (axioms for C alone) and vice versa, the third and fourth groups of axioms (axioms for ≪ alone and mixed axioms) are closed under duality, for instance the axiom (M C1) is dually equivalent to the axiom (M C1), and (M ≪ 2) is dually equivalent to (M ≪ 1).
(≪ 1) 0 ≪ 0, (≪ 2) 1 ≪ 1, (≪ 3) If a ≪ b, then a ≤ b, (≪ 4) If a ′ ≤ a ≪ b ≤ b ′ , then a ′ ≪ b ′ ,
Relational models of EDC-lattices
In order to prove that the axioms of EDC-lattices are true in contact algebras we will introduce a relational models of EDCL which are slight modifications of the relational models of contact algebras introduced in [7] and called there discrette contact algebras. The model is defined as follows.
Let (W, R) be a relational system where W is a nonempty set and R is a reflexive and symmetric relation in W and let a, b be arbitrary subsets of W . Define a contact relation between a and b as follows (Def C R ) aC R b iff ∃x ∈ a and ∃y ∈ b such that xRy. Then any Boolean algebra of subsets of W with thus defined contact is a contact algebra, and moreover, every contact algebra is isomorphic to a contact algebra of such a kind [7].
We will modify this model for EDCL as follows: instead of Boolean algebras of sets we consider only families of subsets containing the empty set ∅ and the set W and closed under the set-union and set-intersection which are bounded distributive lattices of sets. Hence we interpret lattice constants and operations as follows:
0 = ∅, 1 = W , a · b = a ∩ b, a + b = a ∪ b.
For the contact relation we preserve the definition (Def C R ). This modification is just a model of distributive contact lattice studied in [9].
Having in mind the definitions a Cb ↔ def a * Cb * and a ≪ b ↔ def aCb * ) in Boolean algebras, we introduce the following definitions for C and ≪ (for some convenience we present the definition of the negation of ≪):
(Def C R ) a C R b iff ∃x ∈ a and ∃y ∈ b such that xRy, and (Def ≪ R ) a ≪ R b iff ∃x ∈ a and ∃y ∈ b such that xRy. Lemma 2.4 Let (W, R) be a relational system with reflexive and symmetric relation R and let D be any collection of subsets of W which is a bounded distributive set-lattice with relations C, C and ≪ defined as above. Then (D,
C R , C R , ≪ R ) is an EDC-lattice.
Proof. Routine verification that all axioms of EDC-lattice are true. EDC-lattice D = (D, C R , C R , ≪ R ) over a relational system (W, R) will be called discrete EDC-lattice. If D is a set of all subsets of W then D is called a full discrete EDC-lattice. Proof. The proof follows by Lemma refRelationEDCL and the fact that every contact algebra can be isomorphically embedded into a discrete contact algebra over some relational system (W, R) wit reflexive and symmetric relation R [7].
Embedding EDC-lattices into contact algebras
The main aim of this section is the proof a theorem stating that every EDClattice can be embedded into a full discrete EDC-lattice, which, of course is a Boolean contact algebra. As a consequence this will show that the axiomatization program for EDCL is fulfilled successfully. Since all axioms of EDC-lattice are universal first-order conditions, the axiomatization can be considered also as a characterization of the universal fragment of complement-free contact algebras based on the three relations. We will use in the representation theory a Stone like technique developed in [22] for the representation theory of distributive lattices.
Preliminary facts about filters and ideals in distributive lattices
We remaind some basic facts about filters and ideals in distributive lattices, for details see [2,22].
Let D be a distributive lattice. A subset F of D is called a filter in D if it satisfies the following conditions: (f1) 1 ∈ F , (f2) if a ∈ F and a ≤ b then b ∈ F , (f3) if a, b ∈ F then a.b ∈ F . F is a proper filter if 0 ∈ F , F is a prime filter if it is a proper filter and a + b ∈ F implies a ∈ F or b ∈ F . Dually, a subset I of D is an ideal if (i1) 0 ∈ I, (i2) if a ∈ I and b ≤ a then b ∈ I, (i3) if a, b ∈ I then a + b ∈ I. I is a proper ideal if 1 ∈ I, I is a prime ideal if it is a proper ideal and a.b ∈ I implies a ∈ I or b ∈ I.
We will use later on some of the following facts without explicit mentioning.
Facts 3.1 Let D be a bounded distributive lattice and Let F, F 1 , F 2 be filters and I, I 1 , I 2 be ideals.
1. The complement of a prime filter is a prime ideal and vice-versa.
2.
[a) = {x ∈ D : a ≤ x} is the smallest filter containing a;
(a] = {x ∈ D : x ≤ a} is the smallest ideal containing a.
3. F 1 ⊕ F 2 = {c ∈ D : (∃a ∈ F 1 , b ∈ F 2 )(a · b ≤ c)} = {a · b : a ∈ F 1 , b ∈ F 2 } is the smallest filter containing F 1 and F 2 . [a) ⊕ F = {x · y : a ≤ x, y ∈ F } I 1 ⊕ I 2 = {c ∈ D : (∃a ∈ I 1 , b ∈ I 2 )(c ≤ a + b)} = {a + b : a ∈ I 1 , b ∈ I 2 }
is the smallest ideal containing I 1 and I 2 .
(a] ⊕ I = {x + y : x ≤ a, y ∈ I}.
In both cases the operation ⊕ is associative and commutative.
4. [a) ∩ I = ∅ iff a ∈ I If (F ⊕ [a)) ∩ I = ∅ then (∃x ∈ F )(a · x ∈ I), (a] ∩ F = ∅ iff a ∈ F If F ∩ (I ⊕ (a]) = ∅ then (∃x ∈ I)(a + x ∈ F ).
The following three statements are well known in the representation theory of distributive lattices. 1. Filter-extension Lemma. There exists a prime filter F such that F 0 ⊆ F and F ∩ I 0 = ∅.
2.
Ideal-extension Lemma. There exists a prime ideal I such that I 0 ⊆ I and F 0 ∩ I = ∅.
Separation Lemma for filters and ideals.
There exist a (prime) filter F and an (prime) ideal I such that F 0 ⊆ F , I 0 ⊆ I, F ∩ I = ∅, and F ∪ I = D.
Remark 3.3 Note that Filter-extension Lemma is dual to the Ideal-extension
Lemma and that each of the three statement easily implies the other two. Normally they can be proved by application of the Zorn Lemma. The proof, for instance, of Filter-extension Lemma goes as follows. Apply the Zorn Lemma to the set M = {G : G is a filter, F 0 ⊆ G and G ∩ I 0 = ∅} and denote by F one of its maximal elements. Then it can be proved that F is a prime filter, and this finishes the proof. The sketched proof gives, however, an additional property of the filter F , namely (∀x ∈ F )(∃y ∈ F )(x · y ∈ I 0 ), which added to the formulation of the lemma makes it stronger. Since we will need later on this stronger version let us prove this property.
Suppose that x ∈ F and consider the filter
F ⊕ [x). Since F is a maximal element of M , then F ⊕[x)
does not belong to M and consequently F ⊕[x)∩I 0 = ∅. By the Fact 3.1, 4, there exists y ∈ F such that x · y ∈ I 0 . We formulate this new statement below as Strong filter-extension Lemma and its dual as Strong ideal-extension Lemma. We do not know if these two statements for distributive lattices are new, but we will use them in the representation theorem in the next section. 1. Strong filter-extension Lemma. There exists a prime filter F such that F 0 ⊆ F , (∀x ∈ F )(x ∈ I 0 ) and (∀x ∈ F )(∃y ∈ F )(x · y ∈ I 0 ).
Strong ideal-extension Lemma.
There exists a prime ideal I such that I 0 ⊆ I, (∀x ∈ I)(x ∈ F 0 ) and (∀x ∈ I)(∃y ∈ I)(x + y ∈ F 0 ).
Filters and Ideals in EDC-lattices
In the next two lemmas we list some constructions of filters and ideals in EDCL which will be used in the representation theory of EDC-lattices.
Lemma 3.5 Let D = (D, C, C, ≪)
be an EDC-lattice. Then:
1. The set I(xCb) = {x ∈ D : xCb} is an ideal, 2. the set F (x Cb) = {x ∈ D : x Cb} is a filter, 3. the set I(x ≪ b) = {x ∈ D : x ≪ b} is an ideal, 4. the set F (x ≫ b) = {x ∈ D : x ≫ b} is a filter.
Proof. 1. By axiom (C1) 0Cb, so 0 ∈ I(xCb). Suppose x ∈ I(xCb) (hence xCb) and y ≤ x. Then by axiom (C2) yCb). Let x, y ∈ I(xCb), hence xCb and yCb. Then by axiom (C3) and (C4) we get (x + y)Cb which shows that x + y ∈ I(xCb), which ends the proof of this case.
In a similar way one can proof 3. The cases 2. and 4. follow from 1. and 3. respectively by duality.
1. The set I(xCΓ) = {x ∈ D : (∃y ∈ Γ)(xCy)} is an ideal, 2. the set F (x CΓ) = {x ∈ D : (∃y ∈ Γ)(x Cy)} is a filter, 3. the set I(x ≪ Γ) = {x ∈ D : (∃y ∈ Γ)(x ≪ y)} is an ideal, 4. the set F (x ≫ Γ) = {x ∈ D : (∃y ∈ Γ)(x ≫ y)} is a filter.
Proof. Note that the Lemma remains true if we replace Γ by a filter and Γ by an ideal.
1. The proof that I(xCΓ) satisfies the conditions (i1) and (i2) from the definition of ideal is easy. For the condition (i3) suppose x 1 , x 2 ∈ I(xCΓ). Then ∃y 1 , y 2 ∈ Γ such that x 1 Cy 1 and x 2 Cy 2 , Since Γ is a filter then y = y 1 · y 2 ∈ Γ. Since y ≤ y 1 and y ≤ y 2 , then by axiom (C2) we get x 1 Cy and x 2 Cy. Then applying (C3') we obtain (x 1 + x 2 )Cy, which shows that
x 1 + x 2 ∈ I(xCΓ).
In a similar way one can prove 3. The proofs of 2 and 4 follow by duality from 1 and 3, taking into account that Γ is an ideal.
Relational representation theorem for EDC-lattices
Throughout this section we assume that D = (D, C, C, ≪) is an EDC-lattice and let P F (D) and P I(D) denote the set of prime filters of D and the set of prime ideals of D. Let h(a) = {Γ ∈ P F (D) : a ∈ Γ} be the well known Stone embedding mapping. We shall construct a canonical relational structure (W c , R c ) related to D putting W c = P F (D) and defining R c for Γ, ∆ ∈ P F (D) as follows:
ΓR c ∆ ↔ def (∀a, b ∈ D)(a ∈ Γ, b ∈ ∆ → aCb)&(a ∈ Γ, b ∈ ∆ → a Cb)&(a ∈ Γ, b ∈ ∆ → a ≪ b)&(a ∈ Γ, b ∈ ∆ → b ≪ a)
For some technical reasons and in order to use duality we introduce also the dual canonical structure ( W c , R c ) putting W c = P I(D) and for Γ, ∆ ∈ P I(D),
Γ R c ∆ ↔ def ΓR c ∆.
Our aim is to show that the Stone mapping h is an embedding from D into the EDC-lattice over (W c , R c ) (see Section 2.4). First we need several technical lemmas. Lemma 3.7 The canonical relations R c and R c are reflexive and symmetric.
Proof. ( For R c ) Symmetry is obvious by the definition of R c and axioms (C4) and ( C4). In order to prove that ΓR c Γ suppose a ∈ Γ and b ∈ Γ. Then a · b ∈ Γ and since Γ is a prime filter, then a.b = 0. Then by axiom (C5) we obtain aCb, which proves the first conjunct of the definition of R c . For the second conjunct suppose that a ∈ Γ and b ∈ Γ, then, since Γ is a prime filter, a + b ∈ Γ and hence a + b = 1. Then by axiom ( C5) we get a Cb. For the third conjunct suppose a ∈ Γ and b ∈ Γ, which implies that a ≤ b. Then by axiom (≪ 3) we obtain a ≪ b. The proof of the last conjunct is similar.
(For R c ) -by duality.
Lemma 3.8 (i) aCb iff (∃Γ, ∆ ∈ P F (D))(a ∈ Γ and b ∈ ∆ and ΓR c ∆). (ii) a ≪ b iff (∃Γ, ∆ ∈ P F (D))(a ∈ Γ and b ∈ ∆ and ΓR c ∆).
Proof. (i) Note that the proof is quite technical, so we will present it with full details. The reasons for this are twofold: first to help the reader to follow it more easily, and second, to skip the details in a similar proofs.
(⇐) If a ∈ Γ and b ∈ ∆ then by the definition of R c we obtain aCb.
(⇒) Suppose aCb. The proof will go on several steps.
Step 1: construction of Γ. Consider the ideal I(xCb) = {x ∈ D : xCb} (Lemma 3.5). Since aCb, a ∈ {x ∈ D : xCb}. Then [a) ∩ {x ∈ D : xCb} = ∅ and [a) is a filter (see Facts 3.1). By the Strong filter-extension lemma (see Lemma 3.4) there exists a prime filter Γ such that [a) ⊆ Γ and (∀x ∈ Γ)(x ∈ {x ∈ D : xCb} and (∀x ∈ Γ)(∃y ∈ Γ)(x · y ∈ {x ∈ D : xCb}. From here we conclude that Γ satisfies the following two properties:
(#0) a ∈ Γ, (#1) If x ∈ Γ, then xCb, and (#2) If x ∈ Γ, then there exists y ∈ Γ such that (x · y)Cb.
Step 2: construction of ∆. This will be done in two sub-steps.
Step 2.1 Consider the filters and ideals definable by Γ as in Lemma 3.6
I(xCΓ) = {x ∈ D : (∃y ∈ Γ)(xCy)}, F (x CΓ) = {x ∈ D : (∃y ∈ Γ)(x Cy)}, I(x ≪ Γ) = {x ∈ D : (∃y ∈ Γ)(x ≪ y)}, and F (x ≫ Γ) = {x ∈ D : (∃y ∈ Γ)(x ≫ y}.
In order to apply the Separation Lemma we will prove the following condition:
(#3) F (x ≫ Γ) ⊕ F (x CΓ) ⊕ [b) ∩ I(xCΓ) ⊕ I(x ≪ Γ) = ∅. Suppose that (#3) is not true, then for some t ∈ D we have (1) t ∈ F (x ≫ Γ) ⊕ F (x CΓ) ⊕ [b) and (2) t ∈ I(xCΓ) ⊕ I(x ≪ Γ). It follows from (2) that ∃k 1 , k 2 such that (3) k 1 ∈ I(x ≪ Γ) and (4) k 2 ∈ I(xCΓ) and (5) t = k 1 + k 2 .
It follows from (1) that ∃k 4 , k 5 , k 6 ∈ D such that (6) k 4 ∈ F (x ≫ Γ) and
(7) k 5 ∈ F (x CΓ) and (8) k 6 ∈ [b) and (9) t = k 4 · k 5 · k 6 .
From (5) and (9) we get
(10) k 1 + k 2 = k 4 · k 5 · k 6 .
It follows from (3), (4), (6) and (7) that
(11) ∃x 1 ∈ Γ such that k 1 ≪ x 1 , (12) ∃x 2 ∈ Γ such that k 2 Cx 2 , (13) ∃x 3 ∈ Γ such that x 3 ≪ k 4 , (14) ∃x 4 ∈ Γ such that k 5 Cx 4 . Let x = x 1 + x 4 .
Since Γ is an ideal, we obtain by (11) and (14) that (15) x ∈ Γ and x ∈ Γ. Then by (#2) we get (16) ∃y ∈ Γ such that (x · y)Cb.
Let z = x 2 · x 3 · y. Then by (12), (13) and (16) we obtain that (17) z ∈ Γ and by (#1) that (18)zCb.
From x 1 ≤ x and (11) by axiom (≪ 4) we get (14) by axiom ( C2) we obtain (12) by axiom (C2) we get (21) k 2 Cz.
(19) k 1 ≪ x. From x 4 ≤ x and(20) k 5 Cx. From z ≤ x 2 and
From z ≤ x 3 and (13) by axiom (≪ 4) we obtain
(22) z ≪ k 4 .
We shall show that the following holds (23) zC(b · k 1 ).
Suppose for the sake of contradiction that (16).
zC(b · k 1 ). From b · k 1 ≤ k 1 and (19) by axiom (≪ 4) we get (b · k 1 ) ≪ x. From this fact and zC(b · k 1 ) by axiom (M C1) we obtain (b · k 1 )C(z · x). But we also have b · k 1 ≤ b, z · x ≤ y · x, so by axiom (C2) we get bC(y · x) -a contradiction with
The following condition holds (24) zC(b · k 2 ).
To prove this suppose for the sake of contradiction that zC(b · k 2 ). We also have b · k 2 ≤ k 2 , so by axiom (C2) we get zCk 2 -a contradiction with (21).
Suppose that zC(b ·(k 1 + k 2 )). By axiom (C3) we have zC(b ·k 1 ) or zC(b ·k 2 ) -a contradiction with (23) and (24). Consequently zC(b · (k 1 + k 2 )) and by (10) we obtain zC(b · k 4 · k 5 · k 6 ). But b ≤ k 6 (from (8)
), so b · k 4 · k 5 · k 6 = b · k 4 · k 5 . Consequently (25) zC(b · k 4 · k 5 ).
From (18) and (22) by axiom (M C1) we get (26) zC(b · k 4 ).
We shall show that the following condition holds (27)
(z · x)C(b · k 4 )
For to prove this suppose the contrary (z·x)C(b·k 4 ). We also have z·x ≤ y·x, b · k 4 ≤ b, so by axiom (C2) we get (y · x)Cb -a contradiction with (16).
From (25), (26) and (27) by axiom (M C2) we obtain x Ck 5 -a contradiction with (20). Consequently (#3) is true.
Step 2.2: the construction of ∆. Applying the Filter extension Lemma to (#3) we obtain a prime filter ∆ (and this is just the required ∆) such that:
1. F (x ≫ Γ) = {x ∈ D : (∃y ∈ Γ)(x ≫ y} ⊆ ∆, 2. F (x CΓ) = {x ∈ D : (∃y ∈ Γ)(x Cy)} ⊆ ∆, 3. b ∈ ∆, 4. I(xCΓ) = {x ∈ D : (∃y ∈ Γ)(xCy)} ∩ ∆ = ∅, 5. I(x ≪ Γ) = {x ∈ D : (∃y ∈ Γ)(x ≪ y)} ∩ ∆ = ∅.
Step 3: proof of ΓR c ∆. We will verify the four cases of the definition of R c .
• Case 1: y ∈ Γ and x ∈ ∆. We have to show yCx. Suppose yCx. Then xCy and by y ∈ Γ we get x ∈ I(xCΓ). Then by 4.
x ∈ ∆ -a contradiction, hence yCx.
• Case 2: y ∈ Γ and x ∈ ∆. Suppose y ≪ x. Then x ≫ y and y ∈ Γ implies x ∈ F (x ≫ Γ). By (1) x ∈ ∆ -a contradiction, hence y ≪ x.
• Case 3: y ∈ Γ and x ∈ ∆. Suppose x ≪ y. Then x ∈ I(x ≪ Γ) and by 5.
x ∈ ∆ -a contradiction. Hence x ≪ y.
• Case 4: y ∈ Γ and x ∈ ∆. Suppose y Cx. Then x Cy and by 2. we obtain x ∈ ∆ -a contradiction. Hence y Cx.
Thus we have constructed prime filters Γ and ∆ such that: a ∈ Γ, b ∈ ∆ (item 3 from Step 2.2) and ΓR c ∆ (Step 3).
Proof of (ii). (⇐) If a ∈ Γ and b ∈ ∆ then by the definition of R c we obtain a ≪ b.
(⇒) Suppose a ≪ b. The proof, as in (i), will go on several steps.
Step 1: construction of Γ. Consider the ideal I(x ≪ b) = {x ∈ D : x ≪ b} (Lemma 3.5). Since a ≪ b, a ∈ {x ∈ D : x ≪ b}. Then [a) ∩ {x ∈ D : x ≪ b} = ∅ and [a)
is a filter (see FACTS 3.1). By the Strong filter-extension lemma (Lemma 3.4) there exists a prime filter Γ such that [a) ⊆ Γ and (∀x ∈ Γ)(x ∈ {x ∈ D : x ≪ b}) and (∀x ∈ Γ)(∃y ∈ Γ)(x · y ∈ {x ∈ D : x ≪ b}). From here we conclude that Γ satisfies the following properties:
(#0) a ∈ Γ, (#1) If x ∈ Γ, then x ≪ b, and (#2) If x ∈ Γ, then there exists y ∈ Γ such that (x · y) ≪ b.
Step 2: construction of ∆. This will be done in two sub-steps.
Step 2.1 Consider the filters and ideals definable by Γ as in Lemma 3.6
I(xCΓ) = {x ∈ D : (∃y ∈ Γ)(xCy)}, F (x CΓ) = {x ∈ D : (∃y ∈ Γ)(x Cy)}, I(x ≪ Γ) = {x ∈ D : (∃y ∈ Γ)(x ≪ y)}, and F (x ≫ Γ) = {x ∈ D : (∃y ∈ Γ)(x ≫ y}.
In order to apply the Filter-extension Lemma (Lemma 3.2) we will prove the following condition:
(#3) F (x ≫ Γ) ⊕ F (x CΓ) ∩ I(x ≪ Γ) ⊕ I(xCΓ) ⊕ (b] = ∅ Suppose that (#3) is not true. Consequently ∃t such that (1) t = k 1 · k 2 = k 4 + k 5 + k 6 for some k 1 , k 2 , k 4 , k 5 , k 6 ∈ D and (2) ∃x 1 ∈ Γ such that x 1 ≪ k 1 , (3) ∃x 2 ∈ Γ such that k 2 Cx 2 , (4) ∃x 3 ∈ Γ such that k 4 ≪ x 3 , (5) ∃x 4 ∈ Γ such that k 5 Cx 4 , (6) k 6 ≤ b.
Let z = x 2 + x 3 . Then by (3) and (4) we obtain z ∈ Γ. By axiom ( C2) we get (7) k 2 Cz.
By (4) and axiom (≪ 4) we get
(8) k 4 ≪ z.
By z ∈ Γ and (#2) we have
(9) ∃y ∈ Γ such that (z · y) ≪ b. Let x = x 1 · x 4 · y · a.
Then by (#0), (2), (5) and (9) we get x ∈ Γ. By axiom (≪ 4) we get (10) x ≪ k 1 .
By (5), x ≤ x 4 and axiom (C2) we get
(11) k 5 Cx. From x ∈ Γ by (#1) we obtain (12) x ≪ b. From (10) by axiom (≪ 4) we get (13) x ≪ (b + k 1 ) From (7) by axiom ( C2) we obtain (14) z C(b + k 2 ). From (9) by axiom (≪ 4) we get (15) (z · y) ≪ (b + k 2 ).
From (14) and (15) by axiom (M ≪ 1) we obtain y ≪ (b + k 2 ). We also have x ≤ y and by axiom (≪ 4) we get (16)
x ≪ (b + k 2 ).
From (13) and (16) (6)). Thus:
by axiom (≪ 6) we get x ≪ (b + k 1 ) · (b + k 2 ). We have (b + k 1 ) · (b + k 2 ) = b + k 1 · k 2 = b + k 4 + k 5 + k 6 = b + k 4 + k 5 (since k 6 ≤ b from(17) x ≪ (b + k 4 + k 5 ).
Suppose (in order to obtain a contradiction) that x ≪ (b + k 4 ). From (9) and x · z ≤ z · y (which follows from the definitions of x and z) by axiom (≪ 4) we obtain (x · z) ≪ b. Using this fact, (8), x ≪ (b + k 4 ) and axiom (≪ 7) we get x ≪ b -a contradiction with (12). Consequently (18)
x ≪ (b + k 4 ).
From (11) and (17) by axiom (M ≪ 2) we obtain x ≪ (b + k 4 ) -a contradiction with (18). Consequently (#3) is true.
Step 2.2: the construction of ∆. Applying the Filter-extension Lemma to (#3) we obtain a prime filter ∆ (and this is just the required ∆) such that:
1. F (x ≫ Γ) = {x ∈ D : (∃y ∈ Γ)(x ≫ y} ⊆ ∆, 2. F (x CΓ) = {x ∈ D : (∃y ∈ Γ)(x Cy)} ⊆ ∆, 3. b ∈ ∆, 4. I(xCΓ) = {x ∈ D : (∃y ∈ Γ)(xCy)} ∩ ∆ = ∅, 5. I(x ≪ Γ) = {x ∈ D : (∃y ∈ Γ)(x ≪ y)} ∩ ∆ = ∅.
Step 3: proof of ΓR c ∆. The proof is the same as in the corresponding step in (i).
To conclude: we have constructed prime filters Γ, ∆ such that ΓR c ∆, a ∈ Γ and b ∈ ∆, which finishes the proof of the lemma. Lemma 3.9 (i) a Cb iff (∃Γ, ∆ ∈ P I(D))(a ∈ Γ and b ∈ ∆ and Γ R c ∆).
(ii) a Cb iff (∃Γ, ∆ ∈ P F (D))(a ∈ Γ and b ∈ ∆ and ΓR c ∆).
(iii) a ≫ b iff (∃Γ, ∆ ∈ P I(D))(a ∈ Γ and b ∈ ∆ and Γ R c ∆).
(iv) a ≫ b iff (∃Γ, ∆ ∈ P F (D))(a ∈ Γ and b ∈ ∆ and ΓR c ∆).
Proof. (i) by duality from Lemma 3.8. Note that in this case Strong idealextension Lemma is used. The proof can follow in a "dual way" the steps of the proof of Lemma 3.8 (i).
(ii) is a corollary from (i).
(iii) by duality from Lemma 3.8 (ii) with the same remark as above.
(iv) is a corollary from (iii). Proof. It is a well known fact that h is an embedding of distributive lattice into the distributive lattice of all subsets of the set of prime filters P F (D) (see, [22,2]). The only thing which have to be done is to show the following equivalences for all a, b ∈ D:
(i) aCb iff h(a)C R c h(b), (ii) a Cb iff h(a) C R c h(b) (iii) a ≪ b iff h(a) ≪ R c h(b).
Note that these equivalences are another equivalent reformulation of Lemma 3.8 (i) and (ii) and Lemma 3.9 (ii) and (iv). Proof. The theorem is a corollary of Lemma 3.10.
Corollary 3.12 Every EDC-lattice can be isomorphically embedded into a contact algebra.
Proof. Since the lattice of all subsets of a given set is a Boolean algebra, then this is a corollary of Theorem 3.11.
The following theorem states that the axiom system of EDC-lattice can be considered as an axiomatization of the universal fragment of contact algebras in the language of EDC-lattices. Theorem 3.13 Let A be an universal first-order formula in the language of EDC-lattices. Then A is a consequence from the axioms of EDC-lattice iff A is true in all contact algebras.
Proof. The proof is a consequence from Corollary 3.12 and the fact that all axioms of EDC-lattice are universal first-order conditions and that A is also an universal first-order condition.
Relations to other mereotopologies
In this section we will compare EDC-lattices with other two mereotopologies: the relational mereotopology and RCC-8.
Relational mereotopology
Relational mereotopology is based on mereotopological structures introduced in [17]. These are relational structures in the form (W, ≤, O, O, ≪, C, C) axiomatizing the basic mereological relations part-of ≤, overlap O and dual overlap (underlap) O, and the basic mereotopological relations non-tangential part-of ≪, contact C and dual contact C. These relations satisfy the following list of universal first-order axioms: Note that all axioms of mereotopological structures are universal first-order conditions which are true in contact algebras under the standard definitions of the three basic mereological relations.
(≤ 0) a ≤ b and b ≤ a → a = b (≤ 1) a ≤ a, (≤ 2) a ≤ b and b ≤ c → a ≤ c (O1) aOb → bOa ( O1) a Ob → b Oa (O2) aOb → aOa ( O2) a Ob → a Oa (O ≤) aOa → a ≤ b ( O ≤) b Ob → a ≤ b (O ≤) aOb and b ≤ c → aOc ( O ≤) c ≤ a and a Ob → c Ob (O O) aOa or a Oa (≤ O O) cOa and c Ob → a ≤ b (C) aCb → bCa ( C) a Cb → b Ca (CO1) aOb → aCb ( C O1) a Ob → a Cb (CO2) aCb → aOa ( C O2) a Cb → a Oa (C ≤) aCb and b ≤ c → aCc ( C ≤) a Cb and c ≤ b → a Cc (≪≤ 1) a ≪ b → a ≤ b (≪≤ 2) a ≤ b and b ≪ c → a ≪ c (≪≤ 3) a ≪ b and b ≤ c → a ≪ c (≪ O) aOa → a ≪ b (≪ O) b Ob → a ≪ b (≪ CO) aCb and b ≪ c → aOc (≪ C O) c ≪ a
It is proved in [17] that each mereotopological structure is embeddable into a contact algebra (Theorem 26).
The following theorem relates EDC-lattices to mereotopological structures. Proof. Since all axioms of mereotopological structures are universal firstorder sentences true in all contact algebras, then the statement follows from Theorem 3.13.
RCC-8 spatial relations
One of the most popular systems of topological relations in the community of QSRR is RCC-8. The system RCC-8 was introduced for the first time in [11]. It consists of 8 relations between non-empty regular closed subsets of arbitrary topological space. Having in mind the topological representation of contact algebras, it was given in [24] an equivalent definition of RCC-8 in the language of contact algebras:
RCC-8 relations
Looking at this definition it can be easily seen that the RCC-8 relations are expressible in the language of EDC-lattices. Let us note that RCC-8 relations are not expressible in the language of distributive contact algebras from [9].
Additional axioms
In this Section we will formulate several additional axioms for EDC-lattices which are adaptations for the language of EDC-lattices of some known axioms considered in the context of contact algebras. First we will formulate some new lattice axioms for EDC-lattices -the so called extensionality axioms for the definable predicates of overlap -aOb ↔ def a · b = 0 and underlapa Ob ↔ def
a + b = 1. (Ext O) a ≤ b → (∃c)(a · c = 0 and b · c = 0) -extensionality of overlap, (Ext O) a ≤ b → (∃c)(a + c = 1 and b + c = 1) -extensionality of underlap.
We say that a lattice is O-extensional if it satisfies (Ext O) and U-extensional if it satisfies (Ext O). Note that the conditions (Ext O) and (Ext O) are true in Boolean algebras but not always are true in distributive lattices (see [9] for some examples, references and additional information about these axioms).
We will study also the following extensionality axioms.
(Ext C) a = 1 → (∃b = 0)(aCb) -C-extensionality,
(Ext C) a = 0 → (∃b = 1)(a Cb) -C-extensionality.
In contact algebras these two axioms are equivalent. It is proved in [9] that (Ext O) implies that (Ext C) is equivalent to the following extensionality principle considered by Whitehead [30] (EXT C) a ≤ b → (∃c)(aCc and bCc).
Just in a dual way one can show that (Ext O) implies that (Ext C) is equivalent to the following condition (EXT C) a ≤ b → (∃c)(b Cc and a Cc).
Let us note that (EXT C) and (EXT C ) are equivalent in contact algebras. (Con C) a = 0, b = 0 and a + b = 1 → aCb -C-connectedness axiom and (Con C) a = 1, b = 1 and a · b = 0 → a Cb -C-connectedness axiom .
In contact algebras these axioms are equivalent and guarantee topological representation in connected topological spaces.
(Nor 3) a ≪ b → (∃c)(a ≪ c ≪ b).
Let us note that the above three axioms are equivalent in contact algebras and are known by different names. For instance (Nor 1) comes from the proximity theory [23] as Efremovich axiom, (Nor 3) sometimes is called interpolation axiom. We adopt the name normality axioms for (Nor 1), (Nor 2) and (Nor 3) because in topological representations they imply some normality conditions in the corresponding topological spaces. It is proved in [7] that (Nor 1) is true in the relational models (W, R) (see Section 2.2) if and only if the relation R is transitive and that (Nor 1) implies representation theorem in transitive models. In the next lemma we shall prove similar result using all normality axioms. (i) R c is a transitive relation. (ii) D is representable in EDC-lattice over some system (W, R) with an equivalence relation R.
Proof. (i) Let Γ, ∆ and Θ be prime filters in D such that (1) ΓR c ∆ and (2) ∆R c Θ and suppose for the sake of contradiction that (3) ΓR c Θ. By the definition of R c we have to consider four cases.
Case 1: ∃a ∈ Γ, b ∈ Θ such that aCb.
Then by (Nor 1) there exists c, d such that c + d = 1, aCc and bCd.
Since c + d = 1 then either c ∈ ∆ or d ∈ ∆. The case c ∈ ∆ together with a ∈ Γ imply by (1) aCc -a contradiction. The case d ∈ ∆ together with b ∈ Θ imply by (2) bCd -again a contradiction.
Case 2: ∃a ∈ Γ, b ∈ Θ such that a ≪ b.
Then by (Nor 3) ∃c such that a ≪ c and c ≪ b. Consider the case c ∈ ∆. Then a ∈ Γ and (1) imply a ≪ c a contradiction. Consider now c ∈ ∆. Then b ∈ Θ imply c ≪ b -again a contradiction.
In a similar way one can obtain a contradiction in the remaining two cases: Case 3: ∃a ∈ Γ, b ∈ Θ such that b ≪ a and Case 4: ∃a ∈ Γ, b ∈ Θ such that b Ca.
(ii) The proof follows from (i) analogous to the proof of Theorem 3.11.
Another kind of axioms which will be used in the topological representation theory in PART II are the so called rich axioms. (U-rich ≪) a ≪ b → (∃c)(b + c = 1 and aCc), Let us note that U-rich axioms will be used always with the U-extensionality axiom and that O-rich axioms will be used always with O-extensionality axiom.
The following lemma is obvious.
Some good embedding properties
Let (D 1 , C 1 , C 1 , ≪ 1 ) and (D 2 , C 2 , C 2 , ≪ 2 ) be two EDC-lattices. We will write D 1 D 2 if D 1 is a substructure of D 2 , i.e., D 1 is a sublattice of D 2 , and the relations C 1 , C 1 , ≪ 1 are restrictions of the relations C 2 , C 2 , ≪ 2 on D 1 . Since we want to prove embedding theorems, it is valuable to know under what conditions we have equivalences of the form: D 1 satisfies some additional axiom iff D 2 satisfies the same axiom. Definition 5.3 Dense and dual dense sublattice. Let D 1 be a distributive sublattice of D 2 . D 1 is called a dense sublattice of D 2 if the following condition is satisfied: (Dense) (∀a 2 ∈ D 2 )(a 2 = 0 ⇒ (∃a 1 ∈ D 1 )(a 1 ≤ a 2 and a 1 = 0)).
If h is an embedding of the lattice D 1 into the lattice D 2 then we say that h is a dense embedding if the sublattice h(D 1 ) is a dense sublattice of D 2 .
Dually, D 1 is called a dual dense sublattice of D 2 if the following condition is satisfied: a 1 and a 1 = 1)).
(Dual dense) (∀a 2 ∈ D 2 )(a 2 = 1 ⇒ (∃a 1 ∈ D 1 )(a 2 ≤
If h is an embedding of the lattice D 1 into the lattice D 2 then we say that h is a Dual dense embedding if the sublattice h(D 1 ) is a dually dense sublattice of D 2 .
Note that in Boolean algebras, dense and dually dense conditions are equivalent; in distributive lattices this equivalence does not hold (see [9] for some known characterizations of density and dual density in distributive lattices).
For the case of contact algebras [24] and distributive contact lattices [9] we introduced the notion of C-separability as follows. Let D 1 D 2 ; we say that D 1 is a C-separable sublattice of D 2 if the following condition is satisfied:
(C-separable) (∀a 2 , b 2 ∈ D 2 )(a 2 Cb 2 ⇒ (∃a 1 , b 1 ∈ D 1 )(a 2 ≤ a 1 , b 2 ≤ b 1 , a 1 Cb 1 )).
For the case of EDC-lattices we modified this notion adding two additional clauses corresponding to the relations C and ≪ just having in mind the definitions of these relations in contact algebras. Namely Definition 5.4 C-separability. Let D 1 D 2 ; we say that D 1 is a C-separable EDC-sublattice of D 2 if the following conditions are satisfied:
(C-separability for C) - (∀a 2 , b 2 ∈ D 2 )(a 2 Cb 2 ⇒ (∃a 1 , b 1 ∈ D 1 )(a 2 ≤ a 1 , b 2 ≤ b 1 , a 1 Cb 1 )
).
(C-separability for C) -(∀a 2 , b 2 ∈ D 2 )(a 2 Cb 2 ⇒ (∃a 1 , b 1 ∈ D 1 )(a 2 + a 1 = 1, b 2 + b 1 = 1, a 1 Cb 1 )).
(C-separability for ≪) - (∀a 2 , b 2 ∈ D 2 )(a 2 ≪ b 2 ⇒ (∃a 1 , b 1 ∈ D 1 )(a 2 ≤ a 1 , b 2 + b 1 = 1, a 1 Cb 1 )).
If h is an embedding of the lattice D 1 into the lattice D 2 then we say that h is a C-separable embedding if the sublattice h(D 1 ) is a C-separable sublattice of D 2 .
The notion of a C-separable embedding h is defined similarly. The following lemma is analogous to a similar result from [24] (Theorem 2.2.2) and from [9] (Lemma 5).
Lemma 5.5 Let D 1 , D 2 be EDC-lattices and D 1 be a C-separable EDC-sublattice of D 2 . Then:
(i) If D 1 is a dually dense EDC-sublattice of D 2 , then D 1 satisfies the axiom (Ext C) iff D 2 satisfies the axiom (Ext C), (ii) D 1 satisfies the axiom (Con C) iff D 2 satisfies the axiom (Con C), (iii) D 1 satisfies the axiom (Nor 1) iff D 2 satisfies the axiom (Nor 1), (iv) D 1 satisfies the axiom (U-rich ≪) iff D 2 satisfies the axiom (U-rich ≪),
(v) D 1 satisfies the axiom (U-rich C) iff D 2 satisfies the axiom (U-rich C).
Proof. Conditions (i), (ii) and (iii) have the same proof as in Theorem 2.2.2 from [24].
(iv) (⇒) Suppose that D 1 satisfies the axiom (U-rich ≪), a 2 , b 2 ∈ D 2 and let a 2 ≪ b 2 . Then by (C-separability for ≪) we obtain: (∃a 1 , b 1 ∈ D 1 )(a 2 ≤ a 1 , b 2 + b 1 = 1, a 1 Cb 1 ). Since D 1 is a sublattice of D 2 then a 1 , b 1 ∈ D 2 . From a 2 ≤ a 1 and a 1 Cb 1 we get a 2 Cb 1 . Thus we have just proved: (a 2 ≪ b 2 → (∃b 1 ∈ D 2 )(b 2 + b 1 = 1 and a 2 Cb 1 ) which shows that D 2 satisfies (U-rich ≪).
(⇐) Suppose that D 2 satisfies the axiom (U-rich ≪), a 1 , b 1 ∈ D 1 (hence a 1 , b 1 ∈ D 2 ) and let a 1 ≪ b 1 . Then by (U-rich ≪) for D 2 we get: (∃c 2 ∈ D 2 )(b 1 + c 2 = 1, a 1 Cc 2 ). Since a 1 , c 2 ∈ D 2 and a 1 Cc 2 , then by (C-separability for C) we get:
(∃a ′ 1 , b ′ 1 ∈ D 1 )(a 1 ≤ a ′ 1 , c 2 ≤ b ′ 1 , a ′ 1 Cb ′ 1 ).
Combining the above results we get:
1 = b 1 + c 2 ≤ b 1 + b ′ 1 and a 1 Cb ′ 1 .
We have just proved the following:
a 1 ≪ b 1 → (∃b ′ 1 ∈ D 1 )(b 1 + b ′ 1 = 1, a 1 Cb ′ 1 ) which shows that D 1 satisfies (U-rich ≪).
(v) The proof is similar to that of (iv). . The notion of C-separable sublattice can be defined in a dual way as follows:
Definition 5.6 Suppose that D 1 D 2 ; we say that D 1 is a C-separable EDCsublattice of D 2 if the following condition is satisfied:
( C-separability for C) - (∀a 2 , b 2 ∈ D 2 )(a 2 Cb 2 ⇒ (∃a 1 , b 1 ∈ D 1 )(a 1 + a 2 = 1, b 1 + b 2 = 1, a 1 Cb 1 )), ( C-separability for C) - (∀a 2 , b 2 ∈ D 2 )(a 2 Cb 2 ⇒ (∃a 1 , b 1 ∈ D 1 )(a 1 ≤ a 2 , b 1 ≤ b 2 , a 1 Cb 1 )), ( C-separability for ≪) - (∀a 2 , b 2 ∈ D 2 )(a 2 ≪ b 2 ⇒ (∃a 1 , b 1 ∈ D 1 )(a 1 + a 2 = 1, b 1 ≤ b 2 , a 1 Cb 1 )).
The notion of a C-separable embedding h is defined as in definition 5.4.
The following lemma is dual to Lemma 5.5 and can be proved in a dual way. Proof. (i) Note that by Lemma 5.2 B satisfies the axioms (U-rich ≪) and (U-rich C). Then by Lemma 5.5 (iv) and (v) D satisfies the axioms (U-rich ≪) and (U-rich C).
(ii) Similarly to (i) the proof follows from Lemma 5.2 and Lemma 5.7.
PART II: TOPOLOGICAL REPRESENTATIONS OF EXTENDED DISTRIBUTIVE CONTACT LATTICES
The aim of this second part of the paper is to investigate several kinds of topological representations of EDC-lattices. We concentrate our attention mainly on topological representations with some "good properties" in the sense of Section 5.1: dual density and C-separability, and their dual versions -density and C-separability.
Topological models of EDC-lattices
We assume some familiarity of the reader with the basic theory of topological spaces:(see [12]). First we recall some notions from topology. By a topological space we mean a set X provided with a family C(X) of subsets, called closed sets, which contains the empty set ∅, the whole set X, and is closed with respect to finite unions and arbitrary intersections. Fixing C(X) we say that X is endowed with a topology. a and x ∈ b). We say that a is a regular closed set if a = Cl (Int(a)) and a is a regular open set if a = Int (Cl(a)). It is a well known fact that the set RC(X) of all regular closed subsets of X is a Boolean algebra with respect to the relations, operations and constants defined as follows: (Int(a ∩ b), a * = Cl(−a) where −a = X a. If we define a contact C by aCb iff a ∩ b = ∅ then we obtain the standard topological model of contact algebra.
x ∈ Cl(a) iff (∀b ∈ CB(X))(a ⊆ b → x ∈ b), x ∈ Int(a) iff (∃b ∈ OB(X))(b ⊆a ≤ b iff a ⊆ b, 0 = ∅, 1 = X, a + b = a ∪ b, a · b = Cl
Another topological model of contact algebra is by the set RO(X) of regular open subsets of X. The relevant definitions are as follows:
a ≤ b iff a ⊆ b, 0 = ∅, 1 = X, a · b = a ∩ b, a + b = Int(Cl(a ∪ b), a * = Int − a. The contact relation is aCb iff Cl(a) ∩ Cl(b) = ∅.
Note that these two models are isomorphic. Topological model of EDC-lattice by regular-closed sets. Consider the contact algebra RC(X) of regular closed subsets of X. Let us remove the operation a * and define the relations C and ≪ topologically according to their definitions in contact algebra as follows:
a Cb iff Cl(−a) ∩ Cl(−b) = ∅ iff (equivalently) Int(a) ∪ Int(b) = X. a ≪ b iff a ∩ Cl(−b) = ∅ iff (equivalently) a ⊆ Int(b).
Obviously the obtained structure is a model of EDC-lattice. Also any distributive sublattice of RC(X) with the same definitions of the relations C, C and ≪ is a model of EDC-lattice. These models are considered as standard topological models of EDC-lattice by regular closed sets.
Topological model of EDC-lattice by regular-open sets.
Consider the contact algebra RO(X) of regular open subsets of X. Let us remove the operation a * from the contact algebra RO(X) and define the relations C and ≪ topologically according to their definitions in the contact algebra as follows:
a Cb iff Cl(Int(−a) ∩ Cl(Int(−b)) = ∅ iff (equivalently) a ∪ b = X, a ≪ b iff Cl(a) ∩ Cl(Int(−b)) = ∅ iff (equivalently) Cl(a) ⊆ b.
Obviously the obtained structure is another standard topological model of EDC-lattice and any distributive sublattice of RO(X) with the same relations C, C and ≪ is also a model of EDC-lattice.
The main aim of PART II of the paper is the topological representation theory of EDC-lattices related to the above two standard models. The first simple result is the following representation theorem. Proof. It is shown in [6] that every contact algebra is isomorphic to a subalgebra of the contact algebra RC(X) of regular closed subsets of some topological space X, and dually, that it is also isomorphic to a subalgebra of the contact algebra RO(Y ) of the regular open subsets of some topological space Y . Then the proof follows directly from this result and the Corollary 3.12.
The above theorem is not the best one, because it can not be extended straightforwardly to EDC-lattices satisfying some of the additional axioms mentioned in Section 5. That is why we will study in the next sections representation theorems based on embeddings satisfying some of the good conditions described in Section 5.1. Before going on let us remaind some other topological facts, which will be used later on.
A topological space X is called:
• normal if every pair of closed disjoint sets can be separated by a pair of open sets;
• κ-normal [19] if every pair of regular closed disjoint sets can be separated by a pair of open sets;
• weakly regular [10] if it is semiregular and for each nonempty open set a there exits a nonempty open set b such that Cl(a) ⊆ b;
• connected if it can not be represented by a sum of two disjoint nonempty open sets;
• T 0 if for every pair of distinct points there is an open set containing one of them and not containing the other; X is called T 1 if every one-point set is a closed set, and X is called Hausdorff (or T 2 ) if each pair of distinct points can be separated by a pair of disjoint open sets.
• compact if it satisfies the following condition: let {A i : i ∈ I} be a nonempty family of closed sets of X such that for every finite subset J ⊆ I the intersection
{A i : i ∈ J} = ∅, then {A i : i ∈ I} = ∅.
The following lemma relates topological properties to the properties of the relations C, C and ≪ and shows the importance of the additional axioms for EDC-lattices. Lemma 6.2 (i) If X is semiregular, then X is weakly regular iff RC(X) satisfies any of the axioms (Ext C), (Ext C).
(ii) X is κ-normal iff RC(X) satisfies any of the axioms (Nor 1), (Nor 2) and (Nor 3).
(iii) X is connected iff RC(X) satisfies any of the axioms (Con C), (Con C).
(iv) If X is compact and Hausdorff, then RC(X) satisfies (Ext C), (Ext C) and (Nor 1), (Nor 2) and (Nor 3) .
Proof. A variant of the above lemma concerning only axioms (Ext C), (Nor 1) and (Con C) was proved, for instance, in [10]. Having in mind the equivalence of some of the mentioned axioms in RC(X), it is obvious that the present formulation is equivalent to the cited result from [10].
Looking for good topological representations of EDC-lattices
The following topological theorem proved in [9] (Theorem 4) gives necessary and sufficient conditions for a closed base of a topology to be semiregular.
Theorem 6.3 First characterization theorem for semiregularity.
Let X be a topological space and let CB(X) be a closed basis for X. Suppose that "·" is a binary operation defined on the set CB(X) such that (CB(X), ∅, X, ∪, ·) is a lattice. Then:
1. The following conditions are equivalent:
(a) CB(X) is U -extensional. (b) CB(X) ⊆ RC(X).
(c) For all a, b ∈ CB(X), a · b = Cl (Int(a ∩ b)).
(d) (CB(X), ∅, X, ∪, ·) is a dually dense sublattice of the Boolean algebra RC(X).
If any of the (equivalent) conditions (a),(b),(c) or (d) of 1. is fulfilled then:
(a) (CB(X), ∅, X, ∪, ·) is a U -extensional distributive lattice.
(b) X is a semiregular space.
The following is a corollary of the above theorem.
Corollary 6.4 [9] Let X be a topological space, let L = (L, 0, 1, +, ·) be a lattice and let h be an embedding of the upper semi-lattice (L, 0, 1, +) into the lattice C(X) of closed sets of X. Suppose that the set CB(X) = {h(a) : a ∈ L} forms a closed basis for the topology of X. Then:
1. The following conditions are equivalent:
(a) L is U -extensional. (b) CB(X) ⊆ RC(X). (c) For all a, b ∈ L, h(a · b) = Cl(Int(h(a) ∩ h(b))).
(d) h is a dually dense embedding of L into the Boolean algebra RC(X).
If any of the (equivalent) conditions (a),(b),(c) or (d) of 1. is fulfilled then:
(a) L is a U -extensional distributive lattice.
(b) X is a semiregular space.
A dual version of Theorem 6.3 is the following one. (c) For all a, b ∈ OB(X), a + b = Int(Cl(a ∪ b)).
(d) (OB(X), ∅, X, ∩, +) is a dually dense sublattice of the Boolean algebra RO(X). (b) X is a semiregular space.
The following is a corollary of the above theorem. (c) For all a, b ∈ L, h(a + b) = Int(Cl(h(a) ∪ h(b))).
(d) h is a dense embedding of L into the Boolean algebra RO(X).
2. If any of the (equivalent) conditions (a),(b),(c) or (d) of 1. is fulfilled then:
(a) L is a O-extensional distributive lattice.
(b) X is a semiregular space.
Remark 6.7 (i) Let D = (D, C, C, ≪) be an EDC-lattice. Corollary 6.4 shows that if we want to represent D by a dually dense embedding h into the contact algebra RC(X) of some topological space X such that the topology of X to be determined by the set CB(X) = {h(a) : a ∈ D} considered as a closed base for X we must require that the lattice D is U-extensional, i.e. to satisfy the axiom (Ext O) (extensionality of underlap). If in addition we want to apply the good properties of Lemma 5.5 then we must assume that h is also a C-separable embedding into RC(X). But then Corollary 5.8 implies that D must satisfy also the axioms (U-rich ≪) and (U-rich C).
(ii) Similar to the above conclusion is the following. Corollary 6.6 shows that if we want to represent D by a dense embedding h into the contact algebra RO(X) of some topological space X such that the topology of X to be determined by the set OB(X) = {h(a) : a ∈ D} considered as an open base for X we must require that the lattice D is O-extensional, i.e. to satisfy the axiom (Ext O) (extensionality of overlap). If in addition we want to apply the good properties of Lemma 5.7 then we must assume that h is also a C-separable embedding into RO(X). But then Corollary 5.8 implies that D must satisfy also the axioms (O-rich ≪) and (O-rich C). The aim of the next sections is to develop the topological representation theory of U-rich and O-rich EDC-lattices.
Topological representation theory of U-rich EDC-lattices
The aim of this section is to develop a topological representation theory for U-rich EDC-latices. According to Theorem 6.3 we will look for a representation with regular closed sets. To realize this we will follow the representation theory of contact algebras by regular closed sets developed in [6,24], updating the results of Section 4 from [9] to the case of U-rich EDC-lattices. We will consider also extensions of U-rich EDC-lattices with some of the additional axioms mentioned in Section 5. The scheme of the representation procedure is the following: for each U-rich EDC-lattice D from a given class, determined by the additional axioms, we will do the following:
• Define a set X(D) of "abstract points" of D,
• define a topology in X(D) by the set CB(X(D)) = {h(a) : a ∈ D}, considered as a closed base of the topology, where h is the intended embedding of Stone type: h(a) = {Γ : Γ is "abstract point" and a ∈ Γ}. X(D) is called the canonical topological space of D and h is called canonical embedding,
• establish that h is a dual dense embedding of the lattice D into the Boolean algebra RC(X(D)) of regular closed sets of the space X(D).
We will consider separately the cases of representations in T 0 , T 1 and T 2 spaces which requires introducing different "abstract points".
Representations in T 0 spaces
Troughout this section we consider that D = (D, C, C, ≪) is a U-rich EDClattice.
Abstract points of D.
As in [9], we consider the abstract points of D to be clans (see [6] for the origin of this notion). The definition is the following. A subset Γ ⊆ D is a clan if it satisfies the following conditions:
(Clan 1) 1 ∈ Γ, 0 ∈ Γ, (Clan 2) If a ∈ Γ and a ≤ b, then b ∈ Γ, (Clan 3) If a + b ∈ Γ, then a ∈ Γ or b ∈ Γ, (Clan 4) If a, b ∈ Γ then aCb. Γ is a maximal clan if it is maximal with respect to the set-inclusion. We denote by CLAN(D) (MaxCLAN(D) ) the set of all (maximal) clans of D.
The notion of clan is an abstraction from the following natural example. Let X be a topological space and RC(X) be the contact algebra of regular-closed subsets of X and let x ∈ X. Then the set Γ x = {a ∈ RC(X) : x ∈ a} is a clan. Now we will present a construction of clans which is similar to the constructions of clans in contact algebras. First we will introduce a new canonical relation between prime filters. Definition 7.1 Let U, V be prime filters. Define a new canonical relation R C ( R C -canonical relation) between prime filters as follows:
U R C V ↔ def (∀a ∈ U )(∀b ∈ V )(aCb).
Let us note that the relation R C depends only on C and can be defined also for filters. It is different from the canonical relation between prime filters defined in Section 3.3, but the presence of U-rich axioms makes it equivalent to R c as it can be seen from the following lemma. Claim 1: a ∈ U and b ∈ V implies aCb. This is just by the definition of R C . Claim 2: a ∈ U and b ∈ V implies a ≪ b. For the sake of contradiction suppose a ∈ U and b ∈ V but a ≪ b. Then by axiom (U-rich ≪) ( a ≪ b → (∃c)(b+c = 1 and aCc), we obtain b + c = 1 and aCc. Conditions b + c = 1 and b ∈ V imply c ∈ V . But a ∈ U , so aCc -a contradiction. 1. Let F, G be filters and F R C G then there are prime filters U, V such that
F ⊆ U , G ⊆ V and U R C V .
2. For all a, b ∈ D: aCb iff there exist prime filters U, V such that U R C V , a ∈ U and b ∈ V .
In the following lemma we list some facts about clans (see, for instance, [6,9]).
Facts 7.4
1. Every prime filter is a clan.
2. The complement of every clan is an ideal.
3. If Γ is a clan and F is a filter such that F ⊆ Γ, then there is a prime filter U such that F ⊆ U ⊆ Γ. In particular, if a ∈ Γ, then there exists a prime filter U such that a ∈ U ⊆ Γ.
4. Every clan Γ is the union of all prime filters contained in Γ.
5. Every clan is contained in a maximal clan.
6. Let Σ be a nonempty set of prime filters such that for every U, V ∈ Σ we have U R C V and let Γ be the union of the elements of Σ. Then Γ is a clan and every clan can be obtained in this way.
7. Let U, V be prime filters, Γ be a clan and U, V ⊆ Γ,. Then U R C V and U R c V .
Lemma 7.5 Let Γ be a clan and a ∈ D. Then the following two conditions are equivalent:
(i) (∀c ∈ D)(a + c = 1 → c ∈ Γ),(ii)
There exists a prime filter U ⊆ Γ such that a ∈ U .
Proof. (i)→ (ii). Suppose that (i) holds. It is easy to see that the set F = {c : a + c = 1} is a filter. The complement Γ of Γ is an ideal (Facts 7.4) and hence Γ ⊕ (a] is an ideal. We will show that F ∩ Γ ⊕ (a] = ∅. Suppose the contrary. Then there is a c such that a + c = 1 (and hence by (i) c ∈ Γ) and c ∈ Γ ⊕ (a]. Then there is x ∈ Γ such that c ≤ x + a. From here we get: 1 = a + c ≤ a + x + a = x + a, hence x + a = 1 and by (i)x ∈ Γ, contrary to x ∈ Γ. Now we can apply Filter-extension Lemma and obtain a prime filter U extending F such that U ∩ Γ ⊕ (a] = ∅. It follows from here that a ∈ U , U ∩ Γ = ∅ which implies U ⊆ Γ. To show that h is an embedding we use the fact that prime filters are clans and prove that a ≤ b implies h(a) ⊆ h(b). Indeed, from a ≤ b it follows by the theory of distributive lattices (see [2]) that there exists a prime filter U (which is also a clan) such that a ∈ U (so U ∈ h(a)) and b ∈ U (so, U ∈ h(b)), which proves that h(a) ⊆ h(b). Consequently, h is an embedding of the upper semi-lattice (D, 0, 1, +) into the lattice of closed sets of the space X(D). By Corollary 6.4, X(D) is a semiregular space and h is a dually dense embedding of D into the Boolean algebra RC(X). It remains to show that h preserves the relations C, C and ≪. This follows from the following claim. Claim 7.7 (i) Let Γ be a clan and a ∈ D. Then following equivalence holds:
Γ ∈ h(a) iff there exists a prime filter U such that a ∈ U ⊆ Γ.
(ii) Let Γ be a clan and a ∈ D. Then following conditions are equivalent:
(I) (∀c ∈ D)(a + c = 1 → c ∈ Γ), (II) Γ ∈ Cl(−h(a)),
(III) There exists a prime filter U such that a ∈ U ⊆ Γ.
(iii) aCb iff h(a) ∩ h(b) = ∅, (iv) a ≪ b iff h(a) ∩ Cl(−h(b)) = ∅. (v) a Cb iff Cl(−h(a)) ∩ Cl(−h(b)) = ∅,
Proof of the claim. (i) follows easily from Facts 7.4 (3.). (ii) The proof of (I) ↔ (II) follows by the following sequence of equivalences:
Proof of (C-separability for C). Let α, β ∈ RC(X(D)) and α ∩ β = ∅.
b) ⊆ h(d 1 ) ∩ ... ∩ h(d m ). Consequently h(a) ∩ h(b) ⊆ (h(c 1 ) ∩ ... ∩ h(c n ) ∩ (h(d 1 ) ∩ ... ∩ h(d m )) = ∅, so h(a) ∩ h(b) = ∅. Also we have α ⊆ h(c) for all h(c) ∈ A and consequently for all h(c) ∈ A 0 . Hence α ⊆ h(c 1 ) · ... · h(c n ) = h(c 1 · ... · c n ) = h(a), so α ⊆ h(a). Analogously we get β ⊆ h(b).
The following theorem is the main result of this section.
Theorem 7.10 Topological representation theorem for U -rich EDClattices Let D = (D, C, C, ≪) be an U -rich EDC-lattice. Then there exists a compact semiregular T 0 -space X and a dually dense and C-separable embedding h of D into the Boolean contact algebra RC(X) of the regular closed sets of X. Moreover: (i) D satisfies (Ext C) iff RC(X) satisfies (Ext C); in this case X is weakly regular.
(ii) D satisfies (Con C) iff RC(X) satisfies (Con C); in this case X is connected.
(iii) D satisfies (Nor 1) iff RC(X) satisfies (Nor 1); in this case X is κnormal.
Proof. Let X be the canonical space X(D) of D and h be the canonical embedding of D. Then, the theorem is a corollary of Lemma 7.6, Lemma 7.8, Lemma 7.9 and Lemma 6.2.
Note that Theorem 7.10 generalizes several results from [6,10] to the distributive case.
Representations in T 1 spaces
The aim of this section is to obtain representations of some U-rich EDC-lattices in T 1 -spaces extending the corresponding results from [9]. The constructions will be slight modifications of the corresponding constructions from the previous section, so we will be sketchy.
Let D = (D, C, C, ≪) be an U -rich EDC-lattice. In the previous section the abstract points were clans and this guarantees that the representation space is T 0 . To obtain representations in T 1 spaces we assume abstract points to be maximal clans, so for the canonical space of D we put X(D) = M axCLAN (D) and define the canonical embedding h to be h(a) = {Γ ∈ M axCLAN (D) : a ∈ Γ}. The topology in X(D) is defined considering the set CB(X(D)) = {h(a) : a ∈ D} to be the closed base for the space. Note that in general, without additional axioms we can not prove in this case that h is an embedding. In order to guarantee this we will assume that D satisfies additionally the axiom of C-extensionality (Ext C) a = 1 → (∃b = 0)(aCb). Note that in this case, due to U-extensionality (see Section 5), the lattice D satisfies also the axiom (EXT C) a ≤ b → (∃c)(aCc and bCc), which is essential in the proof that h is an embedding. Lemma 7.11 The space X(D) is a semiregular and h is a dually dense embedding of D into the contact Boolean algebra RC(X(D)).
Proof. The proof is similar to the proof of Lemma 7.6, so we will indicate only the differences. First we show that h is an embedding of the upper semilattice (D, 0, 1, +) into the lattice of closed sets of the space X(D). The only new thing which we have to show is: If a ≤ b then h(a) ⊆ h(b). To do this suppose a ≤ b. Then by axiom (EXT C) there exists c ∈ D such that aCc but bCc. Condition aCc implies that there exist prime filters U, V such that U R c V , a ∈ U and c ∈ V . Let Γ 0 = U ∪ V . Γ 0 is a clan and by Facts 7.4 it is contained in a maximal clan Γ. Obviously a, c ∈ Γ, so Γ ∈ h(a). But bCc implies that b ∈ Γ (otherwise we will get bCc). Conditions Γ ∈ h(a) and Γ ∈ h(b) show that h(a) ⊆ h(b). Thus, by Corollary 6.4 h is a dually dense embedding of D into the Boolean algebra RC(X(D)). It remains to show that h preserves the relations C, C and ≪. The proof is almost the same as in the corresponding proof of Lemma 7.6. The only new thing is when we construct a certain clan from prime filters satisfying the relation U R c V in the form U ∪ V , then we extend it into a maximal clan. Note also that Claim 7.7 remains true. We demonstrate this by considering only the preservation of ≪. We have to show:
a ≪ b iff h(a) ∩ Cl(−h(b) = ∅ (⇒) Suppose a ≪ b.
Then by Lemma 3.8 (∃U, V ∈ P F (D))(a ∈ U and b ∈ V and U R c V ). Define Γ 0 = U ∪ V . Γ 0 is a clan containing U and V . Extend Γ 0 into a maximal clan Γ. Then Γ contains a, so Γ ∈ h(a). We have also that b ∈ V ⊆ Γ, so by the Claim 7.7 Γ ∈ Cl(−h(b)).
(⇐) The proof is identical to the corresponding proof from Lemma 7.6.
Lemma 7.12
The space X(D) satisfies the following conditions:
(i) X(D) is T 1 , (ii) X(D) is compact, (iii) h is C-separable embedding.
Proof. (i) Let Γ be an arbitrary maximal clan. The space X(D) is T 1 iff the singleton set {Γ} is closed, i.e. Cl({Γ}) = {Γ}. This follows by the maximality of Γ as follows. Let ∆ be a maximal clan. Then:
∆ ∈ Cl({Γ}) iff (∀c ∈ D)({Γ} ⊆ h(c) → ∆ ∈ h(c)) iff (∀c ∈ D)(Γ ∈ h(c) → ∆ ∈ h(c)) iff (∀c ∈ D)(c ∈ Γ → c ∈ ∆ iff Γ ⊆ ∆) iff Γ = ∆ iff ∆ ∈ {Γ}.
This chain shows that indeed
Cl({Γ}) = {Γ}. (ii)
The proof is similar to the proof of Lemma 7.8 (ii) (iii) follows from (ii) as in the proof of Lemma 7.9.
Theorem 7.13 Topological representation theorem for C-extensional U -rich EDC-lattices Let D = (D, C, C, ≪) be a C-extensional U -rich EDClattice. Then there exists a compact weakly regular T 1 -space X and a dually dense and C-separable embedding h of D into the Boolean contact algebra RC(X) of the regular closed sets of X. Moreover:
(i) D satisfies (Con C) iff RC(X) satisfies (Con C); in this case X is connected.
(ii) D satisfies (Nor 1) iff RC(X) satisfies (Nor 1); in this case X is κnormal.
Proof. The proof follows from Lemma 7.11, Lemma 7.12 and Lemma 6.2.
Representations in T 2 spaces
In the previous section we proved representability in T 1 spaces of U-rich EDClattices satisfying the axiom of C-extensionality (Ext C). The T 1 property of the topological space was guaranteed by the fact that abstract points are maximal clans. In this section we will show that adding the axiom (Nor 1) we can obtain representability in compact T 2 -spaces. The reason for this is that the axiom (Nor 1) makes possible to use new abstract points -the so called clusters, which are maximal clans satisfying some additional properties yielding T 2 separability of the topological space. Clusters have been used in the compactification theory of proximity spaces (see more about their origin in [23]). They have been adapted in algebraic form in the representation theory of contact algebras in [6,26]. In [9] their definition and some constructions are modified for the distributive case. We remaind below the corresponding definition.
Definition 7.14 Let D = (D, C, C, ≪) be an EDC-lattice. A clan Γ in D is called a cluster if it satisfies the following condition:
(Cluster) If for all b ∈ Γ we have aCb, then a ∈ Γ.
We denote the set of clusters in D by CLU ST ER(D).
Let us note that not in all EDC-lattices there are clusters. The following lemma shows that the axiom (Nor 1) guarantees existence of clusters and some important properties needed for the representation theorem. (i) Every cluster is a maximal clan. (ii) If D satisfies (Nor 1) then every maximal clan is a cluster.
(iii) If Γ and ∆ are clusters such that Γ = ∆, then there are a ∈ Γ and b ∈ ∆ such that a + b = 1.
To build the canonical space X(D) we assume in this section that D = (D, C, C, ≪) is an U-rich EDC-lattice satisfying the axioms (Ext C) and (Nor 1). We define X(D) = CLU ST ER(D), h(a) = {Γ ∈ CLU ST ER(D) : a ∈ Γ} and define the topology in X(D) considering the set CB(X) = {h(a) : a ∈ D} as a basis for closed sets in X(D). Since the points of X(D) are maximal clans, just as in Section 7.2 we can prove the following lemma. Lemma 7.16 The space X(D) is a semiregular and h is a dually dense embedding of D into the contact Boolean algebra RC(X(D)). (E-fil 2) If a ∈ Γ and b ∈ Γ, then a Cb. Γ is a minimal E-filter if it is minimal in the set of all E-filters of D with respect to set inclusion.
This definition comes as an abstraction from the following natural example. Let X be a topological space, x ∈ X and RO(X) be the set of all regular-open sets of X. Then the set Γ x = {a ∈ RO(X) : x ∈ a} is an E-filter in the contact algebra RO(X). Note that the definition of E-filter is based not on the relation of contact C, but on the dual contact C.
A general construction of E-filters can be obtain dualizing the construction of clans from Section 7.1. Just to show how this dual construction goes on and how the O-rich axioms works, we will repeat some steps omitting the proofs.
First we will introduce a new canonical relation between prime filters.
Definition 8.2 Let U, V be prime ideals. Define a new canonical relation R C ( R C -canonical relation) between prime ideals as follows:
U R C V ↔ def (∀a ∈ U )(∀b ∈ V )(a Cb).
If U, V are prime filters then we define U R C V ↔ def U R C V .
Let us note that the relation R C depend only on C and can be defined also for ideals. It is different from the canonical relation R c between prime ideals defined in Section 3.3, but the presence of O-rich axioms makes it equivalent to R c as it is stated in the following lemma. The following statement lists some facts about the relation R C .
Facts 8.4 1. Let F, G be ideals and F R C G then there are prime ideals U, V such that F ⊆ U , G ⊆ V and U R C V .
2. For all a, b ∈ D: a Cb iff there exist prime ideals U, V such that U R C V , a ∈ U and b ∈ V .
3. For all a, b ∈ D: a Cb iff there exist prime filters U, V such that U R C V , a ∈ U and b ∈ V .
In the following lemma we list some facts about E-filters.
Facts 8.5 1. Every prime filter is an E-filter.
2. If Γ is an E-filter and a ∈ Γ, then there exists a prime filter U such that Γ ⊆ U and a ∈ U .
3. Every E-filter Γ is the intersection of all prime filters containing Γ.
4. Every E-filter contains a minimal E-filter.
5. Let Σ be a nonempty set of prime filters such that for every U, V ∈ Σ we have U R C V and let Γ be the intersection of the elements of Σ. Then Γ is an E-filter and every E-filter can be obtained in this way.
6. Let U, V be prime filters, Γ be an E-filter, Γ ⊆ U and Γ ⊆ V . Then U R C V and U R c V .
Using the above facts one can prove the following representation theorem.
Theorem 8. 6 Let D = (D, C, C, ≪) be an O-rich EDC-lattice. Then there exists a compact semi-regular space X and a dense and C-separable embedding h from D into the contact algebra RO(X) of regular-open sets of X. Moreover: (i) If D satisfies (Ext C), then X is weakly regular, (ii) If D satisfies (Con C), then X is a connected space, (iii) If D satisfies (Nor 2), then X is κ-normal.
Abstract points for dense representations in T 1 spaces are minimal E-filters and abstract points for dense representations in T 2 spaces are duals of clusters introduced in [6] under the name co-clusters. We adapt this notion for the language of EDC-lattices as follows:
Definition 8.7 An E-filter Γ is called co-cluster if it satisfies the following condition:
(Co-cluster) If (∀b ∈ Γ)(a Cb), then a ∈ Γ. (or, equivalently, if a ∈ Γ, then (∃b ∈ Γ)(a Cb)).
Let us show, for instance, the following statement for co-clusters, which is dual to the corresponding property for clusters as maximal clans: Lemma 8.8 Every co-cluster is a minimal E-filter.
Proof. Suppose that Γ is a co-cluster which is not a minimal E-filter. Then there exists an E-filter ∆ such that ∆ ⊂ Γ, so a ∈ Γ and a ∈ ∆ for some a. Then there exists b ∈ Γ such that a Cb. From here we get b ∈ ∆. Consequently b ∈ Γ -a contradiction.
We left to the reader to formulate and proof the dual analog of Theorem 7.13 and Theorem 7.18.
Concluding remarks
In this paper we generalized the notion of contact algebra by weakening the algebraic part to distributive lattice. One solution of this problem was given in [9] including in the definition only the contact relation. However, the obtained axiomatization in [9] is in a sense "incomplete", because it does not contain the definable in the Boolean case mereotopological relations of dual contact C and non-tangential inclusion ≪ and its dual ≫ and in this sense the system is not closed under duality. We succeed in this paper to axiomatize all these relations considered as primitives on the base of distributive lattices by means of universal first-order axioms. The resulting system is called "extended distributive contact lattice" (EDC-lattice). In this way we obtain, among others, the following two results. First, EDC-lattice is closed under duality, and second, it can be considered as an axiomatization of the universal fragment of contact algebras in the language of distributive lattices with the relations C, C and ≪. We developed topological representation theory of EDC-lattices by means of regular closed and regular open sets generalizing in a quite non-trivial way the corresponding representation theory for contact algebras. Considering this representation theory on a weaker lattice base provided a deeper insight into the interaction of some notions taking place in the representation, which cannot be seen in the Boolean case. For instance we show the role of extensionality of underlap and overlap relations in case of dual dense and dense embeddings.
Our future plans include building of new logics for qualitative spatial representation and reasoning based on EDC-lattices, studying the standard logical problems related to them: axiomatizability, decidability or undecidability, complexity. A good source for possible generalizations and extensions is the paper [3] containing many examples of spatial logics based on contact and precontact algebras.
Acknowledgement
The final publication is available at Springer via http://dx.doi.org/10.1007/s10472-016-9499-5.
C1) If aCb, then a = 0 and b = 0, (C2) If aCb and a ≤ a ′ and b ≤ b ′ , then a ′ Cb ′ , (C3) If aC(b + c), then aCb or aCc, (C4) If aCb, then bCa, (C5) If a.b = 0, then aCb.
PART I: EXTENDED DISTRIBUTIVE CONTACT LATTICES: AXIOMATIZATION AND EMBEDDING IN CONTACT ALGE-BRAS2 Extended distributive contact lattices.
( C1 )
C1If aCb, then a = 0 and b = 0, (C2) If aCb and a ≤ a ′ and b ≤ b ′ , then a ′ Cb ′ , (C3) If aC(b + c), then aCb or aCc, (C4) If aCb, then bCa, (C5) If a.b = 0, then aCb.
( C1 )
C1If a Cb, then a, b = 1, ( C2) If a Cb and a ′ ≤ a and b ′ ≤ b, then a ′ Cb ′ , ( C3) If a C(b · c), then a Cb or a Cc, ( C4) If a Cb, then b Ca, ( C5) If a + b = 1, then a Cb.
(≪ 5 )
5If a ≪ c and b ≪ c, then (a + b) ≪ c, (≪ 6) If c ≪ a and c ≪ b, then c ≪ (a · b), (≪ 7) If a ≪ b and (b · c) ≪ d and c ≪ (a + d), then c ≪ d.
C1) If aCb and a ≪ c, then aC(b · c), (M C2) If aC(b · c) and aCb and (a · d)Cb, then d Cc, (M C1) If a Cb and c ≪ a, then a C(b + c), (M C2) If a C(b + c) and a Cb and (a + d) Cb, then dCc, (M ≪ 1) If a Cb and (a · c) ≪ b, then c ≪ b, (M ≪ 2) If aCb and b ≪ (a + c), then b ≪ c.
Corollary 2. 5
5The axioms of the relations C, C and ≪ are true in contact algebras.
Let F 0 be a filter, I 0 be an Ideal and F 0 ∩ I 0 = ∅. Then:
Lemma 3. 4
4Let F 0 be a filter, I 0 be an Ideal and F 0 ∩ I 0 = ∅. Then:
Lemma 3. 6
6Let D = (D, C, C, ≪) be an EDC-lattice and Let Γ be a prime filter in D. Then:
Lemma 3. 10
10Let (W c , R c ) be the canonical structure of D = (D, C, C, ≪) and h(a) = {U ∈ P F (D) : a ∈ U } be the Stone mapping from D into the distributive lattice of all subsets of W c . Then h is an embedding of D into the EDC-lattice over (W c , R c ).
Theorem 3 .
311 Relational representation Theorem of EDC-latices. Let D = (D, C, C, ≪) be an EDC-lattice. Then there is a relational system W = (W, R) with reflexive and symmetric R and an embedding h into the EDC-lattice of all subsets of W .
and a Cb → c Ob (≪ C O) cCa and c Ob → a ≪ b (≪ CO) cOa and c Cb → a ≪ b.
Theorem 4. 1
1Every EDC-lattice is a mereotopological structure under the standard definitions of the basic mereological relations.
Definition 4. 2
2The system RCC-8.
•
disconnected -DC(a, b): aCb, • external contact -EC(a, b): aCb and aOb, • partial overlap -PO(a, b): aOb and a ≤ b and b ≤ a, • tangential proper part -TPP(a, b): a ≤ b and a ≪ b and b ≤ a, • tangential proper part −1 -TPP −1 (a, b): b ≤ a and b ≪ a and a ≤ b, • nontangential proper part NTPP(a, b): a ≪ b and a = b, • nontangential proper part −1 -NTPP −1 (a, b): b ≪ a and a = b, • equal -EQ(a, b): a = b.
(
Nor 1) aCb → (∃c, d)(c + d = 1, aCc and bCd), (Nor 2) a Cb → (∃c, d)(c · b = 0, a Cc and b Cd),
lemma. Let D = (D, C, C, ≪) be a EDC-lattice satisfying the axioms (Nor1), (Nor 2) and (Nor 3) and let (W c , R c ) be the canonical structure of D (see Section 3.3) Then:
(
U-rich C) a Cb → (∃c, d)(a + c = 1, b + d = 1 and cCd).
(
O-rich ≪) a ≪ b → (∃c)(a · c = 0 and c Cb), (O-rich C) aCb → (∃c, d)(a · c = 0, b · d = 0 and c Cd).
Lemma 5. 2
2The axioms (U-rich ≪), (U-rich C), (O-rich ≪) and (O-rich C) are true in all contact algebras.
Lemma 5. 7
7Let D 1 , D 2 be EDC-lattices and D 1 be a C-separable EDC-sublattice of D 2 ; then: (i) If D 1 is a dense EDC-sublattice of D 2 , then D 1 satisfies the axiom (Ext C) iff D 2 satisfies the axiom (Ext C), (ii) D 1 satisfies the axiom (Con C) iff D 2 satisfies the axiom (Con C), (iii) D 1 satisfies the axiom (Nor 2) iff D 2 satisfies the axiom (Nor 2). (iv) D 1 satisfies the axiom (O-rich ≪) iff D 2 satisfies the axiom (O-rich ≪). (v) D 1 satisfies the axiom (O-rich C) iff D 2 satisfies the axiom (O-rich C). Corollary 5.8 Let D = (D, C, C, ≪) be an EDC-lattice and B = (B, C) be a contact algebra. Then: (i) If h is a C-separable embedding of D into B then D must satisfy the axioms (U-rich ≪) and (U-rich C). (ii) If h is a C-separable embedding of D into B then D must satisfy the axioms (O-rich ≪) and (O-rich C).
A subset a ⊆ X is called open if it is the complement of a closed set. A family of closed sets CB(X) is called a closed basis of the topology if every closed set can be represented as an intersection of sets from CB(X). In a similar way the topology of X can be characterized by the family O(X) of open sets: it contains the empty set, X and is closed under finite intersections and arbitrary unions. A family OB(X) of open sets is called an open basis of the topology if every open set can be represented as an union of sets from OB(X). X is called semiregular space if it has a closed base of regular closed sets or an open base of regular open sets. We remaind the definitions of two important topological operations on sets -closure operation Cl, and interior operation Int. Namely Cl(a) is the intersection of all closed sets of X containing a and Int(a) is the union of all open sets included in a. Note that the operations Cl and Int are interdefinable: Cl(a) = −Int(−a) and Int(a) = −Cl(−a). Using the bases CB(X) and OB(X) the definitions of closure and interior operations have the following useful expressions:
Theorem 6. 1
1Topological representation theorem for EDC-lattices. Let D = (D, C, C, ≪) be an EDC-lattice. Then: (i) There exists a topological space X and an embedding of D into the contact algebra RC(X) of regular closed subsets of X. (ii) There exists a topological space Y and an embedding of D into the contact algebra RO(Y ) of regular open subsets of Y .
Theorem 6. 5
5Second characterization theorem for semiregularity. Let X be a topological space and let OB(X) be an open basis for X. Suppose that + is a binary operation defined on the set OB(X) such that (OB(X), ∅, X, ∩, +) is a lattice. Then: 1. The following conditions are equivalent: (a) OB(X) is O-extensional.(b) OB(X) ⊆ RO(X).
2 .
2If any of the (equivalent) conditions (a),(b),(c) or (d) of 1. is fulfilled then: (a) (OB(X), ∅, X, ∩, +) is an O-extensional distributive lattice.
Corollary 6. 6
6Let X be a topological space, let L = (L, 0, 1, +, ·) be a lattice and let h be an embedding of the lower semi-lattice (L, 0, 1, ·) into the lattice O(X) of open sets of X. Suppose that the set OB(X) = {h(a) : a ∈ L} forms an open basis for the topology of X. Then: (a) L is O-extensional. (b) OB(X) ⊆ RO(X).
Definition 6.8 U-rich and O-rich EDC-lattices. Let D = (D, C, C, ≪) be an EDC-lattice. Then: (i) D is called U-rich EDC-lattice if it satisfies the axioms (Ext O), (U-rich ≪) and (U-rich C). (ii) D is called O-rich EDC-lattice if it satisfies the axioms (Ext O), (O-rich ≪) and (O-rich C).
Lemma 7. 2
2(i) R C is reflexive and symmetric relation.(ii) If D satisfies the axioms (U-rich ≪) and (U-rich C) then R C = R c .Proof. (i) follows from the axioms (C4) and (C5).(ii) The inclusion R c ⊆ R C follows directly by the definition of R c . For the converse inclusion suppose U R C V . To show U R c V we have to inspect the four cases of the definition of R c .
Claim 3 :
3a ∈ U and b ∈ V implies b ≪ a.The proof is similar to the proof of Claim 2.
Claim 4 :
4a ∈ U and b ∈ V implies a Cb. The proof is similar to the proof of Claim 2 by the use of axiom (U-rich C) a Cb → (∃c, d)(a + c = 1, b + c = 1 and cCd).The following statement lists some facts about the relation R C .
(ii)→(i). Suppose(ii) holds: U ⊆ Γ and a ∈ U . Suppose a + c = 1. Then c ∈ U ⊆ Γ, so c ∈ Γ -(i) is fulfilled.Defining the canonical topological space X(D) of D and the canonical embedding h.Define the Stone like embedding: h(a) = {Γ ∈ CLAN (D) : a ∈ Γ} and consider the set CB(X) = {h(a) : a ∈ D} as a closed base of the topology in X(D) = CLAN (D).
Lemma 7. 6
6The space X(D) is semiregular and h is a dually dense embedding of D into the contact Boolean algebra RC(X(D)).Proof. Using the properties of clans, one can easily check that h(0) = ∅, h(1) = X, and that h(a+b) = h(a)∪h(b). This shows that the set CB(X(D)) = {h(a) : a ∈ D} is closed under finite unions and, in fact, it is a closed basis for the topology of X. Also we have the implication: a ≤ b then h(a) ⊆ h(b).
Since α and β are closed sets they can be represented as intersections from the elements of the basis CB(X(D)) = {h(c) : c ∈ D} of X(D). So there aresubsets A, B ⊆ CB(X(D)) such that α = {h(c) : h(c) ∈ A} and β = {h(c) : h(c) ∈ B}. Then α ∩ β = {h(c) : h(c) ∈ A} ∩ {h(c) : h(c) ∈ B} = ∅. By the compactness of X(D) (Lemma 7.8 (ii)), there are finite subsets A 0 ⊆ A and B 0 ⊆ B such that α ∩ β = {h(c) : h(c) ∈ A 0 } ∩ {h(c) : h(c) ∈ B 0 } = ∅. LetA 0 = {h(c 1 ), ..., h(c n )} and B 0 = {h(d 1 ), ..., h(d m )} and let a = c 1 · ... · c n and b = d 1 · ... · d m . Then h(a) ⊆ h(c i ), i = 1...n and from here we get h(a) ⊆ h(c 1 ) ∩ ... ∩ h(c n ). Analogously we obtain that h(
Lemma 7 .
715 [9] Let D = (D, C, C, ≪) be an EDC-lattice. Then:
Lemma 7 .
717 (i) X(D) is T 2 , (ii) X(D) is compact, (iii) h is C-separable embedding.
Lemma 8.3 (i) R C is a reflexive and symmetric relation.(ii) If D satisfies the axioms (O-rich ≪) and (O-rich C), then R C = R c .
Topological representation theory of O-rich EDC-lattices
This section is devoted to the theory of dense representations for O-rich EDClatices (seeDefinition 6.8). According to Theorem 6.5 we will look for dense representations with regular open sets. This case is completely dual to the corresponding theory developed in Section 7. For this reason we will only sketch the main representation scheme and the definitions of abstract points for the T 0 , T 1 and T 2 representations.The representation scheme is dual to the scheme presented in Section 7:• Define a set X(D) of "abstract points" of D,• define a topology in For the case of T 0 dense representation we consider a notion of abstract point which is dual to the notion of clan. This is the so called E-filter (Efremovich filter). E-filters were used in the theory of proximity spaces (see[23]). In the context of contact algebras they were introduced for the first time in[6]. The definition adapted for the language of EDC-lattices is the following.
By Facts 7.4 Γ is a clan, obviously containing a and b, which implies h(a) ∩ h(b) = ∅. (⇐) Suppose h(a) ∩ h(b) = ∅. Then there exists a clan Γ containing a and b, hence aCb. (iv) (⇒ ) Suppose a ≪ b. Then by Lemma 3.8 (ii) there exist prime filters U, V such that U R c V , a ∈ U and b ∈ V . Let Γ = U ∪ V , then Γ is a clan containing U and V . So, a ∈ Γ and hence Γ ∈ h(a). From the condition b ∈ V ⊆ Γ we obtain by (ii) that Γ ∈ Cl(−h(b)) and hence h(a)∩Cl. ∈ U R C V, ∈ V Let Γ = U ∪ V, c ∈ D} is a closed base of the topology of X(D). The equivalence (I) ↔ (III) is just the Lemma 7.5. (iii) (⇒ ) Suppose aCb, then by Lemma 3.8 (i) there exist prime filters U , and V such that. −h(b)) = ∅. (⇐) Suppose h(a) ∩ Cl(−h(b)) = ∅. Then there exists a clan Γ ∈ h(a) andsemi- lattice (D, 0, 1, +) into the lattice of closed sets of the space X(D), the third equivalence uses the fact that the set {h(c) : c ∈ D} is a closed base of the topology of X(D). The equivalence (I) ↔ (III) is just the Lemma 7.5. (iii) (⇒ ) Suppose aCb, then by Lemma 3.8 (i) there exist prime filters U , and V such that U R c V , a ∈ U and b ∈ V . Let Γ = U ∪ V . By Facts 7.4 Γ is a clan, obviously containing a and b, which implies h(a) ∩ h(b) = ∅. (⇐) Suppose h(a) ∩ h(b) = ∅. Then there exists a clan Γ containing a and b, hence aCb. (iv) (⇒ ) Suppose a ≪ b. Then by Lemma 3.8 (ii) there exist prime filters U, V such that U R c V , a ∈ U and b ∈ V . Let Γ = U ∪ V , then Γ is a clan containing U and V . So, a ∈ Γ and hence Γ ∈ h(a). From the condition b ∈ V ⊆ Γ we obtain by (ii) that Γ ∈ Cl(−h(b)) and hence h(a)∩Cl(−h(b)) = ∅. (⇐) Suppose h(a) ∩ Cl(−h(b)) = ∅. Then there exists a clan Γ ∈ h(a) and
It follows by (i) that there exists a prime filter U such that a ∈ U ⊆ Γ and by (ii) we obtain that there exists a prime filter V such that b ∈ V ⊆ Γ. Condition U, V ⊆ Γ implies by Facts 7.4 (7.) that U R c V . Using the properties of the relation R c and a ∈ U and b ∈ V we get a ≪ b. (v) The proof of. Γ ∈ Cl, v) is similar to the proof of (iv) with the use of LemmaΓ ∈ Cl(−h(b)). It follows by (i) that there exists a prime filter U such that a ∈ U ⊆ Γ and by (ii) we obtain that there exists a prime filter V such that b ∈ V ⊆ Γ. Condition U, V ⊆ Γ implies by Facts 7.4 (7.) that U R c V . Using the properties of the relation R c and a ∈ U and b ∈ V we get a ≪ b. (v) The proof of (v) is similar to the proof of (iv) with the use of Lemma
X(D) is compact. X(D) is compact.
The proof is the same as the proof of Lemma 19 from. 9Proof. The proof is the same as the proof of Lemma 19 from [9].
The mapping h is a C-separable embedding of D into RC(X(D)). Lemma, Lemma 7.9 The mapping h is a C-separable embedding of D into RC(X(D)).
Since the definition of C-separability for EDC-lattices uses an extended definition for which the special construction from [9] does not hold. This lemma was proved in [9. in this paper we give a new proof deducing the statement from the compactness of the space X(D). We have to proof the following three statements, corresponding to the three clauses of the condition of C-separability (see Definition 5.4)Proof. This lemma was proved in [9] by a special construction. Since the definition of C-separability for EDC-lattices uses an extended definition for which the special construction from [9] does not hold, in this paper we give a new proof deducing the statement from the compactness of the space X(D). We have to proof the following three statements, corresponding to the three clauses of the condition of C-separability (see Definition 5.4).
-separability for C) (∀α, β ∈ RC(X(D)))(α ∩ β = ∅ → (∃a, b ∈ D)(α ⊆ h(a), β ⊆ h(b). h(a) ∩ h(b)-separability for C) (∀α, β ∈ RC(X(D)))(α ∩ β = ∅ → (∃a, b ∈ D)(α ⊆ h(a), β ⊆ h(b), h(a) ∩ h(b))
. = ∅ , = ∅.
))(Cl(−α) ∩ Cl(−β) = ∅ → (∃a, b ∈ D)(α ∪ h(a) = X(D), β ∪ h(b) = X(D), h(a) ∩ h(b) = ∅). C-separability for C) (∀α, β ∈ RC(X(D(C-separability for C) (∀α, β ∈ RC(X(D))(Cl(−α) ∩ Cl(−β) = ∅ → (∃a, b ∈ D)(α ∪ h(a) = X(D), β ∪ h(b) = X(D), h(a) ∩ h(b) = ∅).
))(α ∩ Cl(−β) = ∅ → (∃a, b ∈ D)(α ⊆ h(a), β ∪ h(b) = X(D), h(a) ∩ h(b) = ∅). C-separability for ≪) (∀α, β ∈ RC(X(DC-separability for ≪) (∀α, β ∈ RC(X(D))(α ∩ Cl(−β) = ∅ → (∃a, b ∈ D)(α ⊆ h(a), β ∪ h(b) = X(D), h(a) ∩ h(b) = ∅).
As an example we shall prove the condition (C-separability for C). The proofs for the other two conditions are similar. As an example we shall prove the condition (C-separability for C). The proofs for the other two conditions are similar.
T 2 suppose that Γ, ∆ are two different clusters. We have to find two disjoint open sets A, B such that Γ ∈ A and ∆ ∈ B. By Lemma 7.15 (iii) there are a, b ∈ D such that a ∈ Γ and b ∈ ∆ such that a + b = 1. Then by Lemma 7.16 we get Γ ∈ h(a), ∆ ∈ h(b) and h(a) ∪ h(b) = X(D), hence −h(a) ∩ −h(b) = ∅. i) To show that the space X(D) is. Define A = −h(a), B = −h(b)Proof. (i) To show that the space X(D) is T 2 suppose that Γ, ∆ are two different clusters. We have to find two disjoint open sets A, B such that Γ ∈ A and ∆ ∈ B. By Lemma 7.15 (iii) there are a, b ∈ D such that a ∈ Γ and b ∈ ∆ such that a + b = 1. Then by Lemma 7.16 we get Γ ∈ h(a), ∆ ∈ h(b) and h(a) ∪ h(b) = X(D), hence −h(a) ∩ −h(b) = ∅. Define A = −h(a), B = −h(b).
Since h(a) and h(b) are closed sets, then A and B are open sets which separate the abstract points Γ and ∆. The proof of (ii) and (iii) is the same as the proof of (ii) and (iii) in Lemma. Since h(a) and h(b) are closed sets, then A and B are open sets which separate the abstract points Γ and ∆. The proof of (ii) and (iii) is the same as the proof of (ii) and (iii) in Lemma
Then there exists a compact T 2 -space X and a dually dense and C-separable embedding h of D into the Boolean contact algebra RC(X) of the regular closed sets of X. Moreover D satisfies (Con C) iff RC(X) satisfies (Con C) and in this case X is connected. D = (d Let, C , C , ≪ ) , Topological representation theorem for U -rich EDClattices satisfying (Ext C) and (Nor 1). The proof follows from Lemma 7.16, Lemma. Let us note that this theorem generalizes several theorems from [6, 24, 27, 26] ReferencesTopological representation theorem for U -rich EDC- lattices satisfying (Ext C) and (Nor 1). Let D = (D, C, C, ≪) be an U -rich EDC-lattice satisfying (Ext C) and (Nor 1). Then there exists a compact T 2 - space X and a dually dense and C-separable embedding h of D into the Boolean contact algebra RC(X) of the regular closed sets of X. Moreover D satisfies (Con C) iff RC(X) satisfies (Con C) and in this case X is connected. Proof. The proof follows from Lemma 7.16, Lemma 7.17 and 6.2. Let us note that this theorem generalizes several theorems from [6, 24, 27, 26] References
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"A Nonparametric Bayesian Approach to Copula Estimation",
"A Nonparametric Bayesian Approach to Copula Estimation",
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"Neil Shephard [email protected] ",
"\nDepartment of Statistics\nDepartment of Economics and Department of Statistics\nHarvard University\n02138CambridgeMA\n",
"\nHarvard University\n02138CambridgeMA\n",
"Shaoyang Ning [email protected] ",
"Neil Shephard [email protected] ",
"\nDepartment of Statistics\nDepartment of Economics and Department of Statistics\nHarvard University\n02138CambridgeMA\n",
"\nHarvard University\n02138CambridgeMA\n"
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"Department of Statistics\nDepartment of Economics and Department of Statistics\nHarvard University\n02138CambridgeMA",
"Harvard University\n02138CambridgeMA",
"Department of Statistics\nDepartment of Economics and Department of Statistics\nHarvard University\n02138CambridgeMA",
"Harvard University\n02138CambridgeMA"
]
| []
| We propose a novel Dirichlet-based Pólya tree (D-P tree) prior on the copula and based on the D-P tree prior, a nonparametric Bayesian inference procedure. Through theoretical analysis and simulations, we are able to show that the flexibility of the D-P tree prior ensures its consistency in copula estimation, thus able to detect more subtle and complex copula structures than earlier nonparametric Bayesian models, such as a Gaussian copula mixture. Further, the continuity of the imposed D-P tree prior leads to a more favorable smoothing effect in copula estimation over classic frequentist methods, especially with small sets of observations. We also apply our method to the copula prediction between the S&P 500 index and the IBM stock prices during the 2007-08 financial crisis, finding that D-P tree-based methods enjoy strong robustness and flexibility over classic methods under such irregular market behaviors. a novel multi-partition Dirichlet-based Pólya tree (D-P tree) prior on the copula. Our D-P tree prior relaxes the binary partition constraints on earlier Pólya-tree-like priors but still preserves the favorable properties of the Pólya tree, including conjugacy and absolute continuity. Based on such a D-P tree prior, we provide a nonparametric Bayesian approach for copula estimation. Its consistency is validated through theoretical analysis.The D-P tree prior overcomes the severe bias problem of previously proposed Pólya-tree-like priors, and the inconsistency issue of family-based nonparametric Bayesian approaches such as the Gaussian copula mixture (Dortet-Bernadet 2005) under model misspecification. Further, compared with classic nonparametric frequentist methods, including the empirical copula estimation and the kernel method, the D-P tree shows a more favorable smoothing effect, especially based on small sets of observations. We illustrate our new method by focusing on copula structure prediction between the S&P 500 daily index and the IBM daily stock prices during the 2007-08 financial crisis. We find that D-P tree-based methods are rather robust and adaptive to irregular market behavior, especially in comparison with commonly-adopted parametric models and the empirical method.Earlier parametric or semi-parametric methods often model copula functions within certain parametric copula families and estimate the parameters by maximum likelihood (ML). For marginals, either parametric or nonparametric estimations are usually adopted(Joe 1997;Jaworski et al. 2010;Chen and Huang 2007;Oakes 1982Oakes 1986Genest et al. 1995). However, these parametric or semiparametric methods suffer from the risk of severe bias when the model is misspecified, thus lack the flexibility to provide accurate estimation for more complex and subtle copula structures. In addition, copula itself is strictly-increasing-transform invariant(Schweizer and Wolff 1981). Thereby, under no further parametric assumptions, the rank statistics of data would preserve sufficient information required for the estimation. In light of these features, nonparametric methods seem to be more natural and coherent for the estimation of copula.Most of the recent studies on nonparametric copula estimation focus on empirical methods(Jaworski et al. 2010;Deheuvels 1979), or kernel-related methods (Scaillet et al. 2007Behnen et al. ). Current nonparametric Bayesian methods focus mainly on an infinite mixture of elliptical copula families such as the Gaussian or the skew-normal(Wu et al. 2014). Yet such models still have limitations: a heavy computational burden as they are implemented through MCMC, and an inconsistency when the model is misspecified, taking the infinite Gaussian copula mixture for a non-symmetric targetwhere p M |Y is the density function of P M |Y . I 2 = J /8 M |p M |Y (x) − c(x)|dx ≤ J /8 M p M |Y (x)dx + J /8 M c(x)dx ≤ I/J /8 M |p M |Y (x) − c(x)|dx + 2 J /8 M c(x)dx. d T V (P M |Y , C) ≤ 2 I/J /8 M |p M |Y (x) − c(x)|dx + 2 J /8 M c(x)dx ≤ 2 I/J /8 M |p M |Y (x) − c(x)|dx +˙ /2. | 10.1080/00949655.2017.1421194 | [
"https://arxiv.org/pdf/1702.07089v1.pdf"
]
| 53,669,003 | 1702.07089 | 1669795fa4baefa7f745d330a44f8b4bec56df91 |
A Nonparametric Bayesian Approach to Copula Estimation
Shaoyang Ning [email protected]
Neil Shephard [email protected]
Department of Statistics
Department of Economics and Department of Statistics
Harvard University
02138CambridgeMA
Harvard University
02138CambridgeMA
A Nonparametric Bayesian Approach to Copula Estimation
copulaPólya treenonparametric BayesGaussian copula mixture modelkernel method
We propose a novel Dirichlet-based Pólya tree (D-P tree) prior on the copula and based on the D-P tree prior, a nonparametric Bayesian inference procedure. Through theoretical analysis and simulations, we are able to show that the flexibility of the D-P tree prior ensures its consistency in copula estimation, thus able to detect more subtle and complex copula structures than earlier nonparametric Bayesian models, such as a Gaussian copula mixture. Further, the continuity of the imposed D-P tree prior leads to a more favorable smoothing effect in copula estimation over classic frequentist methods, especially with small sets of observations. We also apply our method to the copula prediction between the S&P 500 index and the IBM stock prices during the 2007-08 financial crisis, finding that D-P tree-based methods enjoy strong robustness and flexibility over classic methods under such irregular market behaviors. a novel multi-partition Dirichlet-based Pólya tree (D-P tree) prior on the copula. Our D-P tree prior relaxes the binary partition constraints on earlier Pólya-tree-like priors but still preserves the favorable properties of the Pólya tree, including conjugacy and absolute continuity. Based on such a D-P tree prior, we provide a nonparametric Bayesian approach for copula estimation. Its consistency is validated through theoretical analysis.The D-P tree prior overcomes the severe bias problem of previously proposed Pólya-tree-like priors, and the inconsistency issue of family-based nonparametric Bayesian approaches such as the Gaussian copula mixture (Dortet-Bernadet 2005) under model misspecification. Further, compared with classic nonparametric frequentist methods, including the empirical copula estimation and the kernel method, the D-P tree shows a more favorable smoothing effect, especially based on small sets of observations. We illustrate our new method by focusing on copula structure prediction between the S&P 500 daily index and the IBM daily stock prices during the 2007-08 financial crisis. We find that D-P tree-based methods are rather robust and adaptive to irregular market behavior, especially in comparison with commonly-adopted parametric models and the empirical method.Earlier parametric or semi-parametric methods often model copula functions within certain parametric copula families and estimate the parameters by maximum likelihood (ML). For marginals, either parametric or nonparametric estimations are usually adopted(Joe 1997;Jaworski et al. 2010;Chen and Huang 2007;Oakes 1982Oakes 1986Genest et al. 1995). However, these parametric or semiparametric methods suffer from the risk of severe bias when the model is misspecified, thus lack the flexibility to provide accurate estimation for more complex and subtle copula structures. In addition, copula itself is strictly-increasing-transform invariant(Schweizer and Wolff 1981). Thereby, under no further parametric assumptions, the rank statistics of data would preserve sufficient information required for the estimation. In light of these features, nonparametric methods seem to be more natural and coherent for the estimation of copula.Most of the recent studies on nonparametric copula estimation focus on empirical methods(Jaworski et al. 2010;Deheuvels 1979), or kernel-related methods (Scaillet et al. 2007Behnen et al. ). Current nonparametric Bayesian methods focus mainly on an infinite mixture of elliptical copula families such as the Gaussian or the skew-normal(Wu et al. 2014). Yet such models still have limitations: a heavy computational burden as they are implemented through MCMC, and an inconsistency when the model is misspecified, taking the infinite Gaussian copula mixture for a non-symmetric targetwhere p M |Y is the density function of P M |Y . I 2 = J /8 M |p M |Y (x) − c(x)|dx ≤ J /8 M p M |Y (x)dx + J /8 M c(x)dx ≤ I/J /8 M |p M |Y (x) − c(x)|dx + 2 J /8 M c(x)dx. d T V (P M |Y , C) ≤ 2 I/J /8 M |p M |Y (x) − c(x)|dx + 2 J /8 M c(x)dx ≤ 2 I/J /8 M |p M |Y (x) − c(x)|dx +˙ /2.
INTRODUCTION
The copula, as the "link" of a multivariate distribution to its marginals, has attracted growing interest in statistical research since Sklar (1959). By Sklar's Theorem, a copula characterizes the dependence structure between the marginal components. Therefore, the copula plays a central role in multivariate studies and has gained increasing popularity in application to fields such as risk analysis, insurance modeling, and hydrologic engineering (Nelsen 2007;Wu et al. 2014).
The estimation of copulas has been well studied in parametric and semi-parametric settings, but little work has been released on the nonparametric Bayesian inference. In this article, we propose copula as an instance. These motivate us to explore priors with conjugacy and more generality.
Note that here we focus mainly on the bivariate copula case to illustrate our method, and we will discuss higher-dimensional cases towards the end. Also, to concentrate on the estimation of copula structures itself, we assume that the marginals are known or can be accurately estimated. So equivalently, in our simulations, we are concerned mainly with marginally uniform data generated from copula distributions. Such an assumption is reasonable in that: (1) usually we have more information (either parametric or nonparametric) on the marginals of the data for the estimation;
(2) multivariate data are exponentially enriched when considered marginally, providing higher resolution for accurate estimation. Yet we will discuss the scenarios where marginal distributions are to be empirically estimated.
The article is organized as follows: in Section 2, we establish some notation and review previous attempts for copula estimation based on the Pólya tree prior and their limitations. In Section 3, we introduce the proposed D-P tree prior and the procedure for copula inference. In Section 4, we elaborate on properties of the D-P tree. Section 5 provides a simulation-based evaluation of our method in comparison with other common copula estimation methods. In Section 6, we provide an application of our method to the analysis of a bivariate stock-index copula structure. We discuss the copula estimation with unknown marginal distributions and the higher-dimensional cases in Section 7. Section 8 concludes the article.
THE QUASI-PÓLYA TREE PRIOR ON COPULA
The Pólya Tree Prior
Our focus here is on finding Pólya-tree-like priors placed on a copula. The Pólya tree (PT) prior is a tractable case of a tail-free process (Ferguson 1974), which also includes the Dirichlet process (DP) as a special case. But unlike the Dirichlet process, the Pólya tree delivers absolutely continuous measures with probability one by certain choices of the hyper-parameters, which is the attraction for our applications.
Following the definition by Lavine (1992) (Appendix A.2), suppose we have a probability measure P that follows a Pólya tree prior, i.e., P ∼ P T (Π, A). The conjugacy of the Pólya tree follows in that, with one observation Y |P ∼ P, the posterior P|Y still follows a Pólya tree distribution denoted by P T (Π, A|Y ) with the hyper-parameters updated by
α |Y = α + 1 if Y ∈ B , α otherwise.(1)
In practice, to ensure the absolute continuity of measures given by the Pólya tree prior, the hyper-parameters usually take as α = zm 2 at m-th level of the partition, where z is a fixed constant, and the infinite-level Pólya tree is approximated by terminating the sampling process from P T (Π, A) at finite level M .
Therefore, the PT can be intuitively viewed as a smoothed random histogram, and enjoys favorable features such as conjugacy and absolute continuity. Note that Hanson (2006) studied the finite mixture of Pólya trees; Paddock et al. (2003) and Wong et al. (2010) extended the classic PT with randomized partitions to embraces higher flexibility; Filippi and Holmes (in press) applied
Pólya tree to independence test based on the Bayes factor. So it seems promising to start with the PT in search of a more favorable nonparametric prior for Bayesian copula inference.
Dortet-Bernadet's quasi-Pólya tree prior on copula
To our knowledge, Dortet-Bernadet (2005) The Pólya-tree-like probability measure P, which we call a quasi-Pólya tree prior, is defined by independent variables Z = {Z }, hyper-parameters A = {α 0 , α 1 }, where Z ∼ Beta(α 0 , α 1 ) and
P(B = 1 2 ... m ) = m j=1; j =0 or j =2 Z 1 2 ... j−1 /2 m j=1; j =1 or j =3 (1 − Z 1 2 ... j−1 )/2 .
The posterior-like hyper parameters are updated as:
α 0 |Y = α 0 + 1 if Y ∈ B 0 ∪ B 2 , α 0 otherwise; α 1 |Y = α 1 + 1 if Y ∈ B 1 ∪ B 3 , α 1 otherwise.(2)
Unfortunately, Dortet-Bernadet's quasi-Pólya tree prior performs rather unsatisfactorily even in simple bivariate Gaussian copula case. As shown in Figure 2, where we estimate the Gaussian copula with ρ = 0.9 based on N = 10, 000 data points and approximation level M = 10, the "grid" effect is severe for such a quasi-Pólya Tree prior, leading to considerable bias for estimation. In fact, Dortet-Bernadet's Pólya tree prior deviates from the classic Pólya tree in that it mixes a binary partition with a quaternary partition across levels. It does not preserve the features of PT such as the conjugacy, so the posterior-like update is rather ad hoc. Further, it puts strong constraints on its dependence structure by combining the two diagonal dyadic sub-partitions at each level when updating the hyper-parameters for posterior, which causes severe bias when the true copula is heavily asymmetric in the super-partition at the previous level.
OUR APPROACH: DIRICHELET-BASED PÓLYA TREE
3.1 The Dirichlet-based Pólya Tree (D-P tree)
One natural way to remedy the inflexibility in the design of the quasi-Pólya tree is to adopt the more flexible Dirichlet distribution for measure variables (Z ) in place of the much-constrained Beta distribution in the classic PT. Here we first give the Dirichlet-based Pólya tree a general definition: • all the random vectors in Z are independent;
• for every m = 1, 2, . . . and every sequence = 1 2 . . . m , Z = (Z 0 , . . . , Z k ) ∼ Dirichlet(α 0 , . . . , α k ),
with B = ∪ k i=0 B i and k the number of subpartitions in B ;
• for every ,
P(B = 1 2 ... m ) = m j=1 Z 1 2 ... j .
The D-P tree prior still falls into the general class of tail-free process, as the random variables for measures are independent across different partition levels. Yet rather than constraining on binary partitions and beta distributions, the D-P tree adopts a more flexible partition structure and, accordingly, the Dirichlet-distributed variables for the measures, which preserves similar properties to the classic Pólya tree prior.
Conjugacy and Posterior Updating
Adapting the D-P tree prior to bivariate copula estimation, we constrain the D-P tree on Ω = I = [0, 1] × [0, 1], with the quaternary dyadic partition Π = {B 0 , B 1 , B 2 , B 3 }, which repeats Section 2.2, but now the hyper-parameters A = {α 0 , α 1 , α 2 , α 3 } and random variables (Z 0 , Z 1 , Z 2 , Z 3 ) ∼ Dirichlet(α 0 , α 1 , α 2 , α 3 ), as illustrated in Figure 3. From now on, without further specification, we focus only on the D-P tree prior with such a quaternary dyadic partition parametrization, though all results can be generalized.
Such D-P tree prior preserves the conjugacy property of original Pólya tree, thus with P ∼ DP T (Π, A) and an observation Y |P ∼ P, the posterior P|Y can be readily updated.
Proposition 1 (Conjugacy). Let P be a measure on I = [0, 1]×[0, 1], and an observation Y |P ∼ P.
Suppose P follows a D-P tree prior, as P ∼ DP T (Π, A), with the quaternary partition Π = {B } and Dirichlet-distributed random variables Z = {Z } and hyper-parameters A = {α 0 , α 1 , α 2 , α 3 }.
Then the posterior P|Y ∼ DP T (Π, A|Y ), where, for i = 0, 1, 2, 3,
α i |Y = α i + 1 if Y ∈ B i , α i otherwise. Proof: p(Z|Y ) ∝ p(Y |Z)p(Z) ∝ ∞ j=1 Z 1 ... j Z α ∝ Z α +I Y ∈B . For N i.i.d. observations Y = (Y 1 , Y 2 , . . . , Y N )
, the posterior update for multiple observations is rather intuitive and straightforward: at each level of the partitions, the hyper-parameter α associated with the specific partition B is incremented by the number of observations falling in that partition, denoted by n , where n = N i=1 I Y i ∈B . Simply put: α |Y = α + n .
Copula Estimation by the D-P Tree Prior
For the copula estimation, suppose we have
N i.i.d. observations Y = (Y 1 , Y 2 , . . . , Y N ) from an unknown copula distribution C, i.e.,Y 1 , Y 2 , . . . , Y N i.i.d.
∼ C. We assume that C follows a D-P tree prior, i.e., C ∼ DP T (Π, A), where we take Π to be the quaternary partition on the unit square In practice, we approximate the infinite-level D-P tree prior with its M -level approximation P:
Definition 3. For a probability measure P such that P ∼ DP T (Π, A), with the same notation as in Definition 2, its M -level approximation P M is, for any measurable set B ∈ {B = 1 2 ... M },
P M (B) = M j=1 Z 1 2 ... j µ(B) µ(B = 1 2 ... M ) ,
where µ is the uniform measure on Π.
PROPERTIES OF D-P TREE
Equivalence to the Pólya Tree
We first show that, through a re-parametrization, the D-P tree prior on the unit square with the quaternary partition complies with a classic Pólya tree by sequentially combining the quaternary partitions to binary partitions. and Beta-distributedZ = {Z η } and hyper-parametersà = {α η0 ,α η1 } can be constructed as
B η 1 η 2 ...η 2k = B 1 2 ... k ,B η 1 η 2 ...η 2k+1 = B 1 2 ... k (2η 2k+1 ) ∪ B 1 2 ... k (2η 2k+1 +1) , α η 1 η 2 ...η 2k = α 1 2 ... k ,α η 1 η 2 ...η 2k+1 = α 1 2 ... k (2η 2k+1 ) + α 1 2 ... k (2η 2k+1 +1) where k = 0, 1, . . . , i = 2η 2i−1 + η 2i , i = 1, 2, . . . k.
This result follows directly from the property of representing a Dirichlet distribution by independent Gamma distributions, and the independence property between the represented Beta and Gamma distributions. With such equivalence, some of the favorable features of the classic Pólya tree prior can be naturally extended to the D-P tree.
Continuity of D-P Tree Prior
Here we show that the D-P tree prior inherits the feature of generating absolute continuous probability measures under certain constraints on the hyper-parameters A.
Further, with Y = (Y 1 , Y 2 , . . . , Y N )|P i.i.d.
∼ P, P ∼ DP T (Π, A), the posterior DP T (Π, A|Y ) also generates an absolute continuous probability measure with probability one.
The results follow from Theorem 1.121 and Lemma 1.124 in (Schervish 1995). Thereby, as we implied earlier in Section 3.3, the canonical hyper-parameter choice, i.e., α 1 ... m = m 2 will lead to a D-P tree prior that yields absolutely continuous random probability measures, which justifies the smoothing effect of the D-P tree prior in copula estimation.
Consistency of the D-P Tree Posterior
Suppose we have N i.i.d. observations Y = {Y 1 , . . . , Y N } generated from true copula distribution C. For copula estimation, we assume Y i |C i.i.d.
∼ C, with a D-P tree prior C ∼ DP T (Π, A}). Let Notice that here we require that the sample size goes to infinity with a higher order than O(M 3 ), which leaves the variance of our D-P posterior (O( M N )) in a higher order than the empirical copula estimator (O( 1 N )). In fact, by introducing such a D-P tree prior, we sacrifice some asymptotic statistical efficiency in exchange of the continuity of our estimator. Also, as shown in the simulation results later, we gain some advantages in prediction precision with small sets of observations.
If we put smoothness constraints on the target distribution, we can have similar convergence results uniformly on I for the posterior, and further the consistency of the posterior. Note that d T V is the total variation distance between probability measures.
Specifically, we refine the order of convergence for several classic copula distributions, which, in practice, may serve as general guidance for the choice of partition level M based on sample size N .
Proposition 6. The order requirement for the uniform convergence of specific target copulas:
1. For a lower-bounded copula density, i.e., c ≥ ξ > 0, γ(M ) ≥ 2 −2M ξ, thus N ∝ O(M 2+η 2 4M ); 2. For a bivariate Gaussian copula, γ(M ) ≥ Φ 2 ( 1 − |ρ|Φ −1 (2 −M )) 1−|ρ| 1+|ρ| , thus N ∝ O(M 2+η 2 4M ).
Proofs of these results are provided in Appendix B. Such convergence properties ensure the consistency of the estimation based on the D-P tree prior, giving the D-P tree prior advantages over family-based estimation methods under model misspecification.
SIMULATION EXPERIMENTS
Evaluation: Common Copulas
To evaluate the performance of our copula estimation procedure, we conduct simulation studies based on common copulas as listed in Supplementary Material S.1 with various parameter settings, among which Gaussian, Student's and Gumbel are symmetric while the skew-normal is asymmetric.
For each simulation, the procedure is as follows: we first draw i.i.d. data samples from true copula C with the size of N , denoted by Y ; then we follow the procedure described in Section 3.3
for the posterior inference on C; once posterior DP T (Π, A|Y ) is obtained, we draw 10,000 posterior predictive samples from P M |Y to plot the scatterplots, shown in Figure 4-6. Note that without further clarification, all simulations are done with approximation level M = 10.
Posterior Scatterplots
We first report the scatterplots of the posterior predictive draws compared with i.i.d. draws from the target copula distributions based on sample size N = 10, 000, as shown by Figure 5. The plots come in pairs with the left one showing i.i.d. draws from the true copula and the right one i.i.d. predictive draws from the posterior D-P tree to compare. In most cases, our proposed D-P tree prior works well. The difference between our predictive density and the true copula is mild, with exceptions in highly correlated Gumbel case (a = 4) and highly truncated case (skew-normal, ρ = 0.9, α = (−10, 50)).
In light of this, we increase the original sample size to N = 100, 000 ( Figure 6). The overdispersion in low-density area and the grid effect in highly correlated area are eliminated, which corresponds with our asymptotic properties described in Section 4.3. We also explored cases with the more challenging N = 1, 000. Here the asymptotic conditions break, with m = O(1) and N/M 3 = 1. Thereby, for more complex copula structures, the D-P tree prior is prone to loss of sensitivity due to the reduced sample size, which results in the rather unfavorable grid effect as in Figure 4.
Kullback-Leibler Divergence
We further evaluate our method for copula density estimation quantitatively using the Kullback-Leibler (K-L) divergence of our estimates from the true copula:
D KL (C||P M ) = E {log(c/p M )} = log c(u, v) p M (u, v) dC(u, v),(3)
where p M is the density for P M , c is the true copula density, and the expectation is taken over the true copula distribution C .
We vary the sample size N from 0 to 100, 000. For each sample size, we draw 1,000 posterior densities from the D-P tree posterior and calculate the K-L divergence using Monte-Carlo method.
The mean and variance of the K-L divergence are reported in Table 1 for various copula families, as well as the box plots across various sample sizes by Figure 7.
Among all copula families, both mean and variance of the K-L divergence of the D-P tree posterior converge to zero, showing the evidence of consistency in posterior. Specifically the K-L divergence would drop below 0.10 when the sample size is increased to 10,000, and the variance goes to 0, consistent with the convergence claims in section 4.3.
To explore the convergence rate, we fit both D KL ∼ log N and log D KL ∼ log N with linear regression. The fitted curves are shown in Figure 7 by red and green respectively. The green curve gives almost perfect fitting, indicating the convergence rate is in the order of N α , α < 0, though theoretical verification is still required.
Comparison with Existing Methods
Here we compare our method with several existing nonparametric methods for copula estimation.
Comparison with Nonparametric Bayesian Methods
We first compare our method with the infinite Gaussian mixture copula model (Wu et al. 2014).
For copula distribution C, we have the prior Here, we focus on the non-symmetric skew-normal copulas as the data generating copulas. The simulations are carried out with the sample size varying from N = 1, 000 to N = 100, 000, and the K-L divergences of the estimates from the true target copula distribution for both methods are calculated with Monte-Carlo method. We report in Table 2 the cases where the skew-normal copula is highly non-symmetric, and thus the Gaussian mixture model is severely misspecified. More comprehensive simulation results on various target copulas are shown in Supplementary Material S.2. For less non-symmetric copulas (α = (2, 0), (−10, 50), (50, 0)), the Gaussian mixture model dominates due to its parametric nature, yet for these highly non-symmetric cases (α = (100, −100))
C ∼ ∞ i=1 w i C g (ρ i ),
in Table 2, the D-P tree shows a gradually increasing advantage as the data size increases. The results also illustrate the inconsistency issue of the Gaussian mixture model, as its K-L divergence from the data-generating model remains stable (0.17, 0.16) as sample size increases, while the converging trend for the D-P tree posterior is evident.
Comparison with Nonparametric Frequentist Methods
We select three classic nonparametric methods in frequentist settings in comparison with our D-P tree.
Suppose Y i = (U i , V i ) i.i.d.
∼ C: q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q • The empirical estimator:
Ĉ emp (u, v) = 1 N N i=1 I U i ≤u I V i ≤v .
• The histogram estimator:Ĉ hist (B ) = n N , where B is the partition at the highest level. Table 2: Comparison of the K-L divergence between the D-P tree (D-P) and the Gaussian mixture (GM) model for highly non-symmetric skew-normal target copulas.
• The independent Gaussian kernel estimator (Jaworski et al. 2010):
C ker (u, v) = 1 N N i=1 Φ Φ −1 (u) − Φ −1 (U i ) h Φ Φ −1 (v) − Φ −1 (V i ) h ,(4)
where we make the choice of h = N − 1 5 , following Silverman's rule of thumb for the choice of window width.
• The D-P tree posterior mean estimator: for a fair comparison, we use the mean distribution from the D-P tree posterior as the Bayesian estimator by the D-P tree, i.e.,Ĉ D−P = E(C|Y ).
We define several measurements for the distance between the estimator and the target distribution. For density estimation, besides the K-L divergence, we also include the commonly adopted MISE (Mean Integrated Squared Error) based on the averaged L 2 -norm between the estimated density function and the truth:
M ISE(ĉ) = E [0,1]×[0,1] {c(u, v) −ĉ(u, v)} 2 du dv .(5)
Here c is the target copula density andĉ is its estimator.
For the distance measurement of the distribution, we extend the M ISE for density to the
M ISE C (Mean Integrated Squared Error for Cumulative functions)
:
M ISE C (Ĉ) = E [0,1]×[0,1] {C(u, v) −Ĉ(u, v)} 2 du dv ,(6)
whereĈ is the estimated copula function, C is the true copula. We also have a distance measure specifically targeting the grid-based estimation methods, the M SE g :
M SE g (Ĉ) = E 1 2 2M 2 M i,j=1 {C(B ij ) −Ĉ(B ij )} 2 ,
where {B ij } are partitions on [0, 1] × [0, 1], and M is the maximum partition level. Note that all the expectations in the measures defined above are taken over all possible data samples
The simulations are carried out with the sample size varying from N = 10 to N = 10, 000 for a good look at the convergence trend. We again focus mainly on heavily non-symmetric skew-normal copulas (α = (−50, 10), (100, −100)). For each parameter setting, we first draw N i.i.d. samples from the true copula distribution, obtain the copula estimates by three frequentist methods and the D-P tree posterior mean estimator; then we repeat this process 50 times to obtain the Monte-Carlo approximation of the measures as defined above. Note that for the empirical copula estimation, the estimated distribution is discrete, thus the density distance measures not applicable; for the histogram estimator, due to the discrepancy in the supports between the target and the estimated distributions, the K-L divergence is not applicable. To ensure computational efficiency, we report the results based on the approximation level M = 8, and to maintain comparability, we take the same maximum partition level for the histogram estimation method. Comprehensive numeric results are shown in Table S.2-S.5 in Supplementary Material S.2. Here we report mainly the results under the parameter setting ρ = 0.5, α = (100, −100) in Table 3 as exemplary for our conclusions. Table 3: Comparison of various distance measures between the D-P tree posterior mean estimator and frequentist estimators for the skew-normal copula with parameter ρ = 0.5, α = (100, −100).
N K-L √ M ISE D-P
In general, the D-P tree posterior mean estimator performs competitively well compared with all three frequentist nonparametric methods and consistently across various measures. Notably, the D-P tree posterior estimation appears advantageous over other methods with small sets of observations, showcasing a preferably strong smoothing effect induced by the D-P tree prior.
Both the D-P tree and the kernel estimation show drastic advantages in copula density estimation over empirical and histogram methods, as the empirical copula fails to yield density estimator and the histogram estimator gives severely poor density approximation due to the discrepancy in the support. Though both methods take advantage of the smoothing effect in estimation density, under the MISE measurement, the D-P tree dominates kernel method across almost all sample sizes while giving close figures under the K-L divergence.
As for copula distribution estimation, the D-P tree shows a strong advantage over other methods in both measures under scenarios of smaller sample size, which indicates the more favorable continuity feature of the D-P tree prior. When the sample size increases, the neutralizing effect of the D-P tree prior slows down the convergence of the posterior, and thereby, the empirical and histogram estimators catch up in figures. Yet still, up to N = 10, 000, the D-P tree gives close distances as the empirical and the histogram methods, and consistently dominates the kernel method.
REAL DATA APPLICATION
For real data analysis, we apply our method to the S&P 500 daily index and the IBM daily stock prices over the past 20 years (Jan 1, 1994 to Dec 31, 2014, available from https://finance. yahoo.com) and aim to estimate their dependence structure with the copula model. We adopt both the cross-validation and the rolling prediction schemes to evaluate the performance of our method, as described below in detail. We assess three methods under both prediction schemes for comparison: (1) the D-P tree posterior mean with canonical prior; (2) the Gaussian copula; (3) the Student's t copula. For the rolling prediction assessment, we also include (4) the independent Gaussian kernel estimator; (5) the empirical copula estimator, and (6) the D-P tree posterior mean with historic-data-induced prior. Typically, investment groups have focused on using methods (2),
(3) and particularly (5) in practice for risk management.
Cross-validation
We first conduct cross-validation to evaluate the prediction ability of our method. Let the joint daily prices for two stocks be {(y 1 i , y 2 i ), i = 1, . . . , T }, where T = 5, 288, and the returns of log price {r j i = log y j i −log y j i−1 , i = 2 . . . T, j = 1, 2}. Marginally, we fit the commonly adopted GARCH(1,1) model:
r j i = σ j i j i , (σ j i ) 2 = α j 0 + α j 1 (σ j i−1 ) 2 + β j 1 ( j i−1 ) 2 ,F ( 1 i , 2 i ) = C(F 1 ( 1 i ), F 2 ( 2 j )), where ( 1 i , 2 i ) i.i.d.
∼ F , and F 1 and F 2 are the marginal distributions.
We apply the proposed D-P tree (canonical) prior to the copula estimation based on the fitted
innovations from the GARCH model {(ˆ 1 i ,ˆ 2 i )}.
Here we use the empirical estimation for the marginals, as discussed in Section 7.2. Figure 8 compares the scatterplots of the fitted errors (normalized by marginals) and the draws from the D-P tree posterior. We observe no apparent discrepancy between the data and the fitted model. For comparison, we also fit the innovations with the bivariate Gaussian and Student's t copula models respectively, as described in Appendix S.1. The estimates are obtained by maximum likelihood estimation.
To assess the effectiveness of such estimation methods, we adopt the cross-validation scheme to get the estimation of the prediction errors. We randomly divide the marginally GARCH-fitted innovations into 10 sets and each time use one set as the testing set and the rest as the training set.
For prediction errors, we adopt the Monte-Carlo estimation for cross entropy −E c (log(ĉ)), which is equivalent to the K-L divergence of the estimated copula density from the assumed truth up to a constant. Here, c andĉ denote the true copula density and the estimated one respectively. Table 4 gives the mean K-L divergence based on 10-fold cross-validation for the three methods in comparison. As it shows, the D-P tree prior outperforms the Gaussian copula narrowly, while
the Student's t shows an advantage over the other two methods. The result is unsurprising as the distribution of stock returns are notable for the heavy-tail dependence features and the Student's t is thus expected to give good fittings. Nevertheless, when parametric models are misspecified under the Gaussian copula model, nonparametric methods such as the D-P tree prior is still advantageous.
D-P tree Gaussian (ρ=0.59) Student's t (ρ=0.60,ν=6.5) Cross entropy -0.210 -0.209 -0.224 Table 4: Comparison of the mean prediction errors based on the K-L divergence between the D-P tree prior, the Gaussian copula and the Student's t copula. Note that the more negative the numbers, the better the prediction performance.
Rolling Prediction
To mimic the practical prediction scenario, we also evaluate the prediction power of our method under the time-rolling prediction scheme, that is, we predict the future copula structure within a certain window of time based on the most recent observations.
Specifically, we set a training length of T tr , a testing set length of T te , a rolling estimation window of length t e , and a prediction window of length t p . Firstly, we use the daily price time series of the two stocks {y 1 t : t = 1, . . . , T tr } and {y 2 t , t = 1, . . . , T tr } as the training set for the marginal GARCH-model fitting. Consistent with common practical prediction scenarios, we fix such fitted GARCH model and obtain the fitted innovations for the training set {(ˆ 1 t ,ˆ 2 t ), t = 1, . . . , T tr }, and the predicted innovations for the test set {(ˆ 1 t ,ˆ 2 t ), t = T tr + 1, . . . , T tr + T te }. Then, we conduct the rolling prediction of the copula structure based on these estimates. For each rolling step, we apply the proposed D-P tree-based method with both the canonical non-informative prior and the historic-data-induced prior to the most recent t e -fitted/predicted innovations and estimate the future copula structure of length t p . Here we implicitly assume the i.i.d. property of the innovations within the estimation and prediction windows combined of length (t e + t p ). This is reasonable in that the copula structure is usually stable within a certain length of time. We repeat such rolling prediction T te /t p times until the whole testing length (T tr + 1 to T tr + T te ) is covered.
Here we focus on the data of the period covering the 2007-08 financial crisis (i.e., the testing set covering July, 2007 to July, 2009) to highlight the flexibility and robustness of nonparametric methods over traditional parametric models. We set T tr = 500, T te = 500, and vary t e ∈ {10, 20, 50, 100, 250}, t p ∈ {1, 50} and report both the average log-likelihood 1 Tte Tte t=1 logĉ t (equivalent to negative KL divergence plus a constant), and the square root of average M ISE C = 1 Tte Tte t=1 M ISE C (Ĉ t ) as the measures for prediction accuracy ( Table 5). Note that for historicdata-based D-P tree prior, we adopt the posterior of a canonical D-P tree prior updated by the data from testing set (i = 1, . . . , T tr − t e ) with each down-weighted by 0.1. We also carry out the same prediction scheme with other four methods for comparison. Table 5: Comparison of the prediction performance in the average log-likelihood (the higher the numbers, the better the prediction) and the M ISE C (the lower, the better) between various methods: the D-P tree posterior mean with the canonical prior (D-PT), the D-P tree with the historic-data-induced prior (D-PTw), the empirical copula (Emp.), the kernel estimator (Kernel), the Gaussian copula (Gauss.) and the Student's t copula (t) models.
Generally, both the D-P-tree-based methods show strong advantages over other methods by the log-likelihood loss in almost all settings, and by √ M ISE C under a longer prediction window t p = 50 (where the distribution-based measure √ M ISE C is more valid due to multiple testing samples) and a larger prediction set t e ≥ 50. Such results verify the robustness and adaptiveness of the D-P tree-based methods to irregular market behaviors when classic parametric models are terribly misspecified. Further, by incorporating the historic data into the prior, the D-PTw method enjoys a strong boost in prediction accuracy, and dominates other methods in most of the scenarios.
Admittedly, more data are used by the D-PTw for inference than other methods in comparison.
Nevertheless, it is exactly the showcase of the strength of Bayesian methods where historic or empirical information is readily concocted into priors to help.
DISCUSSION
Copula Normalizing
One problem with most nonparametric copula estimation methods including the D-P tree prior is that the posterior marginal does not always follow a uniform distribution. Suppose P ∼ DP T (Π, A|Y ), then marginally P([0, 1/2] × [0, 1]) ∼ Beta(α 0 + n 0 + α 1 + n 1 , α 2 + n 2 + α 3 + n 3 ), which deviates from 0.5 by the randomness. Though, when the sample size N is large, as shown by Proposition 4, the posterior density would have marginals close to uniforms, thus approximate a proper copula density, the issue of normalizing posterior density to proper copula density still needs addressing. Here we provide several methods to carry out the correction.
Ad Hoc Correction
Suppose we have P * ∼ DP T (Π, A|Y ) and P * M is its M-level approximation with a 2 M × 2 M grid density. To normalize its marginals to the uniforms, we need to restrain
P * M ([k/2 M , (k + 1)/2 M ] × [0, 1]) = P * M ([0, 1] × [k/2 M , (k + 1)/2 M ]) = 1/2 M ,(7)
k = 0, 1, . . . , 2 M − 1; i.e., the column sum and row sum of the 2 M × 2 M grid density to be 1/2 M .
One way to realize this is to randomly select 2 · 2 M − 1 grids and manipulate their values to fit (7).
As P * M is close to C when the sample size is large, the marginals of P m would not be too far away from a uniform. Thus the ad hoc correction would not cause severe deviation from the posterior density P * .
Inverse Transform on the Marginals
Higher Dimension
Most of the results of the D-P tree prior on bivariate copulas can be generalized to higher dimensions. Specifically, for a d-dimensional copula, we can generalize the D-P tree prior to C ∼ DP T (Π, A), where Π is a 2 d -partition on the d-dimensional unit cube and the same parametrization for A = {α : α 1 ... m = m 2 }. Those properties of a bivariate D-P tree including conjugacy, continuity and convergence, are still preserved.
However, as the dimension increases, the sparsity of data would cause great difficulty for accurate copula estimation, especially among nonparametric settings including the D-P tree prior.
Further, though the computational complexity is stable, the D-P tree still requires exponentially increasing storage power as the dimension increases. Yet one potentially favorable feature of the D-P tree that we have observed through simulations is its strong smoothing effect and improved estimation accuracy when the sample size is small. Thereby, the D-P tree prior could be the more favorable nonparametric method compared to other alternatives with sparse observations under higher-dimensional scenarios. This could be one potential angle for further studies.
CONCLUSION
The proposed Dirichlet-based Pólya tree (D-P tree) prior preserves properties including conjugacy, continuity and convergence as the classic Pólya tree, which provides a foundation for nonparametric copula estimation under the Bayesian framework. Compared with other Bayesian copula estimation methods, the D-P tree prior exhibits strength in robustness and consistency, remedying the severe bias of the earlier Pólya tree-based prior in copula estimation, and also overcoming the inconsistency issue of the family-based mixture model under misspecification. In comparison with the nonparametric methods under the frequentist settings, the D-P tree posterior mean estimator performs competitively well and rather stably across various distance measures. Notably, with a small sample size, the D-P tree copula estimator is advantageous in estimation accuracy, which may imply its potential in higher-dimensional cases where observations are heavily diluted.
However, there still are issues remaining with the D-P tree prior worthy of further exploration, such as the marginal bias caused by the randomness in the prior and a more efficient application in higher dimensions. Further, in terms of the D-P tree's application to the copula prediction of the stock prices, we have not yet fully exploited the timely nature of the data. The exploration of time-dependent D-P tree prior could be of great future research interest. In addition, alternative priors under nonparametric Bayesian frameworks could also be of future interest to overcome the limitations of the D-P tree prior.
APPENDIX A DEFINITIONS A.1 Copula Definition 4. C : [0, 1] d → [0, 1] is a d-dimensional copula, if C is a joint cumulative distribution function for a d-dimensional random vector on [0, 1] d with uniform marginals. For two-dimensional case, that is, C(u, v) = P (U ≤ u, V ≤ v), where U, V ∼ U nif [0, 1].
And the joint density function c(u, v) is called copula density. (Sklar 1959), if X and Y are random variables with cumulative distribution functions F and G, and a joint distribution function H, then there exists a copula C such that for all (x, y) ∈ R 2 , H(x, y) = C(F (x), G(y)), and for density function, we have h(x, y) = c(F (x), G(y))f (x)g(y), where f and g are marginal density functions and h is the joint density.
Sklar's Theorem
A.2 Pólya Tree
Definition 5. (Lavine 1992) Let Ω be a separable measurable space and Π = {B } be one of its binary tree partitions that generate the measurable sets, where B ∅ = Ω and B = B 0 ∪B 1 . A random probability measure P is said to have a Pólya tree distribution, or Pólya tree prior, with parameters (Π,A), written P ∼ P T (Π, A) , if there exists non-negative numbers A = {α 0 , α 1 , α 00 , . . . } and random variables Z = {Z 0 , Z 1 , Z 00 . . . } such that the following hold:
• all the random variables in Z are independent;
• for every m = 1, 2, . . . and every = 1 2 . . . m , Z ∼ Beta(α 0 , α 1 );
• for every , P(B = 1 2 ... m ) = m j=1; j =0 Z 1 2 ... j−1 m j=1; j =1 (1 − Z 1 2 ... j−1 )
where the first terms in the products are interpreted as Z ∅ ∼ Beta(α 0 , α 1 ) and (1 − Z ∅ ).
B PROOFS B.1 Proof of Proposition 4
We consider P M on the measurable partition {B }. For any B k = B 1 ... k ∈ Π, for M large enough, let B j = B 1 ... j , and B 1 ⊂ B 2 · · · ⊂ B k .
If C(B k ) > 0, E(P M (B k )|Y ) = k j=1 α 1 ... j + n 1 ... j 3 i=0 (α 1 ... j−1 i + n 1 ... j−1 i ) = k j=1 j 2 N + C(B j ) + O( 1 √ N ) 4j 2 N + C(B j−1 ) + O( 1 √ N ) = k j=1 C(B j ) C(B j−1 ) + j 2 − 4j 2 C(B j ) C(B j−1 ) + O( √ N ) 4j 2 + n j−1 ≤ C(B k ) + k j=1 1 + 3j 2 + O( √ N ) 4j 2 + n j−1 − 1 = C(B k ) + exp k j=1 3j 2 + O( √ N ) 4j 2 + n j−1 + O( k j=1 ( 3j 2 + O( √ N ) 4j 2 + n j−1 ) 2 ) − 1 = C(B k ) + O k j=1 3j 2 + O( √ N ) 4j 2 + n j−1 = C(B k ) + O k j=1 3j 2 + O( √ N ) 4j 2 + N C(B j−1 ) + O( √ N ) ≤ C(B k ) + O( k j=1 3j 2 + O( √ N ) N C(B j−1 ) ) = C(B k ) + max{O( M √ N ), O( M 3 N )}. If C(B k ) = 0, suppose l = max i<k {C(B 1 ... i ) > 0}, E(P M (B k )|Y ) = k j=1 α 1 ... j + n 1 ... j 3 i=0 (α 1 ... j−1 i + n 1 ... j−1 i ) = l+1 j=1 j 2 N + C(B j ) + O( 1 √ N ) 4j 2 N + C(B j−1 ) + O( 1 √ N ) 1 4 M −l−1 ≤ C(B l+1 )( 1 4 ) M −l−1 + max{O( M √ N ), O( M 3 N )} 1 4 M −l−1 = 0 + max{O( M √ N ), O( M 3 N )}. var(P M (B k )|Y ) = var( k j=1 Z i ... j |Y ) = var( k j=1 Z j |Y ) = E(var(Z 1 |Y ) k j=2 Z 2 j |Y ) + var(E(Z 1 |Y ) k j=2 Z j |Y ) = var(Z 1 |Y ) k j=2 var(Z 2 j |Y ) + E 2 (Z 1 |Y )var( k j=2 Z j |Y ) ≤ var(Z 1 |Y ) + var( k j=2 Z j |Y ) ≤ k j=1 var(Z j |Y ) = M j=1 (α 1 ... j + n 1 ... j ){ i =j (α 1 ... j−1 i + n 1 ... j−1 i )} { 3 i=0 (α 1 ... j−1 i + n 1 ... j−1 i )} 2 { 3 i=0 (α 1 ... j−1 i + n 1 ... j−1 i ) + 1} ≤ M j=1 1 {4j 2 + n j−1 + 1} ≤ M N C(B k ) = O M N .
Thereby for any measurable set B ⊂ I,
B.2 Proof of Proposition 5
For any
B k = B 1 ... k , k ≥ M , E(P M (B k )|Y ) = M j=1 α 1 ... j +n 1 ... j 3 i=0 (α 1 ... j−1 i +n 1 ... j−1 i ) k j=M +1 1 4 . If C(B 1 ... k ) > 0: E(P M (B k )|Y ) = ( 1 4 ) k−M M j=1 j 2 + n j 4j 2 + n j−1 ≤ ( 1 4 ) k−M (C(B M ) + O( k j=1 3j 2 + O( √ N ) N C(B j−1 ) )) ≤ ( 1 4 ) k−M (C(B M ) + max{O( M √ N γ(M ) ), O( M 3 N γ(M ) )}). For C ∈ C 1 ([0, 1] × [0, 1]): sup |E(P M (B k )|Y ) − C(B k )| ≤ ( 1 4 ) k sup |c(b 1 ... k ) − c(b 1 ... M )| + max{O( M √ N γ(M ) ), O( M 3 N γ(M ) )} ≤ ( 1 4 ) k (1/2) k sup |c | + max{O( M √ N γ(M ) ), O( M 3 N γ(M ) )} = max{O( M √ N γ(M ) ), O( M 3 N γ(M ) )}. If C(B k ) = 0, suppose l = max i<k {C(B 1 ... i ) > 0}: sup E(P M (B k )|Y ) = sup( 1 4 ) k−l−1 l+1 j=1 j 2 N + C(B j ) + O( 1 √ N ) 4j 2 N + C(B j−1 ) + O( 1 √ N ) ≤ sup( 1 4 ) k−l−1 (C(B l+1 ) + O( k j=1 3j 2 + O( √ N ) N C(B j−1 ) )) = 0 + max{O( M √ N γ(M ) ), O( M 3 N γ(M ) )}.
Thereby sup B |E(P M |Y ) − C| → 0.
By the proof of Proposition 4,
sup var(P M (B )|Y ) ≤ sup O( M N C(B k ) ) ≤ O( M N γ(M )
).
Let S δ M = {B 1 ... M : ∃x ∈ B 1 ... M , c(x) < δ}, J δ M = ∪ B∈S δ M B, thereby inf I/J δ M c(x) ≥ δ. By C ∈ C 1 (I), ∀ > 0, for M large enough, ∀B ∈ {B 1 ... M }, x, y ∈ B, |c(x) − c(y)| ≤ /8, taking δ = /4, /4 > sup J /8 M c(x). Therefore, d T V (P M |Y , C) = I |p M |Y (x) − c(x)|dx = I 1 = I/J /8 M |p M |Y (x) − c(x)|dx = I/J /8 M |2 2M P M |Y (B x ) − c(x) + c(b x ) − c(b x )|dx ≤ B∈{B 1 ... M }/S /8 M |P M |Y (B) − C(B)| + B |c(b) − c(x)|dx where B x ∈ {B 1 ... M } such that x ∈ B x , and C(B x ) = c(b x )µ(B x ). ∀B ∈ {B 1 ... M }, b, x ∈ B, |c(b) − c(x)| ≤ /8, we have {B 1 ... M }/S /8 M B |c(b) − c(x)|dx ≤ /8. P {B 1 ... M }/S /8 M |P M |Y (B) − C(B)| > /4 ≤ P max B∈{B 1 ... M }/S /8 M |P M |Y (B) − C(B)| ≥ 2 2M +2 ≤ B∈{B 1 ... M }/S /8 M P (|P M |Y (B) − C(B)| ≥ 2 2M +2 ) ≤ 2 2M +2 ( 2 2M ) 2 sup |E(P M |Y (B)) − C(B)| 2 + sup var(P M |Y (B)) = 2 6M max{O( M √ N γ(M ) ) 2 , O( M 3 N γ(M ) ) 2 , O( M N γ(M ) )}. Note that here r(M ) ∼ min {B 1 ... M }/S /8 M C(B M ) ≥ /2 2M +2 . Thus, by taking N ∝ O(2 10M M 2+η ), P (d T V (P M , C) ≥ |Y ) = O( 1 M η ) → 0. B.3 Proof of Proposition 6 1. For c ≥ ξ > 0, ∀B M , ∃b M ∈ I, such that C(B M ) = 2 −2M c(b M ) ≥ 2 −2M ξ, thereby γ ∼ 2 −2M ,
2. We assume ρ < 0, let α = Φ −1 (2 −M ), by symmetry of Gaussian copula, for fixed ρ,
γ(M ) = α −∞ α −∞ 1 2π 1 − ρ 2 exp − x 2 + y 2 − 2ρxy 2(1 − ρ 2 ) dx dy ≥ α −∞ α −∞ 1 2π 1 − ρ 2 exp − (1 − ρ)(x 2 + y 2 ) 2(1 − ρ 2 ) dx dy = Φ 2 (α 1 + ρ) 1 + ρ 1 − ρ ≈ 2 −2M . SUPPLEMENTARY MATERIAL S.1 COMMON COPULAS S.1.1 Gaussian Copula
The copula density of a bivariate Gaussian copula is given by
c(u, v) = 1 1 − ρ 2 exp − ρ 2 (x 2 + y 2 ) − 2ρxy 2(1 − ρ 2 ) , (S.1)
where ρ ∈ [−1, 1] is the correlation parameter of the copula, x = Φ −1 (u), y = Φ −1 (v), and Φ −1 is the inverse of the standard univariate Gaussian CDF.
S.1.2 Student's t Copula
The copula density of a bivariate Student's t-copula follows
c(u, v) = Γ( ν+2 2 )/Γ( ν 2 ) νπf tν (x)f tν (y) 1 − ρ 2 1 + x 2 + y 2 − 2ρxy ν(1 − ρ 2 ) − ν+1 2 , (S.2)
where the two parameters, the correlation ρ ∈ [−1, 1] and the degree of freedom ν > 0, x = F tν (u), y = F tν (v), and f tν and F tν are the PDF and CDF of the standard univariate Student's t-distribution with the degree of freedom of ν.
S.1.3 Gumbel Copula
The copula density of a bivariate Gumbel copula is given by
c(u, v) = C(u, v)(uv) −1 [{− log(u)} a + {− log(v)} a ] −2+2/a {log(u) log(v)} a−1 (1 + (a − 1)[{− log(u)} a + {− log(v)} a ] −1/a ),
where a ≥ 1 is the dependence parameter.
S.1.4 Skew-normal Copula
A d-dimensional random vector Z = (Z 1 , . . . , Z d ) T follows a skew-normal distribution (Azzalini and Capitanio 1999)
, denoted Z ∼ SN d (Ω, α) if Z = X if X 0 > 0, −X otherwise, where (X 0 , X) T ∼ N d+1 (0, Ω * ), Ω * = 1 δ T δ Ω and α = 1
(1−δ T Ω −1 δ) 1/2 Ω −1 δ. And the density functions for Z is f SN,d (z; Ω, α) = 2φ d (z; Ω)Φ(α T z), where φ d (·; Ω) is the d-dimensional normal density with zero mean and correlation matrix Ω.
For the marginals, suppose Z is partitioned as Z = (Z T 1 , Z T 2 ) T of dimension h and d − h; Ω and α by
Ω = Ω 11 Ω 12 Ω 21 Ω 22 , α = α 1 α 2 ,
then the marginal distribution of Z 1 is SN d (Ω 11 ,ᾱ 1 ), wherē
α 1 = α 1 + Ω −1 11 Ω 12 α 2 (1 + α T Ω 22·1 α 2 ) 1/2 , Ω 22·1 = Ω 22 − Ω 21 Ω −1 21 Ω 12 .
So the bivariate skew-normal copula density is
c(u, v) = φ 2 ((x, y) T , Ω)Φ(α 1 x + α 2 y) 2φ(x)Φ(ᾱ 1 x)φ(y)Φ(ᾱ 2 y) , (S.3)
where Ω = 1 ρ ρ 1 ; ρ ∈ [−1, 1], α 1 ,α 2 are parameters, x = F −1 SN 1 ,ᾱ 1 (u), y = F −1 SN 1 ,ᾱ 2 (v), and F SN 1 ,α is the CDF of Z ∼ SN 1 (1, α). Table S.1: Comparison of the K-L divergence between the D-P tree (left) and the Gaussian mixture (right) estimation for skew-normal target copulas.
S.2 ADDITIONAL SIMULATION RESULTS
N D-P Tree Empirical Kernel Hist, D-P Tree Empirical Kernel Hist. ρ = 0.5, α = (−10, 50) ρ = 0.5, α = (100, −100) 10 0. Table S.5: Comparison of the M SE g between the D-P tree posterior mean estimator and the frequentist estimators for the skew-normal target copulas.
Figure 1 :
1made the first attempt to apply the PT prior to the inference of a bivariate copula on I = [0, 1] × [0, 1]. At each level, each square partition B is split into four sub-partitions {B 0 , B 1 , B 2 , B 3 } by dyadic partitions on its margins. Thereby, a partition of I is obtained by Π = {B }, ∈ {∅, 0, 1, 2, 3, 00, 01, 02, 03 . . . }, demonstrated by the left panel of Figure 1. The quaternary partition (left) on the support [0, 1] 2 of a bivariate copula and the parametrization of Dortet-Bernadet's quasi-Pólya tree prior (right).
Figure 2 :
2Scatterplots comparing the Gaussian copula estimates: the Gaussian copula (left), the quasi-Pólya tree (middle) and the D-P tree (right) priors.
[0, 1 ]
1× [0, 1] and A = {α : α 1 ... m = m 2 }. By Proposition 1, the posterior C|Y ∼ DP T (Π, A|Y ), where A|Y = {α : α 1 ... m = m 2 + n }. Therefore, the D-P tree posterior on copula strongly resembles the construction of a histogram of the observations, but regularized by the imposed prior. Later we will show the choice of hyper-parameters, as in P ∼ DP T (Π, A = {α : α 1 ... m = m 2 }), ensures generating absolutely continuous measures centered on the uniform distribution, and thus the posterior then can be viewed as a shrunk version of the histogram.
Proposition 2 (
2Equivalence to the Pólya tree). Given a D-P tree prior on I = [0, 1] × [0, 1] with the quaternary partition Π = {B } and Dirichlet-distributed random variables Z = {Z } and hyperparameters A = {α 0 , α 1 , α 2 , α 3 }, an equivalent Pólya tree prior with binary partitionΠ = {B η }
Proposition 3 (
3Absolute continuity). A D-P tree prior on I = [0, 1] × [0, 1] with the quaternary partition Π = {B } and Dirichlet-distributed random variables Z = {Z } and hyper-parameters A = {α 0 , α 1 , α 2 , α 3 } generates an absolute continuous probability measure on I with probability one when hyper-parameters on the m-level α 1 ... m ∝ O(m 1+δ ), δ > 0.
P
M be the M-level approximation of C and A be canonical, i.e., the m-level hyper-parameter α 1 ... m = m 2 . For the approximated posterior P M |Y , we have the point-wise convergence to the target copula distribution in terms of any measurable set in the unit square: Proposition 4 (Point-wise convergence). For any measurable set B ⊂ I = [0, 1] × [0, 1], with N ∝ O(M 3+η ), η > 0, then E((P M (B)|Y ) − C(B)) → 0, var(P M (B)|Y ) = O( M N ), therefore P M (B)|Y p → C(B).
Proposition 5 (
5Consistency). If C ∈ C 1 ([0, 1]×[0, 1]), for B ⊂ I measurable, sup B |E(P(B) M |Y )− C| = max{O M √ N γ(M ) , OM 3 N γ(M ) }; sup B var(P(B) M |Y ) = O M N γ(M ) , where γ(M ) ∼ min C(B M )>0 C(B M ). Further, with N ∝ O(2 10M M 2+η ), η > 0, ∀δ > 0 as M → ∞, P (d T V (P M , C) ≥ δ|Y ) → 0.
Figure 4 :Figure 5 :Figure 6 :
456Scatterplots of i.i.d. draws from the true copula distribution (left) vs. the D-P tree posterior (right): sample size N = 1, 000, partition level M = 10. Scatterplots of i.i.d. draws from the true copula distribution (left) vs. the D-P tree posterior (right): sample size N = 10, 000, partition level M = 10. Scatterplots of i.i.d. draws from the true copula distribution (left) vs. the D-P tree posterior (right): sample size N = 100, 000, partition level M = 10.
Figure 7 :
7Box-plots of the K-L divergence of the D-P tree posterior from the target copulas against the sample size N : the solid green line showing a linear fit of log(KL)∼log(N ).
where the innovations { j i } are independent with E( j i ) = 0 and var( j i ) = 1. Further, we assume the distribution of the innovations is time-invariant and put the copula model on their joint distribution
Figure 8 :
8Scatterplots comparing the GARCH-fitted joint return innovations (left, normalized by the empirical marginal distributions) and random draws from the D-P tree estimated copula (right).
E
(P M (B)|Y N ) → C(B), var(P M (B)|Y N ) → 0, P (|P M |Y (B) − C(B)| ≥ ) ≤ E 2 (P M |Y (B) − C(B)) + var(P M (B)|Y ) 2 → 0.
Definition 1. Let Ω be a separable measurable space. We say a partition Π = {B } of Ω is one of its measurable tree partitions if• the subpartitions at level m + 1 {B 1 ... m+1 } is refinement of previous level {B 1 ... m };• Π = {B } generates measurable sets of Ω.Definition 2. Let Ω be a separable measurable space and Π = {B } be one of its measurable tree partitions. A random probability measure P is said to have a Dirichlet-based Pólya tree distribution, or D-P tree prior, with parameters (Π,A), written P ∼ DP T (Π, A), if there exists non-negative numbers A = {α } and random variables Z = {Z } such that the following hold:
where C g indicates the bivariate Gaussian copula, and the weight w i i.i.d. ∼ U [0, 1] and the correlation ρ i i.i.d. ∼ U [−1, 1]. Such a model is the most common one among existing nonparametric Bayesian methods which focus on mixture models based on a specific copula family.Table 1: Estimated K-L divergence of the D-P tree posterior from various targets, with standard errors (SE). Note that we leave out SEs (all 0.00) for N ≥ 100.N
0
10
100 1,000 10,000 100,000
ρ
Gaussian
0.50 0.82(0.19) 0.54(0.05) 0.25 0.13
0.07
0.04
0.90 1.50(0.35) 0.83(0.02) 0.48 0.22
0.10
0.05
ρ
ν
Student's t
0.50
1.00 1.01(0.18) 0.70(0.03) 0.38 0.21
0.10
0.05
0.90
1.00 0.86(0.21) 0.60(0.05) 0.24 0.14
0.07
0.04
0.50
4.00 1.72(0.33) 1.04(0.03) 0.61 0.30
0.13
0.06
0.90
4.00 1.53(0.37) 1.01(0.05) 0.48 0.23
0.10
0.05
a
Gumbel
2.00 1.01(0.21) 0.83(0.07) 0.30 0.16
0.08
0.04
4.00 1.66(0.39) 1.04(0.05) 0.53 0.25
0.11
0.05
ρ
α
skew-normal
0.50
(2,0) 0.72(0.15) 0.44(0.03) 0.23 0.12
0.07
0.04
0.50
(-10,50) 0.91(0.19) 0.52(0.02) 0.31 0.18
0.09
0.05
0.90
(-10,50) 1.22(0.27) 0.65(0.01) 0.40 0.21
0.10
0.05
0.90
(50,0) 1.09(0.22) 0.65(0.04) 0.37 0.16
0.08
0.04
0.50 (100,-100) 1.35(0.30) 0.82(0.02) 0.48 0.26
0.14
0.07
0.90 (100,-100) 2.13(0.39) 1.39(0.04) 0.86 0.46
0.20
0.08
Tree Empirical Kernel Hist. D-P Tree Empirical Kernel Hist.10
0.528
NA
0.528
Inf
1.365
NA
2.190 71.788
20
0.473
NA
0.428
Inf
1.163
NA
2.657 56.726
50
0.386
NA
0.314
Inf
1.050
NA
1.177 36.757
100
0.349
NA
0.261
Inf
1.159
NA
1.347 25.723
500
0.222
NA
0.166
Inf
1.072
NA
1.398 11.665
1,000
0.184
NA
0.136
Inf
0.894
NA
0.703
8.078
5,000
0.112
NA
0.090
Inf
0.703
NA
0.516
3.601
10,000
0.089
NA
0.076
Inf
0.701
NA
0.769
2.600
√
M ISE C
M SE g
10
0.072
0.118
0.091 0.117
0.054
0.321
0.054
0.321
20
0.065
0.082
0.068 0.082
0.054
0.230
0.055
0.230
50
0.044
0.057
0.050 0.057
0.054
0.151
0.054
0.151
100
0.037
0.041
0.038 0.041
0.054
0.113
0.054
0.113
500
0.018
0.017
0.017 0.017
0.053
0.070
0.053
0.070
1,000
0.013
0.012
0.013 0.012
0.053
0.062
0.053
0.062
5,000
0.007
0.006
0.007 0.006
0.053
0.055
0.053
0.055
10,000
0.005
0.004
0.005 0.004
0.053
0.054
0.053
0.054
Table S.2: Comparison of the K-L divergence between the D-P tree posterior mean estimator and the frequentist estimators for skew-normal target copulas.N D-P Tree Empirical Kernel Hist. D-P Tree Empirical Kernel Hist. ρ = 0.5, α = (−10, 50) Table S.3: Comparison of the √ M ISE between the D-P tree posterior mean estimator and the frequentist estimators for skew-normal target copulas. N D-P Tree Empirical Kernel Hist. D-P Tree Empirical Kernel Hist. Table S.4: Comparison of the √ M ISE C between the D-P tree posterior mean estimator and the frequentist estimators for the skew-normal target copulas. ρ = 0.5, α = (−10, 50) ρ = 0.5, α = (100, −100) N D-P Tree Empirical Kernel Hist. D-P Tree Empirical Kernel Hist. = 0.9, α = (−10, 50) ρ = 0.9, α = (100, −100) N D-P Tree Empirical Kernel Hist. D-P Tree Empirical Kernel Hist.242
NA
0.337
Inf
0.528
NA
0.528
Inf
20 0.223
NA
0.247
Inf
0.473
NA
0.428
Inf
50 0.178
NA
0.165
Inf
0.386
NA
0.314
Inf
100 0.161
NA
0.123
Inf
0.349
NA
0.261
Inf
500 0.098
NA
0.064
Inf
0.222
NA
0.166
Inf
1,000 0.085
NA
0.047
Inf
0.184
NA
0.136
Inf
5,000 0.056
NA
0.027
Inf
0.112
NA
0.090
Inf
10,000 0.041
NA
0.020
Inf
0.089
NA
0.076
Inf
ρ = 0.9, α = (−10, 50)
ρ = 0.9, α = (100, −100)
10 0.441
NA
0.449
Inf
1.068
NA
1.099
Inf
20 0.38
NA
0.317
Inf
1.013
NA
0.969
Inf
50 0.328
NA
0.225
Inf
0.836
NA
0.763
Inf
100 0.28
NA
0.146
Inf
0.715
NA
0.619
Inf
500 0.163
NA
0.065
Inf
0.479
NA
0.410
Inf
1,000 0.121
NA
0.044
Inf
0.379
NA
0.345
Inf
5,000 0.068
NA
0.021
Inf
0.209
NA
0.227
Inf
10,000 0.051
NA
0.013
Inf
0.164
NA
0.191
Inf
ρ = 0.5, α = (100, −100)
10 0.773
NA
2.742
93.588 1.365
NA
2.190
71.788
20 0.814
NA
1.926
55.190 1.163
NA
2.657
56.726
50 0.728
NA
1.690
35.150 1.050
NA
1.177
36.757
100 0.918
NA
1.350
25.870 1.159
NA
1.347
25.723
500 0.604
NA
0.526
11.437 1.072
NA
1.398
11.665
1,000 0.528
NA
0.487
8.133
0.894
NA
0.703
8.078
5,000 0.525
NA
0.412
3.617
0.703
NA
0.516
3.601
10,000 0.389
NA
0.263
2.565
0.701
NA
0.769
2.600
ρ = 0.9, α = (−10, 50)
ρ = 0.9, α = (100, −100)
10 1.195
NA
2.365
74.140 2.281
NA
7.372
86.920
20 1.297
NA
1.837
54.820 3.335
NA
6.047
53.928
50 1.521
NA
1.689
34.064 2.217
NA
2.185
36.227
100 1.640
NA
1.427
24.826 2.098
NA
2.380
25.320
500 1.132
NA
0.951
11.515 1.980
NA
1.826
11.277
1,000 0.838
NA
0.994
8.073
1.765
NA
1.535
8.204
5,000 0.700
NA
0.420
3.643
1.540
NA
1.146
3.694
10,000 0.597
NA
0.300
2.575
1.949
NA
1.177
2.843
ρ = 0.5, α = (−10, 50)
ρ = 0.5, α = (100, −100)
10 0.064
0.120
0.083
0.120 0.072
0.118
0.091
0.117
20 0.057
0.083
0.065
0.083 0.065
0.082
0.068
0.082
50 0.044
0.057
0.048
0.057 0.044
0.057
0.050
0.057
100 0.027
0.037
0.029
0.037 0.037
0.041
0.038
0.041
500 0.016
0.018
0.016
0.018 0.018
0.017
0.017
0.017
1,000 0.013
0.013
0.013
0.013 0.013
0.012
0.013
0.012
5,000 0.006
0.006
0.006
0.006 0.007
0.006
0.007
0.006
10,000 0.004
0.004
0.005
0.004 0.005
0.004
0.005
0.004
ρ = 0.9, α = (−10, 50)
ρ = 0.9, α = (100, −100)
10 0.080
0.129
0.102
0.129 0.080
0.121
0.102
0.121
20 0.065
0.089
0.072
0.089 0.075
0.101
0.089
0.101
50 0.055
0.060
0.054
0.060 0.046
0.056
0.051
0.056
100 0.038
0.038
0.037
0.038 0.037
0.036
0.036
0.036
500 0.020
0.018
0.019
0.018 0.021
0.018
0.020
0.018
1,000 0.014
0.014
0.014
0.014 0.014
0.013
0.014
0.013
5,000 0.006
0.006
0.006
0.006 0.007
0.005
0.008
0.005
10,000 0.005
0.004
0.005
0.004 0.005
0.004
0.006
0.004
10 0.026
0.317
0.028
0.317 0.054
0.321
0.054
0.321
20 0.026
0.225
0.028
0.225 0.054
0.230
0.055
0.230
50 0.026
0.144
0.026
0.144 0.054
0.151
0.054
0.151
100 0.026
0.103
0.026
0.103 0.054
0.113
0.054
0.113
500 0.026
0.051
0.026
0.051 0.053
0.070
0.053
0.070
1,000 0.026
0.041
0.026
0.041 0.053
0.062
0.053
0.062
5,000 0.026
0.029
0.026
0.029 0.053
0.055
0.053
0.055
10,000 0.026
0.027
0.026
0.027 0.053
0.054
0.053
0.054
ρ 10 0.010
0.316
0.015
0.316 0.063
0.322
0.065
0.322
20 0.010
0.224
0.014
0.224 0.063
0.232
0.064
0.232
50 0.010
0.142
0.012
0.142 0.063
0.155
0.063
0.155
100 0.010
0.100
0.011
0.100 0.063
0.118
0.063
0.118
500 0.010
0.046
0.009
0.046 0.063
0.077
0.063
0.077
1,000 0.009
0.033
0.009
0.033 0.063
0.070
0.063
0.070
5,000 0.009
0.017
0.009
0.017 0.063
0.064
0.063
0.064
10,000 0.009
0.013
0.009
0.013 0.063
0.063
0.063
0.063
Another way of normalization is to apply the PIT (Probability Inverse Transform) to the marginals of P * M . Factorize the M-level approximate posterior density bywhich is a proper copula distribution.One good property of such normalization is that it preserves the copula structure due to the monotonicity of the transform, i.e., P * m andP * m share the same copula. Further, asymptotically, F x,P * m and F y,P * m converge to the uniforms, leading toP * m p → P * m .Estimation with Unknown MarginalsThroughout this article, especially for the simulations, we focus on the estimation of a copula itself, assuming the marginals are known. Here we address more practical scenarios where the marginals are to be estimated. As we stated earlier, the marginal distributions can be more accurately estimated than the copula as data concentrate to a single dimension. Generally, suppose we have N i.i.d. observations (X i , Y i ), and their marginal distribution estimates are either parametric or nonparametric, denoted byF X andF Y respectively. The inverse transform (F −1 X (X i ),F −1 Y (Y i )) = (Û i ,V i ) is considered copula-distributed observations where the regular D-P tree copula estimation procedure can be applied.
Statistical Applications of the Multivariate Skew Normal Distribution. A Azzalini, A Capitanio, Journal of the Royal Statistical Society: Series B (Statistical Methodology). 61Azzalini, A. and Capitanio, A. (1999), "Statistical Applications of the Multivariate Skew Normal Distribution," Journal of the Royal Statistical Society: Series B (Statistical Methodology), 61, 579-602.
Rank Estimators of Scores for Testing Independence. K Behnen, M Hušková, G Neuhaus, Statistics & Risk Modeling. 3Behnen, K., Hušková, M., and Neuhaus, G. (1985), "Rank Estimators of Scores for Testing Inde- pendence," Statistics & Risk Modeling, 3, 239-262.
Nonparametric Estimation of Copula Functions for Dependence Modelling. S X Chen, T.-M Huang, Canadian Journal of Statistics. 35Chen, S. X. and Huang, T.-M. (2007), "Nonparametric Estimation of Copula Functions for Depen- dence Modelling," Canadian Journal of Statistics, 35, 265-282.
La Fonction de Dépendance Empirique et Ses Propriétés. Un Test Non Paramétrique d'Indépendance. P Deheuvels, Académie Royale de. Belgique. Bulletin de la Classe des Sciences. 6e Série. 65Deheuvels, P. (1979), "La Fonction de Dépendance Empirique et Ses Propriétés. Un Test Non Paramétrique d'Indépendance," Académie Royale de. Belgique. Bulletin de la Classe des Sciences. 6e Série., 65, 274-292.
L Devroye, L Györfi, Nonparametric Density Estimation: the L1 View. New York, NYWiley119Devroye, L. and Györfi, L. (1985), Nonparametric Density Estimation: the L1 View, vol. 119 of Wiley Series in Probability and Statistics, New York, NY: Wiley.
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Estimating the Density of a Copula Function. I Gijbels, J Mielniczuk, Communications in Statistics-Theory and Methods. 19Gijbels, I. and Mielniczuk, J. (1990), "Estimating the Density of a Copula Function," Communi- cations in Statistics-Theory and Methods, 19, 445-464.
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| []
|
[
"RIGHT ℓ-GROUPS ASSOCIATED WITH VON NEUMANN ALGEBRAS",
"RIGHT ℓ-GROUPS ASSOCIATED WITH VON NEUMANN ALGEBRAS"
]
| [
"Carsten Dietzel "
]
| []
| []
| In [Rum18], Rump defined and characterized noncommutative universal groups G(X) for generalized orthomodular lattices X.We give an explicit construction of G(X) in terms of pure paraunitary groups when X is the projection lattice of a von Neumann algebra. The results given here extend some of those in [Die19]. | null | [
"https://arxiv.org/pdf/2001.07750v2.pdf"
]
| 210,859,419 | 2001.07750 | 3d343a9a3d6ba1c8067e7259c25fd013326c5884 |
RIGHT ℓ-GROUPS ASSOCIATED WITH VON NEUMANN ALGEBRAS
30 Jan 2020
Carsten Dietzel
RIGHT ℓ-GROUPS ASSOCIATED WITH VON NEUMANN ALGEBRAS
30 Jan 2020
In [Rum18], Rump defined and characterized noncommutative universal groups G(X) for generalized orthomodular lattices X.We give an explicit construction of G(X) in terms of pure paraunitary groups when X is the projection lattice of a von Neumann algebra. The results given here extend some of those in [Die19].
Introduction
For details on quantum logic, the interested reader can consult, for example, [Coh89].
Hilbert spaces over K = C provide a framework for quantum mechanics in that they contain all possible states a certain physical system can be in.
A very classical Hilbert space in quantum mechanics is the space H = L 2 (R) of squareintegrable complex-valued functions on the real line whose elements represent the states of one-dimensional systems.
Closed subspaces of H represent certain physical properties of the observed system. A simple property in the case of H = L 2 (R), for example, is the property of being of positive parity which mathematically amounts for a wavefunction ψ to be even, i.e. ψ(x) = ψ(−x) for all x. The statement "ψ is of positive parity" is a prototypical example of a proposition in what is called quantum logic.
Closed subspaces, representing quantum-mechanical "propositions" , provide one example of a semantical framework for quantum logic. This logic, however, is an example of a nonclassical logic. A vague explanation for this is that a physical system initially need not be in the state which is being measured but the proposition "ψ is of positive parity" (for example) is a result of the measuring.
A better explanation can be given as follows: classical logic is distributive in the sense that if we are given propositions, say P , Q and R, they fulfil the distributive laws, i.e.
P ∧ (Q ∨ R) ⇔ (P ∧ Q) ∨ (P ∧ R), P ∨ (Q ∧ R) ⇔ (P ∨ Q) ∧ (P ∨ R).
The lattice of closed subspaces of a Hilbert space, however, is not distributive, in general. In fact when the Hilbert space is of infinite dimension it is not even modular, as is discussed in [Hal74,Problem 15].
Quantum logic is thus not classical. In fact, classical logic is contained in quantum logic. We explain this further. A semantic framework for classical logic is provided by Date: January 31, 2020.
the Boolean algebras but how does a semantic framework for quantum logic look like that is independent of the notion of Hilbert space? The answer has been obtained in a series of articles written by Birkhoff,von Neumann et al. ([BvN36], [JvNW34]) and Husimi ([Hus37]) -it is given by the orthomodular lattices:
An orthomodular lattice is a bounded lattice X (with its lowest and greatest elements denoted by 0 and 1) and an order-reversing involution * : X → X such that the following axioms hold:
x ∧ x * = 0 (OL1)
x ∨ x * = 1 (OL2)
x ≤ y =⇒ x ∨ (x * ∧ y) = y (OML)
Typical examples of OMLs are i) Boolean algebras, with * being Boolean negation, ii) the lattice of closed subspaces of a Hilbert space, with * being the operation of taking orthogonal complements or, more generally, iii) the lattice of closed A-invariant subspaces of a Hilbert space H, where A is a von Neumann algebra acting on H, the * -operation being the same as above.
There is a connection between OMLs and group theory which we will investigate in this article:
Let X be an OML. A group-valued measure (which we abbreviate by gvm) on X is a mapping µ : X → G where G is a group, such that µ(x ∧ y) = µ(x)µ(y) whenever y ≥ x * . It is clear that then there must also be a universal gvm on X, i.e. a fixed gvm ι : X → G(X) -where G(X) is the structure group of X -such that the following universal property holds:
For any gvm µ : X → G, there is a unique homomorphism of groups µ : G(X) → G such that µ = µι. [Rum18] that the submonoid S(X) generated by X in G(X) is the negative cone of a right-invariant lattice order on G(X). He furthermore gave a characterization of all possible structure groups in terms of ordered groups. This is one of many of Rump's results connecting logical algebras (L-Algebras) with ordered algebraic structures. A classical precursor of this kind of results is Mundici's theorem on the equivalence of MV-algebras and abelian ℓ-groups with strong order unit ( [Mun86]).
Rump proved in
However, due to the broad variety of OMLs, a general description of their structure groups G(X) is likely to be general as well. In fact, a "generic" G(X) is simply a group defined by generators and relations which are derived from the respective OML.
However, one gets more specific results when restricting to a more specific class of objects. This is essentially what we are doing in this article -namely, we give a rather concrete realization of G(X) whenever X is the OML of A-invariant subspaces for a von Neumannalgebra A on a Hilbert space H.
We will prove:
Theorem. Let A be a von Neumann algebra on a Hilbert space H.
If X(A ′ ) is the OML of A ′ -invariant subspaces where A ′ is the commutant of A, then there is an isomorphism of groups G(X(A ′ )) ∼ = PPU(A), where PPU(A) = ∞ i=−∞ t i ϕ i ∈ A t, t −1 : ∞ i=−∞ t i ϕ i ∞ i=−∞ t −i ϕ * i = 1 ∧ ∞ i=−∞ ϕ i = 1 is the pure paraunitary group of A.
The case when A is a factor of type I n has essentially been covered by the author in the article [Die19] where a realization of G(X) is given in the case when X is the OML associated with a hermitian, anisotropic bilinear form on a finite-dimensional vector space (not necessarily over R or C). Some arguments given in the present article will be similar to the arguments given in the preceeding one. However, we also tried to make some of them more streamlined in the von-Neumann-case and hope that we succeeded in doing so.
The outline of this article is as follows:
Section 1 is rather a collection of definitions and facts: we define orthomodular lattices and explain how their structure groups are constructed. Furthermore, we give the definition of right ℓ-groups and explain Rump's characterization theorem.
In Section 2, we associate with each Banach- * -algebra A a so-called paraunitary group PU(A) and define a certain subgroup -the pure paraunitary group PPU(A). On these groups, we define a right-invariant partial order. To give the reader some intuition, we explicitly calculate the pure paraunitary groups of abelian C * -and von Neumann-algebras.
Section 3 is the heart of this article: we prove that the right-invariant order of PPU(A) is a lattice when A is a von Neumann-algebra. We will essentially do this by establishing an order-isomorphism between PPU(A) and a sublattice of the subspace lattice of a suitable Hilbert space. Thus, the property of the latter being a lattice (which is rather obvious) carries over to PPU(A) (where it is less obvious).
In Section 4 we deduce that PPU(A) is indeed the structure group of the OML X(A ′ ). This will be done by showing that the right ℓ-group PPU(A) fulfils the conditions of Rump's characterization theorem.
In Section 5, we discuss how the pure paraunitary groups might be made "more analytic".
1. Structure groups of orthomodular lattices 1.1. Orthomodular lattices. Let X be a bounded lattice where we denote the lowest and greatest elements by 0 resp. 1.
We call an antitone involution * : X → X an orthocomplementation if for any x ∈ X, we have the identities:
x ∧ x * = 0 x ∨ x * = 1.
We can now introduce the topic of this subsection: Definition 1. An orthomodular lattice (OML, for short) is a bounded lattice X with orthocomplementation * : X → X such that the orthomodular law
(OML) x ≤ y ⇒ x ∨ (x * ∧ y) = y holds.
On each OML, one can define the (co-)orthogonality relations:
x⊥y :⇔ y ≤ x * (⊥) x⊤y :⇔ y ≥ x * . (⊤)
One can easily see that ⊥, ⊤ are symmetric and are connected by the rule x⊤y ⇔ x * ⊥y * .
We define a partial monoid as a set X together with a distinguished element e (its neutral element) and a partial mapping · : X × X ⇀ X such that:
i) for all x ∈ X, e · x and x · e are always defined, and we have e · x = x · e = x, ii) whenever either (x · y) · z or x · (y · z) is defined, so is the other, and in this case we have (x · y) · z = x · (y · z) (partial associativity).
If the existence of x · y implies the existence of y · x, and x · y = y · x holds in this case, we say X is partially commutative.
Each OML is in fact a partial monoid:
Proposition 1. Let X be an OML. The partial operation
x ⊕ y = x ∨ y x⊥y not defined else makes X a partial monoid with e X = 0.
Dually, the partial operation
x ⊓ y =
x ∧ y x⊤y not defined else makes X a partial monoid with e X = 1.
Both partial monoid structures are partially commutative.
Proof. We only prove the first statement for the second statement is proved similarly.
Partial commutativity follows immediately from the symmetry of ⊥ and the commutativity of ∨.
For all x we clearly have 0⊥x, and 0 ∨ x = x, so 0 is indeed a neutral element for ⊕.
It remains to show that the existence of (x ⊕ y) ⊕ z implies that of x ⊕ (y ⊕ z). ∨ is associative, so this already suffices for partial associativity.
So, assume that (x ⊕ y) ⊕ z exists. This means that x⊥y and z⊥(x ⊕ y). Therefore z ≤ (x ∨ y) * ≤ y * which implies z⊥y -this shows that y ⊕ z exists. Similarly, one sees that z⊥x.
From x⊥y, z we deduce x ≤ (y * ∧ z * ) = (y ∨ z) * . This finally shows that x ⊕ (y ⊕ z).
By using partial commutativity (or mimicking the argument from above), one can show that the existence of x ⊕ (y ⊕ z) implies that of (x ⊕ y) ⊕ z, too.
In this article we will be concerned about a certain class of OMLs derived from von Neumann algebras. We first recall some basic definitions and facts about von Neumann algebras (details can be found in [Dix81], for example):
Let H be some complex Hilbert space (with inner product denoted by −, − and norm denoted by . ). Let B(H) be the Banach- * -algebra of bounded operators on H where we define the * -operation, as usual, by taking adjoints, i.e. ϕx, y = x, ϕ * y . A von Neumann algebra on H is a subalgebra A ⊆ B(H) with A ′′ = A that is closed under the * -operation. It is easily seen that A ′ is then a von Neumann algebra, too.
We call π ∈ B(H) a projection when π = π * and π 2 = π. For a closed linear subspace M ⊆ H, we denote by π M the (unique) projection of H onto M.
A closed linear subspace M is called A-invariant whenever AM ⊆ M.
The following duality between projections in A and A ′ -invariant subspaces will play a crucial role later:
Proposition 2. A closed linear subspace M ⊆ H is A ′ -invariant if and only if the or- thogonal projections π M , π M * lie in A.
Proof. We always have 1 H ∈ A. Together with π M * = 1 H − π M this shows that we only have to prove (resp. assume) that π M ∈ A.
First of all, let M be
A ′ -invariant. M * then is A ′ -invariant as well: when y ∈ H fulfils x, y = 0 for all x ∈ M then for any ϕ ∈ A ′ , x ∈ M we have x, ϕy = ϕ * x, y = 0 because ϕ * x ∈ M, since ϕ * ∈ A ′ and M is A ′ -invariant. We must show that π M commutes with all A ∈ A ′ : let A ∈ A ′ . Writing x ∈ H as x = x M + x M * with x M ∈ M, x M * ∈ M * , we have π M Ax = π M (Ax M + Ax M * ∈M * ) = Ax M = Aπ M x which proves that π M ∈ A ′′ = A.
On the other hand, let π M ∈ A. We show that M is invariant under all A ∈ A ′ :
If x ∈ M, we have Ax = Aπ M x = π M Ax from which we conclude that also Ax ∈ M.
Proposition 3. Let A be a von Neumann algebra acting on the Hilbert space H. Then the lattice of A-invariant closed subspaces becomes an OML under the operations:
M ∨ N = M + N M ∧ N = M ∩ N M * = {y ∈ H : ∀x ∈ M : x, y = 0}
Definition 2. We denote the OML described in Proposition 3 by X(A).
Proof of Proposition 3. Intersections and closed sums of A-invariant closed subspaces are again closed A-invariant subspaces. This is proven by standard arguments. It is furthermore clear that M + N and M ∩ N are the least resp. greatest subspaces containing resp. contained in M, N.
It is also clear from the definition that M → M * is antitone as soon as we know that M * is closed and A-invariant whenever M is:
If M is closed and A-invariant, M * is clearly closed. Let y ∈ M * and ϕ ∈ A, then for all x ∈ M we have x, ϕy = ϕ * x, y = 0 because ϕ * x ∈ M due to A being * -closed and M being A-invariant. It is well-known that (M * ) * whenever M ⊆ H is closed. It is also known that M * complements M, i.e. M ∩ M * = 0 and M ⊕ M * = H.
It remains to show Equation OML:
Let M ⊆ N be A-invariant and closed. N can be regarded as a Hilbert space in its own right and it is easily seen that M * ∩ N is the orthogonal complement of M within the Hilbert space N. Therefore, M ⊕ (M * ∩ N) = N which is exactly Equation OML in this framework.
1.2. Right ℓ-groups. As before, we first present the topic of this subsection:
Definition 3. A right-ordered group is a group G, equipped with a partial order -denoted by ≤ -which is invariant under right-multiplication, meaning that the implication
h 1 ≤ h 2 ⇒ h 1 g ≤ h 2 g is valid for all g, h 1 , h 2 ∈ G.
If G is right-ordered, we call it right lattice-ordered, if G becomes a lattice under ≤. We then say, G is a right lattice-ordered group, or, for short, a right ℓ-group.
By a homomorphism of right-ordered groups we will mean a monotone homomorphism of groups. By a homomorphism of right ℓ-groups we will mean a homomorphism of the underlying groups which also respects the lattice operations.
It is well-known that right-ordered groups can be described purely in algebraic terms. To give the description, we need the notion of a positive resp. negative cone:
If G is right-ordered, we define its positive cone as G + := {g ∈ G : g ≥ e} and, similarly, its negative cone as
G − := {g ∈ G : g ≤ e} .
Furthermore, for an arbitrary group G, we call a submonoid P ⊆
G pure if P ∩ P −1 = 1 where P −1 = {g −1 : g ∈ P }.
The following proposition is fundamental in the theory of right-ordered groups:
Proposition 4. Let G be a group. For any right-order on G, the positive cone G + is a pure submonoid of G.
Vice versa, each pure submonoid P ⊆ G gives rise to a right-order with
G + = P via the rule g ≥ h :⇔ gh −1 ∈ P . Proof. See [KM96, Theorem 1.5.1.].
We can now present a class of right ℓ-groups investigated by Rump in [Rum18].
With a partial monoid we can always associate both a structure monoid as a structure group:
Definition 4. Let X be a partial monoid with operation ⊕:
The structure monoid S(X) is a monoid, together with a mapping ι S : X → S(X) such that the following holds: i) ι S is a homomorphism of partial monoids, where S(X) is seen as a partial monoid in the obvious way, ii) if f : X → M is such a homomorphism to another monoid M there is a unique homomorphism of monoidsf :
S(X) → M such that f = ι S •f .
Analogously, the structure group G(X) is a group, together with a mapping ι G : X → G(X) such that the following holds:
i) ι G is a homomorphism of partial monoids, where G(X) is seen as a partial monoid in the obvious way,
ii) if f : X → G is such a homomorphism to another group G there is a unique homo- morphism of groupsf : G(X) → G such that f = ι G •f .
It is easy to see that the structure monoids and groups of a partial monoid are unique and do always exist.
Uniqueness is in both cases a mere categorial fact. Furthermore, S(X) can be constructed as the universal monoid generated by the nonzero elements of X under the relations x ⊕ y = z which are already holding in X. G(X) is constructed in the same way.
Remark 1. It should be noted that it will often happen that the map ι G is not injective. It will not even suffice to assume that S(X) is cancellative: the first example of a cancellative monoid which is not embeddable into any group has been constructed by Malcev [Mal37,§2].
Using the universality of ι S we immediately see that there is a unique homomorphism of
monoids i : S(X) → G(X) with ι G = i • ι S .
Definition 5. If X is an OML, we define its structure monoid S(X) as the structure monoid of the partial monoid (X, ⊕), where ⊕ is defined as in Proposition 1.
Similarly, we define its structure group as the structure group of the partial monoid (X, ⊕).
Rump discovered that structure groups of OMLs can be right lattice-ordered in a natural way:
Theorem 1. [Rum18, Corollary 4.6] Let X be an OML. Then i(S(X)) is the positive cone of a right lattice-order on G(X). Furthermore, ι G : X → G(X) embeds X as an interval of G(X) under this partial order.
Remark 2. The reader familiar with Rump's original article [Rum18] may have noted slight differences in this exposition. We explain these: a) Rump originally defined the structure monoid S(X) by the partial monoid (X, ⊓) which, however, is isomorphic to (X, ⊕) via the isomorphism x → x * . Furthermore, S(X) is embedded as the negative cone into G(X) which does not influence the latticeorderability of the subject. This is justified by the connection of his article to quantum logic: in this context, tautological truth, amounting to the element 1 ∈ X, should be identified with e ∈ G.
In this article we chose to "turn things around" because we found that the definitions are easier accessible when we define S(X) as the positive cone whose multiplication is defined by adding orthogonal elements in X. b) Our definitions of S(X) and G(X) by universal properties are actually a corollary in Rump's article. For we are interested in special structure monoids resp. groups, we also decided in favour of the easier characterizations.
Rump also proved a converse to this theorem. To explain this result, we first need to introduce some special elements of a right ℓ-group.
Definition 6. Let G be a right ℓ-group. Let ∆ > e. We call ∆ a) normal if ∆(g ∨ h) = ∆g ∨ ∆h holds for any g, h ∈ G (that is, left-multiplication by ∆ is isotone), b) singular if for e ≤ x, y ≤ ∆, we always have the implication
xy ≤ ∆ ⇒ yx = x ∨ y,
c) a singular strong order unit if ∆ is normal and singular and, furthermore, for any g ∈ G there exists k ∈ Z with g ≤ ∆ k .
Remark 3. 1) In the classical theory of ℓ-groups (without the predicate left or right), the concept of strong order unit already exists, see [Dar94,Definition 7.4.]. There is also the possibility of defining weak order units as elements g ∈ G + such that g ∧ h = e for all h ∈ G + (see [Dar94,54.3.]) but it is not known to the author if this concept is useful in the one-sided theory. 2) We might define singularity by the implication xy ≤ ∆ ⇒ xy = x ∨ y. It turns out that, using this definition, the theory of right ℓ-groups with singular strong order unit goes in a slightly different direction -see [DRZ19, Section 5].
We use this opportunity to note that singularity is also present in the classical theory, see [Dar94, Definition 6.9.], for example.
The paraunitary group of an involutive Banach algebra
We give an overview on the basic definitions regarding C * -algebras. Details can be found, for example, in [Dix77].
Let K ∈ {R, C}.
A Banach algebra is a Banach space (A, . ) (complex or real) with an associative, continuous bilinear map · : A × A → A; (x, y) → xy which is submultiplicative, i.e. xy ≤ x · y . If there is a neutral element 1 for this multiplication, we call it unital.
An involution on a Banach algebra is a map * : A → A; x → x * which is i) an involution in the classical sense: (x * ) * = x for all x ∈ A, ii) antilinear: for all x, y ∈ A, α ∈ K, (x + y) * = x * + y * (αx) * = αx * (where the latter reduces to mere linearity when K = R), iii) an antihomomorphism: (xy) * = y * x * for all x, y ∈ A, iv) norm-preserving: x * = x .
A Banach algebra with an involution is called a Banach- * -algebra.
If additionally, we have xx * = x * x = x 2 for all x ∈ A, we call A an C * -algebra.
Note that every von Neumann algebra is a C * -algebra under the operator norm and involution given by taking adjoints.
Throughout this section, A will denote an arbitrary Banach * -algebra (over R or C)
We denote by A [t, t −1 ] the set of formal expressions
ϕ = ∞ i=−∞ t i ϕ i with ϕ i ∈ A and ϕ i = 0 for only finitely many i. Furthermore, A [t] (resp. A [t −1 ]) will denote the subsets consisting of all ϕ ∈ A [t, t −1 ] with ϕ i = 0 for i < 0 (i > 0).
There is a natural way of defining a multiplication on A [t, t −1 ], extending that of A: for
ϕ, ψ ∈ A [t, t −1 ], we set ϕψ = ∞ i=−∞ t i ∞ k=−∞ ϕ k ψ i−k .
This is just the ring of finite Laurent series over A which clearly is associative.
The involution * can be extended to elements of A [t, t −1 ] by the rule
ϕ * = ∞ i=−∞ t −i a * i .
Clearly, * is an involution and an anti-homomorphism of A [t, t −1 ], therefore making the latter an involutive K-algebra 1 . Restricting * from A [t, t −1 ] to the subalgebra A, we get the familiar involution on the latter, therefore there will be no danger of confusion when using the same symbol " * " for these operators.
Note that A [t] * = A [t −1 ] and A [t −1 ] * = A [t].
There is a canonical specialization map
ε 1 : A t, t −1 → A ϕ → ∞ i=−∞ ϕ i
which amounts to setting t = 1. The reader can easily convince himself that ε 1 respects the involutive algebra structures in the sense that it is a homomorphism of algebras which additionally fulfils ε 1 (ϕ * ) = ε 1 (ϕ) * .
We can now come to our main definitions:
With A we can associate its unitary group which we define as
U(A) := {x ∈ A : x * x = xx * = 1} .
This is easily seen to become a group under the multiplication of A.
Similarly, we can associate quite another group with A, namely, its paraunitary group
(1) PU(A) := ϕ ∈ A t, t −1 : ϕ * ϕ = ϕϕ * = 1 .
Let ϕ ∈ PU(A). We then have ε 1 (ϕ) * ε 1 (ϕ) = ε 1 (ϕ * )ε 1 (ϕ) = ε 1 (ϕ * ϕ) = ε 1 (1) = 1.
1 Which, however, is not a Banach * -algebra, due to a lack of completeness Similarly, one shows ε 1 (ϕ)ε 1 (ϕ) * = 1, thus proving that ε 1 (ϕ) ∈ U(A).
On the other hand, we have a canonical embedding of rings
ι : A → A t, t −1 x → t 0 x
which is a morphism of rings and clearly commutes with the respective * -operations.
From this one easily sees that ι(x) ∈ PU(A) whenever x ∈ U(A) and that these elements comprise the totality of constant paraunitary polynomials.
Furthermore, ε 1 ι = 1 A which proves Proposition 5. There is a split exact sequence of groups
1 −→ ker ε 1 −→ PU(A) ε 1 −→ U(A) −→ 1.
with ι as a right inverse to ε 1 .
We can now introduce the main character of this section: Remark 4. 1) The designation of the elements of ker ε 1 as pure is due to the following analogy with braid groups ([KT08, Section 1.3]):
Denoting the braid group with n strands by B n and the symmetric group on n elements by S n there is a canonical specialization map ε : B n → S n fitting into a short exact sequence 1 → P n → B n ε → S n where P n := ker(ε) is the so-called pure braid group. 2) ε 1 , however, is not the only possibility of defining a pure paraunitary group: for any z ∈ C with |z| = 1 there is a specialization map
ε z : A t, t −1 → A ϕ → ∞ i=−∞ z i ϕ i
which amounts to setting t = z. One can show that each ε z restricts to a homomorphism PU(A) → U(A).
If we denote the respective kernels by PPU z (A) one can show that each PPU z (A) are isomorphic to PPU(A) by the isomorphism
α z : PPU(A) → PPU z (A) ∞ i=−∞ t i ϕ i → ∞ i=−∞ t i z −i ϕ i .
Therefore, if one knows one member of the family PPU z (A) one knows them all.
It may, however, lead to interesting questions when one applies ε z ′ to PPU z (A) when z, z ′ are not necessarily equal. For an instance of the case z = 1, z ′ = −1, see [ Before diving into our analysis of the pure paraunitary groups for general von Neumann algebras we give two motivating examples:
Example 1. 1) By means of the Gelfand isomorphism, any commutative, unital C *algebra A is isomorphic (in the sense of C * -algebras) to the C * -algebra C(X) of complex-valued continuous functions on some compact Hausdorff space X, the *operation being complex conjugation ([Dix77, Theorem 1.4.1.]). By this isomorphism, each ϕ ∈ A [t, t −1 ] can be seen as a family of finite Laurent polynomials ϕ x (t) ∈ C [t, t −1 ] (x ∈ X) such that the coefficients of ϕ x (t) vary continuously in x and there is a global bound for the degrees of the least and the greatest non-zero coefficients of each ϕ x .
The paraunitarity condition now reads as ϕ x (t) ·φ x (t −1 ) = 1 which is only possible when ϕ x (t) = z x t nx with |z x | = 1 and n x ∈ Z. Thus, the coefficients of ϕ x (t) are either 0 or have absolute value 1 from which we infer that n x must be locally constant, which is equivalent to saying that n x is continuous when Z carries the discrete topology.
Setting t = 1 in ϕ x (t) = z x t nx , one sees that ϕ x (t) represents an element of PPU(A) if and only if z x = 1 for all x. In this case, ϕ x (t) = t nx .
Let C(X, Z) be the (additive) group of continuous functions n : X → Z. Sending such a function n to the family ϕ x (t) = t nx establishes an isomorphism of abstract groups C(X, Z) ∼ = PPU(A).
The elements of PPU + (A) are then represented by families ϕ x (t) = t nx with n x ∈ Z + 0 varying continuously. This shows that the isomorphism in the above paragraph identifies the submonoid C(X, Z + 0 ) of C(X, Z) with PPU + (A). We conclude that, as a right-ordered group, PPU(A) is isomorphic to the additive group C(X, Z) under the pointwise partial order.
2) If A is a commutative von Neumann algebra, A is * -isomorphic to the algebra of essentially bounded, complex-valued, measurable functions on some measure space (Y, µ) [Dix81, I.7.3.,Theorem 1]. By a similar argument as the one given above, we can identify the right-ordered group PPU(A) with L ∞ (Y, Z), that is, the functions in L ∞ (Y ) which are integral µalmost everywhere, and where we define f ≤ g if and only if f (x) ≤ g(x) µ-almost everywhere.
The pure paraunitary group of a von Neumann algebra
In this section, A will always be assumed to be a von Neumann algebra acting on a complex Hilbert space H. Both will be fixed throughout this section.
The aim of this section is to give a proof of the following Theorem 3. If A is a von Neumann algebra, the right-invariant order on PPU(A) defined by the positive cone PPU + (A) makes the former a lattice.
Preliminary constructions. We begin by constructing several Hilbert spaces for the rings
A ′ [t, t −1 ] , A ′ [t] , A ′ [t −1 ] to act on.
First of all, we set
H t, t −1 := x = ∞ i=−∞ t i x i : x i ∈ H, ∞ i=−∞ x i 2 < ∞ H [[t, t −1 ]
] becomes a Hilbert space as follows:
For x = ∞ i=−∞ t i x i and y = ∞ i=−∞ t i y i in H [[t, t −1 ]]
we define their product as ]] together with its inner product is immediately seen to be equivalent to the infinite direct sum ⊕ ∞ i=−∞ H. It is well-known that the result of this construction is also a Hilbert space.
(2) x, y H[[t,t −1 ]] = ∞ i=−∞ x i , y i H .
We define the closed subspaces H
[[t]] , H [[t −1 ]] ⊆ H [[t, t −1 ]] as the collection of all x ∈ H [[t, t −1 ]] fulfilling x i = 0 for i < 0 (resp. i > 0). H [[t, t −1 ]] becomes an A ′ [t, t −1 ]-module in a natural way: for ϕ ∈ A ′ [t, t −1 ] , x ∈ H [[t, t −1 ]], we simply set ϕx = ∞ i=−∞ t i ∞ k=−∞ ϕ k x i−k
When defining a module action on a Hilbert space, it should be proved that the multiplication maps are continuous, i.e.
Proposition 7. For each ϕ ∈ A [t, t −1 ] or ϕ ∈ A ′ [t, t −1 ], the map x → ϕx is continuous on H [[t, t −1 ]].
Proof. It clearly suffices to assume ϕ ∈ A [t, t −1 ].
x → tx and x → t −1 x are just shift operators on H [[t, t −1 ]] which are clearly isometries and therefore continuous.
For ϕ 0 ∈ A, we have t 0 ϕx = ∞ i=−∞ t i ϕ 0 x i = ∞ i=−∞ ϕ 0 x i 2 1 2 ≤ ϕ 0 ∞ i=−∞ x i 2 1 2 = ϕ 0 · x ,
so t 0 ϕ 0 acts continuously, too. The action of an arbitrary element of A [t, t −1 ] is a finite combination of these operations and therefore continuous.
The inner product is connected with the * -operation on A [t, t −1 ] by the equation
(3)
x, ϕy = ϕ * x, y which can be reduced to the case ϕ = t k ϕ k , using linearity. This case follows from the calculation
x, t k ϕ k y = ∞ i=−∞ t i x i , ∞ i=−∞ t i ϕ k y i−k = ∞ i=−∞ x i , ϕ k y i−k = ∞ j=−∞ ϕ * k x j+k , y j = ∞ j=−∞ t j ϕ * k x j+k , ∞ j=−∞ t j y j = t −k ϕ * k x, y .
It turns out, that H [[t]] (resp. H [[t −1 ]]) is invariant under the action of
A ′ [t] (resp. A ′ [t −1 ]).
How does PPU(A) come into play? This question is partly answered by the following proposition: = ϕ * ϕx, y = x, y .
Remark 5. A reader who is interested in the analytical aspects may feel uneasy with the rather "algebraic" definition of PPU(A) in terms of finite Laurent series. (Pure) paraunitary groups with a rather "analytic" flavour will be discussed in Section 5.
We can now introduce a family of lattices which will play a crucial role in this section:
Definition 8. For integers m ≤ n we define sub(H) n m as the lattice of all From now on, when speaking of PPU(A) or PPU + (A) as ordered sets, we will always mean the right-invariant orders defined by ϕ ≤ ϕ ′ whenever there is a ψ ∈ PPU + (A) such that ϕ ′ = ψϕ.
A ′ [t −1 ]-invariant closed subspaces M ⊆ H [[t, t −1 ]] such that t m H [[t −1 ]] ⊆ M ⊆ t n H [[t −1 ]].
Another protagonist in this section is the mapping Ω which is defined as
Ω : PPU(A) → sub(H) ∞ −∞ ϕ → ϕH t −1
which has a sibling Ω + which is similarly defined as
Ω + : PPU + (A) → sub(H) ∞ 0 ϕ → ϕH t −1 .
First of all, we have to show that Ω and Ω + are indeed mappings, that is:
Proposition 9. Ω and Ω + are well defined and monotone.
Proof. We start by showing that
H [[t −1 ]] ⊆ ϕH [[t −1 ]] for ϕ ∈ PPU + (A). This is equiva- lent to ϕ −1 H [[t −1 ]] ⊆ H [[t −1 ]] which follows from ϕ −1 = ϕ * ∈ A [t −1 ].
Take ϕ, ϕ ′ ∈ PPU + (A) with ϕ ≤ ϕ ′ . There is a ψ ∈ PPU + (A) with ϕ ′ = ϕψ. From what we have shown before it follows that
ϕ ′ H t −1 = ϕψH t −1 ⊇ ϕH t −1 .
We can now show that ϕ ∈ PPU
+ (A) implies ϕH [[t −1 ]] ⊆ t n H [[t −1 ]]
for some integer n: take a non-negative n with t n ϕ * ∈ PPU + (A). Then ϕt n ϕ * = t n , therefore
ϕ ≤ t n ⇒ ϕH [[t −1 ]] ⊆ t n H [[t −1 ]].
So, Ω + is well-defined and monotone. Assuming ϕ, ϕ ′ ∈ PPU(A) (but still ψ ∈ PPU + (A)) in the proof of monotonicity for Ω + , we get a proof for the monotonicity of Ω.
The aim of the following two subsections will be the proof of Theorem 4. Ω and Ω + are bijective.
Proof. In the following subsections we will show that Ω + is both injective (Proposition 10) and surjective (Proposition 12). This proves the second part.
The bijectivity of Ω is then an easy corollary:
Let ϕ, ϕ ′ ∈ PPU(A) fulfil ϕH [[t −1 ]] = ϕ ′ H [[t −1 ]].
Chose an integer n with t n ϕ, t n ϕ ′ ∈ PPU + (A). Clearly, we also have t n ϕH
[[t −1 ]] = t n ϕ ′ H [[t −1 ]]
. From the injectivity of Ω + it follows that t n ϕ = t n ϕ ′ and finally ϕ = ϕ ′ . This proves that Ω is injective.
Take M ∈ sub(H) ∞ −∞ . Then there is an integer n with t n M ∈ sub(H) ∞ 0 . Ω + is surjective, therefore there is some ϕ ∈ PPU + (A) with ϕH [[t −1 ]] = t n M, showing that t −n ϕH [[t −1 ]] = M.
Remark 6. Ω and Ω + may be bijective and order-preserving but it does not already follow that they are order-isomorphisms -this will be shown later (Lemma 3).
3.2. Ω + is injective. We begin with a simple lemma:
Lemma 1. Let ψ ∈ PPU(A) fulfil ψH [[t −1 ]] ⊆ H [[t −1 ]]. Then ψ ∈ PPU − (A): Proof. Let ψH [[t −1 ]] ⊆ H [[t −1 ]] and assume ψ / ∈ PPU − (A). Then there is a k > 0 such that ψ = k i=−∞ t i ψ i with ψ k = 0.
Take an x ∈ H with ψ k x = 0.
Then t 0 x ∈ H [[t −1 ]] but ψ(t 0 x) = k i=−∞ t i ψ i x which has the nonzero k'th coefficient ψ k x. Therefore, ψ(t 0 x) / ∈ H [[t −1 ]] which is a contradiction.
It is now easy to show:
Proposition 10. Ω + is injective. Proof. Take ϕ, ϕ ′ ∈ PPU + (A) with ϕH [[t −1 ]] = ϕ ′ H [[t −1 ]]. From this it follows that ϕ −1 ϕ ′ H [[t −1 ]] = H [[t −1 ]] and ϕ ′−1 ϕH [[t −1 ]] = H [[t −1 ]]. Lemma 1 implies that ϕ −1 ϕ ′ ∈ PPU − (A) and (ϕ −1 ϕ ′ ) −1 = ϕ ′−1 ϕ ∈ PPU − (A). But PPU − (A) is a negative cone (Proposition 6), therefore ϕ −1 ϕ ′ = 1 which implies ϕ = ϕ ′ .
3.3. Ω + is surjective. We can define, for any M ∈ X(A ′ ), an element p M := tπ M +π M * ∈ A [t] where the latter membership is guaranteed by Proposition 2.
We can even say more about these elements:
Proposition 11. For each M ∈ X(A ′ ), we have p M ∈ PPU + (A).
Proof. Using the self-adjointness of projections, we calculate:
p * M p M = t −1 π M + π M * (tπ M + π M * ) = t −1 π M π M * =0 +π 2 M + π 2 M * + t π M * π M =0 = π M + π M * = 1.
Similarly, one calculates p M p * M = 1, thus showing that p M is indeed paraunitary.
Finally, ε 1 (p M ) = π M + π M * = 1.
For M ∈ X(A ′ ), we define for an integer i the closed subspace
t i M := t i x : x ∈ M ⊆ H t, t −1 .
It turns out that we can already specify what Ω(p M ) is:
Lemma 2. For M ∈ X(A ′ ), Ω(p M ) = H [[t −1 ]] ⊕ tM. Proof. tM is contained in tH [[t]] = H [[t −1 ]] * therefore the sum is indeed orthogonal. We must show p M H [[t −1 ]] = H [[t −1 ]] ⊕ tM. Take x = 0 i=−∞ t i x i ∈ H [[t −1 ]]
, then an easy calculation shows that
p M x = tπ M x 0 + 0 i=−∞ t i (π M x i−1 + π M * x i ) ∈ H t −1 ⊕ tM,Let x = tx 1 + 0 i=−∞ t i x i ∈ H [[t −1 ]] ⊕ tM, that is, x 1 ∈ M. Furthermore p −1 M = p * M = t −1 π M + π M * . We calculate: p −1 M x = t π M * x 1 =0 + 0 i=−∞ t i (π M x i+1 + π M * x i ) ∈ H t −1
which proves the other inclusion.
Proposition 12. Ω + is surjective.
Proof. Take M ∈ sub(H) ∞ 0 . We will show that there is a ϕ ∈ PPU + (A) with ϕ −1 M = H [[t −1 ]].
That clearly proves our proposition.
There is an n ≥ 0 with M ∈ sub(H) n 0 . The proof will be by induction over this n.
If n = 0 then M = H [[t −1 ]] = 1 · H [[t −1 ]].
Assume that for fixed N ≥ 0 we have shown that for each M ∈ sub(H) N 0 there is a
ϕ ∈ PPU + (A) with ϕ −1 M = H [[t −1 ]].
The structure group of a von Neumann algebra
Here we will prove that the notions of structure group and pure paraunitary group of a von Neumann algebra actually coincide.
We need a lemma before. Recall that, for M ∈ X(A ′ ), we defined p M = tπ M + π M * . Lemma 4. Let ϕ ∈ PPU + (A). Then ϕ ≤ t if and only if ϕ = p M for some M ∈ X(A ′ ).
Proof. By Corollary 2, 1 ≤ ϕ ≤ t is equivalent to H [[t −1 ]] ⊆ ϕH [[t −1 ]] ⊆ tH [[t −1 ]].
By the orthomodular law, On the other hand, for M ∈ X(A ′ ), we have tp * M ∈ PPU + (A) and tp * M p M = t, from which p M ≤ t follows.
ϕH t −1 = H t −1 ⊕ (H t −1 * ∩ ϕH t −1 ) = H t −1 ⊕ (
We proceed with a lemma:
Lemma 5. For M, N ∈ X(A ′ ) with M⊥N we have p M p N = p M ⊕N .
Proof. For M⊥N, it is known that π M π N = 0 and π M + π N = π M ⊕N .
First of all, we deduce π M * π N * = (1 − π M )(1 − π N ) = 1 − (π M + π N ) + π M π N = 1 − π M ⊕N = π (M ⊕N ) * and π M π N * = π M (1 − π N ) = π M − π M π N = π M . Similarly, π M * π N .
Using these formulae, we calculate: p M p N = (tπ M + π M * )(tπ N + π N * ) = t 2 π M π N + t(π M π N * + π M * π N ) + π M * π N * = t(π M + π N ) + π (M ⊕N ) * = tπ M ⊕N + π (M ⊕N ) * = p M ⊕N . Proof. We show that t is a singular strong order unit in PPU(A):
Clearly, t is positive. t is also normal: right-multiplication by t is clearly order-preserving, due to PPU(A) being right-ordered, and so is left-multiplication because t commutes with each element of A [t, t −1 ].
Let ϕ ∈ PPU(A). We show that it right-divides some power of t: let n be the greatest integer with ϕ n = 0. Then ψ := t n ϕ * ∈ PPU + (A). Therefore,
ψϕ = t n ϕ * ϕ = t n .
The last step is to show that t is singular:
The divisors of t are exactly the p M with M ∈ X(A ′ ) (Lemma 4). For p M , p N we calculate p M p N = (tπ M + π M * )(tπ N + π N * ) = t 2 π M π N + t(π M π N * + π M * π N ) + π M * π N * .
The t 2 -coefficient is zero if and only if M⊥N. In this case, from Lemma 5 follows p N p M = p M ⊕N = p M ∨ p N . So, t is indeed singular.
Rump's Theorem 2 now tells us that [1, t] -with the lattice-order inherited by PPU(A) -is an OML under the complementation given by ϕ ⊛ = ϕ −1 t and the natural embedding [1, t] ֒→ PPU(A) identifies the latter with its structure group (we use ⊛ for the orthocomplementation because * will cause confusion in what follows).
By Corollary 3, we already know that [1, t] ∼ = X(A ′ ) with respect to the isomorphism M → p M . The only left task is to show that the orthocomplementation is preserved, too:
Using paraunitarity,
We define the commutant of some subset M ⊆ B(H) by M ′ := {ϕ ∈ B(H) : ∀m ∈ M : ϕm = mϕ} .
Now we can state the promised converse to Theorem 1:Theorem 2.[Rum18, Theorem 4.10] For any OML X, the image ι G (1) is a singular strong order unit in the right ℓ-group G(X).Conversely, if G is a right ℓ-group with singular strong order unit ∆, the interval [e, ∆] becomes an OML under the lattice operations inherited by G and the orthocomplementation defined by g * = g −1 ∆.The embedding [e, ∆] ֒→ identifies G as a structure group for the OML [e, ∆].
Definition 7 .
7The pure paraunitary group associated with A is the group PPU(A) := ker ε 1 . By a classical group theoretic argument one deduces from Proposition 5 the Corollary 1. There is a semidirect product decomposition PU(A) = PPU(A) ⋊ U(A) where we identify U(A) with the constant paraunitary polynomials.
PPU(A) are ordered in a very natural way: We can define the subsets PPU + (A) := PPU(A)∩A [t] and PPU − (A) := PPU(A)∩A [t −1 ] which have the following pleasant property: Proposition 6. The subsets PPU + (A), PPU − (A) are the positive respectively negative cone of a right-invariant order on PPU(A). Proof. We first prove that PPU + (A) is a positive cone. By Proposition 4, we have to show that it is a pure submonoid: First of all, PPU + (A) is a submonoid of PPU(A) which comes from the fact that both PPU(A) and A [t] contain 1 and are multiplicatively closed in A [t, t −1 ]. On the other hand, for ϕ ∈ PPU + (A) we have, by (1), that ϕ −1 = ϕ * ∈ PPU − (A). Therefore PPU + (A) ∩ PPU + (A) −1 = PPU + (A) ∩ PPU − (A), the latter consisting only of constant paraunitary polynomials, i.e. those of the form t 0 x (x ∈ A). But these have to fulfil 1 = ε 1 (t 0 x) = x. Thus PPU + (A) ∩ PPU + (A) −1 = 1. Finally, from PPU + (A) −1 = PPU − (A) one sees that PPU − (A) is the negative cone of the right-invariant order defined by the positive cone PPU + (A).
Proposition 8 .
8Each ϕ ∈ PPU(A) acts as a unitary operator -with respect to the inner product given by (2) -on H [[t, t −1 ]]. Proof. Let ϕ ∈ PPU(A), x, y ∈ H [[t, t −1 ]], then ϕx, ϕy (3)
see each sub(H) n m as a sublattice in the lattice of all closed subspaces of H [[t, t −1 ]].
For
general ϕ ∈ PPU(A) chose an integer m such that t m ϕ ∈ PPU + (A). From the arguments above follows H [[t −1 ]] ⊆ t m ϕH [[t −1 ]] ⊆ t n H [[t −1 ]] for some non-negative integer n. Therefore, t −m H [[t −1 ]] ⊆ ϕH [[t −1 ]] ⊆ t n−m H [[t −1 ]], proving that Ω is welldefined.
Corollary 3 .
3The map defined by Γ : X(A ′ ) → [1, t] ⊆ PPU(A) M → p M is an isomorphism of lattices. Proof. We construct the inverse map: By Lemma 4, [1, t] consists exactly of the p M with M ∈ X(A ′ ). Ω + gives an isomorphism between the intervals [1, t] and [Ω(1), Ω(t)]. But each A ′ [t −1 ]invariant subspace between Ω(1) = H [[t −1 ]] and Ω(t) = tH [[t −1 ]] is of the form H [[t −1 ]] ⊕ tM with M ∈ X(A ′ ). Therefore [Ω(1), Ω(t)] ∼ = X(A ′ ) as lattices. It follows that mapping p M to Ω(p M ) = H [[t −1 ]]⊕tM (Lemma 2) and then to M provides us with Γ −1 . Theorem 5. PPU(A) with the right-invariant lattice-order defined by the positive cone PPU + (A), is isomorphic -in the sense of right-ordered groups -to G(X(A ′ )).
t = p * M t = π M * + tπ M = p M * , i.e. Γ(M * ) = Γ(M) ⊛ meaning that Γ is even an isomorphism of OMLs. Therefore, PPU(A) ∼ = G([1, t]) ∼ = G(X(A ′ )).
We will suppress the subscript H [[t, t −1 ]] of the inner product whenever it is clear if we are concerned with elements of H or H [[t, t −1 ]]. Clearly, H [[t, t −1
thus proving p M H [[t −1 ]] ⊆ H [[t −1 ]] ⊕ tM. order to prove p M H [[t −1 ]] ⊇ H [[t −1 ]] ⊕ tM we must show p −1 M (H [[t −1 ]] ⊕ tM) ⊆ H [[t −1 ]].In
tH [[t]] ∩ ϕH t −1 ) and tH [[t]] ∩ ϕH [[t −1 ]] ⊆ tH [[t]] ∩ tH [[t −1 ]] = tH. tH [[t]] ∩ ϕH [[t −1 ]] is easily seen to be closed and invariant under A ′ ⊆ A ′ [t, t −1 ] and therefore is of the form tM where M ⊆ H is A ′ -invariant. Therefore ϕH [[t −1 ]] = H [[t −1 ]] ⊕ tM = p M H [[t −1 ]] (Lemma 2), and therefore ϕ = p M by Proposition 10.
Acknowledgements I am very grateful to Wolfgang Rump for suggesting the research topic and giving valuable advice on the presentation of this work.Now take M ∈ sub(H) N +1 0 . M is invariant under multiplication by t −1 and contains H [[t −1 ]], so:There is a closed A ′ -invariant subspace M 1 such that M ∩ tH [[t −1 ]] = H [[t −1 ]] ⊕ tM 1this follows from the following consideration:The orthomodular law for Hilbert spaces, together with HThe second summand can be determined further:]. Using Equation 4, we calculate:By our induction hypothesis, there is a3.4. Proof of Theorem 3. We are finished with our proof of Theorem 3 as soon as we can pull back the lattice structure of sub(H) ∞ −∞ to PPU(A). We need the followingProposition 10 now tells us that ψ = ϕ −1 ϕ ′ resp. ϕψ = ϕ ′ which is exactly what we wanted to show.Proof of Theorem 3. Theorem 4, Proposition 9 and Lemma 3 together say that Ω : PPU(A) → sub(H) ∞ −∞ is a bijective embedding of ordered sets, i.e. it is isotone. But sub(H) ∞ −∞ is a lattice, and so is PPU(A).We will need the result later, therefore we cite the following part of the proof given above as a corollary:Corollary 2. Ω : PPU(A) → sub(H) ∞ −∞ is an isomorphism of lattices.Possible generalizationsOur definition of PPU(A) is rather "algebraic" in nature, in that it is constructed as a ring of finite Laurent series over A.Clearly, this restriction is necessary for PPU(A) being a structure group. However, one might wonder if there is a slightly bigger group which is a structure group of X(A ′ ) in another, more analytic, sense.One more or less obvious extension is the following:We first definewhere . is the operator norm. This becomes a Banach algebra under convolution and can be made a Banach- * -algebra by defining f * by f * (i) = (f (−i)) * .We can define an analytic pure paraunitary group bySpecializing at 1 still can be made sense in this framework, and so we could define an analytic pure paraunitary group bywhich is still a subgroup which is also right-ordered by the positive cone ppu + (A), consisting of the f ∈ ppu(A) with f (i) = 0 for i < 0. One possible property could be that M fulfilsHowever, it might be that a definition by means of l 1 (A) is too naive and a more subtle kind of convergence is needed here.In each case, an analytic pure paraunitary group -together with a compatible definition of "analytic" structure groups for general OMLs -could help understanding completion processes in lattice-ordered groups, may they be one-or two-sided.
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E-mail address: [email protected] Institute of algebra and number theory. Pfaffenwaldring. 5770569University of StuttgartE-mail address: [email protected] Institute of algebra and number theory, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
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"Towards Generating Virtual Movement from Textual Instructions A Case Study in Quality Assessment",
"Towards Generating Virtual Movement from Textual Instructions A Case Study in Quality Assessment"
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"Himangshu Sarma [email protected] ",
"Robert Porzel [email protected] ",
"Jan Smeddinck [email protected] ",
"Rainer Malaka [email protected] ",
"\nDigital Media Lab\nTZI\nUniversity of Bremen Bibliothekstr\n\n",
"\n28359BremenGermany\n"
]
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"Digital Media Lab\nTZI\nUniversity of Bremen Bibliothekstr\n",
"28359BremenGermany"
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| []
| Many application areas ranging from serious games for health to learning by demonstration in robotics, could benefit from large body movement datasets extracted from textual instructions accompanied by images. The interpretation of instructions for the automatic generation of the corresponding motions (e.g. exercises) and the validation of these movements are difficult tasks. In this article we describe a first step towards achieving automated extraction. We have recorded five different exercises in random order with the help of seven amateur performers using a Kinect. During the recording, we found that the same exercise was interpreted differently by each human performer even though they were given identical textual instructions. We performed a quality assessment study based on that data using a crowdsourcing approach and tested the inter-rater agreement for different types of visualizations, where the RGBbased visualization showed the best agreement among the annotators. | null | [
"https://arxiv.org/pdf/2006.03846v1.pdf"
]
| 49,528,467 | 2006.03846 | db53f2fe62107608ba6d83af62c4540895f6e962 |
Towards Generating Virtual Movement from Textual Instructions A Case Study in Quality Assessment
Himangshu Sarma [email protected]
Robert Porzel [email protected]
Jan Smeddinck [email protected]
Rainer Malaka [email protected]
Digital Media Lab
TZI
University of Bremen Bibliothekstr
28359BremenGermany
Towards Generating Virtual Movement from Textual Instructions A Case Study in Quality Assessment
Many application areas ranging from serious games for health to learning by demonstration in robotics, could benefit from large body movement datasets extracted from textual instructions accompanied by images. The interpretation of instructions for the automatic generation of the corresponding motions (e.g. exercises) and the validation of these movements are difficult tasks. In this article we describe a first step towards achieving automated extraction. We have recorded five different exercises in random order with the help of seven amateur performers using a Kinect. During the recording, we found that the same exercise was interpreted differently by each human performer even though they were given identical textual instructions. We performed a quality assessment study based on that data using a crowdsourcing approach and tested the inter-rater agreement for different types of visualizations, where the RGBbased visualization showed the best agreement among the annotators.
Introduction
Assessing the quality of human body movement performances is an important task in many application areas, ranging from sports to therapy, learning by demonstration in robotics, automated systems for generative animation, and many more. For example, the manual transformation of physical therapy exercises into computer-supported playful exercises in the form of so-called exergames or levels of exergames requires a lot of time and effort, making it impractical for therapists or smaller practices to transform their preferred sets of therapeutic exercises into exergames to be used by their patients. Motivated by our use-case of automatically generating movement patterns to be used in motionbased games for the support of physiotherapy, rehabilitation, and prevention, we thus set out to explore the potential of crowd-based quality of motion assessments, as a necessary intermediate step in the extraction and validation of motions. The human-computation approach is promising in this regard, since the task involves many aspects that are easy for humans, but difficult for machines (Krause and Smeddnick Copyright © 2015, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. 2011). Since it is known that even human experts in quality of movement judgements share little inter-rater agreement (Pomeroy et al. 2003), we set out to explore whether it is possible to achieve a level of inter-rater reliability that would even allow for quality of motion-assessment, if a later cross-validation is projected and to explore which type of a motion-visualization would support the best inter-rater reliability, whereby we hypothesized that the video-based modality would yield the highest inter-annotator reliability. With this work we contribute to human computation by exploring the novel area of quality of motion assessment, where successful human computation could prove beneficial to a large number of application scenarios, and we address the relevant related independent variable of motion visualization.
State of the Art
Motion-based games for health are subject to a growing body of research and development. In a series of studies, (Uzor and Baillie 2014) have shown that these playful tools can provide a number of benefits compared to traditional instruction by exercise sheets, especially when used to augment unsupervised exercising at home. We summarize these areas to be motivation (to perform repetitive exercises), feedback (regarding the current exercise execution and summarizing developments), and customization (by manual adaptations of automatic adaptivity) (Smeddinck, Herrlich, and Malaka 2015). Such games can be created in a modular fashion, where the specific exercises to be supported are arbitrary, yet require manual effort for a successful implementation. There are thousands of different exercises employed by different practices, thus automated extraction methods could provide a great benefit to this area. Furthermore, the reliable objective assessment of quality of motion, even when supervised by a therapist is a challenge, since inter-rater variance is notably high (Pomeroy et al. 2003). Again, automated, or human computation supported methods could be of great benefit in this area. Both the generation of movements from textual descriptions that are accompanied by images, as well as the validation of movement quality lend themselves as tasks for human computation, since they involve most of the typical strongholds of human computation, including intuitive decisions, aesthetic judgement, contextual reasoning, and embodiment issues (Krause and Smeddnick 2011).
Based on the current state of the art, we set out to establish a human computation based pipeline for extracting validated movements from instruction sheets, with the goal to then explore the potential of further automating the different steps involved in that pipeline, starting with a focus on the step of quality of motion assessment.
Data Collection and Results
At the early stage of the work we developed a Physical Exercise Instruction Sheet Corpus (PEISC) of around 1000 physical exercise instructions drawn from a number of publicly available databases. On the basis of different bodily actions we categorized it into different categories such as standing, seating. For our first case study, we chose five exercises (Table 1) which do not require any additional equipment. Using a Kinect device, we recorded five exercises from seven participants (3 male and 4 female; 15 to 35 years of age, M=25, SD=5). Ten iterations of every exercise were recorded from each participant in a random order. During the recording of those exercises we only provided instruction sheets and asked the participants to perform their interpretation of the exercises without any priming regarding how to perform them. We also collected basic demographic data and after every exercise we collected responses regarding the comprehensibility of the instruction sheets. Analyzing the answers from the questionnaires, we found that the exercise instruction sheets were difficult to understand for some yet seemed easy for others. Furthermore, the same exercise was sometimes performed differently by the participants. With these findings we can claim that instruction sheets are not the optimal way to instruct people to perform exercises.
We have developed 4 different categories of videos from the collected Kinect data (i.e., RGB, Depth, Skeleton, Virtual Reality) and developed a survey application, aiming to crowdsource the assessment of the quality of exercise executions and to determine the best visualization modality for high inter-rater agreement. Following the quality assessment survey, we provided a questionnaire to gather comparative responses regarding the best visualization type, movement quality of different body parts during the performance of the exercises, and to acquire additional demographic data.
In the survey, we asked the participants to read each instruction sheet followed by asking them to watch the videos of all 7 performances in all 4 categories, where all exercises and categories appeared in a random order on the screen. The participants' task was to delete the worst one and repeat that procedure until the best one remained. In total, 20 participants took part in the survey and questionnaire.
Results
With the help of Kappa statistics (Carletta 1996), we calculated the best performer of all 5 exercises (shown in Table 1) and the best visualization type (displayed in Figure 1).
Conclusion and Future Work
In this paper, we have presented a precursory step towards an automated pipeline to go from texts to virtual motions. Here, we have shown a case study for five different exercises Table 1: Best performer per exercise and the inter-rater agreement on the positioning performed by seven different humans with the help of typical exercise instruction sheets. Using a crowdsourcing approach, we have found that assessing the quality of the performed exercises is not an easy task for humans and that the RGB-type (regular video) of visualization yields the most reliable ratings, which could be expected, since it was the visualization modality that subjects were likely most familiar with. In the future, we will aim at developing an automated system that produces virtual motions based on typical exercise instruction sheets as an input. A follow-up to the study presented in this abstract will help inform the decision on the visualization modality that will be employed for those automatically generated exercise executions for an intermediate judgement of the quality of these generated executions. We are hoping to use insights from that step to, in turn, inform the further automation of the overall pipeline.
Figure 1 :
1Agreement
Assessing agreement on classification tasks: The kappa statistic. J Carletta, Comput. Linguist. 222[Carletta 1996] Carletta, J. 1996. Assessing agreement on classification tasks: The kappa statistic. Comput. Linguist. 22(2):249-254.
Human computation-a new aspect of serious games. Handbook of Research on Serious Games as Educational, Business and Research Tools: Development and Design. M Krause, J Smeddnick, Pomeroy, Clinical rehabilitation. 173Agreement between physiotherapists on quality of movement rated via videotape[Krause and Smeddnick 2011] Krause, M., and Smeddnick, J. 2011. Human computation-a new aspect of serious games. Handbook of Research on Serious Games as Ed- ucational, Business and Research Tools: Development and Design. [Pomeroy et al. 2003] Pomeroy, V.; Pramanik, A.; Sykes, L.; Richards, J.; and Hill, E. 2003. Agreement between physio- therapists on quality of movement rated via videotape. Clin- ical rehabilitation 17(3):264-272.
Exergames for physiotherapy and rehabilitation: A medium-term situated study of motivational aspects and impact on functional reach. Herrlich Smeddinck, J D Smeddinck, M Herrlich, R Malaka, S Uzor, Baillie , L , Proceedings of the 33rd Annual ACM Conference on Human Factors in Computing Systems. the 33rd Annual ACM Conference on Human Factors in Computing SystemsACMProceedings of the SIGCHI Conference on Human Factors in Computing SystemsSmeddinck, Herrlich, and Malaka 2015] Smeddinck, J. D.; Herrlich, M.; and Malaka, R. 2015. Exergames for phys- iotherapy and rehabilitation: A medium-term situated study of motivational aspects and impact on functional reach. In Proceedings of the 33rd Annual ACM Conference on Human Factors in Computing Systems, 4143-4146. ACM. [Uzor and Baillie 2014] Uzor, S., and Baillie, L. 2014. In- vestigating the long-term use of exergames in the home with elderly fallers. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, 2813-2822. ACM.
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"Searching for structural predictors of plasticity in dense active packings",
"Searching for structural predictors of plasticity in dense active packings"
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| [
"Julia A Giannini \nDepartment of Physics\nSyracuse University\n13244SyracuseNew YorkUSA\n\nBioInspired Institute\nSyracuse University\n13244SyracuseNew YorkUSA\n",
"Ethan M Stanifer \nDepartment of Physics\nUniversity of Michigan\n48109Ann ArborMichiganUSA\n",
"M Lisa Manning \nDepartment of Physics\nSyracuse University\n13244SyracuseNew YorkUSA\n\nBioInspired Institute\nSyracuse University\n13244SyracuseNew YorkUSA\n"
]
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"Department of Physics\nSyracuse University\n13244SyracuseNew YorkUSA",
"BioInspired Institute\nSyracuse University\n13244SyracuseNew YorkUSA",
"Department of Physics\nUniversity of Michigan\n48109Ann ArborMichiganUSA",
"Department of Physics\nSyracuse University\n13244SyracuseNew YorkUSA",
"BioInspired Institute\nSyracuse University\n13244SyracuseNew YorkUSA"
]
| []
| In amorphous solids subject to shear or thermal excitation, so-called structural indicators have been developed that predict locations of future plasticity or particle rearrangements. An open question is whether similar tools can be used in dense active materials, but a challenge is that under most circumstances, active systems do not possess well-defined solid reference configurations. We develop a computational model for a dense active crowd attracted to a point of interest, which does permit a mechanically stable reference state in the limit of infinitely persistent motion. Previous work on a similar system suggested that the collective motion of crowds could be predicted by inverting a matrix of time-averaged two-particle correlation functions. Seeking a first-principles understanding of this result, we demonstrate that this active matter system maps directly onto a granular packing in the presence of an external potential, and extend an existing structural indicator based on linear response to predict plasticity in the presence of noisy dynamics. We find that the strong pressure gradient necessitated by the directed activity, as well as a self-generated free boundary, strongly impact the linear response of the system. In low-pressure regions the linearresponse-based indicator is predictive, but it does not work well in the high-pressure interior of our active packings. Our findings motivate and inform future work that could better formulate structure-dynamics predictions in systems with strong pressure gradients. | 10.1039/d1sm01675j | [
"https://arxiv.org/pdf/2111.12848v2.pdf"
]
| 244,709,085 | 2111.12848 | add90432045491e54ecd2c9f04e18ee870fa0a9f |
Searching for structural predictors of plasticity in dense active packings
Julia A Giannini
Department of Physics
Syracuse University
13244SyracuseNew YorkUSA
BioInspired Institute
Syracuse University
13244SyracuseNew YorkUSA
Ethan M Stanifer
Department of Physics
University of Michigan
48109Ann ArborMichiganUSA
M Lisa Manning
Department of Physics
Syracuse University
13244SyracuseNew YorkUSA
BioInspired Institute
Syracuse University
13244SyracuseNew YorkUSA
Searching for structural predictors of plasticity in dense active packings
(Dated: January 31, 2022)
In amorphous solids subject to shear or thermal excitation, so-called structural indicators have been developed that predict locations of future plasticity or particle rearrangements. An open question is whether similar tools can be used in dense active materials, but a challenge is that under most circumstances, active systems do not possess well-defined solid reference configurations. We develop a computational model for a dense active crowd attracted to a point of interest, which does permit a mechanically stable reference state in the limit of infinitely persistent motion. Previous work on a similar system suggested that the collective motion of crowds could be predicted by inverting a matrix of time-averaged two-particle correlation functions. Seeking a first-principles understanding of this result, we demonstrate that this active matter system maps directly onto a granular packing in the presence of an external potential, and extend an existing structural indicator based on linear response to predict plasticity in the presence of noisy dynamics. We find that the strong pressure gradient necessitated by the directed activity, as well as a self-generated free boundary, strongly impact the linear response of the system. In low-pressure regions the linearresponse-based indicator is predictive, but it does not work well in the high-pressure interior of our active packings. Our findings motivate and inform future work that could better formulate structure-dynamics predictions in systems with strong pressure gradients.
In amorphous solids subject to shear or thermal excitation, so-called structural indicators have been developed that predict locations of future plasticity or particle rearrangements. An open question is whether similar tools can be used in dense active materials, but a challenge is that under most circumstances, active systems do not possess well-defined solid reference configurations. We develop a computational model for a dense active crowd attracted to a point of interest, which does permit a mechanically stable reference state in the limit of infinitely persistent motion. Previous work on a similar system suggested that the collective motion of crowds could be predicted by inverting a matrix of time-averaged two-particle correlation functions. Seeking a first-principles understanding of this result, we demonstrate that this active matter system maps directly onto a granular packing in the presence of an external potential, and extend an existing structural indicator based on linear response to predict plasticity in the presence of noisy dynamics. We find that the strong pressure gradient necessitated by the directed activity, as well as a self-generated free boundary, strongly impact the linear response of the system. In low-pressure regions the linearresponse-based indicator is predictive, but it does not work well in the high-pressure interior of our active packings. Our findings motivate and inform future work that could better formulate structure-dynamics predictions in systems with strong pressure gradients.
I. INTRODUCTION
Dense amorphous solids -including powders, granular systems, foams, structural glasses, and colloidal assemblies -are ubiquitous in nature [1][2][3]. These materials exhibit unique mechanical and dynamic features that emanate from their disordered structure [4,5]. Similarly, in some cases, active matter comprised of self-propelled agents remains disordered as it achieves very high densities; examples of such systems include bacterial assemblies [6], cellular tissues [7,8], and groups of animals [9,10]. Although active matter is relatively well-studied at low and intermediate densities [11,12], an important open question is whether the emergent mechanical properties of dense active matter are similar to, or different from, their non-active counterparts [13][14][15][16].
One starting point for answering this question is to analyze properties of inherent or reference states of the amorphous solid that underlies a given dense active material [17][18][19][20]. In this framework, one considers how structural information from a static snapshot of the system, usually the positions and sizes of individual particles and the potential energy with which they interact, can provide insight into dynamic yielding behavior when the system is subject to external deformation or activity [21][22][23][24]. A large body of work explores structure-dynamics predictions in sheared, athermal disordered solids. In a recent article (Ref. 25 In this work, we focus on linear-response-based structural metrics, which utilize the spectrum of vibrational modes of a solid computed in the harmonic approximation of the total potential energy. As shown in Ref. 25 and other works, these metrics are surprisingly good at identifying soft spots, or localized microstructural instabilities, in sheared amorphous solids [26][27][28][29]. A primary goal of our work is to extend this class of structural indicators to active solids. Thus, a first challenge is to identify an active material with a time-invariant, well-defined reference state, as most active systems are "self-shearing" and not mechanically stable [22,30]. Here, we consider assemblies of active particles that are infinitely persistent in a radial direction towards a central point of interest. As we will show, the symmetry of this biased activity permits a force-balanced steady state, and allows us to exactly map the relevant non-Hamiltonian self-propulsion forces onto an effective external potential. This choice also necessarily introduces a strong interaction pressure gradient and a self-generated free boundary as depicted in Fig. 1.
Previously, a similar geometry and set of dynamical equations was studied by Bottinelli and Silverberg in a computational model for dense human crowds [31,32]. Their study sought to predict density waves or localized excitations that are thought to correspond to dangerous collective behaviors such as trampling or crowd-crush events. Predicting these phenomena from basic structural information or dynamics is an important first step toward avoiding or controlling crowd disasters. Toward this goal, the authors adopted techniques that have previously been deployed in colloidal systems to estimate the system's linear response [33][34][35], where the dynami- Properties of dense packings of active particles directed towards a central point of interest. At mechanical equilibrium, these packings feature a gradient in interaction pressure that has azimuthal symmetry. The main panel shows the mean per-particle interaction pressure as a function of radius from the edge of the system for an ensemble of 25 packings with N = 2048 and v0 = 0.5. The shaded region shows the standard deviation of the interaction pressure at different locations in the packings. The inset shows an example N = 2048, v0 = 0.5 system with particles colored by the magnitude of their interaction pressures.
cal matrix is estimated from long-time averages of twoparticle correlation functions. In addition to analyzing particle trajectories from simulated crowds, the same authors applied these techniques to video footage of real human crowds and were indeed able to predict wave-like collective motion, albeit over a very short time window [9,36].
A significant challenge associated with this framework, which approximates the linear response of the system, is that the equivalence between the dynamical matrix and two-time correlation functions only holds under certain assumptions: namely, that i) the correlation functions are averaged over long time intervals; ii) the dynamics of the system are thermal; and iii) there are no changes to the underlying contact network during the relevant time intervals. In real crowds or self-propelled particle models, none of these assumptions hold. Therefore, the appropriate analogue of the dynamical matrix in systems whose microscopic details are non-Hamiltonian remains unclear.
Our work is also informed by previous research on thin films and other materials with free surfaces, as we expect that the free boundary alone might alter the mobility or linear response of a disordered packing. For example, a study by Sussman et. al. [37] examines the vibrational modes of unstressed spring networks derived from partially periodic jammed packings with free boundaries, and finds a population of low frequency modes that exhibit an exponential decay in magnitude away from the edges. In contrast, distinct work by Sussman and col-laborators [38] finds that there is a decoupling between structure and dynamics near the edge of glassy thin films, where an attractive interaction generates the free boundary. Specifically, the authors use a machine learning approach to show that there are no special structural features near the edge of the material, even though the mobility is higher there. Taken together, this suggests that there may be some material-dependent subtleties in whether the structure and vibrational properties of a solid predict particle rearrangements near a free boundary.
Here, we build the beginnings of a framework for predicting localized rearrangements in dense active matter. We first demonstrate that "point-of-interest" model systems have well-defined solid reference states, which allow us to map the active forces onto an effective potential that can be encoded in an augmented Hessian or dynamical matrix. Next, we add noisy dynamics to the system to perturb it away from its reference state, and study whether the vibrational spectrum can be used to predict changes in structure. We find that the strong pressure gradients in the system may limit the predictive power of this extended linear response. Ultimately, our results highlight that more sophisticated methods such as nonlinear-response-based structural metrics may be required to identify the microstructural entities that determine the stability of active packings.
II. METHODS
A. Model
We study an active particle model in two dimensions with overdamped dynamics. Stable packings of N discs are formed by evolving the following single-particle equation of motion from a randomized initial state until force balance is reached:˙
r i = 1 Γ F i,int + v 0ni .(1)
Here, r i contains the positional degrees of freedom of particle i, Γ is a viscous damping coefficient set to unity, F i,int is the net interaction force on particle i by its neighbors, v 0 is the magnitude of the self-propulsion velocity (which is the same for all particles),n i is a unit vector pointing in the direction of self propulsion, and( ·) denotes a time derivative. Pairwise repulsive forces between the particles are determined via the Hertzian soft sphere potential:
F ij,int (r ij ) = k Rij 1 − rij Rij 3/2 if r ij < R ij 0 else,(2)
where k is the interaction stiffness constant, R ij = R i + R j is the sum of the radii of particles i and j, and r ij = | r j − r i | is the distance between i and j. We employ 50:50 binary mixtures with a 1:1.4 ratio between the small and large particle radii to discourage crystallization. The direction of the force F ij,int exerted on particle i by j is parallel to the line that connects j's center to i's. In the limit of infinite persistence, in which we form initial reference configurations,n i always points toward a "point of interest" located at the origin in the plane.
In the presence of translational noise, in which we examine active dynamics initialized from each static reference configuration, the single-particle equation of motion is given by˙
r i = 1 Γ F i,int + v 0ni + η i ,(3)
where η i is white noise with zero mean and magnitude σ.
We study dynamics with different levels of noise by examining simulations at different temperatures T = σ 2 Γ 2 . The stochastic differential equations (Eq. 3) are integrated via the velocity Verlet algorithm with a stable timestep determined by examining the relative magnitudes of typical interparticle and self propulsion forces. See Appendix A for more details of our implementation. An example movie of the formation of a static reference configuration and ensuing thermal dynamics is available in the supplemental material.
The dynamics of the polarization directionn i = (cos (θ i ), sin (θ i )) of particle i in our noisy simulations are governed by the angular equation of motion,
θ i = 1 τ ∆θ i ,(4)
where ∆θ i is the (smallest) angular distance betweenn i and the vector that points from i's center to the point of interest in a given simulation time step, and τ is a characteristic turning time set to unity.
In the discussion that follows, we present results from an ensemble of static packings with N ∈ {256, 512, 1024, 2048, 4096} and v 0 ∈ {0.25, 0.5, 1.0}, and corresponding noisy dynamics with T ∈ {0.125, 0.142, 0.165, 0.197, 0.244, 0.320}. For each state point, we consider 25 duplicate simulations. Via a simple toy model which we describe in the next section and in Appendix B, we choose the parameter k such that the maximum packing fraction in the largest configurations does not exceed approximately φ ∼ 1.3. As we will see, while simplistic, this model creates mechanically stable reference configurations in the infinitely persistent limit, and interesting glass-like (heterogeneous) dynamics in the presence of thermal noise.
B. 1D toy model
To obtain estimates for appropriate simulation parameters and gain intuition for the expected steady-state behavior of our self propelled particle model in the limit of infinite persistence, we now consider a simple onedimensional toy model. In a one-dimensional packing ofÑ monodisperse particles with radius R and selfpropulsion velocity v 0 , take particle i = 0 to be fixed at the origin. The other particles lie in the positive half of the number line and are governed by the equation of motionẋ i = 1 Γ j F i,int − v 0 , where variables are defined similarly to above. The particles with i > 0 have persistent velocities toward the origin.
Given the condition for mechanical equilibrium in this toy (and our full) system, that the self-propulsion forces balance the repulsive interparticle forces, we can derive an expression for the typical pair overlap (1 − rij Rij ) as a function of distance from the edge of the packing. Since this overlap is directly related to the local packing fraction, we choose simulation parameters such as the time step and k to satisfy a constraint on the maximum packing fraction of the system, which occurs near the origin. Further, we predict the approximate form of an interaction pressure gradient which reveals the dependence of the static configurations on the simulation parameters N and v 0 . As will be discussed further below, this prediction for the interaction pressure also allows us to form a scaling relation for the stiffness associated with localized excitations in the interior of the active packings. The formulation of this toy model highlights that the distance χ from the free boundary is the most natural variable with which to examine structural gradients in the system. Considering the circular geometry of the packings generated by the full model (see Fig. 1), we can approximate a radial slice of the 2D system using the 1D toy model. Further details regarding these calculations can be found in Appendix B.
C. Linear response and augmented Hessian framework
Linear-response-based structural metrics are computed from curvatures of the potential energy landscape around a metastable minimum. For a material composed of N interacting particles in d dimensions, the total energy U int ( X) is a function of the N d-dimensional vector X representing points in coordinate space. Thus, the curvatures can be characterized by the Hessian, the matrix of second partial derivatives of the potential energy with respect to particle degrees of freedom:
M = ∂ 2 U int ∂ X∂ X .(5)
The dynamical matrix, used to compute the linear response, is computed strictly with respect to deformations from a stable reference configuration. Therefore, it is only well defined if such a stable reference configuration exists. However, when it is defined, the dynamical matrix is equivalent to the Hessian as derivatives with respect to particle positions and those with respect to deformations are identical. The eigenvectors and eigenvlaues of the Hessian constitute the spectrum of vibrational modes and associated stiffnesses of a solid if all par-ticle masses are unity. Previous work on the mechanics of sheared athermal amorphous solids has demonstrated that a low-frequency population of these harmonic eigenmodes become quasi-localized under certain conditions, featuring a disordered core of large putative displacements on tens of particles decorated by a quadrupolar field which decays in magnitude as r −(d−1) . These excitations are thus termed quasi-localized modes (QLMs), and identify glassy defects that become unstable under applied shear, generating structural rearrangements and non-affine motion [4,24,28]. For our initial "point-of-interest" crowd simulations, the self-propulsion forces of the particles are infinitely persistent in the radial direction. Thus, there is an extra contribution to the total energy of the system which is exactly equivalent to a constant force spring potential pulling the particles toward the origin:
U ext ( X) = Γv 0 i r i ,(6)
where r i is the distance between particle i and the origin. Therefore, we compute an "augmented Hessian", where the energy has the usual contributions from interparticle interactions in addition to those from activity, which occur on the on-diagonal entries of the matrix:
M aug = ∂ 2 (U ext ( X) + U int ( X)) ∂ X∂ X .(7)
See Appendix C for details. In contrast to methods that probe the linear response and stability of active particle packings using approximations of the Hessian, our augmented Hessian framework is exact and requires only a snapshot of a static reference configuration.
D. Static quantities
Next, we describe two metrics for characterizing the static structure of the stable packings derived from the infinitely persistent limit of our self propelled particle model, interaction pressure and vibrability. Interaction pressure quantifies the distribution of forces in the active solid using a well-established Irving-Kirkwood description of the stress tensor [39,40]. Vibrability is a linearresponse-based structural metric that is used to quantify the propensity for local regions of the solid to deform under external deformation or active forcing [25,27,41]. The interaction pressure on particle i is given by the trace σ αα int of the interaction stress tensor whose components are a sum over the repulsive forces generated between i and its neighbors:
σ αβ int = 1 V i ij F α ij r β ij ,(8)
where the sum is over (unique) neighbors of i, F α ij is the α component of the force of j on i, r β ij is the β component of the distance vector pointing from j to i, and V i is the volume associated with i in a radical Voronoi tessellation of the system [42]. Since the Voronoi volumes of particles on the free boundary of the system are unbounded, they are excluded in the results that follow. Vibrability was first defined in Ref. 41 and uses the vibrational spectrum of the Hessian of a jammed packing to describe the susceptibility of particles to excitation and rearrangement. The vibrability of particle i is given by
Ψ i = dN −d l 1 ω 2 l | ψ l,i | 2 ,(9)
where the sum is over nonzero vibrational modes of the Hessian, ω l is the frequency of mode l, and | ψ l,i | 2 is the squared magnitude of the polarization of particle i in mode l. It was shown in Refs. 27 and 25 that vibrability is a good predictor of localized plastic rearrangements in sheared athermal computer glasses. In the augmented Hessian framework, we compute vibrability as in Eq. 9, but take the sum over the dN − (d + 1) nontrivial vibrational modes (as we discuss below) of the system.
E. Dynamic quantities
We next explore the connection between the static structure of our active packings and their dynamics under small amounts of translational noise. In sheared amorphous solids at zero temperature, it is well-established that a population of microstructural defects are directly spatially correlated with future plastic deformation [21,25,29,43,44]. In contrast, in thermalized or active glasses it is generally difficult to demonstrate such a direct spatial correlation, except in non-molecular systems where the thermal fluctuations can be vanishingly small [34]. This is not unexpected; given a large population of underlying defects, various subsets of that population can be excited by thermal fluctuations or active forcing at any given time. Thus, resulting rearrangements of unstable regions occur sporadically, and so at any given time point regions with high mobility do not necessarily correlate strongly with structural indicator fields. To address this challenge, Schoenholz and collaborators [5,38,[45][46][47] have developed a method that searches for structure-dynamics correlations by analyzing whether an indicator of structural softness defines a set of energy barriers that accurately predict the rate of rearrangements.
We adopt this methodology here, analyzing particle rearrangement probabilities as a function of temperature T and the structural indicator vibrability Ψ (Eq. 9). As in previous work [45,[48][49][50], we use a hop indicator to identify rearranging regions of the system. The indicator at a given time t is computed directly from particle trajectory information with respect to two time intervals A = [t − t R /2, t] and B = [t, t + t R /2]. We take t R = 10 in simulation time units, consistent with the work of Refs. 5, 38, 48 which chose t R ∼ 10 to correspond to the typical time taken for the system to complete a rearrangement in their simulations of Lennard-Jones polymer and bidisperse Kob-Andersen glasses. We have verified this choice independently by examining distributions of rearrangement times (determined as described below) in a representative range of simulations, where t R ∼ 10 constituted a reasonable upper bound for rearrangement time. Thus, p i,hop (t) for particle i at time t is given by:
p i,hop (t) = ( r i − r i B ) 2 A ( r i − r i A ) 2 B ,(10)
where · A and · B denote time averages over the specified intervals. We identify particle rearrangement events at the locations/times in which the hop indicator exceeds a threshold value, p thresh = 0.2. Since we seek to identify rearrangements that result in irreversible structural changes, this threshold was chosen such that, for a representative set of example noisy simulations, the contact networks vary nontrivially for inherent structures computed with respect to configurations directly preceding and succeeding the times where p thresh is crossed.
In thermal systems, one expects that rearrangement rates are an Arrhenius function of energy barrier heights:
P R (S, T ) = P 0 (S) exp − E(S) T ,(11)
where T is temperature, P 0 is a rearrangement attempt frequency, and E is the energy barrier to rearrangement. Both P 0 and E are generically functions of the structural indicator field S. In Ref. 45, S is taken to be a machine-learning-derived softness field and the authors demonstrate that, after segmenting the system into bins of constant S, the dynamics are indeed Arrhenius with energy barriers that scale linearly with softness. In this work, we take S to be the vibrability field, Ψ. Consistent with Ref. 45 and related works, in the discussion that follows we take an Arrhenius relationship between P R (Ψ) and 1/T in a given region to indicate that vibrability has successfully estimated the corresponding rearrangement energy barrier.
Additionally, since there are strong spatial gradients in hop indicator (as we demonstrate below) in our active packings, we compute P R (Ψ) by averaging particle rearrangement counts over multiple timesteps:
P R (Ψ) = N R (Ψ) N (Ψ) · ∆t R ,(12)
where N R is the number of rearranging particles with vibrability Ψ in the time interval ∆t R , N (Ψ) is the number of particles with vibrability Ψ, and ∆t R is the time between rearrangements of particles with vibrability Ψ. ln(P R (Ψ)) measurements are computed and averaged for each selected value of Ψ (computed from the appropriate static reference configuration) over the duration of each noisy simulation and over duplicate simulations with the same parameter (N , v 0 , and T ) choices. To study the material properties and stability of our ensemble of reference configurations, we first examine structural features and gradients. First, we computed the interaction pressure σ αα int as a function of distance from the exterior of the packings. The 1D toy model introduced above and detailed in Appendix B predicts that interaction pressure should increase monotonically with distance from the exterior of the system, χ. The toy model further predicts that the scaling of interaction pressure with χ √ N R should be approximately independent of simulation parameter choice, N and v 0 , when rescaled by the quantity Γv 0 √ N R (where R ≈ 1.2 is the average particle radius and √ N R is a good estimate for the radius of the packings). Thus, we definẽ Fig. 2, we show the mean interaction pressure as a function of χ as well as the rescaled mean interaction pressureσ αα int as a function ofχ. As shown in the inset of the figure, these rescaled variables produce an approximate collapse of the data across the parameter range of our ensemble. The collapse is especially effective near the exterior (smallχ) -for largeχ, there is noticeable deviation in the data that depends systematically on system size N . This feature is due to contributions to the interaction pressure σ αα int by the Voronoi volume V (see Eq. 8), which de-creases monotonically withχ. In our definition of the rescaled variablesσ αα int andχ motivated by the 1D toy model detailed in Appendix B, these contributions from the Voronoi volume are neglected for simplicity. Inspired by studies that utilize linear-response-related metrics to form structure-dynamics predictions as discussed above, we compute and diagonalize the augmented Hessian, examining the spectra of vibrational modes of our static reference configurations. We identify one rotational zero mode and two low-frequency trivial translational modes in the spectra of our packings. Typically, the vibrational spectra of solids have d translational modes with zero frequency, but in our framework these modes have finite frequency due to the presence of the external potential (Eq. 6). In Fig. 3, we show an example nontrivial low-frequency augmented Hessian eigenmode that is representative of typical soft modes for these systems. The mode exhibits wave-like motion emanating from the center of the packing as well as increased surface mobility. However, this vibrational mode does not show any characteristics of QLMs, as the putative collective displacement primarily involves a large number of particles on the exterior of the system and lacks the quadrupolar structure that has been shown to represent instabilities in traditional glassy systems [4,24,28]. Through direct examination, we have confirmed that such quasilocalized excitations are not commonly realized in the low-frequency modes of the augmented Hessians of our systems, especially near the interior of the packings. This is consistent with intuition we detail below regarding the stiffness of QLMs as a function of local pressure (see Appendix D). Further, it suggests that vibrability, a weighted sum over the soft modes of the augmented Hessian (Eq. 9), may struggle to predict rearrangement events on the high-pressure interiors of our packings.
χ = χ √ N R andσ αα int = σ αα int Γv0 √ N R . In
The observation that excitations in the low-frequency regime of augmented Hessian spectra are concentrated near the edge of the system is highly reminiscent of previous work on jammed packings with free boundaries (Ref. 37). Since the focus of our study is primarily to identify localized modes that predict plastic rearrangement in a specific class of modeled active solids, we do not develop such in-depth mode analysis here. However, we emphasize that our model differs significantly from previously analyzed systems, as our packings feature strong pressure gradients and those in Ref. 37 have homogeneous overall pressure. We explore the decay in vibrational magnitude exhibited in the low-frequency modes of the augmented Hessian in Appendix E.
We next use our augmented Hessian spectra to develop structural indicator fields. Fig. 4b shows a static configuration with N = 2048 and v 0 = 0.5 where each particle is colored by its vibrability, as defined by Eq 9. Similar to the low frequency vibrational modes themselves, the vibrability is large on the exterior and decreases quickly approaching the center of the packing. This trend is also depicted in the inset of Fig. 4a where we show the mean vibrability as a function ofχ for our ensemble of packings.
Each vibrability profile reaches a plateau value at largẽ χ in the interior. We hypothesize that this plateau value, Ψ plat , is dominated by contributions from localized excitations whose vibrational frequencies depend on local pressure and thus on the simulation parameters N and v 0 . By combining the prediction for the pressure from the 1D toy model detailed in Appendix B with a scaling relation for the stiffness of QLMs as a function of local pressure, we are able to generate a prediction for the vibrability of QLMs near the center (χ ∼ 1) of the packings, Ψ QLM , as a function of N and v 0 . Details of this argument are discussed in Appendix D.
When the data are rescaled according to this prediction, as shown in the main panel of Fig. 4a, the vibrabilty profiles exhibit an approximate collapse near the center of the packings (χ ∼ 1). On the exterior, there is more significant variation among the curves, which agrees with the interpretation that the vibrability in this regime has significant contributions from low-frequency, spatially decaying surface vibrations as depicted in Fig. 3.
Additionally, the large-χ collapse is poorer for systems with the largest local pressures (e.g. the dark magenta curve in Fig. 4a corresponding to systems with N = 4096 and v 0 = 1.0). To quantify the quality of this collapse as a function of N and v 0 , we plot the measured vibrability plateau values Ψ plat vs. the predicted values Ψ QLM in Fig. 4c (see Appendix D for details). As expected, this analysis highlights deviations of our scaling prediction from the actual vibrability plateau values Ψ plat at the largest values of N and v 0 . These deviations are likely due to increased multi-body interactions and larger local pressure fluctuations (that scale as N 1/2 and v 1 0 similarly to the pressure itself) which are not accounted for in our simple 1D model and vibrability scaling argument. In Appendix F, we confirm that deviations from our scaling predictions are smaller in a system with higher particle stiffness k, consistent with this expectation.
B. Dynamics in presence of translational noise
Next, we examine the dynamics that result when thermal noise is added to the particle trajectories. Starting from the stable reference configurations discussed above, noise is added with magnitude controlled by the temperature T ∈ {0.125, 0.142, 0.165, 0.197, 0.244, 0.320}. Using the hop indicator (Eq. 10 above) as a measure of particle mobility, we compare the rearrangement dynamics at different locations in the packings. As shown in Fig. 5, for our ensemble of systems with N = 2048 and v 0 = 0.5, there is a dramatic decrease in the mean hop indicator as a function ofχ for all temperatures. These results are similar for other parameter (N and v 0 ) choices. This dynamic profile is reminiscent of the vibrability profiles presented above, which predict increased mobility near the edge of the packings.
Even though the majority of rearrangement events occur on or near the exterior of the packings, it is important to note that particles on the interior of the system do undergo occasional rearrangement. This can be seen from the snapshot in the inset of Fig. 5, where particles are colored grey if p i,hop < p thresh , and colored according to the magnitude of the hop indicator if p i,hop ≥ p thresh . Similarly, the grey curve in Fig. 5 shows the maxiumum of the hop indicator in different regions of the lowest temperature systems, which consistently exceeds p thresh .
The data we present here highlight important similarities to and differences from results regarding the structure and dynamics glassy thin films. Arrhenius plot for ensemble of packings with N = 2048, v0 = 0.5, and the full range of temperatures we examined. The average log of the rearrangement probability in different bins of constant vibrability is plotted as a function of inverse temperature. There are 10 bins of vibrability ranging from Ψ ≈ 0.18 (bottom) to Ψ ≈ 0.73 (top). For vibrability bins where the relationship is well approximated by a linear fit (determined by an associated chi-squared value of less than 0.05 ) (solid lines, cool colors), the rearrangement dynamics are Arrhenius, whereas bins exhibiting nonlinear trends represent sub-Arrhenius regions of the system (dashed lines, warm colors).
Lastly, we study rearrangement probabilities in our thermalized packings as a function of temperature and reference configuration vibrability Ψ in order to determine whether rearrangement energy barriers are wellrepresented by Ψ. In Fig. 7, we show the mean of the natural log of the rearrangement probabilities in n bin = 10 bins of approximately constant vibrability (independent of χ) as a function of inverse temperature for our ensemble of systems with N = 2048, v 0 = 0.5, and T ∈ {0.125, 0.142, 0.165, 0.197, 0.244, 0.320}. Strikingly, the behavior is Arrhenius for large values of Ψ 0.22, but sub-Arrhenius for small values of Ψ 0.22. This result is qualitatively independent of N and v 0 and varies significantly from the results of Refs. 5, 38, and 47, which identified Arrhenius behavior for individual values of softness in both bulk, thin film, and active/biological systems. Recalling the static structural gradients (vibrability and interaction pressure) above, we notice that the sub-Arrhenius portions of the system lie in the interior of the packing, where the interaction pressure is high and soft modes are suppressed.
Taken together, our results indicate that dynamic rearrangements in the interior of packings are generally not well-predicted by our augmented Hessian framework and that the vibrability alone is not a good structural indicator in the interior of this system. We note that useful information may still exist, for instance, in local variations in vibrability (which we preliminarily examined and found that it did not correlate well with rearrangement events), but our work suggests that other approaches such as non-linear-response-based metrics will be more fruitful, as discussed below.
IV. DISCUSSION AND CONCLUSIONS
In this work, we studied results from computer simulations of a soft active particle model in two dimensions with directed self-propulsion in the overdamped regime. We analyzed static structures that are formed in the infinitely persistent limit of the activity and found a strong pressure gradient that is consistent with a simple 1D toy model. We then developed an augmented Hessian to capture the active forces in our analysis of the vibrational properties of the system. Further, we used these eigenspectra and a structural indicator, vibrability, to estimate rearrangement energy barriers in analogy to previous work on the dynamics of supercooled liquids and sheared amorphous solids. Then, we observed the resulting dynamics when simulations with translational noise are initiated from the static configurations. Similar to other particle-based systems with free boundaries, we measured a gradient in mobility that persists through the depth of the packings and features enhanced mobility near the free boundaries. We found that, near the boundary of the packings, vibrability is a good structural indicator of energy barrier heights and rearrangement probabilities, but it fails to represent these features in the interior of the systems.
This failure is surprising, as previous work by Bottinelli and collaborators [9,31,32,36] suggested that vibrational analyses of real crowds and crowd models, estimated from a matrix of time-averaged two-particle correlation functions, were able to forecast localized rearrangements and wave-like motion. Similary to the results which we present here, in Ref. 31, the authors note that simulated half-circular point-of-interest crowds exhibit approximately linearly increasing pressure approaching a point of interest. However, by examining a small number of low-frequency vibrational modes (∼ 3% of the spectrum) derived from an approximation of the Hessian, Bottinelli et. al. identified localized soft regions on the interior of their packings which directly spatially correlated with increased noise-induced particle mobility. Distinctly, our results suggest that such correlations are quite difficult to draw from linear-response-based analyses in regions of high local pressure.
Our analysis suggests an updated interpretation of the results of Refs. 31 and 36. Recently, it has been shown in both theoretical works and experiments that the dynamics of active systems do not obey the fluctuationdissipation theorem (FDT) a priori [51][52][53][54]. Furthermore, to construct the correlation functions necessary to approximate the Hessian in the framework of Bottinelli and coworkers, one must indeed time-average over dynamics where the contact network underlying the system has changed; thus, averages are taken over multiple metastable states. As we outlined in Sec. I, work by Henkes et. al. [17] suggests that this approximation method only holds under specific conditions regarding the dynamics of a system (that they satisfy FDT) and the existence of a well-defined, time-invariant solid reference state underlying its structure. When these conditions are not met, the correspondence of the approximation to the real linear response of the system might actually be quite poor.
Further, our study demonstrates that the real (augmented) Hessian cannot predict rearrangements in high pressure regions. Therefore, we speculate that the timeaveraged approximation picks up dynamic features that are not present in the exact Hessian itself, and that these features are important for the predictive capability of the method of Bottinelli et. al.. Future work might further compare these approaches to enhance our overall forecasting capability surrounding the structure and dynamics of complex active solids. In fact, the model examined in our study would be appropriate for a direct comparison between these approximate and exact approaches.
Additionally, we note that even though the choice of the simple soft sphere model described above (and in Ref. 31) is rather artificial and may not accurately represent the interactions in real human crowds, these studies serve as important initial explorations of the connection between structure and dynamics in active solids. If our framework is to be used to understand and control the behavior of real crowds, more consideration should be given to determining "effective potentials" that might govern pairwise interactions between human beings as well as interactions between humans and external stimuli (such walls, points of interest, and other environmental factors) [55,56].
Although our work here focused on "point-of-interest" active crowds, since the directed activity can be mapped onto an external potential, our observations are likely relevant to other classes of systems with pressure gradients and self-generated boundaries. For example, particle aggregates formed under microgravity conditions exhibit a spherical profile, gradients in density, and a free boundary [57,58]. Similarly, recent studies investigating the relaxation of active colloidal glasses attained sedimentation by inclining the experimental set-up at a small angle. As a result of this geometry, Klongvessa et. al. measured a gradient in density at all levels of particle activity and resultant gradients in mobility [59,60] Additionally, a number of works study granular shear flow and shear banding in cylindrical Couette-like geometries under varying gravitational strength and/or confining pressure [61,62]. Experimental and simulated systems with this set-up exhibit localized particle rearrangements in the presence of density heterogeneity. Further, experiments on particulate systems that are driven by external magnetic fields or vibrations exhibit enhanced surface mobility and glassy dynamics characterized by the coexistence of populations of particles with arrested dynamics and those that undergo large displacements via occasional neighbor exchanges (referred to as "dynamical heterogeneity" in some literature) [62][63][64]. Lastly, we note that the structure and dynamics of sand and grain piles are largely dominated by the presence of pressure gradients [65,66].
A relevant question to our discussion is whether facilitated dynamics occur in systems with free boundaries, where enhanced mobility and frequent structural changes on the exterior of the packings could facilitate other nearby rearrangements that propagate in toward the center over time. In Ref. 38, Sussman et. al. measure a softness propagator which suggests that facilitation does not explain enhanced surface mobility in glassy thin films. A similar analysis might be interesting in systems that also have strong pressure gradients.
In general, our results suggest that an augmented Hessian framework could be directly applied to forming structure-dynamics predictions in any solid-like system for which one can i) define suitable a reference configuration and ii) write down a twice-differentiable augmented potential energy that completely captures the characteristics of any internally-generated active forces or external applied fields. Still, as we have shown, there are materialdependent subtleties that effect the predictive power of our technique such as the influence of global pressure gradients and boundary conditions on material stability. While most analyses involving linear-response-based structural metrics have been applied to mechanically stable systems where the Hessian is positive-definite, recent work investigating avalanche dynamics in sheared amorphous solids suggests that Hessians describing unstable systems may also be useful for predicting dynamics [67].
Last, we note that our methods are likely applicable to systems with different types of noise. While we focused here on a system with translational (additive) noise -the stochastic term η i in the equation for particle positions, Eq. 3 -many studies of active matter focuses on systems with rotational (multiplicative) noise, where a stochastic term is instead added to the angular dynamics, Eq. 4. In the absence of interactions, such dynamics generate particles that execute persistent random walks. We have performed some preliminary simulations indicating that the dynamics in dense active crowd simulations with rotational noise are remarkably similar to those presented here for systems translational noise, especially when the noise magnitude is not too large. This suggests that our methods may be used to analyze systems with finite persistence times, which provides an interesting avenue for future work.
Given our observation that linear-response-based structural indicators fail in systems with strong pressure gradients, an obvious next question is how to formulate a better-performing predictive framework. In a number of recent works, a class of novel non-linear-response-based structural indicators have been constructed that address many of the shortcomings of simple linear-response-based metrics [4,24,25,[68][69][70]. Namely, these so-called nonlinear plastic modes (NPMs) and their approximations have been shown to be robust representations of QLMs, the microstructural entities that control rearrangements in disordered solids. Importantly, these methods can quantify the asymmetry of the energy landscape. Thus, even if a mode has very high curvature -so that the mode does not appear in the low-frequency harmonic spectrum -it can still have a low energy barrier provided the mode is highly asymmetric.
Therefore, NPMs are a very promising future avenue for constructing a non-linear-response-based structural metric that successfully predicts rearrangements in systems with gradients in interaction pressure. As we detail in Appendix D, a simple scaling argument can be constructed which suggests that the stiffness of rearrangement-inducing excitations (QLMs), increases quickly with local pressure. This provides a potential explanation as to why vibrability derived from the augmented Hessian is not sensitive to QLMs that exist in the interior of our active packings, and highlights why NPMs are promising. Alternate methods for computing structural indicators could include machine learning approaches, where it will be important to determine how best to handle the strong gradients in pressure during the supervised learning phase. Overall, our work has elucidated that structural indicators for systems with pressure gradients should not be based on linear response alone. wrote the original draft of the manuscript. All authors contributed to reviewing and editing the manuscript.
CONFLICTS OF INTEREST
There are no conflicts to declare. The equations of motion described above for the dynamics of our self propelled particle model in the limit of infinite persistence and in the presence of translational noise (Eqs. 1 and 3) were integrated using the velocity Verlet method. To ensure numerical stability, during the formation of static reference configurations (in the absence of noise), we used a variable timestep proportional to the maximum unbalanced force in the system. These static simulations were run until the maximum unbalanced force reached a threshold of | F unbalanced | < 10 −8 . In the noisy simulations, random numbers were drawn from a Gaussian with zero mean and unit variance, and rescaled by the variance of the noise, proportional to σ √ dt where dt = 10 −3 is the simulation time step, which was held constant [71,72]. See the description of the 1D toy model below for a description of how this time step was chosen. In both static and dynamic simulations, particles whose positions were very close (within the precision of the simulation) to the central point of attraction were pinned to that location to prevent trivial fluctuating dynamics and numerical instability in the case of the static simulations. The simulation was implemented in Python, and just-in-time compiled with Numba [73] to increase performance.
Appendix B: 1D toy model
In this appendix, we closely examine the one dimensional toy model mentioned in the main text which we use to pick appropriate simulation parameters and make general predictions about the structural features of our static packings. The model consists of a one-dimensional packing ofÑ monodisperse particles with radius R and self-propulsion velocity v 0 . The 0th particle is fixed at the origin, and the other particles lie in the positive half of the number line and are governed by the equation of
motionẋ i = 1 Γ j F int i,j − v 0 .
Similarly to our full model, the force between two overlapping particles, F int i,j , is given by
F int i,j = − ∂φ int ∂xi with φ int (x ij ) = k α (1 − xij 2R ) α where x ij = |x j −x i |
is the distance between the particle centers and α = 2.5 for Hertzian soft spheres. Using the condition for force balance in the system, that the interparticle forces must cancel the (cumulative) self-propulsion forces, we obtain an expression for the force F int i−1,i between two adjacent particles:
F int i−1,i = Γv 0 (Ñ − i).(B1)
This expression also highlights that χ, the distance from the exterior of the packing to a particle's center, is a natural variable in which to express structural and dynamic gradients of the system. We will first estimate the maximum overlap in this toy system to identify an appropriate choice for the simulation parameter k. By setting i = 1 and takingÑ to be large, we can approximate the maximum overlap
γ i,j = 1 − xij 2R in the 1D packing, as F int 0,1 = k 2R (γ 0,1 ) α−1 ≈ Γv 0Ñ implies that γ 0,1 ≈ 2RΓv 0Ñ k 1 α−1 .
(B2)
For a packing of crystalline monodisperse spheres of radius R in 2D, the packing fraction φ can be estimated for
γ < 0.5 via φ ≈ π 2 √ 3 1 (1−γ) 2 .
Applying the estimate for γ 0,1 to a circular packing in 2D, we first assume that radius of such a packing is approximately 2Ñ R.Ñ is again the number of particles in an analogous 1D packing representing a radial slice of the 2D system. Thus, the area of the 2D packing is A ≈ 4πÑ 2 R 2 . We can also approximate the 2D packing area by A ≈ N πR 2 where N is the total number of particles in the system. Equating these, we obtainÑ ≈ √ N 2 and γ 0,1 ≈ ( ΓRv0
√ N k ) 1 α−1 .
Using this relation for the overlap and the above approximation of the packing fraction, we choose k such that the maximum packing fraction in the center of the largest system, N = 4096, does not exceed ∼ 1.3, yielding k ∼ 1500. We note here that the above arguments suggest that √ N R is a good approximation for the overall radius of our 2D packings, which motivates the definitions of the rescaled variablesχ,Π, andσ αα int . Next, using the above expressions, we can compute Π i = F i−1,i x i,i−1 as a function of i, which corresponds approximately to individual contributions to interaction pressure as a function of distance from the exterior of the packing in this toy model. Using Eqs. B1 and B2 (slightly modified to express the overlap as a general function of Toy-model analog to Fig. 9 below. Π, the approximated interaction pressure, is shown as a function of χ.
In the inset, Π and χ are rescaled toΠ = i), we have:
Π i = F i−1,i x i−1,i = 2RΓv 0 (Ñ − i) 1 − 2RΓv 0 (Ñ − i) k 1 α−1 . (B3)
Considering simple geometric arguments to transform this into a function of χ, we finally obtain:
Π(χ) = Γv 0 χ 1 − Γv 0 χ k 1 α−1 . (B4)
This function is plotted for a realistic range in χ and for appropriate parameter choices in Fig. 8. For direct comparison to our simulation data, Fig. 9 shows Π (Π) as a function of χ (χ) for the same ensemble of simulations as Fig. 2 in the main text. Clearly, the toy model (including the relevant rescaled variablesχ,Π, andσ αα int ) succeeds in capturing the behavior of the static packings produced in our full 2D model.
The definition of Π in Eq. B4 above differs from Eq. 8 for σ αα int by a factor of the Voronoi volume V associated with a given particle. Since σ αα int is an intensive variable commonly examined in literature studying jammed packings and active systems, we focus on it primarily in the main text, despite the simplicity of Π in our toy model. Thus, here, we estimate particle Voronoi volume in the framework of our 1D toy model by examining typical interparticle distances x i−1,i . We obtain:
V (χ) = R 2 + Γv 0 χ k 1 α−1 −1 − 1 − 2 R χ 1 α−1 . (B5) boundary center FIG. 9.
Simulation data corresponding to the toy-model prediction depicted in Fig. 8 above. Color map as in Fig. 2 in the main text. Π, the approximated interaction pressure, is shown as a function of χ. The inset shows the rescaled approximate pressureΠ as a function ofχ.
The full estimate for the interaction pressure σ in our 1D toy model is thus given by the quotient of Eqs. B4 and B5. This expression for V suggests that in our systems, the Voronoi volume associated with a particle decreases monotonically withχ for all N and v 0 . Further, larger packings achieve smaller overall values of V (due to increased particle density) asχ → 1. Thus, σ αα int grows with N nearχ = 1 faster than Π does. This trend can be seen in Fig. 10, where we show σ(χ) andσ(χ) computed in our toy model, in direct comparison to Fig. 2 in the main text. Toy-model analog to Fig. 2 in the main text. σ, the interaction pressure, is shown as a function of χ. In the inset, σ and χ are rescaled toσ = Last, we choose a stable simulation time step by considering the maximum force generated between two particles in one timestep. In our toy model of monodisperse spheres in one dimension, if the 0th particle and 1st particle satisfy x 01 = 2R at time t, the largest amount of overlap that can be generated via particle 1's self propulsion at time t + dt is given by v0dt 2R . Thus, if we demand that the corresponding force generated by this overlap be less than some multiple of the self-propulsion force, we obtain an inequality for the simulation time step, dt < ( v0γ2R k ) 1/(α−1) ( 2R v0 ). For our choice of simulation parameters and ∼ 1%, this gives a timestep of dt ∼ 10 −3 .
Appendix C: Augmented Hessian
As discussed in the main text, a key result of our work is the formulation of the augmented Hessian framework. By exactly mapping directed self propulsion in our static packings to an external potential, we account for the contributions of active forces to the energy of the system. We compute M aug from this total potential energy, and examine the corresponding soft modes. In this appendix, we compute the augmented Hessian for a general external potential.
Consider the total potential energy of the system, given by U ( X) = U int ( X)+U ext ( X) as described above. Taking the second derivative of U with respect to two degrees of freedom x iα and x jβ (with Latin indices corresponding to particles and Greek indices corresponding to spatial coordinates), we obtain a general expression for an element of the augmented Hessian.
M aug,ijαβ = ∂ 2 U ∂x iα ∂x jβ = ij ∂φ int ∂r ij ∂ 2 r ij ∂x iα ∂x jβ + ∂ 2 φ int ∂ 2 r ij ∂r ij ∂x iα ∂r ij ∂x jβ + k,γ,λ ∂ 2 φ ext ∂x kγ ∂x kλ δ ki δ γα δ kj δ γβ , (C1) where U int = ij φ int (r ij )
is a sum over energies of interacting pairs and U ext = k φ ext (x kγ ) is a sum over external potential energies of individual particles. Given the form of the second term of this equation, it is clear that the external potential only has nonzero contributions to the augmented Hessian on the block diagonal terms of the matrix.
Appendix D: QLM stiffness scaling relation
To provide intuition for the utility of NPMs in future work that seeks to identify localized instabilities in disordered and active packings with unique structural features such as pressure gradients and free boundaries, we formulate a scaling relation for the stiffnesses associated with quasi-localised excitations (QLMs) in systems with varying homogeneous pressure based off of a body of work that studies the micromechanics of computer glasses. In Ref. 70, Gartner et. al. define κ z ≡ M : z z ∼ ω 2 z , the stiffness associated with the mode z. Next, in Refs. 74 and 75, the authors examined local deformations in model glasses and identified a characteristic energy scale associated with quasi-localized excitations, which can be given by ω QLM ∼ ω g ∼ cs ξg where ξ g is a glassy length scale and c s is the shear wave speed that scales with the overall pressure p of jammed packings as c s ∼ p 1/4 . Last, through examining sample-to-sample fluctuations in the shear moduli of computer glasses with short-range attractive potentials, González-López et. al. showed in Ref. 76 that the length scale ξ g changes with pressure as ξ g ∼ p −1/2d where d is the number of spatial dimensions of the glass.
Combining the above, we finally obtain a scaling prediction for ω QLM with pressure, ω QLM ∼ p (d+2)/4d . Thus, κ QLM ∼ p for QLMs in 2D systems with homogeneous pressure. Taken in the context of our results above which suggest that this type of localized instability is difficult to identify in the harmonic approximation for systems with strong pressure gradients, we expect that rearrangements could be more effectively predicted in future studies by searching for modes with very high asymmetry in the potential energy landscape that may be quite stiff relative to typical soft modes. In passing, we note that numerical studies investigating the vibrational modes of glasses approaching the unjamming transition have predicted that the density of QLMs decreases sharply with increasing overall pressure [77]. Now, we use this scaling relation for ω QLM to formulate the prediction for the plateau vibrabilities shown in Fig. 4. Recall Eq. 9 above for the vibrability. Given the spatial features of QLMs, namely that they feature large polarization vectors on a small number of particles, we conclude that the squared polarization magnitudes in the sum for vibrability are of order one for QLMs. Thus, contributions to vibrability by QLMs are dominated by their inverse squared frequencies. Since we showed above that ω QLM ∼ p 1/2 in 2D, we conclude that Ψ QLM ∼ p −1 . Using Eq. B4 as an estimation of the local pressure, we compute Ψ QLM for the appropriate parameters in our model whenχ ∼ 1. Since low-frequency, wave-like vibrational modes decay quickly in magnitude away from the exterior of our packings (see Fig. 3 and Appendix E below), it is likely that Ψ QLM is the most dominant contribution to vibrability in the interior. Note that this prediction is thus not valid for smallχ.
Appendix E: Mode analysis
Similarly to the analysis of Sussman et. al. in Ref. 37, in this appendix, we study the spatial characteristics of low-frequency vibrational modes of the augmented Hessian. Fig. 11 shows the mean squared vibrational mag- FIG. 11. Spatial decay of vibrational magnitudes as a function of disatance from the exterior χ for systems with v0 = 0.5 and varying system size. Color map is similar to that of Fig. 2. The black dashed lines show | ψ| 2 ∼ χ −1 as a guide to the eye.
nitude as a function of χ for systems with v 0 = 0.5 and N ∈ {256, 512, 1024, 2048, 4096}. The average was taken over modes with ω ≤ 0.4 and over simulation duplicates. The results for systems with v 0 = 0.25, 1.0 are very similar. Contrasting the results of Ref. 37, we do not observe an exponential decay in the vibrational magnitude for any of the systems we examined. Rather, it appears that there is a plateau in | ψ| 2 for small χ 4.5, followed by a | ψ| 2 ∼ χ −1 power law decay. Further, we can identify a lengthscale χ * associated with the onset of this χ −1 scaling. For the ensemble of vibrational modes studied here, χ * ∈ (2.0, 4.5) and increases monotonically with N . This analysis is consistent with the results we presented above for the vibrability of our packings, which reaches a plateau for large χ.
Appendix F: Higher-order interactions and role of large particle overlaps In Figs. 4ac of the main text, it is clear that our prediction for the dominant contribution to vibrability near the center of the packings, Ψ QLM , deviates from the measured vibrability plateau values, Ψ plat , for systems with large values of N and v 0 . In this appendix, we study systems with a higher value of inter-particle interaction stiffness than the one shown in the main text. We expect that a higher value of stiffness k will suppress the magnitude of overlaps and higher-order interactions, where more than two particles overlap each other. As these features are inherently present in two-dimensional packings and absent in our one-dimensional toy model for interaction pressure, we hypothesize that these effects contribute to disagreement with our scaling prediction and that increasing k will therefore reduce the observed deviations. Figs. 4ac of the main text reproduced with additional rescaled vibrability data from a small (5-duplicate) ensemble of systems with N = 4096, v0 = 0.5, and k = 3000. The original k = 1500 data (dashed) has the same color map as in Fig. 4a, and the additional k = 3000 data (solid) is plotted in different shades of red corresponding to v0 values ranging from 0.25 (light, bottom) to 1.0 (dark, top). The data in the inset has a similar color map, where star markers correspond to the k = 3000 data. Fig. 12 shows the mean vibrability rescaled by Ψ QLM (see the main text and Appendix D above for details) as a function ofχ for the ensemble of (k = 1500) packings discussed in the main text (dashed lines) as well as a small ensemble with N = 4096, v 0 ∈ {0.25, 0.5, 1.0}, and k = 3000 (solid lines) illustrating a better approximate collapse for largeχ. The inset to Fig. 12 shows Ψ plat vs. Ψ QLM for the same expanded dataset, confirming that indeed deviations from the scaling prediction are smaller in the systems with larger k.
), Richard et. al. compare the performance of several classes of structural indicators in identifying localized instabilities or defects in computer glasses which * [email protected] † [email protected] forecast plastic rearrangements under shear strain.
FIG. 1. Properties of dense packings of active particles directed towards a central point of interest. At mechanical equilibrium, these packings feature a gradient in interaction pressure that has azimuthal symmetry. The main panel shows the mean per-particle interaction pressure as a function of radius from the edge of the system for an ensemble of 25 packings with N = 2048 and v0 = 0.5. The shaded region shows the standard deviation of the interaction pressure at different locations in the packings. The inset shows an example N = 2048, v0 = 0.5 system with particles colored by the magnitude of their interaction pressures.
pressure as a function of distance from the exterior of packings. Individual lines correspond to the mean pressure for 25 packing ensembles with N ∈ {256, 512, 1024, 2048, 4096} and v0 ∈ {0.25, 0.5, 1.0}. Each color represents a different system size N (increasing from green (left) to magenta (right)) and each saturation level represents a different self propulsion velocity v0 (increasing from light (bottom) to dark (top)). Inset: Data rescaled according to the toy model discussed in the main text and Appendix B, demonstrating an approximate collapse for all choices of simulation parameters N and v0.
FIG. 3 .
3Sample low-frequency vibrational mode of the augmented Hessian for a static packing with N = 2048 and v0 = 0.5. The mode shows a large amount of collective motion around the exterior of the system compared to the interior.
FIG. 4 .
4(a) Mean rescaled vibrability as a function of rescaled distance from the exterior of the packings, color scale as in Fig. 2. The inset shows the unscaled data and a sample value of Ψ plat as a guide to the eye. (b) The same configuration as in Fig. 3 with each particle colored by its vibrability. Particles on the interior are much less susceptible to rearrangements than those near the free boundary. (c) Measured vibrability plateau values Ψ plat compared to corresponding predicted plateau values ΨQLM. The dashed line of slope 1 indicates direct proportionality between ΨQLM and Ψ plat . Color scale as in (a), and each marker represents a different system size ranging from N = 4096 (diamonds, left) to N = 256 (circles, right).
Fig 6 shows the average interaction pressure, vibrability, and hop indicator as a function ofχ, similarly to Fig. 1 in Ref. 38. Notably, there are upticks in both hop indicator and vibrability near the free boundary, whereas the thin films studied in boundary center increasing FIG. 5. Mean hop indicator as a function of rescaled distance from the exterior of the 25-duplicate ensemble of packings with N = 2048 and v0 = 0.5. Each color represents a different temperature ranging from T = 0.125 (bottom, dark) to 0.320 (top, light). The horizontal dashed line is placed at p thresh = 0.2 to show the hop indicator threshhold which represents particle rearrangements. The grey line shows the maxiumum hop indicator over time and simulation duplicates for systems with N = 2048, v0 = 0.5, and T = 0.125. The inset shows an example configuration with T = 0.197 colored by threshholded hop indicator. Particles with p hop < p thresh are colored grey and those with p hop ≥ p thresh are colored according to the magnitude of the hop indicator. Notably, the maximum hop indicator profile and the snapshot in the inset show that rearrangements indeed occur throughout the entire depth of the packings. Ref. 38 exhibit only an analogous uptick in hop indicator. The thin film systems also exhibit little-to-no gradient in pressure. These differences highlight the utility of our augmented Hessian framework, and suggest that the pressure gradient in our active packings contributes boundary center FIG. 6. Structural and dynamic gradients as a function of rescaled distance from the exterior of the packings. Mean (rescaled) interaction pressure (dotted, black) and mean vibrability (solid, blue) are measured from the ensemble of static structures with N = 2048 and v0 = 0.5 and mean hop indicator (dashed, red) from the corresponding dynamics for T = 0.197.significantly to their overall mechanical behavior.
FIG. 7 .
7FIG. 7. Arrhenius plot for ensemble of packings with N = 2048, v0 = 0.5, and the full range of temperatures we examined. The average log of the rearrangement probability in different bins of constant vibrability is plotted as a function of inverse temperature. There are 10 bins of vibrability ranging from Ψ ≈ 0.18 (bottom) to Ψ ≈ 0.73 (top). For vibrability bins where the relationship is well approximated by a linear fit (determined by an associated chi-squared value of less than 0.05 ) (solid lines, cool colors), the rearrangement dynamics are Arrhenius, whereas bins exhibiting nonlinear trends represent sub-Arrhenius regions of the system (dashed lines, warm colors).
contributed to the overall conceptualization of this work. J.A.G. and E.M.S. wrote and developed the simulation and analysis code. J.A.G. ran the simulations and prepared the figures. J.A.G. and M.L.M.
FIG. 8. Toy-model analog to Fig. 9 below. Π, the approximated interaction pressure, is shown as a function of χ. In the inset, Π and χ are rescaled toΠ =
according to Eq. B4, showing a collapse close to the free boundary of the system. The color map is the same as that ofFig. 2in the main text.
FIG. 10. Toy-model analog to Fig. 2 in the main text. σ, the interaction pressure, is shown as a function of χ. In the inset, σ and χ are rescaled toσ =
according to Eq. B4, showing a collapse close to the free boundary of the system.
FIG. 12. Figs. 4ac of the main text reproduced with additional rescaled vibrability data from a small (5-duplicate) ensemble of systems with N = 4096, v0 = 0.5, and k = 3000. The original k = 1500 data (dashed) has the same color map as in Fig. 4a, and the additional k = 3000 data (solid) is plotted in different shades of red corresponding to v0 values ranging from 0.25 (light, bottom) to 1.0 (dark, top). The data in the inset has a similar color map, where star markers correspond to the k = 3000 data.
ACKNOWLEDGEMENTSWe acknowledge a fruitful conversation with David Richard suggesting a scaling relationship between quasilocalized modes and the vibrability. We thank
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[
"UNIQUENESS OF THE SCATTERER FOR ELECTROMAGNETIC FIELD WITH ONE INCIDENT PLANE WAVE",
"UNIQUENESS OF THE SCATTERER FOR ELECTROMAGNETIC FIELD WITH ONE INCIDENT PLANE WAVE"
]
| [
"Genqian Liu [email protected] \nDepartment of Mathematics\nBeijing Institute of Technology\nPeople's100081Beijing, Re-public of China\n"
]
| [
"Department of Mathematics\nBeijing Institute of Technology\nPeople's100081Beijing, Re-public of China"
]
| []
| In this paper, we solve a longstanding open problem for determining the shape of an obstacle from the knowledge of the electric (or magnetic) far field pattern for the scattering of time-harmonic electromagnetic field. We show that the electric (or magnetic) far field patten E ∞ (β, α 0 , k 0 ) (or H ∞ (β, α 0 , k 0 )), known for all β ∈ S 2 , where S 2 is the unit sphere in R 3 , α 0 ∈ S 2 is fixed, k 0 > 0 is fixed, determines the obstacle D and the boundary condition on ∂D uniquely. The boundary condition on ∂D is either the perfect conductor or the impedance one.1991 Mathematics Subject Classification. 35P25, 35R30, 78A25, 78A46. | null | [
"https://arxiv.org/pdf/1606.06529v4.pdf"
]
| 119,132,163 | 1606.06529 | bad3594d0535906d3bc38fe3dbacc67474883f20 |
UNIQUENESS OF THE SCATTERER FOR ELECTROMAGNETIC FIELD WITH ONE INCIDENT PLANE WAVE
7 Aug 2017
Genqian Liu [email protected]
Department of Mathematics
Beijing Institute of Technology
People's100081Beijing, Re-public of China
UNIQUENESS OF THE SCATTERER FOR ELECTROMAGNETIC FIELD WITH ONE INCIDENT PLANE WAVE
7 Aug 2017
In this paper, we solve a longstanding open problem for determining the shape of an obstacle from the knowledge of the electric (or magnetic) far field pattern for the scattering of time-harmonic electromagnetic field. We show that the electric (or magnetic) far field patten E ∞ (β, α 0 , k 0 ) (or H ∞ (β, α 0 , k 0 )), known for all β ∈ S 2 , where S 2 is the unit sphere in R 3 , α 0 ∈ S 2 is fixed, k 0 > 0 is fixed, determines the obstacle D and the boundary condition on ∂D uniquely. The boundary condition on ∂D is either the perfect conductor or the impedance one.1991 Mathematics Subject Classification. 35P25, 35R30, 78A25, 78A46.
Introduction
Throughout this paper, D is assumed to be a bounded domain with boundary ∂D of class C 2 and with the connected complement R 3 \D. The time-harmonic electromagnetic waves in the homogeneous isotropic medium R 3 \D must satisfy the reduced Maxwell equations
∇ × E − ikH = 0 in R 3 \D, ∇ × H + ikE = 0 in R 3 \D. (1.1)
Here E and H denote the space dependent parts of the electric field 1 √ ǫ E(x)e −iωt and the magnetic field 1 √ µ H(x)e −iωt respectively, k is the positive wave number given by k = √ ǫµ ω in terms of the frequency ω, the electric permittivity ǫ and the magnetic permeability µ. The scattering of time-harmonic electromagnetic waves by an impenetrable bounded obstacle D in R 3 yields the exterior boundary value inverse scattering problems for the Maxwell equations. Therefore, the total electromagnetic wave E, H is decomposed E = E i + E s , H = H i + H s into the given incident wave E i , H i and the unknown scattered wave E s , H s which is required to satisfy the Silver-Müller radiation condition lim |x|→∞ (H s × x − |x|E s ) = 0 (or lim |x|→∞ (E s × x + |x|H s ) = 0) (1.2) uniformly with respect to all directions. On the boundary ∂D, the total field has to satisfy a boundary condition of the form T (E, H) = 0 on ∂D (1. 3) with the operator T depending on the nature of the scatterer D. For a perfect conductor we have T (E, H) = ν × E, where ν denotes the unit normal to the boundary ∂D pointing out of D, i.e., the total electric field has a vanishing tangential component
ν × E = 0 on ∂D. (1.4)
The scattering by an obstacle that is not perfectly conducting but that does not allow the electromagnetic wave to penetrate deeply into the obstacle is modeled by an impedance boundary condition
ν × (∇ × E) − iψ(ν × E) × ν = 0 on ∂D (1.5)
with a positive function ψ, that is, T (E, H) = ν × (∇ × E) − iψ(ν × E) × ν. It is wellknown that the existence and well-posedness of the Silver-Müller radiating solution for the above exterior boundary value problems of the Maxwell equations have been established by boundary integral equations (see, e.g. [20], [5] or [2]), and the scattering field E s , H s has the asymptotic form
E s (x) = e ik|x| |x| E ∞ x) + O 1 |x| , |x| → ∞, (1.6) H s (x) = e ik|x| |x| H ∞ x) + O 1 |x| , |x| → ∞, (1.7)
uniformly in all directionsx = x |x| where the vector fields E ∞ and H ∞ defined on the unit sphere S 2 are known as the electric far field pattern and magnetic far field patten, respectively. They satisfy H ∞ = ν × E ∞ and ν · E ∞ = ν · H ∞ = 0 with the unit outward normal ν on S 2 . An important cases of incident fields are plane waves
E i (x, α, k, p) = e ikα·x p, H i (x, α, k, p) = e ikα·x (α × p) (1.8)
with propagation direction α ∈ S 2 , wave number k and polarization vector p. The corresponding scattered waves and far field patterns (or scattering amplitudes) are denoted by E s (x, α, k, p), H s (x, α, k, p) and E ∞ ( x |x| , α, k, p), H ∞ ( x |x| , α, k, p), respectively. Because of the linearity of the scattering problem with respect to the incident waves, we see that the scattered waves and the far field patterns are both linear respect to the polarization vector p. Therefore we can write E s (x, α, k, p) as E s (x, α, k)p, and so forth. The scattering amplitudes E ∞ ( x |x| , α, k) and H ∞ ( x |x| , α, k) are 3 by 3 matrices, which are physics quantities and can be measured experimentally. It follows from [4,5] that for smooth bounded obstacles the far field patterns E ∞ (β, α, k) and H ∞ (β, α, k) are analytic matrices of β and α on the unit sphere S 2 . For a fixed α ∈ S 2 , if E ∞ (β, α, k) as a matrix of β is known on an open subset of S 2 , it is uniquely extended to all of S 2 by analyticity. The same is true for H ∞ (β, α, k).
The basic inverse problem in scattering theory is to determine the shape of the scatterer D from a knowledge of the electric far field pattern E ∞ ( x |x| , α, k)p (or the magnetic far field pattern H ∞ ( x |x| , α, k)p) for one or several incident plane waves with incident directions α and polarizations p. The study of inverse scattering problem for electromagnetic wave is of fundamental important to many areas of science and technology, such as radar, sonar, geophysical exploration, medical imaging and nondestructive testing.
Until the 1980's, very little was known concerning the mathematical properties of far field patterns (cf. [5]). However, in the past three decades results have been obtained for the inverse electromagnetic problems. In [5], based on the ideas of Kirsch and Kress [7], D. Colton and R. Kress proved that for perfect conductor, one fixed incident direction α and polarization p, and all wave number contained in some interval 0 < k 1 < k < k 2 < ∞ can determine D. It has been shown by Liu, Yamamoto and Zou [15] that a perfectly conducting polyhedron is uniquely determined by the far field pattern for plane wave incidence with one direction α and two polarizations p 1 and p 2 . D. Colton and R. Kress proved (see [5]) that if D 1 and D 2 are two scatterers with boundary conditions T 1 and T 2 such that for a fixed wave number the far field patterns coincide for all incident directions α, all polarizations p, and all observation directions x |x| , then D 1 = D 2 and T 1 = T 2 . In [5], D. Colton and R. Kress also showed that a ball and its boundary condition (for constant impedance ψ) is uniquely determined by the far field pattern for plane wave incidence with one direction α and p. We refer to [12], [3], [13], [9], [14], [22] for a review of this topic. In the inverse acoustic obstacle scattering (i.e., the Helmholtz equation), by using a completely new technique the author [10] showed that the scattering amplitude for one single incident direction and one wave number uniquely determines the acoustic obstacle.
However, it has been a challenging open problem (see p. 6 of [1] or p. 4 of [12]) that for a fixed wave number k, a fixed incident direction α and a fixed polarization direction p, whether the electric (or magnetic) far field pattern can uniquely determine the general scatterer D and its boundary condition?
In this paper, using a novel idea and an elementary means by discussing all possible positions of two scatterers and applying the electric (or magnetic) eigenvalue theory, we solve the above inverse scattering problem for the electromagnetic field. Our main result is the following: Theorem 1.1. Assume that D 1 and D 2 are two scatterers with boundary condition T 1 and T 2 such that for a fixed wave number k 0 , a fixed incident direction α 0 , and a fixed polarization p 0 the electric (or magnetic) far field patten of both scatterers coincide (i.e.,
E ∞ 1 (β, α 0 , k 0 )p 0 = E ∞ 2 (β, α 0 , k 0 )p 0 (or H ∞ 1 (β, α 0 , k 0 )p 0 = H ∞ 2 (β, α 0 , k 0 )p 0 ) for all β in an open subset of S 2 ). Then D 1 = D 2 and T 1 = T 2 .
Let us point out that our method is completely new. In particular, we subtlety apply three basic tools: the property of the eigenfunction in a bounded domain, the interior analyticity of the solutions for the time-harmonic Maxwell equations, and the asymptotic property of the scattered waves as |x| → ∞.
Remark 1.2.
For the Maxwell equations, we only need to be concerned with the study of three-dimensional inverse scattering problems since the two-dimensional case can be reduced to the two-dimensional Helmholtz equation that has been solved by the author in [10]. This paper is organized as follows. In Section 2, we present some known results. In Section 3, we prove a key lemma (Lemma 3.1) which shows that the electric (or magnetic) far field pattern determines the total electromagnetic scattering wave in the unbounded connected component of R 3 \ (D 1 ∪ D 2 ). Section 4 is devoted to the proof of the main result.
Preliminaries
Let g(x) be a real-valued function defined in an open set Ω in R n . For y ∈ Ω we call g real analytic at y if there exist a γ ∈ R 1 and a neighborhood U of y (all depending on y) such that
g(x) = γ a γ (x − y) γ for all x ∈ U , where γ = (γ 1 , · · · , γ n ) is a multi-index (a set of non-negative integers), |γ| = n j=1 γ j , and (x − y) γ = (x 1 − y 1 ) γ1 · · · (x n − y n ) γn .
We say g is real analytic in Ω, if g is real analytic at each y ∈ Ω.
Lemma 2.1 (Unique continuation of real analytic function, see, for example, p. 65 of [8]).
Let Ω be a connected open set in R n , and let g be real analytic in Ω. Then g is determined uniquely in Ω by its values in any nonempty open subset of Ω.
Lemma 2.2 (The interior real analyticity of the solutions for real analytic elliptic equations, see [16], [17], [18] or [19]). Let Ω ⊂ R n be a bounded domain, and let L be a strongly elliptic linear differential operator of order 2m
Lu = |γ|≤2m a γ (x)D γ u(x).
If the coefficients a γ (x), |γ| ≤ 2m, and the right-hand side f (x) of the equation Lu = f are real analytic with respect to x = (x 1 , · · · , x n ) in the domain Ω, then any solution u of this equation is also real analytic in Ω.
C 2 . Let E, H ∈ C 1 (R 3 \D) ∩ C(R 3 \ D) be a solution to the Maxwell equations ∇ × E − ikH = 0, ∇ × H + ikE = 0 in R 3 \D
satisfying the Silver-Müller radiation conditions (1.2). Then the radiating solutions E, H to the Maxwell equations automatically satisfy
E(x) = O( 1 |x| ), H(x) = O( 1 |x| ), |x| → ∞, (2.1)
uniformly for all directions x |x| .
Lemma 2.6 (see p. 198 of [5]). Let E, H be a solution to the Maxwell equations in R 3 satisfying the Silver-Müller radiation conditions. Then E, H must vanish identically in R 3 .
∇ × E − ikH = 0 in R 3 \D, ∇ × H + ikE = 0 in R 3 \D, E = E i + E s , H = H i + H s in R 3 \D,
where E i and H i are defined in (1.8), and E s and H s satisfy the Silver-Müller radiation condition, such that
ν × E = ν × H = 0 on Γ. (2.2) Then E ≡ 0 and H ≡ 0 in R 3 \D.
Let Ω ⊂ R 3 be a bounded domain of class C 2 . Consider the following three boundary-value problems:
∇ × E − ikH = 0, ∇ × H + ikE = 0 in Ω, ν × E = 0 on ∂Ω, (2.3) ∆E + τ 2 E = 0, ν × E = 0, ∇ · E = 0 on ∂Ω, (2.4) ∆H + η 2 H = 0, ((∇ × H) × ν) × ν = 0, ν · H = 0 on ∂Ω. (2.5)
The problems (2.3), (2.4) and (2.5) are said to be the Maxwell, electric and magnetic eigenvalue problems, respectively. It is well-known (see [21] or p. 125 of [4]) that there exists for each of the Maxwell, electric and magnetic problems a countable set of positive wave numbers k (respectively, τ , η) called eigenvalues, accumulating only at infinity for which the homogeneous problem has nontrivial solutions. Moreover (see also p. 125 of [4] Lemma 2.8. Let Ω ⊂ R 3 be a bounded domain with piecewise C 2 -smooth boundary. Let E (respectively, H) be the electric (respectively, magnetic) eigen-field in Ω corresponding to the electric (respectively, magnetic) eigenvalue τ (respectively, η). Then E (respectively, H) is real analytic vector-field in Ω.
Proof. Since the electric (respectively, magnetic) eigenvalue τ (respectively, η) is positive number, and since the electric (respectively, magnetic) eigen-equation (2.4) (respectively, (2.5)) is also real analytic in Ω with real vector-valued boundary conditions. The desired result immediately follows from Lemma 2.2.
The following Lemma will be needed in the proof of Lemma 3.1.
Lemma 2.9 (Rellich's lemma, see p. 33 of [5] or p. 178 of [23]). Assume the bounded domain D is the open complement of an unbounded domain and let v ∈ C 2 (R 3 \D) be a solution to the Helmholtz equation (∆ + k 2 )v = 0 satisfying ∂Br(0) |v| 2 ds → 0 as r → ∞, where ∂B r (0) is the sphere {x ∈ R 3 |x| = r}. Then v(x) = 0 for x ∈ R 3 \D.
Uniqueness of scattering solutions in the exterior of two scatterers
We consider the scattering of electromagnetic plane waves with incident direction α ∈ S 2 and polarization vector p as described by the matrices E i (x, α, k) and H i (x, α, k) defined by
E i (x, α, k)p := e ikα·x p, H i (x, α, k)p := e ikα·x (α × p).
Let D j be a bounded domain in R 3 with a connected boundary ∂D j of class C 2 (j = 1, 2). Let E j (x, α, k)p, H j (x, α, k)p be the solution of the scattering problem in R 3 \D j , i.e.,
E j (x, α, k)p := E i (x, α, k)p + E s j (x, α, k)p, H j (x, α, k)p := H i (x, α, k)p + H s j (x, α, k)p, j = 1, 2 satisfy the Maxwell equations ∇ × E j − ikH j = 0, ∇ × H j + ikE j = 0 in R 3 \D j , E j = e ikα·x p + E s j , H j = e ikα·x (α × p) + H s j in R 3 \D j T (E j , H j ) = 0 on ∂D j (3.1) and H s j × x |x| − E s j = o( 1 |x| ), E s j × x |x| + H s j = o( 1 |x| ), as |x| → ∞
uniformly for all direction x |x| . As pointed out in Section 1, we can write
E j (x, α, k)p = e ikα·x p + e ik|x| |x| E ∞ j (β, α, k)p + O( 1 |x| 2 ), (3.2) as |x| → ∞, β = x |x| , H j (x, α, k)p = e ikα·x (α × p) + e ik|x| |x| H ∞ j (β, α, k)p + O( 1 |x| 2 ) (3.3)
as |x| → ∞, β = x |x| , where E ∞ j (β, α, k)p and H ∞ j (β, α, k)p are the electric and magnetic far field patterns for the exterior domains R 3 \D j , j = 1, 2 with polarization p, respectively. Now, we have the following basic lemma:
Lemma 3.1. Let E j (x, α 0 , k 0 )p 0 , H j (x, α 0 , k 0 )p 0 be the solution of the scattering problem for Maxwell equations in R 3 \D j (j = 1, 2). If E ∞ 1 (β, α 0 , k 0 )p 0 = E ∞ 2 (β, α 0 , k 0 )p 0 (or H ∞ 1 (β, α 0 , k 0 )p 0 = H ∞ 2 (β, α 0 , k 0 )p 0 ) for all β = x |x| ∈ S 2 ,
a fixed α 0 ∈ S 2 , a fixed k 0 ∈ R 1 and a fixed p 0 ∈ R 3 , then
E 2 (x, α 0 , k 0 )p 0 = E 1 (x, α 0 , k 0 )p 0 for x ∈ D 12 , (3.4)
and
H 2 (x, α 0 , k 0 )p 0 = H 1 (x, α 0 , k 0 )p 0 for x ∈ D 12 , (3.5)
where D 12 is the unbounded connected component of R 3 \ (D 1 ∪ D 2 ).
Proof. For each j and any boundary condition T (E j , H j ), by (3.2) and (3.3) we have
E 2 (x, α 0 , k 0 )p 0 − E 1 (x, α 0 , k 0 )p 0 = e ik 0 |x| |x| E ∞ 2 (β, α 0 , k 0 )p 0 − E ∞ 1 (β, α 0 , k 0 )p 0 (3.6) +O( 1 |x| 2 ), as |x| → ∞, β = x |x| , H 2 (x, α 0 , k 0 )p 0 − H 1 (x, α 0 , k 0 )p 0 = e ik 0 |x| |x| H ∞ 2 (β, α 0 , k 0 )p 0 − H ∞ 1 (β, α 0 , k 0 )p 0 (3.7) +O( 1 |x| 2 ), as |x| → ∞, β = x |x| . In view of E ∞ 1 (β, α 0 , k 0 )p 0 = E ∞ 2 (β, α 0 , k 0 )p 0 for all β ∈ S 2 , (or H ∞ 1 (β, α 0 , k 0 )p 0 = H ∞ 2 (β, α 0 , k 0 )p 0 for all β ∈ S 2 ), we obtain E 1 (x, α 0 , k 0 )p 0 − E 2 (x, α 0 , k 0 )p 0 = O( 1 |x| 2 ), as |x| → ∞, β = x |x| , (3.8) (or H 1 (x, α 0 , k 0 )p 0 − H 2 (x, α 0 , k 0 )p 0 = O( 1 |x| 2 ), as |x| → ∞, β = x |x| ). (3.9)
With the aid of Lemma 2.3, we get that E 1 − E 2 (or H 1 − H 2 ) satisfies the vector Helmholtz equations, i.e,
∆ E 1 (x, α 0 , k 0 )p 0 − E 2 (x, α 0 , k 0 )p 0 + k 2 0 E 1 (x, α 0 , k 0 )p 0 − E 2 (x, α 0 , k 0 )p 0 = 0 in D 12 , (or ∆ H 1 (x, α 0 , k 0 )p 0 − H 2 (x, α 0 , k 0 )p 0 + k 2 0 H 1 (x, α 0 , k 0 )p 0 − H 2 (x, α 0 , k 0 )p 0 = 0 in D 12 )
. It follows from (3.8), (3.9) and Lemma 2.9 (Rellich's lemma) that
E 1 (x, α 0 , k 0 )p 0 − E 2 (x, α 0 , k 0 )p 0 = 0 for x ∈ D 12 , (or H 1 (x, α 0 , k 0 )p 0 − H 2 (x, α 0 , k 0 )p 0 = 0 for x ∈ D 12 ).
Furthermore, by applying any one of the above two relations to the Maxwell equations
∇ × E j − ikH j = 0, ∇ × H j + ikE j = 0 in R 3 \D j ,
we see that (3.4) and (3.5) hold simultaneously.
Proof of main theorem
Proof of theorem 1.1. For convenience, we assume below the obstacle has the perfect conductor boundary condition, but our proof is valid for the impedance boundary condition as well. Also, we only discuss unique determination of the scatterer by the electric far field pattern because the magnetic case can be similarly dealt with. It is an obvious fact that if two bounded domains D 1 and D 2 of class C 2 satisfying D 1 = D 2 , then either D 1 = D 2 and D 1 ∩ D 2 = ∅, or D 1 = D 2 and D 1 ∩ D 2 = ∅. We will show that the above two cases can never occur.
= ∅. Since E ∞ 1 (β, α 0 , k 0 )p 0 = E ∞ 2 (β, α 0 , k 0 )p 0 for all β ∈ S 2
in an open subset of S 2 , we immediately get that the above relation is still true for all β ∈ S 2 by analyticity. From Lemma 3.1 we get that
E 1 (x, α 0 , k 0 )p 0 = E 2 (x, α 0 , k 0 )p 0 and H 1 (x, α 0 , k 0 )p 0 = H 2 (x, α 0 , k 0 )p 0 ) for all x ∈ D 12 ,
where E j (x, α 0 , k 0 )p 0 , H j (x, α 0 , k 0 )p 0 is the solution of scattering problem for the Maxwell equations in R 3 \D j (j = 1, 2), and D 12 is the unbounded connected component of R 3 \ (D 1 ∪ D 2 ). Note that the real part and imaginary part of cartesian components of E j , H j are both real analytic in R 3 \D j (j = 1, 2) by Lemma 2.4. Since E 1 (x, α 0 , k 0 )p 0 , H 1 (x, α 0 , k 0 )p 0 is defined in D 2 and satisfies there the Maxwell equations, the unique continuation property implies that E 2 (x, α 0 , k 0 )p 0 , H 2 (x, α 0 , k 0 )p 0 can be defined in D 2 and satisfies there the Maxwell equations. Consequently, E 2 (x, α 0 , k 0 )p 0 , H 2 (x, α 0 , k 0 )p 0 is defined in R 3 , it is a smooth function that satisfies the Maxwell equations in R 3 , and the same is true for
E 1 (x, α 0 , k 0 )p 0 , H 1 (x, α 0 , k 0 )p 0 . Therefore the scattered parts E s 1 (x, α 0 , k 0 )p 0 , H s 1 (x, α 0 , k 0 )p 0 and E s 2 (x, α 0 , k 0 )p 0 , H s 2 (x, α 0 , k 0 )p 0 of the scattering solutions E 1 (x, α 0 , k 0 )p 0 , H 1 (x, α 0 , k 0 )p 0 and E 2 (x, α 0 , k 0 )p 0 , H 2 (x, α 0 , k 0 )p 0 satisfy the Maxwell equations ∇ × E − ikH = 0, ∇ × H + ikE = 0 in R 3
and have the Silver-Müller radiation conditions. It follows from Lemma 2.6 that E s
1 (x, α 0 , k 0 )p 0 = E s 2 (x, α 0 , k 0 )p 0 = 0, H s 1 (x, α 0 , k 0 )p 0 = H s 2 (x, α 0 , k 0 )p 0 = 0 in R 3 and hence E 1 (x, α 0 , k 0 )p 0 = E 2 (x, α 0 , k 0 )p 0 = e ik0α0·x p 0 , H 1 (x, α 0 , k 0 )p 0 = H 2 (x, α 0 , k 0 )p 0 = e ik0α0·x (α 0 × p 0 ) in R 3 . This is impossible since ν × E j (x, α 0 , k 0 )p 0 = 0 on ∂D j , j = 1, 2, while e ik0α0·x (ν × p 0 )
can not vanish identically for all x ∈ ∂D j . Thus, we must have D 1 = D 2 . Case 2. Suppose by contradiction that D 2 = D 2 and D 1 ∩ D 2 = ∅. Then either (R 3 \ D 1 ) ∩ (R 3 \D 12 ) or (R 3 \D 2 ) ∩ (R 3 \D 12 ) has only finitely many connected components, and each of them adjoins the unbounded domain D 12 by sharing a common C 2 -smooth surface, where D 12 is the unbounded connected component of R 3 \ (D 1 ∪ D 2 ). Let us assume that Ω be any one of the above connected components. Clearly, Ω is a bounded domain with piecewise C 2 -smooth boundary. Without loss of generality, we let Ω ⊂ R 3 \D 1 . Since E ∞ 1 (β, α 0 , k 0 )p 0 = E ∞ 2 (β, α 0 , k 0 )p 0 for all β ∈ S 2 by analyticity, applying Lemma 3.1 once more we find that
E 1 (x, α 0 , k 0 )p 0 = E 2 (x, α 0 , k 0 )p 0 , H 1 (x, α 0 , k 0 )p 0 = H 2 (x, α 0 , k 0 )p 0 ) for all x ∈ D 12 ,
where E j (x, α 0 , k 0 )p 0 , H j (x, α 0 , k 0 )p 0 is the solution of scattering problem for the Maxwell equations in R 3 \D j (j = 1, 2). Note that (ν × E j ) ∂Dj = 0, j = 1, 2, and (ν × E 1 ) ∂D12 = (ν × E 2 ) ∂D12 = 0 . It is easy to see from this and the definition of Ω that the restriction of
E 1 (x, α 0 , k 0 )p 0 , H 1 (x, α 0 , k 0 )p 0 to Ω satisfies ∇ × E − ikH = 0 in Ω, ∇ × H + ikE = 0 in Ω, ν × E = 0 on ∂Ω,(4.1)
i.e., the restriction of E 1 (x, α 0 , k 0 )p 0 , H 1 (x, α 0 , k 0 )p 0 to Ω is a Maxwell eigen-field corresponding to the Maxwell eigenvalue k. We find by Lemma 2.4 that Re E 1 (x, α 0 , k 0 )p 0 and Im E 1 (x, α 0 , k 0 )p 0 (respectively, Re H 1 (x, α 0 , k 0 )p 0 and Im H 1 (x, α 0 , k 0 )p 0 ) are both real analytic vector-valued function in R 3 \D 1 , where Re E 1 (x,α 0 , k 0 )p 0 and Im E 1 (x, α 0 , k 0 )p 0 (respectively, Re H 1 (x,α 0 , k 0 )p 0 and Im H 1 (x, α 0 , k 0 )p 0 ) are the real part and imaginary part of the electric field E 1 (x, α 0 , k 0 )p 0 (respectively, the magnetic field H 1 (x, α 0 , k 0 )p 0 ), i.e., E 1 (x, α 0 , k 0 )p 0 = Re E 1 (x, α 0 , k 0 )p 0 + i Im E 1 (x, α 0 , k 0 )p 0 (respectively, H 1 (x, α 0 , k 0 )p 0 = Re H 1 (x, α 0 , k 0 )p 0 +i Im H 1 (x, α 0 , k 0 )p 0 ). By the definition of the electric field E 1 (x, α 0 , k 0 )p 0 , we have that for all x ∈ R 3 \D 1 ,
E 1 (x, α 0 , k 0 )p 0 = e ik0α0·x p 0 + E s 1 (x, α 0 , k 0 )p 0 (4.2) = (cos(k 0 α 0 · x) + i sin(k 0 α 0 · x))p 0 + (Re E s 1 (x, α 0 , k 0 ) + i Im E s 1 (x, α 0 , k 0 ))p 0 = cos(k 0 α 0 · x) + Re E s 1 (x, α 0 , k 0 ) p 0 + i sin(kα 0 · x) + Im E s 1 (x, α 0 , k 0 ) p 0 .
Combining Lemma 2.3, (4.1) and (2.4), we see that the electric field E 1 (x, α 0 , k 0 )p 0 of the Maxwell eigen-field in Ω are also an electric eigen-field in Ω corresponding to the same eigenvalue k > 0. It follows from Lemma 2.8 that the electric eigen-field E 1 (x, α 0 , k 0 )p 0 must be a real analytic vector-valued function in Ω. From this and (4.2), we get that sin(k 0 α 0 · x)p 0 + Im E s 1 (x, α 0 , k 0 )p 0 must vanish identically for all x ∈ Ω, i.e.,
Im E s 1 (x, α 0 , k 0 )p 0 = − sin(k 0 α 0 · x)p 0 for all x ∈ Ω. (4.3)
With the aid of Lemma 2.1, we know that the real analytic vector-valued function Im E s 1 (x, α 0 , k 0 )p 0 is uniquely determined in (Ω∪D 12 ∪((∂Ω)∩(∂D 12 ))) • by its values in the subset domain Ω, where (Ω∪D 12 ∪((∂Ω)∩(∂D 12 ))) • is the interior of Ω∪D 12 ∪((∂Ω)∩(∂D 12 )). Let us remark that (Ω∪D 12 ∪((∂Ω)∩(∂D 12 ))) • is still a unbounded connected component (i.e., a unbounded domain in R 3 ). Note also that the real analytic vector-valued function − sin(k 0 α 0 ·x)p 0 defined for x ∈ Ω has just a unique real analytic extension to (Ω ∪ D 12 ∪ ((∂Ω) ∩ (∂D 12 ))) • , that is, − sin(k 0 α 0 · x)p 0 for x ∈ (Ω ∪ D 12 ∪ ((∂Ω) ∩ (∂D 12 ))) • . Thus, we have that for all x ∈ (Ω ∪ D 12 ∪ ((∂Ω) ∩ (∂D 12 ))) • , Im E s 1 (x, α 0 , k 0 )p 0 = − sin(k 0 α 0 · x)p 0 . (4.5)
Since E s 1 (x, α 0 , k 0 )p 0 is the electric scattering solution of the Maxwell equations in R 3 \D 1 satisfying the Sommerfeld radiation condition, by (2.1) of Lemma 2.5 we get lim |x|→∞ |E s 1 (x,α 0 ,k 0 )p 0 | = 0. On the other hand, from (4.5) we see that |E s 1 (x, α 0 , k 0 )p 0 | = |Re E s 1 (x, α 0 , k 0 )p 0 | 2 + |Im E s 1 (x, α 0 , k 0 )p 0 | 2 1/2 = |Re E s 1 (x, α 0 , k 0 )p 0 | 2 + | sin(k 0 α 0 · x)p 0 | 2 1/2 ≥ | sin(k 0 α 0 · x)p 0 | for all x ∈ ((Ω ∪ D 12 ) ∪ ((∂Ω) ∩ (∂D 12 ))) • , and so |E s 1 (x, α 0 , k 0 )p 0 | can't tend to zero as |x| → ∞. Here |b| denotes the Euclidean norm of a vector b in R 3 . This is a contradiction, which implies that any domain Ω mentioned above can never appear. Therefore we must have D 1 = D 2 .
Finally, denoting D = D 1 = D 2 , E = E 1 = E 2 , and H = H 1 = H 2 , we assume that we have different boundary condition T 1 (E, H) = T 2 (E, H). For the sake of generality, consider the case where we have impedance boundary conditions with two different continuous impedance functions ψ 1 = ψ 2 . Then, from ν × H − iψ j (ν × E) × ν = 0 on ∂D for j = 1, 2 we observe that i(ψ 1 −ψ 2 )(ν ×E)×ν = 0 on ∂D. Therefore for the open set Γ := {x ∈ ∂D ψ 1 (x) = ψ 2 (x)} we have that (ν × E) × ν = 0 on Γ so that (ν × E) = 0 on Γ. Consequently, we obtain ν × H = 0 on Γ by the boundary condition. Hence, by Holmgren's uniqueness theorem for the Maxwell equations (see Lemma 2.6), E = H = 0 in R 3 \ D, which implies that the scattered wave E s , H s is an entire solution of the Maxwell equations, and E s and H s satisfy the the Silver-Müller radiation condition. But the incident field E i , H i doesn't satisfy the Silver-Müller radiation condition. This is a contradiction. Hence ψ 1 = ψ 2 . The case where one of the boundary conditions is the perfect boundary condition can be treated analogously.
Lemma 2.3 (see Theorem 6.4 of [5]). Let E, H be a solution to the Maxwell equations ∇ × E − ikH = 0, ∇ × H + ikE = 0. Then E and H are divergence free (i.e., ∇ · E = 0 and ∇ · H = 0) and satisfy the vector Helmholtz equation ∆E + k 2 E = 0 and ∆H + k 2 H = 0. Conversely, let E (or H) be a solution to the vector Helmholtz equation satisfying ∇ · E = 0 (or ∇ · H = 0). Then E and H := 1 ik ∇ × E (or H and E := − 1 ik ∇ × H) satisfy the Maxwell equations.
Lemma 2.4 (see Theorem 6.3 of[5]). Any continuously differentiable solution to the Maxwell equations has analytic cartesian components. In particular, the cartesian components of solutions to the Maxwell equations are automatically two times continuously differentiable.
Lemma 2. 5
5(see Theorem 6.7 of [5]). Assume the bounded domain D is the open complement of an unbounded domain of class
Lemma 2. 7 (
7Holmgren's uniqueness theorem for the scattering total solutions of the Maxwell equations, see Theorem 6.5 of[5]). Let D be a bounded domain with C 2 -smooth boundary ∂D and let Γ ⊂ ∂D be an open subset with Γ ∩ (R 3 \ D) = ∅. Assume that E, H is a solution of the scattering problem for the Maxwell equations
), one has E = M ∪ D, H = M ∪ N, where D, N , M, E, H denote the set of eigenvalues of the Dirichlet Laplacian, Neumann Laplacian, Maxwell, electric and magnetic problems, respectively.
Case 1 .
1Suppose by contradiction that D 1 = D 2 and D 2 ∩D 1
AcknowledgmentsThis research was supported by NNSF of China (11171023/A010801) and NNSF of China (11671033/A010802).
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| []
|
[
"Machine learning classification for field distributions of photonic modes",
"Machine learning classification for field distributions of photonic modes"
]
| [
"Carlo Barth \nHelmholtz-Zentrum Berlin für Materialien und Energie\nKekuléstr. 512489BerlinGermany\n",
"Christiane Becker [email protected] \nHelmholtz-Zentrum Berlin für Materialien und Energie\nKekuléstr. 512489BerlinGermany\n"
]
| [
"Helmholtz-Zentrum Berlin für Materialien und Energie\nKekuléstr. 512489BerlinGermany",
"Helmholtz-Zentrum Berlin für Materialien und Energie\nKekuléstr. 512489BerlinGermany"
]
| []
| Machine learning techniques can reveal hidden structure in large data amounts and can potentially extent or even replace analytical scientific methods. In nanophotonics, modes can increase the light yield from emitters located inside the nanostructure or near the surface. Optimizing such systems enforces to systematically analyze large amounts of three-dimensional field distribution data. We present a method based on finite element simulations and machine learning for the identification of modes with large field energies and specific spatial properties. By clustering we reduce the field distribution data to a minimal subset of prototypes. The predictive power of the approach is demonstrated using an analysis of experimentally measured fluorescence enhancement of quantum dots on a photonic crystal surface. The clustering method can be used for any optimization task that depends on three-dimensional field data, and is therefore relevant for biosensing, quantum dot solar cells or photon upconversion. | 10.1038/s42005-018-0060-1 | [
"https://arxiv.org/pdf/1803.08290v1.pdf"
]
| 4,956,206 | 1803.08290 | 95ddf197ce6c7fabc5fb9c9854d93df9847e0621 |
Machine learning classification for field distributions of photonic modes
22 Mar 2018
Carlo Barth
Helmholtz-Zentrum Berlin für Materialien und Energie
Kekuléstr. 512489BerlinGermany
Christiane Becker [email protected]
Helmholtz-Zentrum Berlin für Materialien und Energie
Kekuléstr. 512489BerlinGermany
Machine learning classification for field distributions of photonic modes
22 Mar 2018(Dated: 23 March 2018)1 arXiv:1803.08290v1 [physics.optics]Photonic crystalsLeaky modesMachine learningUnsupervised learn- ingClassificationClusteringFluorescence enhancementExcitation enhancementBiosensing
Machine learning techniques can reveal hidden structure in large data amounts and can potentially extent or even replace analytical scientific methods. In nanophotonics, modes can increase the light yield from emitters located inside the nanostructure or near the surface. Optimizing such systems enforces to systematically analyze large amounts of three-dimensional field distribution data. We present a method based on finite element simulations and machine learning for the identification of modes with large field energies and specific spatial properties. By clustering we reduce the field distribution data to a minimal subset of prototypes. The predictive power of the approach is demonstrated using an analysis of experimentally measured fluorescence enhancement of quantum dots on a photonic crystal surface. The clustering method can be used for any optimization task that depends on three-dimensional field data, and is therefore relevant for biosensing, quantum dot solar cells or photon upconversion.
Machine learning is a rapidly developing discipline, which uses statistical approaches to learn from data without explicitly rule-based programming. Driven by today's massive increase in data amounts, the related techniques are extended and improved at a fast pace 1 .
Machine learning is currently applied to all aspects of science, from health sciences and psychology 2,3 , to biology 4-6 , to environmental 7 and material sciences 8,9 . But also to matters of everyday-life, from online security, to finance and insurance. While supervised learning has led to breakthroughs in computer vision 10 and speech recognition 11 , unsupervised learning is expected to become far more important in the future 12 . The latter techniques, such as clustering [13][14][15] , allow for the recognition of patterns in unlabeled data and can therefore reveal a hidden structure. They have been successfully applied to e.g. anomaly detection 16,17 or genetics 18 .
In the field of nanophotonics, increasing computer power, storage space and data throughput, as well as improvements in modeling techniques, greatly accelerated all-numerical system design. For nanostructures the following typical optimization tasks are met:
• Simple design: Scalar parameters → scalar output (e.g. lengths/refractive indices → reflectivity)
• Inverse design:
Multivariate parameters → scalar output (e.g. permittivity distribution → reflectivity)
• Qualitative design: Scalar parameters → multivariate output (e.g. lengths/refractive indices → 3D field distribution) Simple design tasks are generally solved by simulating the system for many different parameter combinations (i.e. grid search), or by applying function minimization routines.
More sophisticated techniques such as the reduced basis method 19 for finite element method (FEM) simulations have successfully been applied to speed-up this optimization process for large parameter spaces. Inverse design tasks introduce a high-dimensional input parameter space, typically by allowing for arbitrary changes in the permittivity distribution (r) of the nanostructure. Machine learning techniques have successfully been applied for this purpose, mainly using genetic algorithms [20][21][22][23][24] . Simple and inverse tasks have in common that they possess a scalar measure of success, i.e. they can be seen as minimization problems. The machine learning approach in inverse design therefore belongs to the field of supervised learning (more specifically regression). The third design task introduced above substantially differs in the way that the system should be optimized for a high-dimensional output. Due to the inaccessibility of a scalar success metric, we denote this problem as qualitative design. This is for example the case if the 3-dimensional spatial distribution of the electromagnetic fields has to be taken into account. Usually, such problems are solved by appropriate visualizations. But since any change in the input parameters leads to a change in the high-dimensional output, the data amounts quickly become extremely large. We will demonstrate below that machine learning, or more specifically clustering, is able to overcome these issues by reducing the output dimensionality.
As indicated before, an example of qualitative design is to optimize a photonic nanostructure, e.g. a photonic crystal (PhC), for an appropriate spatial field distribution. This is of high relevance whenever an interaction of the field with a (potentially vague) particle distribution is present, e.g. for emitters on nanophotonic surfaces or emitters embedded into the nanostructure. PhC slabs exhibit a phenomenon called leaky modes: resonances that can be excited using external radiation [25][26][27][28][29] . Leaky modes have been used to improve various applications (e.g. light trapping in photovoltaic devices [30][31][32][33][34] , light-emitting diodes 35,36 ), but can also affect near-surface emitters, such as QDs, atoms or molecules. Especially in the life sciences, the applications range from PhC enhanced microscopy and single molecule detection to enhanced live cell imaging, DNA sequencing and gene expression analysis [37][38][39][40] .
Besides the rather well-investigated extraction enhancement effect [26][27][28]35,[41][42][43][44][45] , the excitation enhancement effect 39,46-51 increases the stimulated emission rate of the emitters by enhanced near-field energy densities of leaky modes in the absorption wavelength range. To optimize photonic nanostructures for excitation enhancement it is therefore inevitable to take the 3D spatial electromagnetic field distribution into account.
In this study we present a powerful technique based on machine learning for the classification of 3D electromagnetic field distribution data. This method can be of avail in any case where large amounts of electromagnetic field (or energy) distribution data should be reduced to a minimal subset of typical distributions. We will refer to these as distribution prototypes. We directly apply the technique to a specific dataset of our previous publication on fluorescence enhancement of lead sulfide (PbS) quantum dots (QDs) on a silicon PhC slab surface 52 , however, without loss of generality. A similar setup was used in previous studies 19,53,54 .
The effect is sketched in Fig. 1(a), depicting emitters (black dots) that interact with a leaky mode of the PhC excited by an external laser source. The illumination conditions introduce four parameters: the laser wavelength λ, the laser polarization P (TE or TM), the polar angle θ with the plane normal, and the azimuthal angle φ used to define the high symmetry direction (Γ − M or Γ − K). The latter is also indicated in the scanning electron microscope image of the sample (without emitters) in Fig. 1(b). An example of an electric field distributions E(r) of a leaky mode is depicted in Fig. 1(a). As mentioned, the energy density of the electric field of the leaky modes, w lm (r), can be larger compared to the energy density of the incident plane wave, w pw , known as field energy enhancement (w lm (r)/w pw > 1). To study this effect in large parameter spaces we usually define the volume-integrated field energy enhancement
E + = 1 w pw V sup Vsup w lm (r) dV sup ,(1)
where V sup is the volume of interest. In our case V sup is the superspace of the computational domain, as indicated by the yellow dashed line in Fig. 1(a). The energy density of the plane wave has no spatial dependence and is proportional to the amplitude of the electric field, E pw,0 , and the refractive index n of the surrounding medium, i.e.
w pw = 0 4 n 2 E pw,0 2 .(2)
In the figure, a uniform random distribution of emitters is shown as an example. But depending on the coating process, emitters might have a very specific spatial distribution in a real application, e.g. a monolayer attached to the surface, or a higher concentration inside the holes, or at the plateaus between the holes. Consequently, the spatial distribution of the energy density w lm (r) becomes a determining factor and, therefore, the integrated field energy enhancement E + is not sufficient to quantify the effect on the emitters.
An optimized design for an application as sketched in Fig. 1(a) can hence be achieved by identifying a mode which has (i) a large volume-integrated field energy enhancement E + and (ii) an appropriate spatial field energy density distribution overlapping the locations of the emitters, at the same time. Task (i) is a "simple design" task, as defined in the introduction, while (ii) is a "qualitative task", i.e. an optimization of a multivariate output.
The results of the fluorescence enhancement measurements of our previous study 52 are shown again in the upper row of Fig. 2, as well as the results for the volume-integrated field energy enhancement E + in the center row. The latter results solve task (i), as described above. Task (ii) potentially enforces to take into account 3D field distribution data of all combinations of the illumination condition parameters λ, P, θ and φ. If the number of considered wavelengths N λ and the number of angles N θ becomes large, it is no longer feasible to directly visualize all the 3-dimensional field distributions for all points in the λ-θ-maps shown in Fig. 2. It is hence necessary to reduce the amount of field distribution data in an appropriate way. One possibility to achieve this reduction is to pitch on specific wavelengths and incident angles for which the field distribution is evaluated, as it was done in the previous study 52 . This way, however, information is mainly gained at random, so that general trends might be overseen.
A more systematical approach is to cluster field distributions which are similar, and to therefore derive typical distributions (i.e. "distribution fingerprints"). It is known that a certain undisturbed photonic band in the leaky mode regime will not significantly change its symmetry properties when crossing the λ-θ space 29,55 , as will be explained in more detail below. As a result, the entirety of field distributions are composed of a finite set of patterns which are basically caused by the finite number of bands. This feature space can efficiently be partitioned into the typical patterns using machine learning clustering techniques.
In the following, we will first reconsider the experimental and numerical results of the previous study 52 , highlighting aspects which were left unexplained by the prior analysis technique. Afterwards, we introduce the clustering technique and apply it to systematically analyze the 3D field energy distribution properties. The distributions are classified by as-signing them to distribution prototypes, which are consulted as representative solutions to fully explain the effects observed in the experiment. We further consider a mathematical method based on silhouette coefficients 56 to assess the clustering result. Based on these analyses we will explain how the method enables to solve complex optimization tasks with high-dimensional output, as indicated in the introduction. The all-numerical technique is of relevance for the design of nanophotonic structures for any application in which emitters interact with the electromagnetic field, e.g. highly-sensitive biosensors 37 , quantum dot solar cells 57-60 , or up-conversion devices 61,62 .
RESULTS
The upper two rows of Fig
Introduction and justification of the clustering technique
The E + -maps given in the center row of Fig. 2 only provide information about the volumeintegrated field enhancement over a characteristic volume V sup , marked by the yellow dashed line in Fig. 1(a). Therefore, regions of high E + can be regarded as a necessary condition for fluorescence enhancement, but not as a sufficient one. A high E + without a corresponding fluorescence enhancement F + hence indicates a lack in the spatial overlap of the emitters with the regions of enhanced field energy density.
However, it is known that bands of the photonic crystal have well-behaved spatial properties when varying the k-vector between two high-symmetry points of the irreducible Brillouin zone 29,55 . More specifically, the modes belong to the same symmetry point group as the system seen from the point in k-space, i.e. they exhibit the same spatial symmetry.
Consequently, it is theoretically justified to expect that the spatial properties of the bands only change smoothly with θ. For a fixed high-symmetry direction, e.g. Γ − K, we only expect two types of solutions, which are 1. regions that correspond to leaky-mode bands, and 2. regions that are off any photonic band, and therefore corresponding to the continuum of radiation modes.
The regions of the radiation modes are expected to exhibit solutions that resemble plane waves, i.e. show oscillatory behavior in the the exterior domain. All things considered, only a small number of different spatial symmetry types is expected, which is of the order of the number of bands that cross the parameter scan window. This is where machine learning comes into play. If we consider the specific 3D electric field distribution for a single illumination setting as a sample, and the electric field values of each point in the considered volume as features, then clustering techniques are able to subdivide the entirety of field distributions into a finite number of field distribution prototypes. This approach is reasonable because the data range is expected to contain a finite number of typical field patterns, and each of the real field patterns can be identified with one of those prototypes. Moreover, these prototypes have a sufficient "uniqueness", e.g. they considerably differ in their symmetry properties. The methods section "Clustering of electric field data"
. . −0.5 . 0.0 . 0.5 . 1.0 . Silhouette coefficient . Cluster label . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . Γ − K, TM . −0.5 . 0.0 . 0.5 . 1.0 . Silhouette coefficient . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . Γ − K, TE . −0.5 . 0.0 . 0.5 . 1.0 . Silhouette coefficient . 1 . 2 . 3 . 4 . 5 . 6 . 7 . Γ − M, TM . −0.5 . 0.0 . 0.5 . 1.0 . Silhouette coefficient . 1 . 2 . 3 . 4 . 5 . 6 . 7 . Γ − M,
gives a detailed description of how the clustering is performed. In a nutshell, for each illumination condition, i.e. a set of (P, φ, θ, λ), the electric field strength E = (E x , E y , E z ) is derived from a FEM simulation. It is sufficient to export the fields on symmetry planes to reduce the data volume, for which we use the xy, xz and yz planes marked in Fig. 1(c). The validity of this approach was tested using a comparison to full 3D exports using a smaller dataset. Note that, in contrast to the the volume-integrated field energy enhancement E + which is calculated in the volume V sup (Fig. 1(a)), the fields for the clustering are also considered in the dielectric materials (silicon PhC and glass substrate, Fig. 1(c)). To account for the different cluster sizes (narrow bands) and unknown cluster shapes in the data set, the flexible Gaussian mixture model (GMM) clustering technique is used (see the methods section "Gaussian mixture model clustering" for details), implemented in the Python library Sckit-learn 63 . From the clustering itself two characteristics can directly be gained: the classification, which labels each observation with a cluster index i, and the distribution prototypes, usually denoted as "cluster centers" in the general clustering literature.
The latter are the average of the electric field distributions (on the chosen planes) of all samples that belong to a specific cluster i. We note that we averaged the normalized input data, i.e. the exact data used for the clustering, to calculate the prototypes. The prototypes therefore represent the actual mathematical cluster centers, with the tradeoff that the absolute field amplitude information is lost, because the samples are normalized individually. Another possibility would be to average the unnormalized fields, so that the amplitude information would be conserved, with the tradeoff that the prototypes derived that way are not exactly the cluster centers. We settled for the normalized fields, as the amplitude information is essentially included in the E + maps.
As in most clustering techniques, the number of clusters must be specified in GMM clustering, so that the appropriateness of this choice has to be validated. This aspect will be covered shortly.
Classification maps
The classification can be visualized by assigning each point (θ j , λ k ) to a different color that corresponds to its label i. Recall that the clustering is carried out individually for each The procedure of determining the number of clusters will be explained in the next subsection.
When comparing the classification maps to the E + maps above, a striking accordance can be observed. The narrow bands of high field enhancement in the E + maps correspond to narrow areas at the same positions in the classification maps. Note that the E + maps and the classification maps are based on very different data sets: the former are derived from a spatial integration over the electric field energy density distribution w lm (r) in the superspace volume V sub only (Eq. (1) In all cases, we observe that the saturation decreases at the border of two clusters. This is expected, as silhouette scores close to 0 indicate a sample which is in fact close to the border of the neighboring cluster. It is apparent from this phenomenon, and important to stress here, that the clustering technique is a tool. The field distribution data is not categorical, so we expect superposed solutions which are badly represented by "pure" modes. That said, these intermediate parts are small, as it is seen from the saturation distributions, so that the clustering is still a valid and effective approach.
Silhouette analysis and the number of clusters
Before we investigate the field distribution prototypes, the quality of the clustering itself is evaluated using a mathematical analysis in the following. This can be done using the silhouette coefficients, using a scheme known as silhouette analysis 56 . Figure 3 depicts socalled silhouette plots for each combination using the same column order as in Fig. 2
Field distribution prototypes
As the second essential outcome, the clustering procedure yields the field distribution prototypes. As the input data for the GMM algorithm has been electric field values on three planes, namely xy, xz and yz, the prototype data is available on these planes as well.
The prototypes for all clusters of each combination, and on all three planes, are depicted in Fig. 4. For each direction/polarization combination a prototype map is shown in one of the four horizontal panels. Each panel consists of three rows for the xz, yz and xy planes, respectively (from top to bottom). Each of these rows has the same number of columns that accounts for the number of clusters, and each column has a colored edge in the top-most row that corresponds to the color used for that label in the classification maps shown in the lower row of Fig. 2. The cluster label is further given in the title of the xz-row. Each distribution plot depicts the electric field energy distribution E 2 in the respective plane.
The distribution plots further feature semitransparent markings for the glass superstrate Fig. 4). In contrast, clusters that belong to radiation modes have energy distributions that increase away from the PhC, e.g. cluster 2 of the Γ − K, TE case (dark blue, see panel 2, column 2). Further comparisons will be considered below. FIG. 4. Prototype maps for the different direction/polarization combinations. For each direction/polarization combination (4 horizontal panels) a prototype (i.e. cluster center) map is shown, which consists of 3 rows for the xz, yz and xy planes, respectively (from top to bottom). Each of these rows has a number of columns that accounts for the number of clusters, and each column has a colored edge in the top-most row that corresponds to the color used for that label in the classification maps shown in the lower row of Fig. 2. The cluster label is further given in the title of the xz-row. Each distribution plot depicts the electric field energy distribution
Putting the pieces together
To explain the measured fluorescence enhancement effects shown in the upper maps of Fig. 2, it is necessary to combine all information gained from the numerical analysis.
This is, the volume-integrated field energy enhancement maps (E + , Fig. 2, center row), the classification maps (Fig. 2, lower row) and the prototype maps (Fig. 4). A guide on how the different aspects of the results can be connected to yield a complete interpretation may read as follows:
1. Select a feature in the volume-integrated field energy enhancement (E + ) maps. For these features the simulation suggests a possible excitation enhancement effect. the red one can be identified with a radiation mode. The energy is therefore less well confined to the surface in the red case, which is exactly seen in the fluorescence maps, where only at the location of a broad green region a fluorescence enhancement can be observed, but no enhancement is seen at the prosecution of the mode at higher incident angles. The green band has a node in the yz plane, while the energy in x-direction is strongly localized at the flanks. It is therefore likely to be seen in the fluorescence enhancement, when comparing to the previous results. However, a fluorescence enhancement for the green band is mainly seen in the interaction zone. This can be explained when considering the energy distribution of the cyan band: this band does not have the node in the yz plane. In the interaction zone, the two bands basically overlap, so that the node is partially erased.
Therefore, a stronger effect in the fluorescence enhancement is expected, just as it is observed. Interestingly, there is moreover a small effect at ≈ 1080 nm for large angles in the F + map. The clustering reveals another band with strong localization at the flanks: the orange region with label 4 (and the similar yellow one which couples more strongly to the radiation modes). It is likely that the measured effect is actually caused by this band, as it has a similar energy distribution as other bands which are clearly seen.
To give a clear idea of the 3D energy density distribution for three selected modes, and also to show how well the clusters match the actual physical fields, Fig. 5 shows full-3D renderings 64 . The images depict multiple periods of the photonic crystal as a grayish metallike material, without showing a superspace material. The upper row shows a topview of the volume-rendered electric field energy density color-coded using a heat map, which is not comparable between the figures. The lower row shows a closer view and indicates a random distribution of QDs as bright small spheres, emitting white light with an intensity proportional to the field energy density at their specific positions. The QD distribution is the same for all three images. The columns relate to three different modes of the Γ−K, TE case, denoted as A, B and C. They correspond to clusters 8, 3 and 6, as also marked in Fig. 4.
The modes are the actual solutions from the finite-element solver that have the smallest deviations from the assigned prototype (i.e. cluster center), determined using the silhouette coefficients. Incident angle θ and wavelength λ for each mode are given in the headings.
Note that these images have an illustrative character, but can be very helpful to imagine the actual physical situation. Modes A and B are the ones which have been discussed recently, and it is clearly seen that the former concentrates its energy at the plateaus, while the latter has high energy densities at the flanks of the holes. A third type is shown with mode C, which focusses the energy directly inside the holes. The illustrations in the lower row give a notion of how these modes activate different QDs, depending on their position. Only a small density of QDs is used for the images for purposes of visibility, and they are randomly distributed in a layer that fills the holes and extents 100 nm in z-direction. Mode A very efficiently excites QDs at the plateaus, just as expected, while modes B and C do the same at the flanks and inside the holes, respectively. Consequently, these renderings completely confirm the results of the clustering approach.
DISCUSSION
The aim of the numerical approach presented here is the systematic identification of suitable leaky modes of nanophotonic structures for interaction with near-surface emitters. For instance, (a) a monolayer of emitting species attached at the surface of the nanophotonic structure is expected to strongly interact with a leaky mode with shallow field distribution.
In contrast, an experiment with (b) emitters in a coating on top of the photonic nanostructure, the interaction with a shallow leaky mode will be rather small due to the limited spatial overlap of the mode volume with the emitting material. Here, a leaky mode with an energy density enhancement in a large volume outside the photonic nanostructure would be better suited. In a third scenario, (c), emitters fill the voids of the photonic nanostructure, for example if they are solved in a liquid solution and dropped onto the structure. In that case, leaky modes with strong field enhancement inside these voids are expected to cause the strongest effects. In the chosen dataset, the fluorescence enhancement experiment of PbS quantum dots on a silicon PhC slab in nanohole geometry, the distribution of QDs resembles a mixture of case (b) and (c). The clustering technique revealed that the modes which have the best spatial overlap with the QD distribution effectively cause the strongest fluorescence enhancement effects in the measurements.
In the previous study 52 we used a selection of a small number of points for which the field energy distributions was analyzed. The clustering technique confirmed the results that were achieved this way, but it also helps to explains complicated details, e.g. as caused by the superposition of two modes. Therefore, the clustering approach gave a much more coherent and detailed explication of the underlying physical phenomena. It emphasizes the interesting parts automatically and systematically, e.g. by revealing regions of rapidly changing field distributions through individual clusters, or through large deviations from the assigned prototype. Moreover, the clustering technique seems to be applicable to even more complicated cases, e.g. in windows with more bands for which an analysis using selected points is not reasonable any more.
The presented technique composed of (i) the field energy enhancement maps and (ii) the 3D electric field distribution clustering provides a versatile tool for the analysis and design of photonic nanostructures for applications that utilize near-field enhancement effects for increased emission. For any known distribution of near-surface emitters that should be affected by leaky modes, optimum values for all relevant parameters can in principle be determined. It is e.g. possible to define a wavelength range for the excitation of the emitters by considering their absorption properties, and to numerically calculate the field energy enhancement E + and field values in 3D for clustering (as provided in the center and lower row of Fig. 2, here). By choosing the mode with the largest spatial overlap of high field energy with the emitter distribution from the prototypes (as in Fig. 4), an optimum mode can systematically be determined. This process can moreover be repeated for possible geometrical parameters of the photonic nanostructure, e.g. the lattice constant, slab thickness or hole radius. Alternatively, if the geometrical parameters should be varied extensively, the technique could be applied for an initial set of geometrical parameters to select a potential mode and to reduce the wavelength and angle window. Successively, only the field energy enhancement E + may be calculated in the scan over the possible geometrical parameters to determine the absolute maximum of the enhancement.
The clustering technique is extremely flexible. It is not limited to uniformly sampled feature spaces as shown in our example application. It would also have been possible to choose arbitrary snapshot points in the θ-λ space, e.g. with a higher density in regions of high field energy enhancement E + . It is further not limited to the shown number of feature parameters, i.e. we could have added a variation of the hole diameter or other geometrical parameters as well. But the method is even more powerful, because the trained classifier can be used to classify field distributions that it has not "seen" yet, known as prediction. In contrast to the clustering itself, this is a computationally cheap process, and the classifier can even be persistently stored on disk for later use. To make these considerations more clear, it would have been possible to choose a smaller number of possibly non-uniformly sampled points in the θ-λ space for efficient clustering. The silhouette analysis can be used to make sure that the number of samples is sufficient to reach an appropriate clustering result. From this clustering the prototype field distributions can be derived and the classifier can be stored to disk. Afterwards, an e.g. uniform scan over θ, λ, and other parameters that are expected to not change the field distributions considerably, (e.g. hole diameter, slab thickness, refractive indices, . . . ) could be performed. The resulting new solutions could then be assigned to the prototypes using the classifier from disk with minimal computational effort.
Numerous applications could benefit from these optimization abilities. In the field of biosensing, photonic nanostructures have become an important platform for e.g. label-free biosensing or for the enhancement of the output of photon emitting tags used in the life sciences and in vitro diagnostics. A recent review article 37 shows that nanophotonic enhanced biosensors are yet extremely relevant, even commercially and potentially on industrial scale. Exploiting leaky modes with large Q-factors enables for narrow bandwidths (< 1 nm) and extremely high sensitivities, e.g. for detection of disease biomarkers in serum with concentrations of ∼ 1 pg ml −1 . The numerous applications that are described in the mentioned review article have in common that the nanophotonic structure is designed for a very specific mode, i.e. a specific illumination condition and a determinable distribution of the molecules/cells/virus particles in question. This is where the technique presented here could be utilized for a systematic optimization in the design process, and hence to further increase the sensitivities of related sensors. Photon upconversion 65,66 in biomedical imaging and solar energy is another application that could benefit from the discussed all-numerical design abilities. Recent publications 61,62 demonstrate upconversion using thin emitter layers, which as well could potentially be improved using specifically tailored nanophotonic structures.
In summary, we have developed a numerical method that allows to systematically optimize nanophotonic structures pertaining to the 3D field distribution and field energy enhancement of modes. The method uses a combination of FEM simulations and postprocessing using machine learning clustering. We showcased the modelling power of the method by explaining experimentally measured fluorescence enhancement of QDs on a photonic crystal slabs surface. The method yielded information that was not easily accessible using e.g. a visualization-based analysis for selected parameter combinations, and which allowed to fully explain the experimental results. Consequently, the presented technique could be of great avail for applications that utilize effects that depend on the spatial field distribution of nanophotonic modes, such as in the fields of biosensing 37,66 , quantum dot solar cells 57-60 , or up-conversion in solar energy 61,62,65 .
METHODS
Clustering of electric field data
The clustering is executed on an input matrix X of shape N s × N f , where N s is the number of samples and N f the number of features. A sample is the solution for a specific set of input parameters, in our case incident angle θ and wavelength λ. The features, in the present case, are absolute values of the electric field components E j with j ∈ {x, y, z} for a number of points r i ∈ R 3 , i.e. of the form |E j (r i )|. Consequently, if the field is evaluated at N p points, these are N f = 3N p features. To avoid exporting the electric field on a full Cartesian grid in 3D, which would cause huge amounts of data when trying to achieve a reasonable resolution, data is only exported on the symmetry planes marked in Fig. 1(c), respectively. More symmetry planes could be used as well, but based on these three planes a reasonable classification can be reached, as tested using smaller data sets and comparing to a full 3D field output. A field pattern of a single simulation holds data for each of the 3 spatial directions, and for each component j of the electric field (altogether a 4D data set). As each sample X i must be a 1D row vector with observations of single scalar values x 0 , . . . , x N f −1 , it is necessary to flatten these data sets in always the same way, yielding "1D representations" of the fields. The data is moreover normalized by scaling each sample to unit norm individually. The field export is performed for each point in each map of Fig. 2, center row, so that the samples are unique simulations for a given direction/polarization combination, wavelength λ and incident angle θ. The number of samples for a single map is given by N s = N λ · N θ . To give an expression for the complete input matrix X we abbreviate
X = E 0,0,0 x (3)
For the wavelength and angle resolution values of 0.5 nm and 0.3 • have been used, respectively. For each clustering procedure the input matrix X had a size of N s × N f = 47 034 × 8616. This is a comparably large problem size, especially because the large feature dimensionality (N f ), so that the procedure took more than 10 hours on a hexa-core workstation with roughly 40 GB of memory consumption.
Gaussian mixture model clustering
Simple clustering techniques, such as the k-means algorithm 67 , can be extremely robust, but also have their disadvantages. E.g. k-means assumes that the clusters are circular, i.e. representable by a (hyper-)sphere in feature space. The center of this sphere defines the cluster center (i.e. prototype), while the radius acts as a hard boundary used to decide which samples belong to the cluster. In contrast, the GMM 67,68 is a so-called soft method. That is, a "score" for each cluster is assigned to the samples, which account for the probability that the sample belongs to a specific cluster. In GMM clustering, the clusters are represented by Gaussian distributions of the dimensionality of the features space (i.e. N f ).
In general, a superposition of N multivariate Gaussian distributions of the form
p(x) = N i=1 c i N i (x)(4)
can be used to approximate almost any continuous density to arbitrary accuracy (this is intuitive with 1D Gaussians, which can fit almost any 1D signal if enough Gaussians are superimposed). Here, the N i (x) are multivariate Gaussian distributions of the form 67
N (x) = 1 (2π) D/2 | Σ| 1/2 exp − (x − µ) T 2 Σ −1 (x − µ)(5)
for a D-dimensional vector x, the D-dimensional mean-vector µ, and the D × D covariance matrix Σ with determinant | Σ|. Equation (4) is called a Gaussian mixture, the N i (x) are called components of the mixture, and the c i are weight factors. Loosely speaking, the distribution of sample points is "fitted" using a set of high-dimensional Gaussians. A GMM can therefore represent much more complex data sets and can be seen as a generalization of the k-means algorithm for non-circular clusters. One can imagine that it would be straight-forward to fit the multivariate Gaussians to a data set for which the labels are known. With unlabeled data the case is more difficult, and enforces to take into account another step. In the literature, this problem is commonly denoted as to find out which (latent) component is "responsible" for a certain sample, -which is somehow a different way of asking to which cluster the sample belongs. But it underlines that the GMM clustering is a probabilistic approach, because it calculates the probability that the sample was generated by cluster i for all clusters. These probabilities, which are also called responsibilities, are simply the weight factors c i of Eq. (4). In the implementation that was utilized here, the cluster assignment is solved using a method known as expectation-maximization 69,70 . This algorithm starts with a random Gaussian mixture (i.e. random components), which is typically initialized using a prior application of k-means to improve the convergence. In the next step it determines for each sample the probability of being generated by each component of the mixture. Based on these probabilities, the parameters of the Gaussian distributions are fitted to give the best approximation of the data by maximizing their likelihood 67 . This process is executed iteratively and is guaranteed to converge to a local optimum.
Solution quality rating using silhouette coefficients
To give a definition of the silhouette coefficient, let X k i be a sample that was assigned to the cluster k and a(i) be the average dissimilarity of X k i to all other members X k j =i of this cluster. The measure for the dissimilarity is usually the Euclidian distance. Let d(i, m) be the average dissimilarity of X k i to all members of the cluster m = k and b(i) be the minimum of d(i, m) for these clusters, i.e. The cluster m for which this minimum is obtained is called the neighboring cluster of X i . If the number of clusters N k is > 1, we can define the silhouette coefficient s(i) for the sample X i by
s(i) = b(i) − a(i) max{a(i), b(i)} = 1 − a(i)/b(i), if a(i) < b(i) 0, if a(i) = b(i) b(i)/a(i) − 1, if a(i) > b(i) .(6)
From this definition it is seen that the silhouette coefficient s is in the range −1 ≤ s ≤ 1. Values near 1 indicate that the sample is far away from the neighboring cluster and accordingly fits well into its own cluster. A value of 0 indicates that the sample is on or very close to the boundary between its own and the neighboring cluster, and negative values indicate that it might have been assigned to the wrong cluster. A sorted diagram of all silhouette coefficients can thus be used to visualize the representation quality of a clustering. In addition, the average silhouette coefficient for all samples -usually denoted as silhouette score -can be used to compare the representation quality for different clusterings, e.g. using different N k -values. It hence even provides a single numeric value for solution quality assessment.
FIG. 1 .
1Overview of the nanostructure. (a) Light incident on a silicon photonic crystal (PhC, gray) on glass (cyan) excites a leaky mode that exhibits enhanced electromagnetic near-field energies in the superspace volume (marked by the yellow dashed line). Emitters (black dots) in the vicinity of the PhC surface interact with the local electric field distribution. (b) Scanning electron microscopy image of the PhC sample with denoted high-symmetry directions Γ − K and Γ − M . (c) A unit cell of the PhC system as used in the simulation. Yellow, green and red rectangles mark the planes used for the field export.
FIG. 2 .
2Comparison of measured quantum dot fluorescence enhancement F + , simulated volume-integrated field energy enhancement E + , and corresponding classification maps. (Upper row) Measured fluorescence enhancement F + as a function of vacuum wavelength and incident angle θ of the laser source (logarithmic color scale; see Fig. 1(a) which indicates θ; see supplementary material of Ref. 52 for experimental setup). The columns correspond to the four combinations of sample orientation (Γ − M and Γ − K) and source polarization (TE and TM). (Center row) Simulated volume-integrated electric field energy enhancement E + for the same conditions as in the upper row. For a definition of the volume V sup see Fig. 1(a). The white lines mark the experimental data limits. (Lower row) Classification maps depicting the cluster assignments (labels) using different colors independently for each plot, and the respective silhouette coefficients using alpha-blending with a black background (color bar omitted; see Fig. 3 for value ranges). More saturated colors denote larger silhouette coefficients. (Note: Upper and center rows of the panel repeat the same results as already shown in Ref. 52 for a larger angle and wavelength range.)
. 2 repeat the main findings of the prior fluorescence enhancement study 52 . The upper row shows the fluorescence enhancement (F + ) maps obtained by tilting the QD-coated PhC sample along the respective high-symmetry directions of the irreducible Brillouin zone (Γ−M or Γ−K, adjusted using φ), and by using transverse-electric (TE) or transverse-magnetic (TM) polarization P of the incident laser radiation. Each measured spectrum (for a single incident angle) was first integrated over the fluorescence peak from λ = 1200 nm to λ = 1700 nm and normalized to the measured incident laser power and the absorption profile of the QDs, yielding the fluorescence F . A minimum estimate for the fluorescence enhancement F + is obtained from dividing by the minimal value in each of the maps. The maps feature regions of enhanced fluorescence. The measured fluorescence enhancement is caused by increased energy densities of the fields at the emitter positions. Concerning task (i) of the introduction, the center row of Fig. 2 maps the electric field energy enhancement E + integrated over the simulated superspace volume V sup , which contains the QDs (see Eq. (1)). The E + maps exhibit clearly visible bands of strong field energy enhancement, which partly correspond to regions of high measured fluorescence F + . Some deviations are caused by a Q-factor mismatch between the spectral bandwidths of the leaky modes and the excitation laser source. However, a few features of the measured F + maps remained unexplained, for example: • Γ − K, TE: The declining band after the anticrossing point, which is visible in the corresponding E + -map, but missing in the experimental fluorescence enhancement F + . • Γ − M , TE: The elongated bright spot of high fluorescence enhancement at about (10 • , 1100 nm -1140 nm). Please consult Ref. 52 for further details of the comparison.
FIG. 3 .
3Silhouette analysis plots for the different direction/polarization combinations. In each of the silhouette plots (i.e. columns), silhouette coefficients for each sample are plotted as a bar in x-direction with a length corresponding to its value. The samples are sorted by their silhouette coefficients, with smaller values being located at smaller y-positions; and grouped and color-coded using the same colors as inFig. 2. Red dashed lines mark the silhouette scores.
combination of polarization P and azimuthal angle φ (=high symmetry direction Γ − M or Γ − K). Plotted in the same fashion as the E + maps of the center row of Fig. 2, we denote the resulting figures as classification maps. These classification maps are shown in the lower row of the figure. The color scale relates the colors to the labels and, hence, identify the corresponding cluster. Note that the classification maps cannot be compared among each other, although the same colors have been used. The clusterings for the Γ − K-cases used 8 clusters, while the Γ − M -cases only required 7 (i.e. there is no gray region in these maps).
), while the latter uses electric field patterns E(r) on planes that include the PhC and glass domains. When observing the regions off the leaky-mode bands, i.e. the domains of the radiation modes, it is seen that these regions are multiply subdivided in some cases; e.g. Γ − K, TM: bottom left. In contrast, other parts are homogenous over large ranges, such as Γ − M , TE: top right.Another detail of these plots are the different levels of saturation used for each point, obtained by alpha-blending with a black background. This additional layer of information illustrates the representation quality of the local solution by the assigned cluster, as determined using so-called silhouette coefficients 56 . The silhouette coefficients provide a way to assess the initial choice of the number of clusters, and how well the samples lie in their respective clusters, at the same time. The silhouette coefficient rates how well a sample fits into its own cluster. If it is far away from all other clusters and very close to the cluster center (i.e. prototype), the sample gets a positive rating. If the distances to a different cluster and its own cluster are comparable, it is rated with values close to zero. Finally, if it is much closer to a different cluster, a negative rating is assigned. See the methods section "Solution quality rating using silhouette coefficients" for a severe definition.
. In each of the plots, the silhouette coefficients for each sample are plotted as a bar in x-direction with a length corresponding to its value (negative values point into the −x-direction). The samples are sorted by their silhouette coefficients, with smaller values being located at smaller y-positions. In addition, the samples are grouped for each cluster k and color-coded using the same colors as inFig. 2(lower row). The red dashed lines mark the average of all silhouette coefficients, which is a measure for the absolute quality of the representation denoted as silhouette score. The results are the typical "sails" or "shark fins". The width of each fin in the silhouette plots is proportional to the area of the correspondingly labelled points in the classification maps.Considering the distribution of the silhouette coefficients, fins which are not too sharp are observed, i.e. having broad plateaus of high silhouette coefficients. There is only a minimum number of values with negative coefficients. Both arguments together give a validation for the fact that the number of clusters is not underestimated: negative values would occur if there were to few clusters, leaving back samples which do not fit in one of the classes (s ∼ −1). Too many clusters could be identified by a large fluctuation in the fin widths.But this does not fully apply here, as the areas occupied by the bands and the residual parts are unequal. Therefore, equally broad classes are not expected. A slightly too large number of clusters can be seen as unproblematic, because it would basically subdivide the radiation mode regions further, which are of limited relevance for the interpretation. Another point that suggests a good representation is that there are few clusters with below average silhouette scores. Using this reasoning, the optimum number of clusters was determined for each case by comparing the silhouette plots for different number of clusters (results omitted).
(
blue) and the silicon of the PhC (gray) in the case of xz and yz; and a white circle indicating the hole circumference in the case of xy. Recall that the color scales do not give absolute values, as the prototypes are based on normalized data and, therefore, cannot be compared with respect to their absolute amplitudes.For each prototype, the field energy plots on the three planes give a notion of the 3D field energy distribution. The solutions with the same label (color) in the classification maps ofFig. 2all share this distribution type. Lower saturations quantify how much the individual solutions deviate from the prototype. Clusters that correspond to leaky mode bands with strong field enhancement, such as cluster 6 of the Γ − K, TM case (pinkish), have strongly localized energy distributions (see panel 1, column 6 in
E 2
2in the respective plane. The distribution plots feature semitransparent markings for the glass superstrate (blue) and the silicon of the PhC (gray) in the case of xz and yz; and a white circle indicating the hole circumference in the case of xy. (Color scales do not give absolute values, as the prototypes are based on normalized data.)
2 .FIG. 5 .
25Check whether there is an according feature in the experimental fluorescence enhancement (F + ) maps.3. Afterwards, observe the corresponding region in the classification maps and determine the cluster label from the color using the color bar.4. Using this label or color, locate the related column in the prototype map that belongs to the direction/polarization combination (the prototype maps are ordered according to the columns inFig. 2, from left to right). Check if the field energy distributions on the three planes can explain the observed fluorescence enhancement (this may necessitate to take into account all cases, because the QD distribution is unknown).For ease of comprehension, we will analyze the results for selected cases in order of increasing complexity.Γ − M , TM. The experimental fluorescence enhancement (F + ) map features a single stripe of increased fluorescence with a high contrast. This stripe excellently corresponds to the single leaky-mode band causing a high volume-integrated field enhancement in the E + map. The classification map reveals this band accordingly with label 4 (orange), for which the field distributions are shown in column 4 of the prototype map. The xz and yz patterns show that the energy of this band is accumulated at the plateaus between the holes.As the QDs in this experiment are distributed inside the holes and particularly in an about 100 nm -300 nm thick film on top of the structure, they overlap with the leaky mode volume very well. When observing other columns of the prototype map, there are other interesting patterns which could potentially increase the emission of the QDs, for instance in columns 1 and 5. These two modes gather their energy in the center of the hole and at the flanks, respectively. However, when looking at the classification maps again, these correspond to the red and yellow regions, which are outside the measurement window. Another important point is given by the patterns of solutions that are related to radiation modes. These would be expected at regions off any band, e.g. in the dark blue and cyan regions with labels 2 and 7. Returning to the prototype maps, these modes in fact have energy distributions that increase with the distance from the PhC surface, but less dominant in case of the the prototype of cluster 2. Moreover, these modes do not fulfill the necessary condition 2 of the guide given above, i.e. they do not exhibit an integrated field energy enhancement (E + ).Γ − K, TM. We observe a steep band of high fluorescence and a large field energy enhancement (condition 2) at a similar position. The clustering approach found the band as well, labeled as 6 (pinkish). Column 6 of the prototype map shows that this band localizes its energy at the center of the hole (slightly above the PhC in z-direction) but also at the plateaus in the xz plane. It therefore potentially affects QDs relatively independently of whether they are gathered inside the hole or on the plateaus. Again, as the emitters are distributed inside the hole and in a bulk film on top of the structure in the experiment, the overlap with the leaky mode field distribution is good. Returning to the clustering, it is seen that a second, much broader band running from top left to bottom right is seen in the E + maps. The classification maps reveal that the field distribution of this band undergoes a change when crossing the pinkish band, from label 7 (cyan) to label 1 (red). This band is basically not seen in the fluorescence enhancement.Γ−K, TE. The TE cases both feature a more complicated band structure. In the Γ−K, TE case, there are two very clear bands that show anticrossing, a steeper band crossing the complete wavelength range from roughly 20 • to 40 • , and a shallower one coming from top left. The former is very clearly seen in the clustering by the gray region with label 8.From the prototype map it is observed that this band has a node along the x-direction and concentrates its energy at the flanks and the plateaus in y-direction. The energy distribution is therefore comparable to the one of the orange band in the Γ−M , TM case, which is clearly seen in the experimentally measured fluorescence as well. The shallower mentioned band undergoes a transition from the green cluster (label 3) to the red cluster (label 1) in the classification maps. For the green parts the energy is strongly localized at the flanks, while Full-3D volume renderings of selected modes for the Γ−K, TE case. Semi-artistic ray tracing images depicting multiple periods of the photonic crystal as a grayish material. The upper row shows a topview of the full-3D E-field energy density, color-coded using a heat map (not comparable between the figures). The lower row shows a closer view and indicates the same random distribution of quantum dots (bright small spheres), emitting white light with an intensity proportional to the field energy density at their specific positions. The columns relate to three different modes of the Γ − K/TE case, denoted as modes A, B and C, as marked inFig. 4. The modes are the actual solutions from the finite-element solver that have the smallest deviations from the assigned prototype (i.e. cluster center), determined using the silhouette coefficients. Incident angle θ and wavelength λ for each mode are given in the headings. The figures use real physical proportions.
Γ
− M , TE. The E + -maps cover two bands that show anticrossing at the long wavelength/steep angle end of the data window (top left). The lower band and most parts of the interaction zone are labeled as cluster 3 (green), while the upper band has label 7 (cyan).
=
|E j (r i , θ m , λ l )|, where the additional indices m = 0 . . . N θ and l = 0 . . . N λ have been introduced, and where the hat denotes the absolute value and normalization. The input matrix then reads
ACKNOWLEDGMENTSThe authors thank Klaus Jäger from Helmholtz-Zentrum Berlin for useful discussions.
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| []
|
[
"Learning to Recognize Dialect Features",
"Learning to Recognize Dialect Features"
]
| [
"Dorottya Demszky [email protected] \nStanford Linguistics\n\n",
"Devyani Sharma [email protected] \nQueen Mary University of London\n\n",
"Jonathan H Clark [email protected] \nGoogle Research\n\n",
"Vinodkumar Prabhakaran \nGoogle Research\n\n",
"Jacob Eisenstein [email protected] \nGoogle Research\n\n"
]
| [
"Stanford Linguistics\n",
"Queen Mary University of London\n",
"Google Research\n",
"Google Research\n",
"Google Research\n"
]
| [
"Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies"
]
| Building NLP systems that serve everyone requires accounting for dialect differences. But dialects are not monolithic entities: rather, distinctions between and within dialects are captured by the presence, absence, and frequency of dozens of dialect features in speech and text, such as the deletion of the copula in "He ∅ running". In this paper, we introduce the task of dialect feature detection, and present two multitask learning approaches, both based on pretrained transformers. For most dialects, largescale annotated corpora for these features are unavailable, making it difficult to train recognizers. We train our models on a small number of minimal pairs, building on how linguists typically define dialect features. Evaluation on a test set of 22 dialect features of Indian English demonstrates that these models learn to recognize many features with high accuracy, and that a few minimal pairs can be as effective for training as thousands of labeled examples. We also demonstrate the downstream applicability of dialect feature detection both as a measure of dialect density and as a dialect classifier.Janneke Van Hofwegen and Walt Wolfram. 2010. Coming of age in African American English: A longitudinal study. | 10.18653/v1/2021.naacl-main.184 | [
"https://www.aclweb.org/anthology/2021.naacl-main.184.pdf"
]
| 225,066,916 | 2010.12707 | 5a9c0c322ed2006a4952f6acfcfa1ac52c8540c8 |
Learning to Recognize Dialect Features
June 6-11, 2021
Dorottya Demszky [email protected]
Stanford Linguistics
Devyani Sharma [email protected]
Queen Mary University of London
Jonathan H Clark [email protected]
Google Research
Vinodkumar Prabhakaran
Google Research
Jacob Eisenstein [email protected]
Google Research
Learning to Recognize Dialect Features
Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies
the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language TechnologiesJune 6-11, 20212315
Building NLP systems that serve everyone requires accounting for dialect differences. But dialects are not monolithic entities: rather, distinctions between and within dialects are captured by the presence, absence, and frequency of dozens of dialect features in speech and text, such as the deletion of the copula in "He ∅ running". In this paper, we introduce the task of dialect feature detection, and present two multitask learning approaches, both based on pretrained transformers. For most dialects, largescale annotated corpora for these features are unavailable, making it difficult to train recognizers. We train our models on a small number of minimal pairs, building on how linguists typically define dialect features. Evaluation on a test set of 22 dialect features of Indian English demonstrates that these models learn to recognize many features with high accuracy, and that a few minimal pairs can be as effective for training as thousands of labeled examples. We also demonstrate the downstream applicability of dialect feature detection both as a measure of dialect density and as a dialect classifier.Janneke Van Hofwegen and Walt Wolfram. 2010. Coming of age in African American English: A longitudinal study.
Introduction
Dialect variation is a pervasive property of language, which must be accounted for if we are to build robust natural language processing (NLP) systems that serve everyone. Linguists do not characterize dialects as simple categories, but rather as collections of correlated features (Nerbonne, 2009), such as the one shown in Figure 1; speakers of any given dialect vary regarding which features they employ, how frequently, and in which contexts. In comparison to approaches that classify speakers or documents across dialects (typically using metadata such as geolocation), the feature-based perspective has several advantages: (1) allowing for fine-grained comparisons of speakers or documents * Work done while at Google Research. within dialects, without training on personal metadata;
(2) disentangling grammatical constructions that make up the dialect from the content that may be frequently discussed in the dialect; (3) enabling robustness testing of NLP systems across dialect features, helping to ensure adequate performance even on cases other than "high-resource" varieties such as mainstream U.S. English (Blodgett et al., 2016); (4) helping to develop more precise characterizations of dialects, enabling more accurate predictions of variable language use and better interpretations of its social implications (e.g., Craig and Washington, 2002;Van Hofwegen and Wolfram, 2010). The main challenge for recognizing dialect features computationally is the lack of labeled data. Annotating dialect features requires linguistic expertise and is prohibitively time-consuming given the large number of features and their sparsity. In dialectology, large-scale studies of text are limited to features that can be detected using regular expressions of surface forms and parts-of-speech, e.g., PRP DT for the copula deletion feature in Figure 1; many features cannot be detected with such patterns (e.g. OBJECT FRONTING, EXTRANEOUS ARTICLE). Furthermore, part-of-speech tagging is unreliable in many language varieties, such as re-gional and minority dialects (Jørgensen et al., 2015;Blodgett et al., 2016). As dialect density correlates with social class and economic status (Sahgal and Agnihotri, 1988;Rickford et al., 2015;Grogger et al., 2020), the failure of language technology to cope with dialect differences may create allocational harms that reinforce social hierarchies (Blodgett et al., 2020).
In this paper, we propose and evaluate learningbased approaches to recognize dialect features. We focus on Indian English, given the availability of domain expertise and labeled corpora for evaluation. First, we consider a standard multitask classification approach, in which a pretrained transformer (Vaswani et al., 2017) is fine-tuned to recognize a set of dialect features. The architecture can be trained from two possible sources of supervision: (1) thousands of labeled corpus examples, (2) a small set of minimal pairs, which are hand-crafted examples designed to highlight the key aspects of each dialect feature (as in the "typical example" field of Figure 1). Because most dialects have little or no labeled data, the latter scenario is more realistic for most dialects. We also consider a multitask architecture that learns across multiple features by encoding the feature names, similar to recent work on few-shot or zero-shot multitask learning (Logeswaran et al., 2019;Brown et al., 2020).
In Sections 4 and 5, we discuss empirical evaluations of these models. Our main findings are:
• It is possible to detect individual dialect features: several features can be recognized with reasonably high accuracy. Our best models achieve a macro-AUC of .848 across ten grammatical features for which a large test set is available.
• This performance can be obtained by training on roughly five minimal pairs per feature. Minimal pairs are significantly more effective for training than a comparable number of corpus examples.
• Dialect feature recognizers can be used to rank documents by their density of dialect features, enabling within-dialect density computation for Indian English and accurate classification between Indian and U.S. English.
Data and Features of Indian English
We develop methods for detecting 22 dialect features associated with Indian English. Although India has over 125 million English speakers -making it the world's second largest English-speaking population -there is relatively little NLP research focused on Indian English. Our methods are not designed exclusively for specific properties of Indian English; many of the features that are associated with Indian English are also present in other dialects of English. We use two sources of data in our study: an annotated corpus ( § 2.1) and a dataset of minimal pairs ( § 2.2). For evaluation, we use corpus annotations exclusively. The features are described in Table 1, and our data is summarized in Table 2.
Corpus Annotations
The International Corpus of English (ICE; Greenbaum and Nelson, 1996) is a collection of corpora of world varieties of English, organized primarily by the national origin of the speakers/writers. We focus on annotations of spoken dialogs (S1A-001 -S1A-090) from the Indian English subcorpus (ICE-India). The ICE-India subcorpus was chosen in part because it is one of the only corpora with large-scale annotations of dialect features. To contrast Indian English with U.S. English ( § 4), we use the Santa Barbara Corpus of Spoken American English (Du Bois et al., 2000) that constitutes the ICE-USA subcorpus of spoken dialogs.
We work with two main sources of dialect feature annotations in the ICE-India corpus:
Lange features. The first set of annotations come from Claudia Lange (2012), who annotated 10 features in 100 transcripts for an analysis of discoursedriven syntax in Indian English, such as topic marking and fronting. We use half of this data for training (50 transcripts, 9392 utterances), and half for testing (50 transcripts, 9667 utterances).
Extended features. To test a more diverse set of features, we additionally annotated 18 features on a set of 300 turns randomly selected from the conversational subcorpus of ICE-India, 2 as well as 50 examples randomly selected from a secondary dataset of sociolinguistic interviews (Sharma, 2009) to ensure diverse feature instantiation. We selected our 18 features based on multiple criteria: 1) prevalence in Indian English based on the dialectology literature, 2) coverage in the data (we started out with a larger set of features and removed those with fewer than two occurrences), 3) diversity of linguistic phenomena. The extended
Minimal Pairs
For each of the 22 features in Table 1, we created a small set of minimal pairs. The pairs were created by first designing a short example that demonstrated the feature, and then manipulating the example so that the feature is absent. This "negative" example captures the envelope of variation for the feature, demonstrating a site at which the feature could be applied (Labov, 1972 For most features, each minimal pair contains exactly one positive and one negative example. However, in some cases where more than two variants are available for an example (e.g., for the feature INVARIANT TAG (isn't it, no, na)), we provide multiple positive examples to illustrate different variants. For Lange's set of 10 features, we provide a total of 113 unique examples; for the 18 extended features, we provide a set of 208 unique examples, roughly split equally between positives and negatives. The complete list of minimal pairs is included in Appendix D.
Models and training
We train models to recognize dialect features by fine-tuning the BERT-base uncased transformer architecture (Devlin et al., 2019). We consider two strategies for constructing training data, and two architectures for learning across multiple features.
Sources of supervision
We consider two possible sources of supervision:
Minimal pairs. We apply a simple procedure to convert minimal pairs into training data for classification. The positive part of each pair is treated as a positive instance for the associated feature, and the negative part is treated as a negative instance. Then, to generate more data, we also include elements of other minimal pairs as examples for each feature: for instance, a positive example of the RESUMPTIVE OBJECT PRONOUN feature would be a negative example for FOCUS only, unless the example happened to contain both features (this was checked manually). In this way, we convert the minimal pairs into roughly 113 examples per feature for Lange's features and roughly 208 examples per feature for the extended features. The total number of unique surface forms is still 113 and 208 respectively. Given the lack of labeled data for most dialects of the world, having existing minimal pairs or collecting a small number of minimal pairs is the most realistic data scenario.
Corpus annotations. When sufficiently dense annotations are available, we can train a classifier based on these labeled instances. We use 50 of the ICE-India transcripts annotated by Lange, which consists of 9392 labeled examples (utterances) per feature. While we are lucky to have such a large resource for the Indian English dialect, this highresource data scenario is rare.
Architectures
We consider two classification architectures:
Multihead. In this architecture, which is standard for multitask classification, we estimate a linear prediction head for each feature, which is simply a vector of weights. This is a multitask architecture, because the vast majority of model parameters from the input through the deep BERT stack remain shared among dialect features. The prediction head is then multiplied by the BERT embedding for the [CLS] token to obtain a score for a feature's applicability to a given instance.
DAMTL.
Due to the few-shot nature of our prediction task, we also consider an architecture that attempts to exploit the natural language descriptions of each feature. This is done by concatenating the feature description to each element of the minimal pair. The instance is then labeled for whether the feature is present. This construction is shown in Figure 2. Prediction is performed by learning a single linear prediction head on the [CLS] token. We call this model description-aware multitask learning, or DAMTL.
Model details. Both architectures are built on top of the BERT-base uncased model, which we fine-tune by cross-entropy for 500 epochs (due to the small size of the training data) using the Adam optimizer (Kingma and Ba, 2014), batch size of 32 and a learning rate of 10 −5 , warmed up over the first 150 epochs. Annotations of dialect features were not used for hyperparameter selection. Instead, the hyperparameters were selected to maximize the discriminability between corpora of Indian and U.S. English, as described in § 5.2. All models trained in less than two hours on a pod of four v2 TPU chips, with the exception of DAMTL on corpus examples, which required up to 18 hours.
Regular Expressions
In dialectology, regular expression pattern matching is the standard tool for recognizing dialect features (e.g., Nerbonne et al., 2011). For the features
Results on Dialect Feature Detection
In this section, we present results on the detection of individual dialect features. Using the features shown in Table 1, we compare supervision sources (corpus examples versus minimal pairs) and classification architectures (multihead versus DAMTL) as described in § 3. To avoid tuning a threshold for detection, we report area under the ROC curve (ROC-AUC), which has a value of .5 for random guessing and 1 for perfect prediction. 5
Results on Lange Data and Features
We first consider the 10 syntactic features from Lange (2012), for which we have large-scale annotated data: the 100 annotated transcripts from the ICE-India corpus are split 50/50 into training and test sets. As shown in situations are by far the most common data scenario among the dialects of the world.
The multihead architecture outperforms DAMTL on both corpus examples and minimal pairs. In an ablation, we replaced the feature descriptions with non-descriptive identifiers such as "Feature 3". This reduced the Macro-AUC from to .80 with corpus examples, and to .76 with minimal pairs (averaged over five random seeds). We also tried longer feature descriptions, but this did not improve performance.
Unsurprisingly, the lexical features (e.g., FOCUS itself ) are easiest to recognize. The more syntactical features (e.g., COPULA OMISSION, RESUMP-TIVE OBJECT PRONOUN) are more difficult, although some movement-based features (e.g., LEFT
DISLOCATION, RESUMPTIVE SUBJECT PRONOUN)
can be recognized accurately.
Qualitative model comparison. We conducted a qualitative comparison of three models: regular expressions and two versions of the multihead model, one trained on corpus examples and another trained on minimal pairs. Table 4 includes illustrative examples for the Lange data and features where models make different predictions. We find that the minimal pair model is better able to account for rare cases (e.g. use of non-focus "only" in Example 1), likely as it was trained on a few carefully selected set of examples illustrating positives and negatives. Both multihead models are able to account for disfluencies and restarts, in contrast to regular expressions (Example 2). Our analysis shows that several model errors are accounted for by difficult examples (Example 3: "is there" followed by "isn't"; Example 6: restart mistaken for left dislocation) or the lack of contextual information available to the model (Example 4 & 7: truncated examples). Please see Appendix B for more details and random samples of model predictions.
Learning from fewer corpus examples. The minimal pair annotations consist of 113 examples; in contrast, there are 9392 labeled corpus examples, requiring far more effort to create. We now consider the situation when the amount of labeled data is reduced, focusing on the Lange features (for which labeled training data is available). As shown in Figure 3, even 5000 labeled corpus examples do not match the performance of training on roughly 5 minimal pairs per feature. Corpus examples stratified by feature. One reason that subsampled datasets yield weaker results is that they lack examples for many features. To enable a more direct comparison of corpus examples and minimal pairs, we created a set of "stratified" datasets of corpus examples, such that the number of positive and negative examples for each feature exactly matches the minimal pair data. Averaged over ten such random stratified samples, the multihead model achieves a Macro-AUC of .790 (σ = 0.029), and DAMTL achieves a Macro-AUC of .722 (σ = .020). These results are considerably worse than training on an equivalent number of minimal pairs, where the multihead model achieves a Macro-AUC of .848 and DAMTL achieves a Macro-AUC of .783. This demonstrates the utility of minimal pairs over corpus examples for learning to recognize dialect features.
Results on Extended Feature Set
Next, we consider the extended features, for which we have sufficient annotations for testing but not training (Table 1). Here we compare the DAMTL and multihead models, using minimal pair data in both cases. As shown in Table 5, performance on these features is somewhat lower than on the Lange features, and for several features, at least one of the recognizers does worse than chance: DIRECT OB-JECT PRO-DROP, EXTRANEOUS ARTICLE, MASS NOUNS AS COUNT NOUNS. These features seem to require deeper syntactic and semantic analysis, which may be difficult to learn from a small number of minimal pairs. On the other extreme, features with a strong lexical signature are recognized with high accuracy: GENERAL EXTENDER and all, FO-CUS itself , FOCUS only. These three features can also be recognized by regular expressions, as can However, for a number of other features, it is possible to learn a fairly accurate recognizer from just five minimal pairs: (Benor, 2010). This necessitates a more nuanced description for speakers and texts than a discrete dialect category. Following prior work (e.g., Van Hofwegen and Wolfram, 2010) we construct dialect density measures from feature detectors by counting the predicted number of features in each utterance, and dividing by the number of tokens. For the learningbased feature detectors (minimal pairs and corpus examples), we include partial counts from the detection probability; for the regular expression detectors, we simply count the number of matches and dividing by the number of tokens. In addition, we construct a DDM based on a document classifier: we train a classifier to distinguish Indian English from U.S. English, and then use its predictive probability as the DDM. These DDMs are then compared on two tasks: distinguishing Indian and U.S. English, and correlation with the density of expert-annotated features. The classifier is trained by fine-tuning BERT, using a prediction head on the [CLS] token.
Ranking documents by dialect density
One application of dialect feature recognizers is to rank documents based on their dialect density, e.g. to identify challenging cases for evaluating downstream NLP systems, or for dialectology research. We correlate the dialect density against the density of expert-annotated features from Lange (2012), both measured at the transcript-level, and report the Spearman rank-correlation ρ.
As shown in Table 6, the document classifier performs poorly: learning to distinguish Indian and U.S. English offers no information on the density of Indian dialect features, suggesting that the model is attending to other information, such as topics or entities. The feature-based model trained on labeled examples performs best, which is unsurprising because it is trained on the same type of features that it is now asked to predict. Performance is weaker when the model is trained from minimal pairs. Minimal pair training is particularly helpful on rare features, but offers far fewer examples on the high-frequency features, which in turn dominate the DDM scores on test data. Regular expressions perform well on this task, because we happen to have regular expressions for the highfrequency features, and because the precision issues are less problematic in aggregate when the DDM is not applied to non-dialectal transcripts.
Dialect Classification
Another application of dialect feature recognizers is to classify documents or passages by dialect (Dunn, 2018). This can help to test the performance of downstream models across dialects, assessing dialect transfer loss (e.g., Blodgett et al., 2016), as well as identifying data of interest for manual dialectological research. We formulate a classification problem using the ICE-India and the Santa Barbara Corpus (ICE-USA). Each corpus is divided into equal-size training and test sets. The training corpus was also used for hyperparameter selection for the dialect feature recognition models, as described in § 3.2.
The dialect classifier was constructed by building on the components from § 5.1. For the test set, we measure the D ("D-prime") statistic (Macmillan and Creelman, 1991),
D = µ IN − µ US 1 2 (σ 2 IN + σ 2 US )
.
(1)
This statistic, which can be interpreted similarly to a Z-score, quantifies the extent to which a metric distinguishes between the two populations. We also report classification accuracy; lacking a clear way to set a threshold, for each classifier we balance the number of false positives and false negatives. As shown in Table 6, both the document classifier and the corpus-based feature detection model (trained on labeled examples) achieve high accuracy at discriminating U.S. and Indian English. The D discriminability score is higher for the document classifier, which is trained on a cross-entropy objective that encourages making confident predictions. Regular expressions suffer from low precision because they respond to surface cues that may be present in U.S. English, even when the dialect feature is not present (e.g., the word "only", the phrase "is there").
Related Work
Dialect classification. Prior work on dialect in natural language processing has focused on distinguishing between dialects (and closely-related languages). For example, the VarDial 2014 shared task required systems to distinguish between nationlevel language varieties, such as British versus U.S. English, as well as closely-related language pairs such as Indonesian versus Malay (Zampieri et al., 2014); later evaluation campaigns expanded this (2015) designed lexical patterns to identify non-standard spellings that match known phonological variables from AAVE (e.g., sholl 'sure'), demonstrating the presence of these variables in social media posts from regions with high propor-tions of African Americans. Blodgett et al. (2016) use the same geography-based approach to test for phonological spellings and constructions corresponding to syntactic variables such as habitual be; Hovy et al. (2015) show that a syntactic feature of Jutland Danish can be linked to the geographical origin of product reviews. These approaches have focused mainly on features that could be recognized directly from surface forms, or in some cases, from part-of-speech (POS) sequences. In contrast, we show that it is possible to learn to recognize features from examples, enabling the recognition of features for which it is difficult or impossible to craft surface or POS patterns. (2020) to improve data efficiency, but is methodologically closer to probing work that uses minimal pairs to represent specific linguistic features.
Conclusion
We introduce the task of dialect feature detection and demonstrate that it is possible to construct dialect feature recognizers using only a small number of minimal pairs -in most cases, just five positive and negative examples per feature. This makes it possible to apply computational analysis to the many dialects for which labeled data does not exist. Future work will extend this approach to multiple dialects, focusing on cases in which features are shared across two or more dialects. This lays the groundwork for the creation of dialectbased "checklists" (Ribeiro et al., 2020) to assess the performance of NLP systems across the diverse range of linguistic phenomena that may occur in any given language.
Ethical Considerations
Our objective in building dialect feature recognizers is to aid developers and researchers to effectively benchmark NLP model performance across and within different dialects, and to assist social scientists and dialectologists studying dialect use. The capability to detect dialectal features may enable developers to test for and mitigate any unintentional and undesirable biases in their models towards or against individuals speaking particular dialects. This is especially important because dialect density has been documented to correlate with lower socioeconomic status (Sahgal and Agnihotri, 1988). However, this technology is not without its risks. As some dialects correlate with ethnicities or countries of origin, there is a potential dual use risk of the technology being used to profile individuals. Dialect features could also be used as predictors in downstream tasks; as with other proxies of demographic information, this could give the appearance of improving accuracy while introducing spurious correlations and imposing disparate impacts on disadvantaged groups. Hence we recommend that developers of this technology consider downstream use cases, including malicious use and misuse, when assessing the social impact of deploying and sharing this technology.
The focus on predefined dialect features can introduce a potential source of bias if the feature set is oriented towards the speech of specific subcommunities within a dialect. However, analogous issues can arise in fully data-driven approaches, in which training corpora may also be biased towards subcommunities of speakers or writers. The feature-based approach has the advantage of making any such bias easier to identify and correct. A Regular Expressions Table 7 shows the regular expressions that we used for the five features, where such patterns were available.
B Sample Outputs
The examples below represent a random sample of the multihead models' outputs for Lange's features, comparing the one that is trained on corpus examples (CORPUS) to the one that is trained on minimal pairs (MINPAIR). We show true positives (TP), false positives (FP) and false negatives (FN). We randomly sample three examples for each output type (TP, FP, FN) and model (BOTH, CORPUS only, MINPAIR only). Our manual inspection shows a few errors in the human annotation by Lange and that certain false positives should be true positives, especially for FO-CUS only. We highlight such examples in green . Among the rest of the false positives and false negatives, a large proportion of errors can be explained by contextual information that is not available to the models. For example, without context it is ambiguous whether "we possess only" is an example of FOCUS only. Inspection of context shows that it is a truncated utterance, representing a standard use of only, hence it is correctly characterized as a false positive. Another source of confusion to the model is missing punctuation. For example "Both girls I have never left them alone till now" could be construed as OBJECT FRONTING with RESUMP-TIVE OBJECT PRONOUN. However, in the original context, the example consists of multiple sentences: "Two kids. Both girls. I have never left them alone till now." We removed punctuation from examples, since in many cases automatic ASR models do not produce punctuation either. However, this example demonstrates that punctuation can provide valuable information about clause and phrase boundaries, and should be included if possible.
Figure 1 :
1Feature area: Agreement Typical example: He Ø a good teacher.176. Deletion of copula be: before NPs An example dialect feature from the Electronic World Atlas of Varieties of English (eWAVE). 1
Figure 2 :
2Conversion of minimal pairs to labeled examples for DAMTL, using two minimal pairs.
Figure 3 :
3Performance of the multihead model as the number of corpus examples is varied. Box plots are over 10 random data subsets, showing the 25th, 50th, and 75th percentiles; whiskers show the most extreme points within ±1.5 times the inter-quartile range.
Minimal pairs in NLP. A distinguishing aspect of our approach is the use of minimal pairs rather than conventional labeled data. Minimal pairs are well known in natural language processing from the Winograd Schema(Levesque et al., 2012), which is traditionally used for evaluation, butKocijan et al. (2019) show that fine-tuning on a related dataset of minimal pairs can improve performance on the Winograd Schema itself. A similar idea arises in counterfactually-augmented data (Kaushik et al., 2019) and contrast sets(Gardner et al., 2020), in which annotators are asked to identify the minimal change to an example that is sufficient to alter its label. However, those approaches use counterfactual examples to augment an existing training set, while we propose minimal pairs as a replacement for large-scale labeled data. Minimal pairs have also been used to design controlled experiments and probe neural models' ability to capture various linguistic phenomena(Gulordava et al., 2018; Ettinger et al., 2018; Futrell et al., 2019; Gardner et al., 2020; Schuster et al., 2020). Finally, Liang et al. (2020) use contrastive explanations as part of an active learning framework to improve data efficiency. Our work shares the objective of Liang et al.
B. 1
1Focus itself [TP:BOTH] We are feeling tired now itself [TP:BOTH] Coach means they should be coached from when they are in nursery UKG itself [TP:BOTH] I'm in final year but like they have started from first year itself [TP:CORPUS] And she got a chance of operating also during her internship itself nice and because that Cama hospital is for ladies only so she has lot of experience [TP:MINPAIR] But even if they women is are working as much as a man she is earning the same monthly saving as a man itself [TP:MINPAIR] You go around say one O'clock and then go for a movie and come back in the evening itself you see you [FP:MINPAIR] And primarily you know the the issue orders were issued on fifth that is on the election day itself [FP:MINPAIR] That is to we take on the coughs our human blood itself [FP:MINPAIR] Now since you are doing the PGCT now after going back is it possible for you to use simple English in the classroom itself [FN:BOTH] All the sums were there in the text book itself but still they have not done properly in the exam [FN:BOTH] And thinking about dissection hall itself they really get scared and that also in the midnight [FN:BOTH] Means what do you think that the basic itself is not good or now they are getting interest in maths [FN:CORPUS] But even if they women is are working as much as a man she is earning the same monthly saving as a man itself [FN:CORPUS] You go around say one O'clock and then go for a movie and come back in the evening itself you see you [FN:MINPAIR] And she got a chance of operating also during her internship itself nice and because that Cama hospital is for ladies only so she has lot of experience
Table 1 :
1Features of Indian English used in our evaluations and their counts in the two datasets we study.Dialect features
Unique annotated examples
Feature set
Count Corpus ex.
Min. pair ex.
Lange (2012)
10
19059
113
Extended
18
367
208
Table 2 :
2Summary of our labeled data. All corpus examples for the Lange features are from ICE-India; for the Extended feature set, examples are drawn from ICE-India and the Sharma data.features overlap with those annotated by Lange,
yielding a total set of 22 features. Annotations
were produced by consensus from the first two
authors. To measure interrater agreement, a third
author (JE) independently re-annotated 10% of the
examples, with Cohen's κ = 0.79 (Cohen, 1960). 3
ARTICLE OMISSION: chair is black → the chair is black FOCUS only: I was there yesterday only → I was there just yesterday. NON-INITIAL EXISTENTIAL: every year inflation is there → every year there is inflation.). Consequently,
negative examples in minimal pairs carry more in-
formation than in the typical annotation scenario,
where absence of evidence does not usually im-
ply evidence of absence. In our minimal pairs, the
negative examples were chosen to be acceptable
in standard U.S. and U.K. English, and can thus
be viewed as situating dialects against standard
varieties. Here are some example minimal pairs:
Table 3 :
3ROC-AUC scores on the Lange feature set, av-
eraged across five random seeds. Asterisk (*) marks
features that can be detected with relatively high accu-
racy (> 0.85 ROC-AUC) using regular expressions.
described in Table 1, we were able to design reg-
ular expressions for only five. 4 Prior work some-
times relies on patterns that include both surface
forms and part-of-speech (e.g., Bohmann, 2019),
but part-of-speech cannot necessarily be labeled
automatically for non-standard dialects (Jørgensen
et al., 2015; Blodgett et al., 2016), so we consider
only regular expressions over surface forms.
Table 3 ,
3Features: FOCUS itself , FOCUS only, NON-INITIAL EX-ISTENTIAL, INVARIANT TAG (isn't it, no, na), and GENERAL EXTENDER and all. Table 7 lists all regular expressions. 5 Results for area under the precision-recall (AUPR) curve are shown in Appendix C. According to this metric, minimal pairs are less effective than the full training set of corpus examples, on average.it is possible to
achieve a Macro-AUC approaching .85 overall with
multihead predictions on minimal pair examples.
This is promising, because it suggests the possi-
bility of recognizing dialect features for which we
lack labeled corpus examples -and such low-data
4
Table 4 :
4Example model predictions from the Lange data and feature set, comparing regular expressions with two versions of the multihead model, one trained on corpus examples and another on minimal pairs. 'Gold label' indicates whether the feature was manually labeled as present in the original Lange data. Green and red indicate correct and incorrect predictions, respectively.
ARTICLE OMISSION, INVERSION IN EMBEDDED CLAUSE, LEFT DISLOCATION, LACK OF INVER-SION IN WH-QUESTIONS.Dialect feature
DAMTL Multihead
ARTICLE OMISSION
0.581
0.658
DIRECT OBJECT PRO-DROP
0.493
0.563
EXTRANEOUS ARTICLE
0.546
0.465
FOCUS itself *
1.000
0.949
FOCUS only*
0.998
0.775
HABITUAL PROGRESSIVE
0.439
0.718
INVARIANT TAG
0.984
0.901
INVERSION IN EMBEDDED CLAUSE
0.719
0.884
LACK OF AGREEMENT
0.543
0.674
LACK OF INVERSION IN WH-QUESTIONS
0.649
0.660
LEFT DISLOCATION
0.758
0.820
MASS NOUNS AS COUNT NOUNS
0.443
0.465
NON-INITIAL EXISTENTIAL*
0.897
0.885
OBJECT FRONTING
0.722
0.789
PREPOSITION OMISSION
0.500
0.648
PP FRONTING WITH REDUCTION
0.655
0.697
STATIVE PROGRESSIVE
0.645
0.789
GENERAL EXTENDER and all
0.994
0.991
Macro Average
0.698
0.741
Table 5 :
5ROC-AUC results on the extended feature set, averaged across five random seeds. Because labeled corpus examples are not available for some features, we train only on minimal pairs. Asterisk (*) marks features that can be detected with relatively high accuracy (> 0.85 ROC-AUC) using regular expressions.4.3 Summary of Dialect Feature DetectionMany dialect features can be automatically recognized with reasonably high discriminative power, as measured by area under the ROC curve. However, there are also features that are difficult to recognize: particularly, features of omission (such as DIRECT OBJECT PRO-DROP and PREPOSITION OMISSION), and the more semantic features such as MASS NOUNS AS COUNT NOUNS. While some features can also be identified through regular expressions (e.g., FOCUS only), there are many features that can be learned but cannot be recognized by regular expressions. We now move from individual features to aggregate measures of dialect density. Measuring Dialect Density A dialect density measure (DDM) is an aggregate over multiple dialect features that tracks the vernacularity of a passage of speech or text. Such measures are frequently used in dialectology(Van Hofwegen and Wolfram, 2010), and are also useful in research on education (e.g.,Craig and Washington, 2002). Recently, a DDM was used to evaluate the performance of speech recognition systems by the density of AAVE features(Koenecke et al., 2020). The use of DDMs reflects the reality that speakers construct individual styles drawing on linguistic repertoires such as dialects to varying degrees5
Table 6 :
6Performance of dialect density measures at the
tasks of ranking Indian English transcripts by dialect
density (quantified by Spearman ρ) and distinguishing
Indian and U.S. English transcripts (quantified by accu-
racy and D discriminability).
set to other varieties (Zampieri et al., 2017). In
general, participants in these shared tasks have
taken a text classification approach; neural architec-
tures have appeared in the more recent editions of
these shared tasks, but with a few exceptions (e.g.,
Bernier-Colborne et al., 2019), they have not out-
performed classical techniques such as support
vector machines. Our work differs by focusing
on a specific set of known dialect features, rather
than document-level classification between dialects,
which aligns with the linguistic view of dialects as
bundles of correlated features (Nerbonne, 2009)
and tracks variable realization of features within
dialect usage.
Discovering and detecting dialect features.
Machine learning feature selection techniques have
been employed to discover dialect features from
corpora. For example, Dunn (2018, 2019) induces
a set of constructions (short sequences of words,
parts-of-speech, or constituents) from a "neutral"
corpus, and then identifies constructions with dis-
tinctive distributions over the geographical subcor-
pora of the International Corpus of English (ICE).
In social media, features of African American Ver-
nacular English (AAVE) can be identified by corre-
lating linguistic frequencies with the aggregate de-
mographic statistics of the geographical areas from
which geotagged social media was posted (Eisen-
stein et al., 2011; Stewart, 2014; Blodgett et al.,
2016). In contrast, we are interested in detecting
predefined dialect features from well-validated re-
sources such as dialect atlases.
Along these lines, Jørgensen et al. (2015) and
Jones
Baldridge, Dan Jurafsky, Slav Petrov, Jason Riesa, Kristina Toutanova, and especially Vera Axelrod. Thanks also to the anonymous reviewers. Devyani Sharma is supported in part by a Google Faculty Research Award.References
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Feature
Regular expression
FOCUS itself
\bitself\b
FOCUS only
\bonly\b
NON-INITIAL EXISTENTIAL
\bis there\b|\bare there\b
INVARIANT TAG (isn't it, no, na) \bisn't it\b|\bis it\b|\bno\b|\bna\b
GENERAL EXTENDER and all
\band all\b
Table 7 :
7Regular expressions we used, for the features that such patterns were available.
B . 2
.Focus only [TP:BOTH] All the types only [TP:BOTH] Hey you sur be like that only [TP:BOTH] suddenly it will be become perfect only [TP:CORPUS] That is I like dressing up I told you at the beginning only [TP:CORPUS] Because today only he had come and I've got up today at nine thirty [TP:CORPUS] Actually from childhood only I was brought up in the same atmosphere like if Papa still has shifted to another place I would have got the feeling of not having comfortable in a particular language but on the whole I think it doesn't matter exactly how we go about chosing or selecting a language [TP:MINPAIR] it was bit it was difficult only [TP:MINPAIR] I'm one minute I've got it in front of me only [TP:MINPAIR] He is in our college only [FP:BOTH] Because we are supposed to perform well there only then [FP:BOTH] Ho Ho Hollywood Hollywood after Hollywood it seems India only [FP:BOTH] No he'll be there in the campus only [FP:CORPUS] Oh God there only it's happening so and forget about [FP:CORPUS] The thing is that it is rural area only but the people are from all over india they are staying here [FP:CORPUS] Not much work these days because first week and last week only we've quiet good business [FP:MINPAIR] Only in India there is manual work [FP:MINPAIR] Film hits only [FP:MINPAIR] So Bharati Vidya Bhavan people have such type of persons only [FN:BOTH] If they be in always that this is there are not improve no improvement only [FN:BOTH] When we were living when I was living in Kashmir no I was brought up there only and everything is [FN:BOTH] This is the first phase then in the second phase we have some clinical subjects in which we come in direct contact with the patients but it's on two basis like when we see the patients at the same time we study about the pathology only the pathology and then we learn about some of the drugs which are to be which are used for their treatment [FN:CORPUS] No you must put apply science only [FN:CORPUS] Actually they are good only [FN:CORPUS] it was bit it was difficult only [FN:MINPAIR] My both the parents are farmers only [FN:MINPAIR] Because today only he had come and I've got up today at nine thirty [FN:MINPAIR] That is I like dressing up I told you at the beginning only B.3 Invariant Tag (isn't it, no, na) [TP:BOTH] Very difficult once the school starts na very difficult [TP:BOTH] I am okay rainy season no [TP:BOTH] Oh yours your head is not reeling any more no ? [TP:CORPUS] Kind of but it would be better than an indoor game no [TP:CORPUS] We'll ask that person no that Sagar you can tell [TP:CORPUS] Nothing at all that's why you got scratching on that day I know that no that's why I asked [TP:MINPAIR] I'm not fair no [TP:MINPAIR] Husband no I'll do I'll prepare it [TP:MINPAIR] He could have agreed no what is that [FP:BOTH] TELCO deta hai to kuch problem nahi na [FP:BOTH] I think once you have got in you no [FP:BOTH] I didn't go anywhere no [FP:CORPUS] Or two hundred rupees that no [FP:CORPUS] Know when we go back no I think we'll get a rosy welcome home welcome there [FP:CORPUS] I like straight and perspiration then only I feel at home otherwise no [FP:MINPAIR] No got it repaired [FP:MINPAIR] No no he is here [FP:MINPAIR] Okay no but [FN:BOTH] I just go out for tea isn't [FN:BOTH] Hey you you like serious movies is it you like serious movies [FN:BOTH] See no the scene exactly happened you know the other day what happen I was reading baba [FN:CORPUS] I'm not fair no [FN:CORPUS] I think no [FN:CORPUS] Tell me no why you can't tell [FN:MINPAIR] Yeah then it's first time first time it was new to me no [FN:MINPAIR] That is the main thing na here that would again the main thing that they don't take at all interest in the their children at all [FN:MINPAIR] So culture nahi hai there is I don't follow culture religion nothing na B.4 Lack of Copula [FP:CORPUS] Which October first I think [TP:BOTH] This principal she is very particular about it [TP:BOTH] Vilas and Ramesh they they make noise man [TP:BOTH] That's why those Muslims they got very angry [TP:CORPUS] And med medium class they can't understand soon [TP:CORPUS] That will become difficult and common people they don't understand [TP:CORPUS] And now the Kukis they refused to pay any more [TP:MINPAIR] It's because of this some other participant they complained about this and then they started they started this particular [TP:MINPAIR] We've lot of fun in theatres you know we always take the back seat and all that for this guys distinct one we keep teasing them [TP:MINPAIR] My post graduation degree I finished it in mid June nineteen eighty-six [FP:BOTH] But whereas when they really come to know the people they like to help the people [FP:BOTH] It's actually some of them like to see it really so huge and long and bigger snakes they are in all closed and all there it is nice to see it [FP:BOTH] But generally the educated people I don't find much variation but in accent there may be a variation [FP:CORPUS] Everytime he keeps speaking you know they get irritated and say aram se [FP:CORPUS] What happened is they will change programme and the fifty guys they'll just keep quite [FP:CORPUS] Whereas Hyderabad the people are more conservative and like they don't like to go out even or at the first move they don't like to talk with people also [FP:MINPAIR] And the songs now once we hear it afterwards when some other famous songs comes that we forget the last ones [FP:MINPAIR] But when we approach since it seems they they put lot of conditions yes that you fed up with those people and [FP:MINPAIR] so that's why we missed we that missed that holiday it being a Sunday [FN:BOTH] Administration it is all done by Bharati Vidya Bhavan [FN:BOTH] Oh our Joshi okay II got got him [FN:BOTH] Yes yes it is true but our constitution makers [FN:CORPUS] and he has used the the place where the palace once palace might be there and that portion and the remaining part he built an antenna he has fixed it there at the top [FN:CORPUS] Not exactly but Calcutta sweets I think they do have a little flavour and that I haven't got anywhere in India [FN:CORPUS] Computer it was in the first semester [FN:MINPAIR] And med medium class they can't understand soon [FN:MINPAIR] Shireen she was excellent at that [FN:MINPAIR] Yeah arti arti students they loiter about in the corridor B.6 Non-initial Existential X is / are there [TP:BOTH] Libraries are there [TP:BOTH] only specimen like operated cases like supposing a is there [TP:BOTH] Problems are there problems are there what [TP:CORPUS] to assist there some teachers are there and together we conduct the classes [TP:CORPUS] It's there but it's common no [TP:CORPUS] Yeah I think Varlaxmi is there [FP:BOTH] My husband is there mother is there [FP:CORPUS] Come no Shaukat is here Natalie is here even if Savita is not there they two are there na [FP:CORPUS] Actually there the thing is that you know for example [FP:CORPUS] Any thing is there produced materials which do not require much resource personnel [FP:MINPAIR] Ph D degree is awarded there [FN:BOTH] Yeah the royalties too there they're there and we've the king [FN:BOTH] Okay somebody else's some somebody else is there [FN:BOTH] In that you know everything is about nature I'll tell you yeah it's very lovely means very nice lovely what but and small children were there in that [FN:MINPAIR] American and all other capitalist nations were also there [FN:MINPAIR] Nice movie yaar that song is there no hai apna dil to awara [FN:MINPAIR] It's not there B.7 Object Fronting [TP:BOTH] Just typing work I have to do [TP:CORPUS] writing skills there are so many you can teach them [TP:CORPUS] Each other and so many things we have learnt [TP:CORPUS] My birthday party you arrange [FP:CORPUS] Formalities I will come [FP:CORPUS] Mar Marxism you were [FP:MINPAIR] Other wise we have to [FN:BOTH] That also I'm not having just I jump jumped jumped I came studies also [FN:BOTH] Yes Hawa Mahal we heard [FN:BOTH] About ten to twenty books I'll read that's all [FN:MINPAIR] Small baby very nice it was [FN:MINPAIR] But more keen she is [FN:MINPAIR] And camera handling actually outdoor landscaping that landscape shot I have taken and actually the close ups and some parts of your architectural shots of that building Ganesh took my husband took and close ups of the faces my husband and Ganesh took B.8 Resumptive Object Pronoun [TP:MINPAIR] and he has used the the place where the palace once palace might be there and that portion and the remaining part he built an antenna he has fixed it there at the top [TP:MINPAIR] Yeah also pickles we eat it with this jaggery and lot of butter [TP:MINPAIR] My post graduation degree I finished it in mid June nineteen eighty-six [FP:MINPAIR] Having humurous something special I would love it to join it [FP:MINPAIR] I see a number of people I like them very much [FP:MINPAIR] Old and ancient things in carving we get it so beautifully [FN:BOTH] Oh our Joshi okay II got got him [FN:BOTH] Normaly no we don't overdrawn on account but haan haan whatever is balance you know yeah help them give them suppose cheque books and all we are supposed to keep them yeah two fifty balance [FN:BOTH] He is in a that's what he was telling me today see I want your draft like draft draft by January by the month of January by the end of January so that II might rectify it and then I will do it I will give it back to you by mid Febraury so that you can get it final draft by by the end of Febraury [FN:CORPUS] and he has used the the place where the palace once palace might be there and that portion and the remaining part he built an antenna he has fixed it there at the top [FN:CORPUS] Yeah also pickles we eat it with this jaggery and lot of butter [FN:CORPUS] My post graduation degree I finished it in mid June nineteen eighty-six B.9 Resumptive Subject Pronoun [TP:CORPUS] Like those terrorists they wanted us to to accompany them in the revolt against India [TP:CORPUS] And one more thing another thing how I rectified myself because all almost all all of us all my brother and sisters we have read in English medium school [TP:CORPUS] Dr this Mr V he was totally changed actually because he was the concepts are clear not clear to us [FP:CORPUS] There are so many people they can they could shine like anything [FP:CORPUS] Kolhapur he had come to Guwahati [FP:CORPUS] I don't know what he whenever whenever I see those guys they they nicely speak to me [FP:MINPAIR] His house he is going to college KK diploma electronics [FN:BOTH] they I thought that another one Patil is there a horrible he is I thought that Patil [FN:BOTH] Computer it it plays a great role because we are having computers in each field nowa-days [FN:BOTH] You know that a woman she is a apprehensive about many things [FN:MINPAIR] Like those terrorists they wanted us to to accompany them in the revolt against India [FN:MINPAIR] Whereas in Hyderabad they still have the old cultures and so many things that even the parents they don't even let the girls talk with the guys [FN:MINPAIR] And the students who come out with a degree MMSI understand that there is a report that has been received from different firms that the students of BITS Pilani specially MMS candidates they are prepared to soil their hands B.10 Topicalized Non-argument Constituent [TP:CORPUS] for Diwali you went I know that [TP:CORPUS] So very long time we have not travelled together [TP:CORPUS] Pooja vacation also we used to conduct some classes practical classes [TP:MINPAIR] In pooja day some important days we stay back [FP:CORPUS] In Jaipur then we have also we have a Birla [FP:CORPUS] Like that we [FP:CORPUS] Everytime we have some work to do [FP:MINPAIR] Aa i i initial periods I did very difficult but I [FN:BOTH] I mean here in Hyderabad the people are it's okay they are nice [FN:BOTH] And that old ones again we put them we feel like hearing again [FN:BOTH] But in drama we'll have to be very different [FN:CORPUS] In pooja day some important days we stay back [FN:MINPAIR] for Diwali you went I know that [FN:MINPAIR] Pooja vacation also we used to conduct some classes practical classes [FN:MINPAIR] Sir from Monday onwards I too want to take leave sir for four days because total I have five C Ls so from C Average Precision Results[FP:CORPUS] June nineteen eighty-six
[FP:MINPAIR] Construction all before
[FP:MINPAIR] Not in the class
[FP:MINPAIR] The tendency to
[FN:BOTH] you've she said his grandfather still
working
[FN:BOTH] Everybody so worried about the ex-
ams and studies
[FN:BOTH] Again classes bit too long I feel five
O'clock is tiring
B.5 Left Dislocation
Supervision:
Corpus examples
Minimal pairs
Dialect feature
DAMTL Multihead DAMTL Multihead
FOCUS itself *
0.668
0.631
0.665
0.613
FOCUS only*
0.582
0.404
0.344
0.416
INVARIANT TAG
0.876
0.871
0.441
0.495
COPULA OMISSION
0.029
0.015
0.012
0.036
LEFT DISLOCATION
0.425
0.383
0.149
0.232
NON-INITIAL EXISTENTIAL*
0.887
0.906
0.556
0.510
OBJECT FRONTING
0.238
0.202
0.031
0.083
RES. OBJECT PRONOUN
0.052
0.020
0.046
0.061
RES. SUBJECT PRONOUN
0.460
0.409
0.078
0.198
TOPICALIZED NON-ARG. CONST. 0.080
0.076
0.021
0.044
Macro Average
0.430
0.392
0.234
0.269
Table 8 :
8Average precision for the Lange features. Scores are in the range [0, 1], with 1 indicating perfect performance. Asterisks mark features that can be recognized with a regular expression.LACK OF INVERSION IN WH-QUESTIONSDialect feature
DAMTL Multihead
ARTICLE OMISSION
0.210
0.308
DIRECT OBJECT PRO-DROP
0.044
0.057
EXTRANEOUS ARTICLE
0.116
0.065
FOCUS itself *
1.000
0.853
FOCUS only*
0.859
0.274
HABITUAL PROGRESSIVE
0.008
0.020
INVARIANT TAG
0.614
0.420
INVERSION IN EMBEDDED CLAUSE
0.106
0.162
LACK OF AGREEMENT
0.084
0.110
0.309
0.106
LEFT DISLOCATION
0.288
0.301
MASS NOUNS AS COUNT NOUNS
0.045
0.034
NON-INITIAL EXISTENTIAL*
0.506
0.397
OBJECT FRONTING
0.147
0.193
PREPOSITION OMISSION
0.064
0.116
PP FRONTING WITH REDUCTION
0.091
0.134
STATIVE PROGRESSIVE
0.267
0.329
GENERAL EXTENDER and all
0.769
0.778
Macro Average
0.307
0.259
Table 9 :
9Average precision for the extended feature set. As described in the main text, corpus training examples are unavailable for these features. ARTICLE OMISSION the person I like the most is from mechanical department 1 1 ARTICLE OMISSION person I like the most is from the mechanical department 1 1 ARTICLE OMISSION person I like most is from the mechanical department 1 1 ARTICLE OMISSION person I like most is from mechanical department 1 1 ARTICLE OMISSION the person I like the most is from the mechanical department 0 2 DIRECT OBJECT PRO-DROP we have two tailors who can make for us 1 6 DIRECT OBJECT PRO-DROP we have two tailors who can make clothes for us 0 6 DIRECT OBJECT PRO-DROP we have two tailors who can make them for us 0 7 DIRECT OBJECT PRO-DROP he didn't give me 1 7 DIRECT OBJECT PRO-DROP he didn't give it to me 0 8 DIRECT OBJECT PRO-DROP in our old age we can go and enjoy 1 8 DIRECT OBJECT PRO-DROP in our old age we can go and enjoy it 0 9 DIRECT OBJECT PRO-DROP DIRECT OBJECT PRO-DROP she doesn't like it 0 10 DIRECT OBJECT PRO-DROP he likes here more 1 10 DIRECT OBJECT PRO-DROP he likes it here more 0 11 FOCUS itself So if you're not good at communication you may get filtered at the first level itself 1 11 FOCUS itself So if you're not good at communication you may get filtered at even the first level 0D Minimal pairs
ID
Feature
Example
https://ewave-atlas.org. Shapes indicate variety type, e.g. creole, L1, and L2 English varieties.
We manually split turns that were longer than two clauses, resulting in 317 examples.
Our annotations will be made available at https:// dialectfeatures.page.link/annotations.
\band all\b, \bitself\b, \bonly\b, \bis there\b|\bare there\b
Acknowledgments. Thanks to Claudia Lange for sharing her annotations, and for discussion of this research. Thanks to Axel Bohmann for sharing information about his work on recognizing dialect features with regular expressions. Valuable feedback on this research was provided by Jason | []
|
[
"Scaling Probe-Based Real-Time Dynamic Global Illumination for Production",
"Scaling Probe-Based Real-Time Dynamic Global Illumination for Production"
]
| [
"Zander Majercik \nNVIDIA\n\n",
"Adam Marrs \nNVIDIA\n\n",
"Josef Spjut \nNVIDIA\n\n",
"Morgan Mcguire \nNVIDIA\n\n"
]
| [
"NVIDIA\n",
"NVIDIA\n",
"NVIDIA\n",
"NVIDIA\n"
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| Figure 1. Image rendered in a pre-release version of Unity with our global illumination technique. Most of the indirect lighting in this scene comes from emissives (the orange monitor screens) which are integrated automatically by our technique.AbstractWe contribute several practical extensions to the probe based irradiance-field-with-visibility representation[Majercik et al. 2019] [McGuire et al. 2017] to improve image quality, constant and asymptotic performance, memory efficiency, and artist control. We developed these extensions in the process of incorporating the previous work into the global illumination solutions of the NVIDIA RTXGI SDK [NVIDIA 2020], the Unity and Unreal Engine 4 game engines, and proprietary engines for several commercial games. These extensions include: a single, intuitive tuning parameter (the "self-shadow" bias); heuristics to speed transitions in 1 the global illumination; reuse of irradiance data as prefiltered radiance for recursive glossy reflection; a probe state machine to prune work that will not affect the final image; and multiresolution cascaded volumes for large worlds. | null | [
"https://arxiv.org/pdf/2009.10796v3.pdf"
]
| 221,856,518 | 2009.10796 | 4ba00fd768eede0a04f05ab0bfa8e3cc05509439 |
Scaling Probe-Based Real-Time Dynamic Global Illumination for Production
22 Jun 2021
Zander Majercik
NVIDIA
Adam Marrs
NVIDIA
Josef Spjut
NVIDIA
Morgan Mcguire
NVIDIA
Scaling Probe-Based Real-Time Dynamic Global Illumination for Production
22 Jun 2021
Figure 1. Image rendered in a pre-release version of Unity with our global illumination technique. Most of the indirect lighting in this scene comes from emissives (the orange monitor screens) which are integrated automatically by our technique.AbstractWe contribute several practical extensions to the probe based irradiance-field-with-visibility representation[Majercik et al. 2019] [McGuire et al. 2017] to improve image quality, constant and asymptotic performance, memory efficiency, and artist control. We developed these extensions in the process of incorporating the previous work into the global illumination solutions of the NVIDIA RTXGI SDK [NVIDIA 2020], the Unity and Unreal Engine 4 game engines, and proprietary engines for several commercial games. These extensions include: a single, intuitive tuning parameter (the "self-shadow" bias); heuristics to speed transitions in 1 the global illumination; reuse of irradiance data as prefiltered radiance for recursive glossy reflection; a probe state machine to prune work that will not affect the final image; and multiresolution cascaded volumes for large worlds.
Introduction
This paper discusses an algorithm to accelerate the evaluation of global illumination. The acceleration happens in two parts. The main part creates and maintains a data structure that allows a query of the form irradiance(location, orientation) (E(X, ω)), which replaces a potentially expensive computation of diffuse global illumination with a O(1) lookup into a data structure for locations anywhere in space. The second part re-uses that data structure to sample weighted average of incident radiance for glossy global illumination ( Γ L(X, ω) · W (X, ω)dω) and combines the result with filtered screen-space and geometric glossy ray tracing.
This paper describes a refinement of a previous version of the diffuse portion of this method [Majercik et al. 2019]. This refinement is the union of what we learned when incorporating that algorithm into several products, including the Unity game engine, the Unreal Engine 4 game engine, the NVIDIA RTXGI SDK version 1.1 [NVIDIA 2020], and several unannounced commercial games. These learnings include changes to the underlying algorithm to improve quality and performance, advice on tuning the algorithm and content, expansion of the algorithm to a complete solution that also accelerates glossy reflections, and system integration best practices for these methods. This was driven by constraints from various platforms, requests from game developers and game artists, and new research on the problem. Because they were developed across several different productization efforts with different vendors, we believe that these learnings are fairly universal and robust, but they should not be construed as describing the features or performance of any one in particular.
A key element of our algorithm is a probe, which stores directional information at a point. Environment maps are a type of probe-they store distant radiance as seen from any point in the scene. Our probes store irradiance, weighted averages of distance, and weighted averages of squared distance (See Table 1 for terms we use in relation to probes) for a 3D grid-like structure of points in the scene.
Our algorithm has several components related to the organization, computation, and querying of probes. The new information described in this paper is indicated in Table 2. In the table, we indicate what is new relative to descriptions of previous versions of this algorithm. In addition, we give a complete description of the full algorithm below so that readers will not need to consult descriptions of previous versions to understand the algorithm.
Term Definition
Probe
A probe stores data at a point with values for directions on the sphere. Probe Query
Trilinear interpolation (bilinear filtering and direction) and a visibility and angle weighted interpolation between multiple probes. The net result is an irradiance value that esimates the irradiance field at a point relative to a normal. Irradiance Incident power per unit area; the cosine-weighted integral of radiance relative to the sample direction. Weighted sum of distance Weighted sum (a weighted average in our implementation) of the distance to the nearest surface seen from a 3D point in a particular direction. In our case we use a cosine raised to a power. Direct lighting Light that is emitted from a light source, reflects from one surface, and then reaches the viewer.
Indirect lighting
Light that reflects off two or more surfaces before reaching the viewer (all lighting that is not direct) Global illlumination
Light that includes both direct and indirect lighting.
Overview of the algorithm
At the core of the algorithm are probes that store weighted sums of color, distance, and squared distance. A 2D version of a probe storing a weighted average of distance to nearest object is shown in Figure 2. These probes are processed as follows.
Build and Initialization
Start by building a 3D grid. From that grid, optimize probe positions by moving them outside of static geometry (Section 5). Then, classify all probes into "Off","Sleeping", "Newly Awake", "Newly Vigilant", "Awake", or "Vigilant" (Section 6). At the end of this stage, all probes are in their final positions and initial states.
Probe Query
Take a 3D point (within the probe volume) and normal direction. For every point within the volume, there are 8 probes (corners of a 3D box) that surround it. Loop over those 8 probes. For each one, compute a weight based combination of:
• trilinear weight from probe position • backface weight (is the probe behind the point relative to the normal?)
• visibility (can the probe see the point?). This includes a self-shadow bias' term for robust occlusion queries (Sec. 4.1).
Sample the value from each probe in the direction of the normal, and sum those using the computed weights. That is the sampled irradiance value. For multiple volumes, do this for each volume, and then weight between the volumes as described in Section 7.3. Volume blending with tracking windows is discussed in Section 7.2.
Probe Update
For each probe that is "Awake" or "Vigilant" (Section 6), trace rays in a spherical fibonacci pattern, rotating the pattern randomly every frame. Shade these ray hits using the normal deferred shading algorithm, including sampling the probe volume to include the irradiance from the probes. A section of an example ray cast, with a texel to which the rays contribute highlighted, is shown in Figure 2. The update then proceeds for both irradiance and mean distance values as follows.
Figure 2.
A 2D probe for illustration. This probe shows one "cell" (texel) as the bold segment of the circle. The bold arrow is the direction associated with the cell. The cell stores the weighted average of the hit distances of each of the sample directions. Note that this weighted average includes directions "outside" the center cell. The weighting function is larger for directions near the cell center, and the resulting weighted average is thus influenced more by the longer directions in this particular example. The bold dotted line is the stored "distance" in the cell. Note that a direction can contribute to more than one cell, and we loop over directions updating any cell that a direction contributes to.
Irradiance Compute a cosine-weighted average of the radiance values of these shaded ray hits relative to the direction of each probe texel. Then, for each probe texel, blend these newly computed values into the probe texel at a rate of (1 − α)-we refer to this alpha term as "hysteresis". We adjust this hysteresis per probe and per texel based on our convergence heuristics, described in Section 4.3.
Mean Distance and Mean Distance-Squared Compute a power-cosine weighted average of the distance values for each ray relative to the direction of each probe texel. For each probe texel, blend these values as with irradiance above. We adjust the hysteresis for mean distance separately from irradiance-details and reasoning are provided in Section 4.3.
We update the probe texels by alpha blending in the new shading results at a rate of 1 − α, where α is a hysteresis parameter that controls the rate at which new irradiance and visibility values override results from previous frames (Eq. 2). We dynamically adapt this hysteresis value per-probe and per-texel (Section 4.3). The upodate equation is as follows:
E [n] = αE[n] + (1 − α) ProbeRays max(0,n ·ω) · L(ω)(1)
Where E is the old irradiance/visibility texel in directionn, E is the new texel value,ω is the direction of the ray, and L(ω) is the radiance transported along the ray.
Related Work
Interactive global illumination has been an active area of research for years. We review the areas most relevant to our work.
Interactive Global Illumination with Light Probes Image-based lighting solutions are ubiquitous in modern video games [Martin and Einarsson 2010;Ritschel et al. 2009;McAuley 2012;Hooker 2016]. A common workflow for such solutions involves placing light probes densely inside the volume of a scene, each of which encodes some form of a spherical (ir)radiance map. Prefiltered versions of these maps can also be stored to accelerate diffuse and glossy runtime shading queries.
Variants of traditional light probes allow artists to manually place box or sphere proxies in a scene. These proxies are used to warp probe queries at runtime in a manner that better approximates spatially-localized reflection variations [Lagarde and Zanuttini 2012]. Similarly, manually-placed convex proxy geometry sets are also used to bound blending weights when querying and interpolating between many light probes at runtime, in order to reduce the light leaking artifacts common to probe-based methods.
Practitioners agree that eliminating manual probe and proxy placement remains an important open problem in production [Hooker 2016]. Without manual adjustment of traditional probes, it is impossible to automatically avoid probe placements that lead to light and dark (i.e., shadow) leaks or displaced reflection artifacts. Majercik et al.'s [2019] light probes avoid light and dark leaking with raytraced visibility information, but placing these probes in a uniform grid still leads to suboptimal probe locations (e.g. probes stuck in walls). To avoid these issues for glossy GI, some engines rely instead on screen-space ray tracing [Valient 2013] for pixel-accurate reflections. These methods, however, fail when a reflected object is not visible from the camera's point of view, leading to inconsistent lighting and view dependent (and so temporally unstable) reflection effects. Light Field probes [McGuire et al. 2017] automatically resolve many light/dark leaking issues (in scenes with static geometry and lighting) by encoding additional information about the scene geometry into spherical probes. A solution for dynamic lighting is presented in Silvennoinen et al. [2017], but this solution only supports coarse dynamic occluders and requires complex probe placement based on static geometry. As mentioned above, the irradiance probes of Majercik et al. [2019] avoid most light/dark leaks in scenes with dynamic lighting and geometry, but probe placement is stilll suboptimal. Suboptimal placement can lead to lighting results that, while believable, are inferior to the correctly sampled result, and sometimes exhibit shadow leaking in cases of complex geometry with acute corners.
Interactive Ray Tracing and Shading. Correct shading with probe-based lighting methods relies on point-to-point visibility queries. At a high-level, one can interpret our ray tracing technique as tracing rays against a voxelized representation of the scene (as in voxel cone tracing), but with a spherical voxelization instead of an octree. Two important differences that contribute to many of the practical advantages of our representation are 1) we explicitly encode geometric scene information (i.e. radial depth and depth squared) instead of relying on the implicit octree structure to resolve local and global visibility details, and 2) that neither our spatial parameterization nor our filtering relies on scene geometry. This prevents light (and dark) leaking artifacts and allows us to resolve centimeter-scale geometry at about the same cost (in space and time) as a voxel cone tracer that operates at meter-scale. As we target true world-space ray-tracing in a pixel shader, and not just screen-space ray tracing, our technique can be seen as a generalization of many previous, e.g., real-time environment map Monte Carlo integration methods [Stachowiak and Uludag 2015;Wyman 2005;Toth et al. 2015;Jendersie et al. 2016] .
Similarly, our extensions to the previously published DDGI algorithm are a guide for adapting it and other probe-based techniques to a production setting. We report real changes that we made to the base algorithm to fit production constraints.
Qualitative Image Improvements
Self-shadow bias for correct visibility
When querying the probe volume at a surface, variance in the visibility estimate will be highest around the mean of the distribution-in other words, at the surface (see Figure 3). To avoid the shadow leaking that results from this, an additional bias away from the mean of the distribution is added to the sample point during probe query. The previous technique [2019] used a combination of scene-tuned biases on the mean of the distribution, the variance of the distribution, and the chebyshev statistical test to move the visibility query to a point of lower variance in the distribution. Intuitively, "a point of lower variance in the distribution" can be thought of as a point slightly offset from the surface (in world space). Thus, we unify these statistical bias parameters into a single self-shadow bias term. The self-shadow bias is a world-space vector pointing away from the initial sample point on the surface and is computed as follows:
BiasV ector = (n * 0.2 + ω o * 0.8) * (0.75 * D) * B(2)
Where n is the normal vector at the sample point, ω o is the direction from the sample point to the camera, 0.2 and 0.8 are empirically determined constants, D is the minimum axial distance between probes, and B is a user-tunable floating point scalar. We add this bias vector to the initial sample point to yield a new point which we use for the visibility test.
Our self-shadow bias is more robust than the previous biases because a default value of the B parameter (0.3f) worked well for most scenes, whereas the previous biases each had to be specifically tuned per scene. In cases where scene specific tuning is necessary, tuning is easier because we present a single tunable parameter instead of three. Generally, a higher self shadow bias is necessary when there is increased variance in the depth estimate, as would be the case when lower ray counts are used to update the probes (as might be done to improve performance).
To further decrease light leaking, probe update rays that hit backfaces record a value of 0 for irradiance and shorten their depth values by 80%. Shortening depth values ensures that the probe will see backface surfaces as shadowed and not light them. We set irradiance to 0 to ensure that any lighting that does come from that probe does not cause light to leak where it should not. We do not set depth values to 0 for two reasons: 1) it would drive the computed chebyshev weight towards 0, which might be driven higher when the weights are normalized and 2) probes that see some backfaces but are not stuck in walls (due to idiosyncrasies of geometry) could have overly skewed average depths if many of them were set to 0.
To minimize the number of probes stuck in walls as much as possible, we offset probe positions using an iterative adjustment algorithm, as described in Section 5.
Perception-based exponential encoding
If the irradiance probes are slow to converge, abrupt lighting changes in a scene can create noticeable lag in the diffuse indirect illumination. The lag is most salient in light-to-dark transitions. To combat this, we accelerate convergence by applying a perception-based exponential gamma encoding to probe irradiance values. This encoding interpolates perceptually linearly during lighting changes-faster to light-todark convergence reads perceptually as a linear drop in brightness. We determined experimentally that an exponent of 5.0f leads to best results (lower does not converge as fast, higher does not converge any faster). See our video supplement for results. Code listing is give in Figure 4.
This perception-based encoding has the additional effect of reducing low frequency flicker due to fireflies-bright flashes in the diffuse GI caused by an update ray hitting a small, bright irradiance source.
Fast Convergence Heuristics
We further accelerate convergence with new heuristic based on per-texel thresholding for irradiance data. Our lower threshold detects changes with magnitude above 25% of maximum value and lowers the hysteresis by 0.15f. Our higher threshold detects changes with magnitude above 80% and lowers the hysteresis to 0.0f-we assume in this case that the distribution the probe is sampling has changed completely. These thresholds are active only for irradiance updates-we found them to be too unstable when updating visibility. See Figure 5.
We also implement scene-dependent, per-probe heuristics that adjust the hysteresis based on lighting or geometry changes. These are as follows: (0); // For the 8 probes in the surrounding cage for (int i = 0; i < 8; ++i): vec3 probeIrradiance = texture(irradianceTexture, texCoord).rgb; // Decode the tone curve, but leave a gamma = 2 curve // to approximate sRGB blending for the trilinear probeIrradiance = pow(probeIrradiance, vec3(irradianceGamma * 0.5));
irradiance += probeWeight * probeIrradiance;
// Go back to linear irradiance irradiance = square(irradiance); return irradiance; • Small lighting change (e.g. player-held flashlight turns on): reduce irradiance hysteresis by 15% for 4 frames.
• Large lighting change (e.g. abrupt time of day shift): reduce irradiance hysteresis by 50% for 10 frames.
• Large object change (e.g. ceiling caves in): reduce irradiance hysteresis by 50% for 10 frames and visibility hysteresis by 50% for 7 frames.
In all our heuristics, we try to avoid low hysteresis for visibility updates as much as possible to achieve the most stable result. In each of the scene dependent heuristics, hysteresis for all probes (not just the probes local to the change) is reduced.
Many effective heuristics exist for adjusting probe hysteresis per-texel and perprobe on a scene dependent basis-we have not explored this space in depth. For example, it would probably be more effective to reduce hysteresis only for probes affected by a lighting or object change rather than for all probes in the scene. While exploring more specific and sensitive heuristics remains a fruitful subject for future work, the heuristics presented here worked well enough for us as we integrated the technique into multiple engines. We never came across content that forced us to adapt them, but our survey was not exhaustive. Note that temporal anti-aliasing (TAA) applies its own hysteresis, so the base hysteresis for our technique can be lower if TAA is applied. In this case, the TAA hysteresis should be adjusted according to scene heuristics just like the probe hysteresis, or else it will always add a large cost to convergence even on a dramatic lighting or object change.
Second Order Glossy
We compute glossy reflections with a half-screen resolution wavefront ray trace. These shaded ray hits are then blurred according to surface roughnes and distance from the camera before being integrated into the indirect radiance computation during the deferred shading pass. These raytraced reflections are more realistic than screenspace reflections, but tracing rays for 2nd through nth order reflections is infeasible on most scenes. We improve reflections by reusing the filtered radiance data in the probess to shade 2nd through nth order glossy reflections, resulting in better image quality with minimal performance overhead. See Figure 6 for an on/off comparison.
It is common practice in production path tracing to reduce noise by roughening surfaces (or otherwise truncating the BSDF evaluation) on recursive bounces [Fascione et al. 2019]. Reusing the irradiance probes for second order reflections is a similar approximation, which here avoids noise by taking advantage of a data structure already available to us. Note, however, that the probe data structure stores cosinefiltered irradiance-not the cosine-weighted integral of radiance over the hemisphere, which is the correct measure for reflectance. These two quantities are equivalent to a factor of 2π, but the units are different: radiance (Ws −1 m −2 ) vs. irradiance (Wm −2 ).
Probe Position Adjustment
The probe visibility information prevents light and shadow leaks from occluded probes, but leaves some probes in total occlusion such that they never contribute to shading. We present a simple, fast optimizer that iteratively shifts probes around static geometry to maximize the number of useful probes and generate good viewpoints. During initialization, our optimizer adjusts each probe through the closest backface it can see, then further adjusts probes away from close front-faces to maximize surface visibility (see Figure 7). Pseudocode is given in Figure 8. We do not move probes around dynamic geometry because this causes instability-a stable result is preferable to an unstable result with lower average error. To correctly light dynamic objects, we leverage the fact that a uniformly sampled probe is an approximation of the full irradiance field at its sample location. If a probe passes through a dynamic object, our backface heuristics (described at the end of Section 4.1) will minimize shadow leaking. When the probe emerges, our convergence heuristics (Section 4.3) will quickly converge its value.
Out of a desire to maintain a uniformly sampled irradiance field representation, we did not implement more complex probe sampling techniques, such as importance sampling, which might speed probe convergence at the cost of stability and on-the-fly generalization to moving geometry. Exploring these update techniques in detail is promising for future work.
The purpose of the optimizer is to increase the number of probes that can contribute to the final image. The following scenario, however, demonstrates that our optimizer can sometimes add additional computation without increasing image quality. Consider an 8-probe cage surrounding a flat wall (Figure 9). The optimizer can cause probes to "double cover" a surface if the 4 probes within the surface are adjusted outside it. This causes the full probe cage to turn on and shade the surface, increasing the number of actively tracing probes without appreciably affecting the image quality ( Figure 9). For our test scenes, this slight inefficiency was worth the added benefit of optimizing probe positions globally.
The probe position optimizer runs for 5 iterations during probe state classification, which is enough for almost all probes to converge their locations. We cap the number of iterations at 5 to prevent probes from moving back and forth (infinitely) through tangent backfaces.
More work is needed to determine the best position optimizer algorithm, and many investigations in this vein exist (see, for example, Wang et al. [2019]). Our optimizer worked well for multiple engines, but is almost certainly not optimal. in int backfaceCount; // number of rays that hit backfaces in vec3 closestBackfaceVector; // direction to closest backface in vec3 farthestFrontfaceVector; // direction to farthest frontface in vec3 closestFrontfaceVector; // direction to closest frontface inout vec3 currentOffset; // Current offset from the grid for this probe.
Probe States
For all but the most basic scene geometry, even after adjustment many probes in a uniform 3D grid will not contribute to the final image. We introduce a robust set of probe states to avoid tracing or updating from such probes to increase performance with the same visual result. Our probe states separate probes that should not update from probes that must, with an additional intermediate state to identify probes that have just appeared (either at scene initialization or with a moving volume-see Section 7.2) and adjust their hysteresis accordingly. The full set of states is shown in Figure 10 and discussed in the following sections.
Off Probes
As noted above, the constraints on probe movement imposed by the 3D grid indexing make it impossible to move all probes out of walls (some probes are too constrained by the grid structure). We identify probes that remain inside static geometry and turn them "Off" (never trace or update). As the optimizer only considers static geometry, probes that happen to spawn inside dynamic geometry are unaffected, and will correctly turn on when appropriate.
Probe Update States
Even probes that are outside static geometry are not used for shading every frame: when no geometry is within probeSpacing of a probe, that probe's value is not used. We set these probes to "Asleep" and wake them up when a surface is about to use them for shading. Note that a probe needs to be "Awake" if and only if it is shading a surface or about to shade one. Lighting changes and camera proximity do not matter if the probe is not shading a surface. The same is true for making probes "Asleep": when the camera can't see a probe, it still needs to be "Awake" if it is shading a surface because it is propagating diffuse irradiance (with 2nd through nth order visibility). Thus, probes that shade static geometry should be "Vigilant" (they should always trace and update). Though probes near geometry must trace to propagate GI, the grid resolution need not be as fine in regions that are far from the camera. Pseudocode for the probe state optimizer is given in Figure 11.
Participating Media and Probe States The probe data structure encodes a 3D irradiance field that is queryable at any point within its volume. Thus, it might be queried at positions in empty space to provide global illumination in participating media. In this case, even probes not shading a surface would need to be "Awake" if they are within the participating medium.
Full Probe Initialization Algorithm
Probe positions and states are computed in a four step pass:
• For all uninitialized probes, trace rays for five frames to determine optimal positioning and initial state. At the end of this pass, all previously uninitialized probes are "Newly Vigilant", "Off", or "Sleeping".
• Extend AABBs for all dynamic objects by a probe grid cell + the self-shadow bias for a conservative estimate. Set all "Sleeping" probes inside the extended AABB of a dynamic object to "Newly Awake".
• Optionally trace a large number of rays for "Newly Vigilant" and "Newly Awake" probes to converge them in a frame, setting hysteresis to 0. Set their states to "Vigilant" and "Awake" respectively.
• Trace rays from "Vigilant" and "Awake" probes to update their values with the normal hysteresis value for the scene. This step can also be used to converge "Newly Vigilant" and "Newly Awake" probe values if the previous step was omitted.
The first step of the algorithm can be greatly accelerated with static geometry bounding boxes, as a probe can be directly adjusted against those bounding boxes rather than relying on distance and backface information from the spherical ray cast. Many probes could be immediately classified "Newly Vigilant" with this approach, though ray tracing would still be necessary to correctly determine which probes should be set to "Off".
Though these passes run every frame, for the majority of frames the first step will not run because no probes will be uninitialized. If the optional convergence pass is omitted, then only the final update step will run for most frames.
Probe Sleeping Performance
Probe sleeping using our probe state scheme leads to a 30-50% average performance improvement ( Figure 12). In addition to the performance improvement (shown in the middle column) we also show corresponding increases in rays cast per probe for the same performance. Casting more rays per probe makes new probe values more stable and allows for a lower global hysteresis, which makes the GI converge faster.
Quantitative performance improvements
Probe Update Shader Optimization
The approach of Majercik et al. [2019] updated probe texels using a pixel shader with a stencil buffer (to avoid processing border texels in the update pass). Border texels were updated in a separate pixel shader pass for correct bilinear interpolation. This approach leverages the graphics hardware for alpha blending results. Despite this, however, faster update can be achieved by using a general purpose GPU (GPGPU) compute operation optimized with GPU compute best practices. We give background and details of this approach below. Modern GPU architectures dispatch thread groups to cover user-specified compute grid dimensions. All threads in a group execute the same code in parallel, so ensuring that threads do not take different control paths in the code (coherent execution) is vital for performance. By ensuring coherent execution, we achieve a 3x performance improvement in the update pass over the pixel shader approach with careful indexing over thread blocks consisting of an integer number of groups. All group execution is fullly coherent. In addition, we store incoming shaded sample ray hits in shared memory buffers so that all threads can read it in parallel when computing a new probe texel value.
Previous work showed the effect of probe resolution on image quality and performance. We maintain image quality while selecting probe resolution (8x8 irradiance, 16x16 visibility) for a combination of bandwidth, memory footprint, fast convolution, efficient index computation, and most important: mapping to SIMD instructions (thread lanes on a GPU) for peak occupancy on our target hardware. At powers of Figure 12. Performance data for probe sleeping. The "Baseline" column shows the time for probe trace and update without probe sleeping (all probes are marked "Vigilant"). The "Equal Quality" and "Time Saved" columns show savings of probe sleeping as a percentage of time and as absolute time respectively. Finally, the "Better Quality" column shows the absolute ray increase achievable by tracing more rays from active probes to match the baseline time.
Higher quality is achieved here by tracing more rays per probe per update pass-this reduces the variance in the estimation and speeds convergence. two, a probe can be updated by an integer number of 32 or 64 thread groups (common hardware-defined minimum sizes) for maximum possible occupation and coherence. Arbitrary resolution values offer the highest flexibility at the cost of efficiency.
(a) Octahedral representation and border copy texels. Colors denote faces on the collapsed octahedron. Letters in border cells denote copy destinations for cells inside the border labeled with the same letter.
(b) Thread block alignment for probe update on an 8x8 irradiance probe (left) and a 16x16 visibility probe (right).
(c) Thread block alignment for probe border copy. One block of 32 threads copies corners for four irradiance and four visibility probes (orange). Four blocks copy edges for four irradiance probes (green). Eight blocks copy edges for four visibility probes (blue). Figure 13 shows details of our compute shader indexing, including an example octahedral probe encoding to illustrate border-texel copy for correct hardware bilinear interpolation. Our optimized compute shader is included alongside the update shader of the previous technique [2019] in the supplemental material.
Tracking Windows
Conceptually, a probe grid covers all space in the scene. In practice, however, we do not have the compute or raytracing budget to update and trace a level-sized, high resolution probe grid as it may contain tens of thousands of probes. To maintain high probe resolution where it is most necessary, we implement a 3D tracking window of probes. We used this window to track the camera, though any object can be tracked with the same strategy. Our window begins centered on the camera. As the camera moves, if it moves further from the center than the distance between two probes in a cage (along any axis), a new plane of probes spawns in front of it (relative to its direction of motion) and the plane furthest behind it disappears. We implement this behavior using a 3D fixed-length circular buffer. When a new probe plane appears and is initialized, its new values are written to the memory of the plane in the last row behind the camera: the probes "leapfrog" over the camera in discrete steps ( Figure 14). A discretely stepping probe window necessitates careful interpolation between multiple probe volumes-our strategy for this is discussed in Section 7.3. The row of probes that moves is colored in green. When the camera passes the center bounding threshold moving in the +X direction, the leftmost row of probes leapfrogs to the +X face of the volume. The newly computed grid index is shown in green. The corresponding phase offset change is shown on the right.
Multiple Probe Volumes
Multiple probe volumes at differing resolutions can be used to efficiently implement progressively decreasing grid resolutions that cascade out from the camera, thus saving performance without effecting image quality. The data for these probe volumes is packed into a single texture as shown in Figure ?? The same approach is used in geoclipmaps [Losasso and Hoppe 2004], light propagation volumes [Kaplanyan and Dachsbacher 2010], and voxel cone volumes [Crassin et al. 2011]. Additional highresolution volumes can also be used to efficiently cover hero assets with complex geometry that require higher resolution diffuse irradiance.
(a) Multiple probe volumes (b) Transition start (marked in green) (c) Dense volume hidden Figure 15. Spheres visualized show a dense volume (smaller sphers) and a sparse volume (larger spheres). The spheres are sized based on the probe spacing within each volume. On the far left, the pink region shows the the area fully shaded by the dense volume, which gradually falls off to blue, the area shaded by the sparse volume. The center image marks the start of this transition. The rightmost image hides the dense probes to make visualizing the transition region easier. Figure 16. Shaded ray hit data for multiple volumes packed into a single texture. This texture is irradiance data taken from our multivolume scene in the supplemental video. The texture includes shaded update rays for the camera locked volume, the city scale volume, and the level scale volume-these are labeled in the figure and delineated within the texture by the red lines (which are not part of the irradiance data).
We blend between volumes by linearly falling off from 1.0-0.0 at the last grid cell (starting at the second-to-last plane of probes) along each axis of the 3D grid (see Figure 15). In the deferred shader, a weight is computed for each volume starting from most to least dense. This is also the sampling order because the most dense volume will have the best approximation of the local lightfield. Volume weights are accumulated at each volume sample. After the weight total reaches 1.0, further volumes are skipped.
The weighted volume blending described above yields smooth transitions for static volumes, but can cause popping in the GI when applied to camera locked volumes. When a volume leapfrogs in front of the camera, some points can go from being fully shaded by a sparse cascade to being heavily shaded by the camera cascade ( Figure 17). When computing blending weights for camera locked volumes, we address this by tightening the transition region by one grid cell (along each axis) then centering it on the camera. When a new plane of probes leapfrogs to the front of a volume, points that are newly within that volume will not immediately be shaded by it. Instead, those points will gradually transition between volumes as the camera moves towards them. Results are shown in our supplemental video.
The prototype multivolume code passes all probe volumes to the deferred shader, and then per-pixel iterates through them to figure out which ones contain the point being shaded. Though not the optimal approach for performance, this provides the highest flexibility in tweaking the blending algorithm to evaluate image quality. For a production implementation, the usual solutions for the deferred shading light loop issue (considering the volumes as lights) are available:
• Do the full brute force light loop-for fewer than 10 volumes, the point-in-OBB test to determine which volumes contain the shaded point is fast to evaluate.
• Make one deferred pass per volume, rasterizing the volume's bounds to find the covered pixels.
• Make a spatial data structure (e.g., octtree, BVH) over the volumes and then traverse that at runtime in the pixel shader to find which volumes the pixel is in. This method requires more bookkeeping and potentially costly data-dependent fetches.
• Use tiles [Olsson et al. 2012] set up on the CPU or with a GPU pass to conservatively approximate one of the previous methods.
For the pure cascaded method, these optimizations are not necessary because volumes are axis-aligned in world space and nested in a regular pattern.
Inline Shading
Previous probe schemes required an extra shader pass to gather the indirect contribution over the frame. We present a simpler framework that optimizes the global illumination gather step to directly sample the probe data structure during shading, 25 yielding reduced bandwidth requirements. Our code is included in the supplemental material in GIRenderer_deferredShade.pix.
(a) Initial camera position. Labels show the blending region for the camera tracking window, the camera boundary that will cause the volume to move, and the volume weights for a point being shaded by the camera volume (brown circles) and a surrounding volume (not visualized).
(b) Camera moves. Without camera-aware blending, volume weights on the point change dramatically in one frame.
(c) Camera moves. With camera-aware blending, the volume weights change slowly over the course of multiple frames, leading to smoother transtions.
Conclusion and Discussion
We present multiple extensions to the dynamic diffuse global illumination algorithm [Majercik et al. 2019] to improve image quality, performance, and ease of deployment in a production setting. These extensions were developed in response to production constraints encountered when integrating the technique into the NVIDIA RTXGI SDK [NVIDIA 2020], the Unity game engine, Unreal Engine 4, and several commercial games.
The base algorithm of Majercik et al. [2019] is inherently practical due to it's image quality and performance. This paper covers the gap between a practical algorithm and one that is ready for production deployment. Extensions like our "self-shadow bias" make the algorithm easier to tune, and our performance optimizations to the update pass make it feasible for the render budget of production games. For all of our extensions, we sought solutions that were robust, easy to understand, and easy to tune without fundamentally changing the algorithm.
Limitations and Future Work
Though our proposed convergence heuristics increase convergence over the previous approach, there is still some ghosting in the indirect illumination for small, bright light sources (like flashlights-see our video supplement at 7:05). This lag could be addressed by intensifying our specific hysteresis-reduction heuristics on small lights known to cause ghosting, though doing this globally may cause instability in other regions of the image. While more specialized methods like reflective shadow maps yield less ghosting [Xu 2016], an advantage of our method is that all light sources can be handled generically to produce global illumination-we trade some quality for generality.
In addition to our performance improvements, a per-frame ray budget could be implemented to allow more control over the render budget of the technique. For our applications, we found that controlling a) the rays per probe and b) the number of probes in a volume was enough to hit our performance targets. A more sophisticated treatment of ray budget would trace different ray amounts on a per-probe basis, adding a lot of complexity to the implementation. We chose simplicity over a more optimized ray budget, but a study of optimal ray apportioning between probes (taking into account lighting and geometry changes, the camera position, etc.) is interesting future work.
Our algorithm covers a large space of rendered effects and thus suggests many possible directions for future work. For instance, our techniqe forces second-order glossy reflections to maximum roughness in order to re-use the irradiance values as cosine-filtered radiance. Increasing roughness over scattering events has precedent as a noise reduction technique in film production [Kulla et al. 2018], although typically not at such an aggressive scale.
Second order glossy reflections could be improved by using multiple higher resolution filtered radiance textures with different cosine power weighting-like the weighting for visibility probes, but with multiple octahedral representations per sample point instead of one. These could be used to render second order glossy reflections of varying roughness.
Figure 3 .
3A night scene from our prototype. The wall entering the alley in the left image shows light leaking due to overly high self-shadow bias. The correct self-shadow bias in the right image computes proper occlusion.
Perception decoding during probe sampling vec3 irradiance = vec3
Figure 4 .
4Perceptual encoding and decoding of probe irradiance during update and sampling.
Figure 5 .
5= maxComponent(result.rgb -oldValue.xyz); // Lower the hysteresis when a large change is detected if (abs(changeMagnitude) Pseudocode for probe update with per-texel hysteresis adjustment.
Figure 6 .
6A shiny robot against a mirror background. Both the mirror background and the robot have high glossy reflectance. The left image shows no second order glossy reflections, while the right image shows second order glossy reflections sampled from probes.
Figure 7 .
7A view of the ceiling on our Greek Villa scene. Spheres are a visualization of the probes. The black probes are correctly dark, but are not contributing to the final image. The acute corner leads to shadow leaking (labeled with a green ellipse) with a default probe grid (left). Our optimizer adjusts probes out of the wall and ceiling to remove the leak (right).
Figure 8 .
8If there's a close backface AND you see more than 25% backfaces, // assume you're inside something. if ((float(backfaceCount) / RAYS_PER_PROBE) > 0.25f) { // Solve for the maximum scaling possible on each axis. vec3 positiveOffset = (-currentOffset.xyz + offsetLimit) / closestBackfaceDirection; vec3 negativeOffset = (-currentOffset.xyz -offsetLimit) / closestBackfaceDirection; vec3 combinedOffset = vec3(max(positiveOffset.x, negativeOffset.x), max(positiveOffset.y, negativeOffset.y), max(positiveOffset.z, negativeOffset.z)); // Slightly bias this point to ensure we stay within bounds. const float epsilon = 1e-3; // Millimeter scale float scaleFactor = (min(min(combinedOffset.x, combinedOffset.y), combinedOffset.z) -epsilon);// If we can't move through the backface, don't move at all. fullOffset = currentOffset.xyz + closestBackfaceDirection * ((scaleFactor <= 1.0f) ? 0.0f : scaleFactor); } else if (!(dot(farthestDirection, randomOrientation * sphericalFibonacci(closestFrontfaceIndex, RAYS_PER_PROBE)) > 0.5f)) { // The farthest frontface is also the closest if the probe can // only see one surface. In this case, don't move the probe. // Move minimum distance possible. vec3 farthestDirection = min(0.2f, farthestFrontfaceDistance) * normalize(randomOrientation * sphericalFibonacci(farthestFrontfaceIndex, RAYS_PER_PROBE)Pseudocode for an iteration of the probe position optimizer operating on a single probe.
Figure 9 .
9A corner of the Greek Villa scene. Spheres are visualizations of the probes, encircled in green to denote the "Vigilant" state. Probes are marked "Vigilant" when the optimizer adjusts them out of surfaces, leading to double coverage of surfaces when all 8 probes of a cage can see the front face of the point they're shading.
Figure 10 .
10Probe states with transitions between each state. α is the hysteresis for the current frame. α is the default hysteresis for the scene.
Figure 11 .
11Pseudocode for probe state computation.
Figure 13 .
13Octahedral probe layout and probe update thread indexing.
Figure 14 .
14Conceptual layout of the camera tracking window indexing with phase offset in 2D.
Figure 17 .
172D illustration of volume blending using the static volume method vs. a camera aware volume blending.
Table 1 .
1Terms and definitions.
Table 2 .
2Evolution of probe based GI showing spatial organization, encoding, initialization, update, and query for the GI computation.
Probe Representation. As in the work byMajercik et. al [2019], we applyCigolle et al.'s [2014] octahedral mapping from the sphere to the unit square to store and query our spherical distributions. This parameterization has slightly less distortion than cube maps and provides easier methods for managing seams. In this work, we select resolutions for octahedral irradiance and mean distance/distance squared for quality and performance. GI in Production: A Motivating Example In both offline and realtime rendering, significant previous work has been devoted to adapting existing global illumination algorithms for production. Path tracing in film, which radically changed both artist workflow and render farm computation load, is a good example. The core path tracing algorithm has remained largely unchanged, but practical considerations of the particular hardware and software systems required specialized updates to the technique[Keller et al. 2015].
AcknowledgementsForemost, we thank Peter Shirley for his invaluable feedback and editing. Thanks to Corey Taylor and Mike Mara for the initial probe implementation. Thanks to Derek Nowrouzezahrai and Jean-Philippe Guertin for their effort in the original DDGI paper. Thanks to Paul Hodgson, Peter Featherstone, Jesper Mortensen, Kuba Cupisz, and the rest of the Unity Copenhagen lighting team for their help with Unity. Thanks to Kelsey Blanton and Alan Wolfe for their work on the NVIDIA RTXGI SDK. Thanks to Pablo Palmier at Ninja Theory for his help with Unreal Engine 4.
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The supplemental materials contain relevent C++ and shader code for our extensions. Where appropriate, we have included code from Majerick. Index of Supplemental Materials The supplemental material contains video results for each of our extensions. The video is available here. et al. [2019] for comparisonIndex of Supplemental Materials The supplemental material contains video results for each of our extensions. The video is available here: https://youtu.be/vbJ2aNI94Ho. The supplemental materials contain relevent C++ and shader code for our extensions. Where appropriate, we have included code from Majerick et al. [2019] for comparison.
| []
|
[
"Generalized Period-halving Bifurcation of a Neuronal Recurrence Equation",
"Generalized Period-halving Bifurcation of a Neuronal Recurrence Equation"
]
| [
"René Ndoundam ",
"Serge Alain Ebélé [email protected] ",
"\nUniversity of Yaounde I\nLIRIMA, Team GRIMCAPE\nUMI 209P.o.Box 812YaoundeCameroon IRD\n",
"\nUMMISCO\nIRD France Nord\nF-93143BondyFrance\n",
"\nSorbonne Unversités\nUniv\nParis 06, UMI 209\n",
"\nUMMISCO\nF-75005ParisFrance\n"
]
| [
"University of Yaounde I\nLIRIMA, Team GRIMCAPE\nUMI 209P.o.Box 812YaoundeCameroon IRD",
"UMMISCO\nIRD France Nord\nF-93143BondyFrance",
"Sorbonne Unversités\nUniv\nParis 06, UMI 209",
"UMMISCO\nF-75005ParisFrance"
]
| []
| We study the effect of the perturbation of the coefficients of a neuronal recurrence equation y(n) of memory size h on its attracting basin. Firstly, we give a characterization of k-chains in 0-1 periodic sequence. Secondly, we characterize the periods of all cycles of some neuronal recurrence equations y(n). Thirdly, we apply a perturbation on the neuronal recurrence equation y(n) to obtain a neuronal recurrence equation z(n, d), and we characterize the structure and the length of all the cycles of the neuronal recurrence equation z(n, d). Based on the structure and the length of all the cycles, we deduce the existence of the generalized period-halving bifurcation. | 10.25088/complexsystems.20.4.325 | [
"https://arxiv.org/pdf/1503.06866v2.pdf"
]
| 1,412,438 | 1110.3586 | c9f94f61a9a6227de5c4f367deb6a44ec38f4833 |
Generalized Period-halving Bifurcation of a Neuronal Recurrence Equation
10 Apr 2015
René Ndoundam
Serge Alain Ebélé [email protected]
University of Yaounde I
LIRIMA, Team GRIMCAPE
UMI 209P.o.Box 812YaoundeCameroon IRD
UMMISCO
IRD France Nord
F-93143BondyFrance
Sorbonne Unversités
Univ
Paris 06, UMI 209
UMMISCO
F-75005ParisFrance
Generalized Period-halving Bifurcation of a Neuronal Recurrence Equation
10 Apr 2015arXiv:1503.06866v2 [cs.NE]Neuronal recurrence equationcycletransientperiod-halving bi- furcationcontrol
We study the effect of the perturbation of the coefficients of a neuronal recurrence equation y(n) of memory size h on its attracting basin. Firstly, we give a characterization of k-chains in 0-1 periodic sequence. Secondly, we characterize the periods of all cycles of some neuronal recurrence equations y(n). Thirdly, we apply a perturbation on the neuronal recurrence equation y(n) to obtain a neuronal recurrence equation z(n, d), and we characterize the structure and the length of all the cycles of the neuronal recurrence equation z(n, d). Based on the structure and the length of all the cycles, we deduce the existence of the generalized period-halving bifurcation.
Introduction
Caianiello and De Luca [3] have suggested that the dynamic behavior of a single neuron with a memory, which does not interact with other neurons can be modeled by the following recurrence equation :
x(n) = 1 k j=1 a j x(n − j) − θ (1)
where :
• x(n) is a variable representing the state of the neuron at t = n;
• x(0), x(1), · · · , x(k − 2), x(k − 1) are the initial states;
• k is the memory length, i.e. the state of the neuron at time t = n depends on the states x(n − 1), . . . , x(n − k) assumed by the neuron at the k previous steps t = n − 1, . . . , n − k;
• a j (j = 1, . . . , k) are real numbers called the weighting coefficients. More precisely, a j represents the influence of the state of the neuron at time n − j on the state assumed by the neuron at time n.
• θ is a real number called the threshold.
• 1[u] = 0 if u < 0, and 1[u] = 1 if u ≥ 0.
The system obtained by interconnecting several neurons is called a neural network. These networks were introduced by McCulloch and Pitts [8], and are quite powerful. Neural networks are able to simulate any sequential machine or
Turing machine if an infinite number of cells is provided. Neural networks have been studied extensively as tools for solving various problems such as classification, speech recognition, and image processing [19]. The field of application of threshold functions is large [1,9,10,19] . The spin moment of the spin glass system is one of the most cited example in solid state physics that has been simulated by neural networks.
Neural networks are usually implemented by using electronic components or are simulated in software on a digital computer. One way in which the collective properties of a neural network may be used to implement a computational task is through of the energy minimization concept. The Hopfield network is a well-known example of such an approach. It has attracted a wide attention in literature as a content-addressable memory [2].
Given a finite neural network, the configuration assumed by the system at time t is ultimately periodic. As a consequence, there is an integer p > 0 called the period (or a length of a cycle) and another integer T ≥ 0 called the transient length such that:
• Y (p + T ) = Y (T ) • ∄ T ′ and p ′ (T ′ , p ′ ) = (T, p) T ≥ T ′ and p ≥ p ′ such that Y (p ′ + T ′ ) = Y (T ′ )
where Y (t) = (x(t), x(t − 1), . . . , x(t − k + 2), x(t − k + 1)). The period and the transient length of the sequences generated are good measures of the complexity of the neuron. A bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system, causes a sudden 'qualitative' or topological change in its behaviour. A period-halving bifurcation in a dynamical system, is a bifurcation in which the system switches to a new behaviour with half the period of the original system.
A great variety of results have been established on recurrence equations modeling neurons with memory [1,4,5,6,11,12,14,17,20]. However some mathematical properties are still very intriguing and many problems are being posed.
For example, the question remains whether there exists one neuronal recurrence equation with transients of exponential lengths [18]. In [13], we give a positive From period's point of view:
• in the papers [5,17,11,12,14], the authors didn't study all the cycles generated by the neuronal recurrence equation;
• in the current paper, we are going to study all the cycles generated by the neuronal recurrence equation {y(n) : n ≥ 0}.
From bifurcation's point of view
• in the paper [13], we studied the dynamic of the sequence {z(n) : n ≥ 0}
from one and only one initial configuration. We characterized only one cycle of the sequence {z(n) : n ≥ 0};
• in the paper [15], for any d ( 0 ≤ d ≤ ρ(m) − 1 ), we studied the dynamic of the sequence {z(n, d) : n ≥ 0} from one and only one initial configuration. We characterized only one cycle of the sequence {z(n, d) :
n ≥ 0};
• in the following paper, for any The technique used in this paper to get the period-halving bifurcation is to modify some parameters ( weighting coefficients and threshold ) of the neuronal recurrence equation. This technique relies on control theory. Controllability is related to the possibility of forcing the system into a particular state by using an appropriate control signal. The ultimate proof of our understanding of complex systems is reflected in our ability to control them.
d ( 0 ≤ d ≤ ρ(m) − 1 ),
The paper is organized as follows: in Section 2, some previous results are presented. Section 3 presents a characterization of k-chains in 0-1 periodic sequence. Section 4 is devoted to the characterization of the period length of all the cycles. In section 5, we studied a bifurcation. Concluding remarks are stated in Section 6.
Previous Results
The first study of bifurcation was done by Cosnard and Goles in [6]. Cosnard and Goles [6] studied the bifurcation of the neuronal recurrence equation in two particular cases of neuronal recurrence equation:
Case 1: geometric coefficients and bounded memory Cosnard and Goles described completely the structure of the bifurcation of the following equation:
x n+1 = 1 θ − k−1 i=0 b i x n−i
when θ varies. They showed that the associated rotation number is an increasing number of the parameter θ.
Case 2: geometric coefficients and unbounded memory Cosnard and Goles described completely the structure of the bifurcation of the following equation:
x n+1 = 1 θ − n i=0 b i x n−i
when θ varies. They showed that the associated rotation number is a devil's staircase.
Cosnard, Tchuente and Tindo [5] show the following Lemma:
Lemma 1 [5]
If there is a neuronal recurrence equation with memory length k that generates sequences of periods p 1 , p 2 , . . . , p r , then there is a neuronal recurrence equation
with memory length kr that generates a sequence of period r × lcm(p 1 , · · · , p r ).
Lemma 1 does not take into account the study of the transient length. One can amend Lemma 1 to obtain the following lemma: kg that generates a sequence of transient length g × max(T 1 , T 2 , . . . , T g ) and of period P er. P er is defined as follows:
First case: ∃ j, 1 ≤ j ≤ r such that p j ≥ 2 P er = r × lcm(p 1 , · · · , p g ).
Second case: p j = 1 ; ∀ j, 1 ≤ j ≤ r. P er is a divisor of g.
3
Characterization of k-chains in 0-1 periodic sequence
We recall the concept of k-chains in 0-1 periodic sequences [1] which it is useful in the study of the limit orbits. Let Y = (y(t) : t ∈ N) be a periodical sequence of 0's and 1's; suppose that the period γ(Y ) ( which is a priori unknown ) divides
T . Thus y(t) ∈ {0 , 1} for any t ∈ Z and y(t) = y(t ′ ) when t ≡ t ′ (mod T ).
In studying period lengths we shall deal with sets invariant under translations[1], so the following notation will be useful: if Γ ⊂ Z T , l ∈ Z, we write:
Γ + l = { t + l ( mod T ) : t ∈ Γ} Let us partition the set Z T into Γ 0 (Y ) = { t ∈ Z T : y(t) = 0} and Γ 1 (Y ) = { t ∈ Z T : y(t) = 1} which is called the support of Y. The period of the set Γ 1 (Y ) is the smallest positive number γ such that Γ 1 (Y )+γ = Γ 1 (Y ).
The following result was established in [1] : the period of the sequence ( i.e. γ(Y )
) is equal to the period of Γ 1 (Y ). It is shown in [1] that:
γ(Y ) divides k if and only if Γ 1 (Y ) + k = Γ 1 (Y )
Now let us define k-chains ( for k ≥ 1 ) contained in the support Γ 1 (Y ). A subset C ⊂ Γ 1 (Y ) is called a k-chain if and only if it is of the form C = { t + kl ( mod T ) : 0 ≤ l ≤ s − 1} for some s ≥ 1. So a k-chain is a subset
C = { t + kl ∈ Z T : 0 ≤ l ≤ s − 1 } such that y(t ′ ) = 1 for any t ′ ∈ C.
We characterize the 0-1 sequence which contains two differents chains.
Lemma 3 If a 0-1 sequence {u(n) : n ≥ 0} contains:
• a ℓ 1 chain
• a ℓ 2 chain
• such that ℓ 1 and ℓ 2 are relatively prime
Then ∃ t ∈ N such that • u(t) = 1 • u(t + ℓ 1 ) = 1 • u(t + ℓ 2 ) = 1
We use the Lemma 3 to characterize all the periods of all the attractors.
Characterization of the periods of all the cycles
Let k be a positive integer. For a vector a ∈ R k , a real number θ ∈ R and a vector φ ∈ {0, 1} k , we define the sequence {x(n) : n ∈ N} by the following recurrence:
x(t) = φ(t) ; t ∈ {0, . . . , k − 1} 1 k i=1 a i x(t − i) − θ ; t ≥ k(2)
We denote by S(a, θ, φ) the sequence generated by equation (2), P er(a, θ, φ) its period.
Let m be a positive integer, we denote the cardinality of the set P = {p :
p prime and 2m < p < 3m} by ρ(m). We also denote by π(x) the number of prime less or equal to x. From estimations of Rosser and Schoenfeld [16], we have π(x) < 1.25506 × x ln(x) , f or 1 < x.
It is easy to deduce that:
ρ(m) < π(3m) < 3.78 × m ln(3m) < 3.78 × m ln(m)
Let us denote by p 0 , p 1 , . . . , p −1+ρ(m) the prime numbers belonging to the set
{2m + 1, 2m + 2, . . . , 3m − 2, 3m − 1}, the sequence {α i : 0 ≤ i ≤ −1 + ρ(m)} is defined as α i = 3m − p i , 0 ≤ i ≤ −1 + ρ(m).
We also suppose that:
p −1+ρ(m) < p −2+ρ(m) < · · · < p i+1 < p i < · · · < p 1 < p 0(3)
Subsequently, we consider only the integers m such that m ≥ e 2 .
It is easy to check that {2m + 1, 2m + 2, . . . , 3m − 2, 3m − 1} contains at most
⌈ m−1 2 ⌉ odd integers. It follows that ρ(m) ≤ m − 1 2 (4) We set k = (6m − 1)ρ(m) and ∀ i ∈ N, 0 ≤ i ≤ −1 + ρ(m), we define : µ(m, α i ) = k 3m − α i β(m, α i ) = k − ((3m − α i ) × µ(m, α i )) From the previous definitions, we have k = ((3m − α i ) × µ(m, α i )) + β(m, α i ). It is clear that ∀ i ∈ N, 0 ≤ i ≤ −1 + ρ(m) 2m + 1 ≤ 3m − α i ≤ 3m − 1 This implies that (6m − 1)ρ(m) 3m − 1 ≤ k 3m − α i ≤ (6m − 1)ρ(m) 2m + 1 Therefore 2ρ(m) ≤ µ(m, α i ) ≤ 3ρ(m) (5) ∀ i ∈ N, 0 ≤ i ≤ −1 + ρ(m),· · · 100 . . . 0 3m−αi · · · 100 . . . 0 3m−αi · · ·(6)
and which describes a cycle of length 3m − α i = p i .
∀ i ∈ N, 0 ≤ i ≤ −1 + ρ(m), let φ αi ∈ {0, 1} k be the vector defined by φ αi (0) . . . φ αi (k − 1) = 0 . . . 0 β(m,αi) 10 . . . 0 pi · · · 10 . . . 0 pi µ(m,αi)pi(7)
In other words, φ αi is defined by:
φ αi (j) = 1 if ∃ ℓ, 0 ≤ ℓ ≤ µ(m, α i ) − 1 such that j = β(m, α i ) + ℓp i 0 otherwise
We define the neuronal recurrence equation {x αi (n) : n ≥ 0 } by the following recurrence:
x αi (t) = φ αi (t) ; t ∈ {0, . . . , k − 1} 1 k j=1ā j x αi (t − j) −θ ; t ≥ k(8)
whereā j is defined as follows:
First case: ρ(m) is even and ∀ i 2 ∈ N, 0 ≤ i 2 ≤ −1 + ρ(m) a j = 2 if j ∈ P os(α i2 ) and j ≤ 3×ρ(m)×pi 2 2 , −2 if j ∈ P os(α i2 ) and j > 3×ρ(m)×pi 2 2 , −4 × k otherwise. (9) Second case: ρ(m) is odd, ρ(m) ≥ 3 and ∀ i 2 ∈ N, 0 ≤ i 2 ≤ −1 + ρ(m) a j = 2 if j ∈ P os(α i2 ) and j ≤ (3ρ(m)−1) 2 × p i2 , −2 if j ∈ P os(α i2 ) and (3ρ(m)+1) 2 × p i2 ≤ j ≤ (2ρ(m) − 2) × p i2 , −1 if j ∈ {(2ρ(m) − 1) × p i2 , 2ρ(m) × p i2 } , −4 × k otherwise.(10)
The parameters P os(α i ),θ and k are defined as follows:
P os(α i ) = {jp i : j = 1, . . . , 2ρ(m)} (11) = {p i , 2p i , . . . , (−1 + 2ρ(m))p i , 2ρ(m)p i }, 0 ≤ i ≤ −1 + ρ(m); (12) D = {i : i = 1, . . . , k} = {1, 2, . . . , k − 1, k}; (13) F = −1+ρ(m) i=0 P os(α i );(14)G = D \ F ; (15) θ = 2 × ρ(m); (16) k = (6m − 1) × ρ(m).(17)
By definition P os(α i ) represents the set of indices j, 1 ≤ j ≤ k such that
x αi (k − j) = 1.
From the definition of P os(α i ) and Equation (7), one can easily verify that
j ∈ P os(α i ) =⇒ x αi (k − j) = 1 (18) j ∈ D \ P os(α i ) =⇒ x αi (k − j) = 0(19)
∀ d ∈ N, 0 < d < p i , we also denote P P os(α i , d) the set of indices j such that x αi (k + d − j) = 1, in other words:
P P os(α i , d) = { j : x αi (k + d − j) = 1 and 1 ≤ j ≤ k } ∀ i, d ∈ N, 0 ≤ i ≤ −1 + ρ(m) and 0 < d < p i , we denote: Q(α i , d) = {d + jp i : j = 0, 1, . . . , µ(m, α i )}, 0 < d ≤ β(m, α i ) Q(α i , d) = {d + jp i : j = 0, 1, . . . , −1 + µ(m, α i )}, β(m, α i ) < d < p i E(α i , d) = Q(α i , d) ∩ F
The neuronal recurrence equation {x αi (n) : n ≥ 0} with memory of length k is defined by Equations (7) and (8).
We will show that the neuronal recurrence equation {x αi (n) : n ≥ 0} evolves as specified in Equation (6).
In the following proposition, we present an important property.
Proposition 1 [14] ∀ i ∈ N, 0 ≤ i ≤ −1 + ρ(m) and ∀ d ∈ N, 1 ≤ d < p i card E(α i , d) ≤ ρ(m) − 1.
The following proposition characterizes the sum of the interaction coefficients a j when j ∈ P os(α i ).
Proposition 2 ∀ i ∈ N, 0 ≤ i ≤ −1 + ρ(m), we have: j∈P os(αi)ā j = 2 × ρ(m).
The following lemma characterizes the evolution of the sequence {x αi (n) :
n ≥ 0} at time t = k.
Lemma 4
x αi (k) = 1.
The values of the sequence {x αi (n) : n ≥ 0} at time t = k + 1, . . . , k − 1 + p i are given by the following lemma.
Lemma 5 ∀ t ∈ N such that 1 ≤ t ≤ 3m − 1 − α i , we have x αi (k + t) = 0.
It is easy to verify that ∀ i ∈ N, 0 ≤ i ≤ −1 + ρ(m), we have:
P P os(α i , j) = Q(α i , j) ∀ j, 1 ≤ j ≤ 3m − 1 − α i Lemma 6
There existsā, φ αi ∈ R k andθ ∈ R such that :
P er(ā,θ, φ αi ) = p i . Lemma 7 ∀ t, i ∈ N, t ≥ k and 0 ≤ i ≤ −1 + ρ(m) µ(m, α i ) ≤ k j=1 x αi (t − j) ≤ 1 + µ(m, α i ).
In order to present some properties of the sequence {x αi (n) : n ≥ 0}, we introduce the following notation:
Notation 1 Let us define S1(α i , n) as:
S1(α i , n) = k j=1ā j x αi (n − j)
and let λ be a strictly negative real number such that:
∀ i, 0 ≤ i ≤ ρ(m) − 1 max { S1(α i , n) −θ : S1(α i , n) <θ and n ≥ k} ≤ λ Lemma 8 ∀ i, n ∈ N such that 0 ≤ i ≤ −1 + ρ(m) and n ≥ k, S1(α i , n) ∈ −4k(1 + µ(m, α i )),θ − 1 ∪{θ}, λ ∈ [−1, 0[.
Let {v αi (n) : n ≥ 0} be the sequence whose first k terms are defined as
follows: v αi (0)v αi (1) . . . v αi (k − 1) = x αi (1) · · · x αi (k − 1)x αi (k),(20)
and the other terms are generated by the following neuronal recurrence equation:
v αi (n) = 1 k j=1ā j v αi (n − j) −θ , n ≥ k. (21) Remark 1 The term x αi (k) is equal to 1, this implies that v αi (k − 1) is equal to 0.
The parametersā j , 1 ≤ j ≤ k andθ used in neuronal recurrence Equation
(21) are those defined in Equations (9), (10) and (16).
Notation 2 Let us note:
R1(l, i) = { j : 1 ≤ j ≤ k and v αi (k + l − j) = 1}; X1(l, i, j) = card R1(l, i) ∩ P os(α j ) ; H1(l, i) = card R1(l, i) ∩ (F \ P os(α i )) .
The following lemma characterize the values of X1(0, i, i) and H1(0, i).
Lemma 9 ∀ i ∈ N, 0 ≤ i ≤ ρ(m) − 1, we have: X1(0, i, i) = 0; H1(0, i) < ρ(m).
The following lemma characterize partially the value of H1(l, i).
Lemma 10 If
• i ∈ N, 0 ≤ i ≤ ρ(m) − 1; • l ∈ N and l ≡ −1 + p i mod p i ; • v αi (k + t) = 0, ∀ t ∈ N, 0 ≤ t ≤ l − 1; Then • X1(l, i, i) = 0; • H1(l, i) < ρ(m).
The preceding lemma and the following lemma characterize completely the value of H1(l, i).
Lemma 11 If
• i ∈ N and 0 ≤ i ≤ ρ(m) − 1;
• l ∈ N and l ≡ −1 + p i mod p i ;
• v αi (k + t) = 0, ∀ t ∈ N, 0 ≤ t ≤ l − 1;
Then v αi (k + l) = 0.
Lemma 12
In the evolution of the sequence {v αi (n) : n ≥ 0}, ∀ t ∈ N, t ≥ k we have: We construct the sequence {u(n) : n ≥ 0} generated by the neuronal recurrence equation
(a) v αi (t) = 0, (b) k j=1ā j v αi (t − j) ≤θ − 1,u(n) = 1 k j=1ā j u(n − j) −θ , n ≥ k(22)
such that the initial terms are defined as follows:
u(0)u(1) · · · u(k − 1) ∈ {0, 1} k .(23)
Let us characterize the attracting basin of the sequence {u(n) : n ≥ 0} by showing the following proposition:
Proposition 3 The sequence {u(n) : n ≥ 0} converges
• to the null sequence i.e. to 0 0 · · · 0 0 · · · 0 0 · · · , or
• to one of the sequences {x αi (n) : n ≥ 0} , 0 ≤ i ≤ ρ(m) − 1.
Notation 3 Let us define the memory length of some neuronal recurrence equation as follows:
h = ρ(m) × k = (6m − 1) × (ρ(m)) 2 .
Let {y(n) : n ≥ 0} be the sequence whose first h terms are defined as follows:
y (i) ∈ {0, 1}, 0 ≤ i ≤ −1 + h,(24)
and the other terms are generated by the following neuronal recurrence equation:
y(n) = 1 h f =1 b f y(n − f ) − θ 1 ; n ≥ h (25) where b f = ā j , if f = ρ(m) × j, 1 ≤ j ≤ k 0 , otherwise.(26)θ 1 =θ.(27)
The parametersā j are those defined in Equations (9) and (10). The parameters θ and k are defined in Equations (16) and (17).
, (16) and (17).
Our aim is to characterize the structure of the attracting basin of the sequence y(n) from a qualitative point of view.
The next theorem gives the period of the sequences {y(n) : n ≥ 0}. We define the sets A(d), B(d) and the integer T ot(d) as follows:
A(d) = {j × p d × ρ(m) : 1 ≤ j ≤ µ(m, α d )}; (28) B(d) = d ℓ=0 A(ℓ) , 0 ≤ d ≤ ρ(m) − 1; (29) T ot(d) = cardA(d) = µ(m, α d ) , 0 ≤ d ≤ ρ(m) − 1;
We also defined the parameters ξ(d), β(d) and θ 2 as follows:
ξ(d) = λ − β(d) 8 , 0 ≤ d ≤ ρ(m) − 1; (30) β(d) = λ T ot(d) , 0 ≤ d ≤ ρ(m) − 1;(31)θ 2 = θ 1 + ξ(−1 + ρ(m)).
We define the coefficients of the neuronal recurrence equation w(n, d) as
follows:
c(f, d) = b f , if 1 ≤ f ≤ h and f / ∈ A(d) b f + β(d) , if 1 ≤ f ≤ h and f ∈ A(d)(32)
The first h terms of the neuronal recurrence equation w(n, d) is defined as follows:
w(f, d) = 1, if f = (β(m, α d ) + (ℓ × p d )) × ρ(m) and 0 ≤ ℓ < µ(m, α d ) 0, otherwise(33)
The other terms of the neuronal recurrence equation w(n, d) is defined as follows:
w(n, d) = 1 h f =1 c(f, d)w(n − f, d) − θ 2 ; n ≥ h.(34)
Notation 4 Let us note:
R2(l, d) = { j : 1 ≤ j ≤ h and w(h + l − j, d) = 1}; (35) X2(l, d, j) = card R2(l, d) ∩ A(j) ;(36)H2(l, d) = card R2(l, d) ∩ (B(−1 + ρ(m)) \ A(d)) .(37)
The following lemma characterizes the value of the sequence {w(n, d) : n ≥ 0} at time h.
Lemma 13
w(h, d) = 0, ∀ d ∈ N, 0 ≤ d ≤ ρ(m) − 1. Lemma 14 ∀ ℓ ∈ N, ℓ ≥ 1 If • w(h + t, d) = 0, ∀ t ∈ N, 0 ≤ t ≤ ℓ − 1; • ℓ ≡ 0 mod ρ(m) × p d ; Then • H2(ℓ, d) < ρ(m); • X2(ℓ, d, d) = 0. Lemma 15 ∀ ℓ ∈ N, ℓ ≥ ρ(m) × p d If • w(h + t, d) = 0 ∀ t ∈ N, 0 ≤ t ≤ ℓ − 1 • ℓ ≡ 0 mod ρ(m) × p d Then w(h + ℓ, d) = 0 Lemma 16 ∀ d, n ∈ N such that n ≥ h and 0 ≤ d ≤ ρ(m) − 1, we have: w(n, d) = 0.
In the next section, we study the behavior of the neuronal recurrence equation z(n, d).
Bifurcation of the neuronal recurrence equation z
The basic idea is to construct a sequence {z(n, d) : n ≥ 0} whose terms are generated by the following neuronal recurrence equation:
z(n, d) = 1 h f =1 c(f, d)z(n − f, d) − θ 2 ,(38)
and whose first h terms belong to the set {0, 1} h , i.e.
z(f, d) ∈ {0, 1} , 0 ≤ f ≤ h − 1.
The sequence {z(n, 0) : n ≥ 0} exploits the instability of the sequences {x αi (n) : n ≥ 0} to converge to one cycle of the attracting basin of sequence {y(n) : n ≥ 0}. The sequence {z(n, d) : n ≥ 0} exploits the behavior of the sequences w(n, 0), w(n, 1), · · · w(n, d) to converge into the cycle of length:
1. ρ(m) × lcm(T d+1 , T d+2 , · · · , T −1+ρ(m) ) such that {T d+1 , T d+2 , · · · , T −1+ρ(m) } ⊆ {p d+1 , p d+2 , · · · , p −1+ρ(m) }, or 2. p where p is the divisor of ρ(m).
It is easy to see that:
• This second item is obtained by the following transformations:
c(f, 1 + d) = c(f, d), if f / ∈ A(d) ∪ A(1 + d) c(f, d) + β(1 + d), if f ∈ A(1 + d) ∩ A(d) c(f, d) − β(d), if f ∈ A(d) ∩ A(1 + d) c(f, d) − β(d) + β(d + 1), if f ∈ A(d) ∩ A(1 + d)(39)
The main results of the paper are:
Theorem 2 ∀m ∈ N such that m ≥ e 2 .
• from any initial configuration, the neuronal recurrence equation
Comment 2:
In other words, the first part of the Theorem 3 can be interpreted as follows:
in some cases, the length of the cycles of the neuronal recurrence equation
Remark 3
The new contribution in this paper with respect to the previous works are:
Firstly, from period's point of view
• in the papers [5,17,11,12,14], the authors didn't study all the cycles generated by the neuronal recurrence equation;
• in the current paper, we studied all the cycles generated by the neuronal recurrence equation {y(n) : n ≥ 0}.
Secondly, from bifurcation's point of view
• in the paper [13], we studied the dynamic of the sequence {z(n) : n ≥ 0}
from one and only one initial configuration. We characterized only one cycle of the sequence {z(n) : n ≥ 0};
• in the paper [15], for any d ( 0 ≤ d ≤ ρ(m)−1 ), we studied the dynamic of the sequence {z(n, d) : n ≥ 0} from one and only one initial configuration. We characterized only one cycle of the sequence {z(n, d) : n ≥ 0};
• in the following paper, for any d ( 0 ≤ d ≤ ρ(m) − 1 ), we studied the dynamic of the sequence {z(n, d) : n ≥ 0} from any initial configurations.
We characterized the length of all cycles (i.e. the attracting basin) of the sequence {z(n, d) : n ≥ 0}.
Example :
In this example, we illustrate the period-halving bifurcation on one initial condition. The construction of the neuronal recurrence equations by perturbation proceed as follows: Let us consider the following initial condition of the sequence {y(n) : n ≥ 0}:
y((j × ρ(m)) + i) = φ αi (j) ; 0 ≤ i ≤ ρ(m) − 1 and 0 ≤ j ≤ k − 1. (40)
The term φ αi (j) used in the above equation is defined in Equation (7). The behaviour of the neuronal recurrence equations can be described briefly as follows:
• From the initial condition defined by Equation (40) It is easy to see that:
• • u(b + (j × ℓ 2 )) = 1 , ∀j ∈ N By hypothesis, we also have that the integers ℓ 1 and ℓ 2 are relatively prime and from the definition of greatest common divisor, we can deduce:
∃ u, v ∈ Z such that u × ℓ 1 + v × ℓ 2 = 1(41)
From Equation (41), we can easily deduce:
u × (b − a) × ℓ 1 + v × (b − a) × ℓ 2 = b − a (42a) a + (u × (b − a) × ℓ 1 ) = b − (v × (b − a) × ℓ 2 ) (42b)
From Equation (42b), it follows that ∃ i 0 , j 0 ∈ N defined as follows:
i 0 = u × (b − a) + 1+ | u(b − a) | + | v(b − a) | ×ℓ 2 (43a) j 0 = −v × (b − a) + 1+ | u(b − a) | + | v(b − a) | ×ℓ 1 (43b)
such that:
a + (i 0 × ℓ 1 ) = b + (j 0 × ℓ 2 )
It suffices to choose t = a + (i 0 × ℓ 1 ).
Proof of Proposition 2
Based on the parity of ρ(m), we consider two cases:
= 2 × 3 × ρ(m) 2 −2 × ρ(m) 2 ; (46) = 2 × ρ(m).(45)
Second Case : ρ(m) is odd
We can easily deduce that: (48)
= 2 × −1 + (3 × ρ(m)) 2 −2 × ρ(m) − 3 2 −2;(49)= 2 × ρ(m).(50)
Proof of Lemma 4
By definition we have:
x αi (k) = 1 k j=1ā j x αi (k − j) −θ ;(51)
Proof of Lemma 5
The result follows from the definition of E(α i , d), D and by application of Proposition 1.
Proof of Lemma 6
The result follows by application of Lemma 4 and Lemma 5.
Proof of Lemma 7
From Equation (6) and Equation (7), we deduce the result.
Proof of Lemma 8
From Lemma 7, we have k j=1 x αi (n − j) ≤ 1 + µ(m, α i ); therefore it follows that −4k(1+µ(m, α i )) ≤ S1(α i , n) because −4k ≤ā j for j such that 1 ≤ j ≤ k.
Sinceā j is an integer, on the basis of Equation (1), Equation (9) and Equation (10), we easily deduce that S1(α i , n) <θ =⇒ S1(α i , n) ≤θ − 1.
From the evolution of the sequence {x αi (n) : n ≥ 0}, we know that for t ∈ {k, k + 1, ..., k + 3m − α i − 2, k + 3m − α i − 1} the only t at which x αi (t) = 1 is k and that S1(α i , k) =θ. From the fact that the sequence {x αi (n) : n ≥ 0} has period 3m − α i , we deduce that S1(α i , n) ≥θ =⇒ S1(α i , n) =θ.
From the fact that ∀ n ∈ N,
S1(α i , n) ∈ [−4k(1 + µ(m, α i )),θ − 1] ∪ {θ}.
It follows from the definition of λ that λ ∈ [−1, 0[.
Proof of Lemma 9
Firstly we want to show that:
X1(0, i, i) = 0.
From Equation (7), Equation (8) It is easy to deduce that:
j ∈ R1(0, i) =⇒ j ≡ 1 mod p i(56)
From the definition of P os(α i ), it is clear that
j ∈ P os(α i ) =⇒ j ≡ 0 mod p i(57)
From Equation (56) and Equation (57), we can easily conclude that:
X1(0, i, i) = 0.
Secondly, we want to prove that:
H1(0, i) < ρ(m).
To show the result, we proceed by contradiction. Let us suppose that
H1(0, i) ≥ ρ(m)(58)
Based on definition of H1(l, i), on Equation (58) and on the Pigeonhole Principle, we easily deduce that ∃ i1 ∈ N, i1 = i and X1(0, i, i1) ≥ 2 (59)
1 ≤ j 1 , j 2 ≤ k (60) 1 ≤ γ 1 , γ 2 ≤ µ(m, α i1 ) (61) 1 ≤ γ 3 , γ 5 ≤ µ(m, α i ) (62) 0 ≤ γ 4 ≤ p i (63) j 1 = (γ 3 × p i ) + γ 4 = γ 1 × p i1 (64) j 2 = (γ 5 × p i ) + γ 4 = γ 2 × p i1(65)
From Equation (64) and Equation (65), we deduce that
(γ 3 − γ 5 ) × p i = (γ 1 − γ 2 ) × p i1(66)
From the fact that p i and p i1 are different prime numbers, we deduce that Equation (66) is true if and only if:
γ 3 − γ 5 is a multiple of p i1 (67) γ 1 − γ 2 is a multiple of p i(68)
From Equation (5), Equation (61), Equation (62) and estimations of Rosser and Schoenfeld [16], we deduce that:
γ 3 − γ 5 ≤ µ(m, α i ) ≤ 3 × ρ(m) < 3.78m ln(m) (69) γ 1 − γ 2 ≤ µ(m, α i1 ) ≤ 3 × ρ(m) < 3.78m ln(m)(70)
From the fact that m ≥ e 2 , from Equation (69) and Equation (70), we easily deduce that:
γ 3 − γ 5 < 2 × m (71) γ 1 − γ 2 < 2 × m(72)
We can conclude that:
• from Equation (67), Equation (71) and the fact that 2m < p i1 < 3m, we deduce a contradiction
• from Equation (68), Equation (72) and the fact that 2m < p i < 3m, we deduce a contradiction It follows that:
H1(0, i) < ρ(m).
Proof of Lemma 10
Firstly we want to show that:
X1(l, i, i) = 0.
From Equation (7), Equation (8) From the hypothesis, we have:
v αi (k + t) = 0, ∀ t ∈ N, 0 ≤ t ≤ l − 1(74)
From Equation (73) and Equation (74), we easily deduce that:
j ∈ R1(l, i) =⇒ j ≡ l + 1 mod p i(75)
From the definition of P os(α i ), it is clear that:
j ∈ P os(α i ) =⇒ j ≡ 0 mod p i(76)
From Equation (75) and Equation (76), we can easily conclude that:
X1(l, i, i) = 0.
Secondly, we want to prove that:
H1(l, i) < ρ(m).
To show the result, we proceed by contradiction. Let us suppose that:
H1(l, i) ≥ ρ(m).(77)
Based on definition of H1(l, i), From Equation (77) and on the Pigeonhole Principle, we easily deduce that
∃ i1 ∈ N, i1 = i and X1(l, i, i1) ≥ 2(78)
From Equation (78), we deduce that ∃j 1 , j 2 , γ 1 , γ 2 , γ 3 , γ 4 , γ 5 ∈ N such that:
1 ≤ j 1 , j 2 ≤ k (79) 1 ≤ γ 1 , γ 2 ≤ µ(m, α i1 ) (80) 1 ≤ γ 3 , γ 5 ≤ µ(m, α i ) (81) 0 ≤ γ 4 < p i (82) j 1 = (γ 3 × p i ) + γ 4 = γ 1 × p i1 (83) j 2 = (γ 5 × p i ) + γ 4 = γ 2 × p i1(84)
From Equation (83) and Equation (84), we deduce that
(γ 3 − γ 5 ) × p i = (γ 1 − γ 2 ) × p i1(85)
From the fact that p i and p i1 are different prime numbers, we deduce that Equation (85) is true if and only if:
γ 3 − γ 5 is a multiple of p i1 (86) γ 1 − γ 2 is a multiple of p i(87)
From Equation (5), Equation (80), Equation (81) and estimations of Rosser and Schoenfeld [16], we deduce that:
γ 3 − γ 5 ≤ µ(m, α i ) ≤ 3 × ρ(m) < 3.78m ln(m) (88) γ 1 − γ 2 ≤ µ(m, α i1 ) ≤ 3 × ρ(m) < 3.78m ln(m)(89)
From the fact that m ≥ e 2 , Equation (88) and Equation (89), we easily deduce that:
γ 3 − γ 5 < 2 × m; (90) γ 1 − γ 2 < 2 × m.(91)
We can conclude that:
• from Equation (86), Equation (90) and the fact that 2m < p i1 < 3m, we deduce a contradiction
• from Equation (87), Equation (91) and the fact that 2m < p i < 3m, we deduce a contradiction It follows that:
H1(l, i) < ρ(m)
Proof of Lemma 11
From Equation (7), Equation (8) From the hypothesis, we have:
v αi (k + t) = 0, ∀ t ∈ N, 0 ≤ t ≤ l − 1 (93) l ≡ −1 + p i mod p i(94)
From Equation (92), Equation (93), we easily deduce that:
v αi (k + l − p i ) = 0.(95)
From Equations (9, 10, 21, 92, 93, 94, 95), we can easily deduce that:
v αi (k + l) = 0.
Proof of Lemma 12
It suffices to prove (a) because (b) and (c) are the consequence of (a). We want to prove by recurrence the part (a) of the Lemma.
Basis Case: t = k
From the Lemma 9, we have:
X1(0, i, i) = 0 (96) H1(0, i) < ρ(m)(97)
The Equation (96) and Equation(97) imply that:
v αi (k) = 0.
Recurrence Hypothesis: we suppose that the property is true at all steps k, k + 1, · · · , k + l − 1, i.e.
∀ t ∈ N such that 0 ≤ t ≤ l − 1, we have v αi (k + t) = 0 (98) and ∀ t ∈ N such that : 0 ≤ t ≤ k − 1, v αi (t) is defined by Equation (20).
Step k + l:
Based on the value of k + l, we can distinguish two cases:
First Case: l ≡ −1 + p i mod p i By application of Lemma 11, we deduce that v αi (k + l) = 0.
Second Case: l ≡ −1 + p i mod p i From the Lemma 10, we deduce that:
X1(l, i, i) = 0 (99) H1(l, i) < ρ(m)(100)
Based on Equation (99), Equation(100) and Equation (21), we deduce that:
v αi (k + l) = 0.
We have shown the part (a) of the Lemma.
Proof of Proposition 3
The proof is divided into two parts.
Firstly, let us suppose that the sequence {u(n) : n ≥ 0} converges to the sequence {u1(n) : n ≥ 0} and that the sequence {u1(n) : n ≥ 0} is not equal to one of the following two sequences:
• to the null sequence i.e. to 0 0 · · · 0 0 0 · · ·
• to one of the sequences {x αi (n) : n ≥ 0} , 0 ≤ i ≤ ρ(m) − 1
We can extract from the sequence {u1(n) : n ≥ 0} a ℓ-chain such that :
ℓ ≡ 0 mod p i , 0 ≤ i ≤ ρ(m) − 1(101)
Without loss of generality, let us assume that:
u1(t 1 ) = 1(102)
From the fact that the sequence {u1(n) : n ≥ 0} admits a ℓ-chain, we can deduce that:
u1(t 1 + ℓ) = 1.(103)
From Equation (101), we can easily deduce that:
a ℓ = −4 × k.(104)
From the fact that:
• u1(ℓ + t 1 ) = 1( k j=1ā j u1(ℓ + t 1 − j) −θ), •ā ℓ = −4 × k,
• u1(t 1 ) = 1.
We deduce that : u1(ℓ + t 1 ) = 0. It follows that we have a contradiction • a ℓ 1 -chain such that ℓ 1 = p i1 , 0 ≤ i1 ≤ ρ(m) − 1;
• a ℓ 2 -chain such that ℓ 2 = p i2 , 0 ≤ i2 ≤ ρ(m) − 1;
• i1 > i2, i.e. ℓ 1 < ℓ 2 .
From the fact that the sequence {u1(n) : n ≥ 0} admits two chains : ℓ 1 -chain and ℓ 2 -chain, we deduce by application of Lemma 3 that there exists t 1 ∈ N which verify:
u1(t 1 ) = 1; (105a) u1(t 1 + ℓ 1 ) = 1; (105b) u1(t 1 + ℓ 2 ) = 1.(105c)
We have : 2 × m ≤ p i1 < p i2 ≤ 3 × m. It follows that:
ı = ℓ 2 − ℓ 1 = p i2 − p i1 < m.(106)
From Equation (9), Equation (10) and Equation (106), we deduce that:
a ı = −4 × k(107)
Based on the facts that:
• u1(t 1 + ℓ 1 ) = 1,
•ā ı = −4 × k, • u1(t 1 + ℓ 2 ) = 1 k j=1ā j u1(t 1 + ℓ 2 − j) −θ .
We deduce easily that u1(t 1 +ℓ 2 ) = 0. This is a contradiction with the Equation (105c). We easily deduce that the sequence {u1(n) : n ≥ 0} contains only and only one chain.
Proof of Theorem 1
Based on the Lemma 2, the Lemma 6 and Proposition 3, we deduce the result.
Proof of Lemma 13
From Equation (34), we deduce that:
w(h, d) = 1( h f =1 c(f, d)w(h − f, d) − θ 2 ); (108) = 1( f ∈P os(α d )ā f + µ(m,α d )
Proof of Lemma 14
Based on the hypothesis, we have:
ℓ ≡ 0 mod ρ(m) × p d ; (113) w(h + t, d) = 0 ∀ t ∈ N, 0 ≤ t ≤ ℓ − 1.(114)
From the definition of R2(l, d) and Equation (114), it is easy to deduce that:
j ∈ R2(ℓ, d) =⇒ j ≡ ℓ mod ρ(m) × p d .(115)
Based on definition of A(d), we easily deduce that:
j ∈ A(d) =⇒ j ≡ 0 mod ρ(m) × p d .(116)
Based on Equation (115) and Equation (116), it is easy to deduce that:
X2(ℓ, d, d) = 0.(117)
To show the following result:
H2(ℓ, d) < ρ(m),(118)
we proceed by contradiction. Let us suppose that:
H2(ℓ, d) ≥ ρ(m).(119)
Based on Equation (119) and on the Pigeonhole Principle, we easily deduce that ∃ d1 ∈ N, d1 = d and X2(ℓ, d, d1) ≥ 2.
From Equation (120), we deduce that ∃j 1 , j 2 , γ 1 , γ 2 , γ 3 , γ 4 , γ 5 ∈ N such that:
1 ≤ j 1 , j 2 ≤ h;(121)
1 ≤ γ 1 , γ 2 ≤ µ(m, α d1 );
From Equation (124) and Equation (125), we deduce that:
ρ(m) × (γ 3 − γ 5 ) × p d = ρ(m) × (γ 1 − γ 2 ) × p d1 .(126)
This implies that:
(γ 3 − γ 5 ) × p d = (γ 1 − γ 2 ) × p d1 .(127)
From the fact that p d and p d1 are different prime numbers, we deduce that Equation (127) is true if and only if:
γ 3 − γ 5 is a multiple of p d1 ;(128)γ 1 − γ 2 is a multiple of p d .(129)
From Equation (122), Equation (123) and estimations of Rosser and Schoenfeld [16], we deduce that:
γ 3 − γ 5 ≤ µ(m, α d ) ≤ 3 × ρ(m) < 3.78m ln(m) ; (130) γ 1 − γ 2 ≤ µ(m, α d1 ) ≤ 3 × ρ(m) < 3.78m ln(m) ;(131)
From the fact that m ≥ e 2 , Equations (130, 131), we easily deduce that:
γ 3 − γ 5 < 2 × m;(132)γ 1 − γ 2 < 2 × m.(133)
We can conclude that:
• from Equations (128, 132) and the fact that 2m < p d1 < 3m, we deduce a contradiction
• from Equations (129, 133) and the fact that 2m < p d < 3m, we deduce a contradiction It follows that:
H2(ℓ, d) < ρ(m).
Proof of Lemma 15
Based on Lemma 13, Equation (32), from the fact that ℓ ≡ 0 mod ρ(m) × p d and the fact that
w(h + t, d) = 0 ∀ t ∈ N, 0 ≤ t ≤ ℓ − 1;
we easily deduce that:
w(h + ℓ, d) = 0.
The proof will be done by strong recurrence on n.
Basic case: n = h From Lemma 13, we can deduce that:
w(h, d) = 0.
Induction Hypothesis: n = h, h + 1, . . . , h − 1 + ℓ
We suppose that ∀ t ∈ N 0 ≤ t ≤ ℓ − 1, we have:
w(h + t, d) = 0 and that w(0, d), w(1, d), · · · , w(h − 1, d) is defined by Equation (33).
Step : n = h + ℓ
Based on the value of n, we can distinguish two cases: Second Case : ℓ ≡ 0 mod (ρ(m) × p d )
By application of Lemma 15, we deduce that:
w(h + ℓ, d) = 0.
Proof of Theorem 2
By application of Lemma 16, we deduce that:
w(n, 0) = 0, ∀n ≥ h
Based on Equation (136) and by application of Theorem 1, we deduce the result.
Proof of Theorem 3
By application of Lemma 16, we deduce that:
w(n, ℓ) = 0, ∀n ≥ h and ℓ = 0, 1, 2 · · · , d
Based on Equation (137) and by application of Theorem 1, we deduce the result.
answer to this question by exhibiting a neuronal recurrence equation with memory which generates a sequence of exponential transient length and exponential period length with respect to the memory length. One question remains: does there exist one neuronal recurrence equation with exponential transient length and fixed point ? We give a positive answer in [15] to this question by constructing from a neuronal recurrence equation {y(n) : n ≥ 0} which describes a cycle of sub exponential length ( respectively to the length of the memory ) a neuronal recurrence equation {z(n) : n ≥ 0} which describe a transient of sub exponential length and a cycle of sub exponential length ( respectivelty to the memory length ).
Lemma 2 [ 15 ]
215If there is a neuronal recurrence equation with memory length k that generates a sequence {x (n) : n ≥ 0} , 1 ≤ ≤ g of transient length T and of period p then there is a neuronal recurrence equation with memory length
we want to construct a neuronal recurrence equation {x αi (n) : n ≥ 0} with memory of length k which evolves as follows : 00 . . . 0 β(m,αi) 100 . . . 0 3m−αi 100 . . . 0 3m−αi
(c) The sequence {v αi (n) : n ≥ 0} describes a transient of length k − p i and a fixed point. The instability of the sequence {x αi (n) : n ≥ 0} occurs as a result of the convergence of the sequence {v αi (n) : n ≥ 0} to 0 0 · · · 0 0.
Remark 2
2(a) The first h terms of the sequence {y(n) : n ≥ 0} are obtained by taking any element of the set {0, 1} h . (b) The coefficients b f of neuronal recurrence equation (25) is obtained by applying the construction of Lemma 1 to the parameters defined by Equations (9),
Theorem 1
1From any initial terms, the sequence {y(n) : n ≥ 0} converges to a cycle of length : • ρ(m) × lcm(elt 1 , elt 2 , . . . , elt s ) where elt i ∈ {p 0 , p 1 , . . . , p −1+ρ(m) } for any i ∈ {1, 2, . . . , s}, or • p where p is a divisor of ρ(m).
by perturbation, we can build the neuronal recurrence equation {z(n, 0) : n ≥ 0} from the neuronal recurrence equation {y(n) : n ≥ 0}; • by perturbation, we can build the neuronal recurrence equation {z(n, d + 1) : n ≥ 0} from the neuronal recurrence equation {z(n, d) : n ≥ 0}.
{y(n) : n ≥ 0} converges to a cycle of length ρ(m) × lcm(elt 1 , elt 2 , . . . , elt s ) where elt i ∈ {p 0 , p 1 , . . . , p −1+ρ(m) } for any i ∈ {1, 2, . . . , s}, or to a cycle of length T where T is a divisor of ρ(m). • by perturbation, we can build the neuronal recurrence equation {z(n, 0) : n ≥ 0} from the neuronal recurrence equation {y(n) : n ≥ 0}. From any initial configuration, the neuronal recurrence equation {z(n, 0) : n ≥ 0} converges to a cycle whose length is of the following form ρ(m) × lcm(elt 1 , elt 2 , . . . , elt s ) where elt i ∈ {p 1 , . . . , p −1+ρ(m) } for any i ∈ {1, 2, . . . , s} or to a cycle of length T where T is a divisor of ρ(m).
Comment 1 :
1We can interpret the second part of the Theorem 2 in other words as follows:in some cases, the length of the cycles of the neuronal recurrence equation{y(n) : n ≥ 0} is divided by p 0 to obtain the length of cycles of the neuronal recurrence equation {z(n, 0) : n ≥ 0}. By perturbation, we can build the neuronal recurrence equation {z(n, d) : n ≥ 0} from the neuronal recurrence equation {z(n, d − 1) : n ≥ 0}. The following result characterizes the attracting basin of the neuronal recurrence equation {z(n, d) : n ≥ 0}. Theorem 3 ∀m, d ∈ N such that m ≥ e 2 and 1 ≤ d ≤ ρ(m) − 1, we construct a set of neuronal recurrence equations whose the behaviour has the following characteristics: • From any initial configuration, the neuronal recurrence equation {z(n, d) : n ≥ 0} converges to a cycle of length ρ(m) × lcm(elt 1 , elt 2 , . . . , elt s ) where elt i ∈ {p d+1 , p d+2 , . . . , p −1+ρ(m) } for any i ∈ {1, 2, . . . , s}, or to a cycle of length T where T is a divisor of ρ(m); • the period of the neuronal recurrence equation {z(n, −1 + ρ(m)) : n ≥ 0} is 1 (i.e. a fixed point).
{z(n, d−1) : n ≥ 0} is divided by p d to obtain the length of cycles of the neuronal recurrence equation {z(n, d) : n ≥ 0}. The role of the term β(d) ( in the definition of c(f, d) ) and of the term ξ(d) is to guarantee that the sequence {z(n, d) : n ≥ 0} converges to any cycle of the form ρ(m) × lcm(elt 1 , elt 2 , . . . , elt s ) where elt i ∈ {p d , p d+1 , . . . , p −1+ρ(m) } for any i ∈ {1, 2, . . . , s}.
1 .
1By perturbation, from the neuronal recurrence equation {y(n) : n ≥ 0} we can build the neuronal recurrence equation {z(n, 0) : n ≥ 0}; 2. For any d, 0 ≤ d ≤ ρ(m) − 2, we can build by perturbation the neuronal equation {z(n, d + 1) : n ≥ 0} from the neuronal equation {z(n, d) : n ≥ 0}.
, the neuronal recurrence equation {y(n) : n ≥ 0} describes a cycle of length ρ(m) × lcm(p 0 , p 1 , · · · , p ρ(m)−1 ; • The neuronal recurrence equation {z(n, 0) : n ≥ 0} describes from any h consecutive terms of the cycle of the sequence {y(n) : n ≥ 0}, a cycle of length ρ(m) × lcm(p 1 , p 2 , · · · , p ρ(m)−1 ; • For any d, 0 ≤ d ≤ ρ(m) − 3, the neuronal recurrence equation {z(n, d + 1) : n ≥ 0} describes from any h consecutive terms of the cycle of the neuronal equation {z(n, d) : n ≥ 0} a cycle of length ρ(m) × lcm(p d+2 , p d+3 , · · · , p ρ(m)−1 ; • The neuronal recurrence equation {z(n, ρ(m)−1) : n ≥ 0} describes from any h consecutive terms of the cycle of the neuronal equation {z(n, ρ(m)− 2) : n ≥ 0} a cycle of length 1 (i.e. a fixed point).
the length of the cycle of the neuronal recurrence equation {z(n, 0) : n ≥ 0} is the divisor of the length of the cycle of the neuronal recurrence equation {y(n) : n ≥ 0}; • the length of the cycle of the neuronal recurrence equation {z(n, d + 1) : n ≥ 0} is the divisor of the length of the cycle of the neuronal recurrence equation {z(n, d) : n ≥ 0}. 6 Conclusion By perturbation, we have built the neuronal recurrence equation {z(n, d) : n ≥ 0} from the neuronal recurrence equation {z(n, d − 1) : n ≥ 0} such that the period of the neuronal recurrence equation {z(n, d) : n ≥ 0} is a divisor of the period of the neuronal recurrence equation {z(n, d − 1) : n ≥ 0}. Our study concerned all the attracting basin. We showed that the neuronal recurrence equation {z(n, −1 + ρ(m)) : n ≥ 0} converges to a fixed point. Thus, we have built a period-halving bifurcation of a neuronal recurrence equation. This result is inscribed in the framework of results on control theory. The structure of the configuration of neuronal recurrence equation can be used in steganography.
,
Lemma 4, and Equation (20) we can easily deduce that: v αi (0)v αi (1) . . . v αi (k − 1) = 0 . . .
, 1
1Lemma 4, and Equation (20) we can easily deduce that:v αi (0)v αi (1) . . . v αi (k −
1
≤ γ 3 , γ 5 ≤ µ(m, α d ); (123)j 1 = (γ 3 × ρ(m) × p d ) + γ 4 = γ 1 × ρ(m) × p d1 ;(124)j 2 = (γ 5 × ρ(m) × p d ) + γ 4 = γ 2 × ρ(m) × p d1 ;
First
Case : ℓ ≡ 0 mod (ρ(m) × p d ) By application of Lemma 14, we deduce that: definition of termsā j , b j , c(f, d), Equation (134) and Equation (135), we easily deduce that: w(h + ℓ, d) = 0.
we are going study the dynamic of the sequence {z(n, d) : n ≥ 0} from any initial configurations. We are going to characterize the length of all cycles (i.e. the attracting basin) of the sequence {z(n, d) : n ≥ 0}.
with Equation (103). We can deduce that there do not exists on the sequence{u1(n) : n ≥ 0} a ℓ-chain which verifies Equation (101). Secondly, let us suppose that on the sequence {u1(n) : n ≥ 0}, there exists at least two different chains. Without loss of generality, let us suppose that there exists on the sequence {u1(n) : n ≥ 0} :
AcknowledgementsThis work was supported by UMMISCO , by LIRIMA and by the University of Yaounde 1.AppendixProof of Lemma 3From the hypothesis, the sequence {u(n) : n ≥ 0} contains two chains, we can deduce that:• u(a + (i × ℓ 1 )) = 1 , ∀i ∈ N
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Automata Networks and Artificial Intelligence. F , Fogelman Soulié, Automata Networks in Computer Science : Theory and Applications. F. Fogelman, Y. Robert, and M. TchuenteManchester University PressF. Fogelman Soulié, and al., "Automata Networks and Artificial Intelli- gence," pages 133-186 in Automata Networks in Computer Science : The- ory and Applications edited by F. Fogelman, Y. Robert, and M. Tchuente (Manchester, Manchester University Press, 1987).
Contributionà l'étude de la Dynamique d'un Automateà Mémoire. D Moumida, University of GrenobleDoctoral dissertationD. Moumida, "Contributionà l'étude de la Dynamique d'un Automateà Mémoire," ( Doctoral dissertation, University of Grenoble, 1989).
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[
"FeSe and the missing electron pocket problem",
"FeSe and the missing electron pocket problem"
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| [
"Luke C Rhodes \nSchool of Physics and Astronomy\nUniversity of St\nKY16 9SSAndrews, St. AndrewsUnited Kingdom\n",
"Matthias Eschrig \nInstitute of Physics\nUniversity of Greifswald\nFelix-Hausdorff-Strasse 617489GreifswaldGermany\n",
"Timur K Kim \nDiamond Light Source\nHarwell Campus\nOX11 0DEDidcotUnited Kingdom\n",
"Matthew D Watson \nDiamond Light Source\nHarwell Campus\nOX11 0DEDidcotUnited Kingdom\n"
]
| [
"School of Physics and Astronomy\nUniversity of St\nKY16 9SSAndrews, St. AndrewsUnited Kingdom",
"Institute of Physics\nUniversity of Greifswald\nFelix-Hausdorff-Strasse 617489GreifswaldGermany",
"Diamond Light Source\nHarwell Campus\nOX11 0DEDidcotUnited Kingdom",
"Diamond Light Source\nHarwell Campus\nOX11 0DEDidcotUnited Kingdom"
]
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| The nature and origin of electronic nematicity remains a significant challenge in our understanding of the iron-based superconductors. This is particularly evident in the iron chalcogenide, FeSe, where it is currently unclear how the experimentally determined Fermi surface near the M point evolves from having two electron pockets in the tetragonal state, to exhibiting just a single electron pocket in the nematic state. This has posed a major theoretical challenge, which has become known as the missing electron pocket problem of FeSe, and is of central importance if we wish to uncover the secrets behind nematicity and superconductivity in the wider ironbased superconductors. Here, we review the recent experimental work uncovering this nematic Fermi surface of FeSe from both ARPES and STM measurements, as well as current theoretical attempts to explain this missing electron pocket of FeSe, with a particular focus on the emerging importance of incorporating the dxy orbital into theoretical descriptions of the nematic state. Furthermore, we will discuss the consequence this missing electron pocket has on the theoretical understanding of superconductivity in this system and present several remaining open questions and avenues for future research. | 10.3389/fphy.2022.859017 | [
"https://arxiv.org/pdf/2201.11702v1.pdf"
]
| 246,294,833 | 2201.11702 | 22ef28665ed84c3a681365406ccc26e539ba6c6b |
FeSe and the missing electron pocket problem
Luke C Rhodes
School of Physics and Astronomy
University of St
KY16 9SSAndrews, St. AndrewsUnited Kingdom
Matthias Eschrig
Institute of Physics
University of Greifswald
Felix-Hausdorff-Strasse 617489GreifswaldGermany
Timur K Kim
Diamond Light Source
Harwell Campus
OX11 0DEDidcotUnited Kingdom
Matthew D Watson
Diamond Light Source
Harwell Campus
OX11 0DEDidcotUnited Kingdom
FeSe and the missing electron pocket problem
(Dated: January 28, 2022)
The nature and origin of electronic nematicity remains a significant challenge in our understanding of the iron-based superconductors. This is particularly evident in the iron chalcogenide, FeSe, where it is currently unclear how the experimentally determined Fermi surface near the M point evolves from having two electron pockets in the tetragonal state, to exhibiting just a single electron pocket in the nematic state. This has posed a major theoretical challenge, which has become known as the missing electron pocket problem of FeSe, and is of central importance if we wish to uncover the secrets behind nematicity and superconductivity in the wider ironbased superconductors. Here, we review the recent experimental work uncovering this nematic Fermi surface of FeSe from both ARPES and STM measurements, as well as current theoretical attempts to explain this missing electron pocket of FeSe, with a particular focus on the emerging importance of incorporating the dxy orbital into theoretical descriptions of the nematic state. Furthermore, we will discuss the consequence this missing electron pocket has on the theoretical understanding of superconductivity in this system and present several remaining open questions and avenues for future research.
I. INTRODUCTION
One of the reasons for the huge interest in FeSe over the past decade has been the sense that it holds the key to the wider understanding of the whole Fe-based superconductor family [1][2][3]. With its minimalistic crystal structure and alluringly simple band structure in the tetragonal phase, alongside the prevalence of high-quality single crystals, it seemed like the ideal test bed to examine in detail the themes that were emerging in the field: strong orbital-dependent correlations [4][5][6], spin fluctuation pairing [7,8], and most pertinently for this review, the so-called nematic phase [9][10][11], where C 4 rotational symmetry is spontaneously broken below 90 K.
The measurement of the momentum-dependence of the superconducting gap in FeSe, between 2016 and 2018, was a particular experimental triumph. The data from both scanning tunneling microscopy (STM) [12] and multiple angle-resolved photoemission spectroscopy (ARPES) measurements [13][14][15][16][17] revealed a clear conclusion: the gap structure is extremely anisotropic, and broadly follows the d yz orbital weight around the Fermi surface. While a twofoldsymmetric gap is of course symmetry-allowed in an orthorhombic system, the fact that such a strong anisotropy was observed implied that the nematic state must also induce a profound anistropic effect on the Fermi surface of FeSe. However due to significant uncertainty as to the correct description of the low-temperature electronic structure, multiple theoretical explanations for the anisotropic gap structure were proposed [12,16,[18][19][20][21].
A critical question required to understand this anisotropic superconducting gap is how does the nematic state influence the the low temperature Fermi surface and electronic structure of FeSe? Given that we have a second-order phase transition [22], and that the lattice distortion |a−b| (a+b) is only ∼0.2%, the natural assumption, from an ab-initio perspective [23], would be that nematicity should only weakly distort the established Fermi surface of the high-temperature tetragonal phase, which ARPES measurements have shown contains two hole pockets and two electron pockets [23][24][25][26][27][28][29][30]. Yet ARPES measurements in the nematic state have revealed sizeable band shifts, of the order of 10-50 meV [30], much larger that what would be predicted from ab-initio calculations [23].
Unfortunately, the precise identification of specific parts of the band structure, the nematic energy scales and even the Fermi surface of FeSe has been complicated by the formation of orthorhombic domains upon entering the nematic state. In an orthorhombic crystal, conventional ARPES experiments measure a superposition of two perpendicularly orientated crystallographic domains, which doubles the number of bands observed in the experimental data and creates ambiguity about which bands arise from which domain. For this reason, a recent focal point of research has involved overcoming this technical challenge of orthorhombic domains, for example by applying uniaxial strain [25,[31][32][33][34][35][36][37] or using NanoARPES [38] or scanning tunneling microscopy [12,[39][40][41]. The conclusion from these measurements have been unanimous, and have revealed that within the nematic state the Fermi surface of FeSe consists of one hole pocket and one electron pocket.
This finding, however, is very surprising and presents a fundamental theoretical conundrum that is at the heart of understanding the nematic and superconducting properties of FeSe. The bands that generate the two electron pockets observed in the tetragonal state form saddle points at the high symmetry M point close to the Fermi level. It is therefore not trivial to deform or shift these saddle points to lift one of these electron pockets away from the Fermi level upon entering the nematic state. This current theoretical challenge has become known as the "missing electron pocket problem" of FeSe and resolving this problem promises deeper insight into the nematic state, and a wider understanding of superconductivity in the ironbased superconductors.
In this review we will overview the recent experimental arXiv:2201.11702v1 [cond-mat.supr-con] 27 Sketch of the temperature evolution of the the lattice constants in FeSe, as described in Ref. [42], highlighting the evolution from a tetragonal (dark grey surface) to orthorhombic system with differently orientated domains (red and blue striped surface). and theoretical work uncovering the Fermi surface of FeSe in the nematic state and tackling the missing electron pocket problem. In section 2 we will briefly introduce the experimental electronic structure of FeSe in the tetragonal state, to use as the foundation for understanding the nematic electronic structure. In section 3 we will discuss the recent experimental data uncovering the electronic structure in the nematic state, in particular focusing on measurements which overcome the technical problems associated with orthorhombic crystals, including ARPES measurements under uniaxial strain, NanoARPES measurements and Scanning tunneling microscopy (STM) measurements. In Section 4 we will review the latest theoretical attempts to resolve this missing electron pocket problem, highlighting the necessity of considering the d xy orbital in the phenomenological description of the nematic state. And in section 5 we will discuss the consequence the updated Fermi surface has on the understanding of the superconducting properties of FeSe. A summary of the electronic structure and missing electron pocket problem of FeSe is presented in Fig. 1.
II. ELECTRONIC STRUCTURE IN THE TETRAGONAL STATE
From both a theoretical and experimental point of view, the electronic structure of the tetragonal state is relatively well understood. Prior to the onset of nematicity at T s = 90 K, FeSe exhibits tetragonal symmetry with a P 4/nmm crystal structure [42]. This structure consists of layers of Fe atoms, in a 2D square lattice configuration, bridged by staggered out-ofplane Se atoms, giving rise to a crystallographic unit cell containing two Fe atoms and two Se atoms. The two Fe atoms are related by a glide-mirror symmetry, which can theoretically half the number of bands and allows for an unfolding to a 1-Fe Brillouin zone used by some authors [43], but here we use the 2-Fe unit cell notation for comparison with ARPES measurements.
The low energy electronic properties are governed by the partially-filled 3d xz , 3d yz and 3d xy orbitals of the two Fe atoms, which in momentum space gives rise to three hole bands around the Γ point and two symmetry-protected saddle point van-Hove singularities around the M point [44] as shown in Fig. 2 Of the three hole bands, two exist as a C 4 symmetric pair exhibiting predominantly d xz and d yz orbital weight (labelled h 1 and h 2 in Fig.2(a)) and the third is dominated by d xy orbital character (h 3 ). h 1 and h 2 would be energy degenerate at the high symmetry point, however spin orbit coupling lifts this degeneracy [45]. As for the van-Hove singularities around the M point, one is a saddle point connecting bands of majority d xz and d yz weight (vH 1 ) and the other is a saddle point connecting two d xy dominated bands (vH 2 ). This general structure is broadly applicable to all P 4/nmm Fe-based superconductors (e.g. Fe(Te,Se,S), LiFeAs, NaFeAs, LaFeAsO), with some modifications for the 122 family due to the I-centering of the lattice.
The experimentally measured Fermi surface of FeSe at 100 K (or more precisely, a map of the experimental spectral function at the chemical potential) at approximately k z = π is shown in Fig. 2(b), revealing a two-hole pocket and two electron pocket Fermi surface. Measurements around the center of the Brillouin zone show that both h 1 and h 2 cross the chemical potential at 100 K, as shown in Fig. 2(c). Their band maximas are separated by ∼20 meV due to spin-orbit coupling [45][46][47]. At k z = 0 these bands have a maxima at approximately h 2 = −13 meV and and h 1 = +7 meV [46], and at k z = π (shown in Fig. 2(c)) the bands have maxima of approximately h 2 = +5 meV and h 1 = +30 meV. The second smaller hole pocket of FeSe is thus only present at finite k z , which highlights an important property of this system. Even though FeSe has a "quasi-2D" structure, i.e the energy shift of the bands as a function of k z is only on the order of 20 meV, this energy scale is actually on the same order of magnitude as the total Fermi energy of this system, and therefore is non-negligible in quantitative descriptions of the physical properties of FeSe. We note in passing that, due to the small Fermi energy of this system, the electronic structure is subject to substantial temperature-dependence of the chemical potential, and the appearance of the "Fermi surface" changes substantially between 100 and 300 K [48], although without any change of the symmetry.
The third d xy hole band, h 3 , is observed to be much flatter and cross both h 1 and h 2 at an energy of approximately -50 meV. In most ARPES data sets, this band has a much lower intensity than the h 1 and h 2 bands, which is a consequence of photoemission-based matrix element effects, which ensures the intensity of photoelectrons originating from d xy states with momentum near |k| = 0 will be suppressed [49]. Nevertheless, h 3 can be identified most clearly near where it hybridises with h 1 and h 2 , and thus acquires some d xz and d yz orbital weight as shown in Fig. 2(c).
Near the corner of the Brillouin zone, both the d xy dominated electron band, connected to vH 2 , and the d xz /d yz electron band, connecting to vH 1 , are observed to cross the Fermi level. Here the outer four-fold symmetric electron pocket is dominated by d xy orbital character while the inner pocket is dominated by d xz and d yz orbital weight [50]. As this is a compensated system, the total Fermi volume of these electron pockets should be equal to that of the hole pockets [26].
These two sets of electron bands connect to the saddle points which have an energy of approximately vH 1 = −20 meV and vH 2 = −40 meV at the high symmetry point. The exact position of these stationary points, however, are masked by the presence of self-energy interactions which give rise to a broadening of the electronic states around the M point. This broadening is also captured in theoretical simulations involving spin and charge fluctuations [51].
The ARPES data presented in Fig. 2 is taken from our own works [26,29], however multiple data sets are available in the literature and are all consistent with the interpretation presented here [23-25, 27, 30, 52]. Indeed, the electronic structure must be constrained by the symmetry based arguments of Fig. 2(a) [44,50,53] and each of the bands observed in the measurements can be mapped to corresponding bands calculated from ab-initio techniques such as density functional theory (DFT) [23,26,50] of the paramagnetic tetragonal phase.
There are, however, serious quantitative issues with DFTbased calculations, which severely limit its use in describing the low energy properties of FeSe. First, DFT-based calculations overestimate the bandwidth of the Fe 3d-bands by a factor of ∼3 [26]. This is a generic finding across all Fe-based superconductors [54], and derives from the fact that electronic correlations are inadequately treated in DFT. It has been often argued that the correlation effects are orbital-dependent and particularly strong for the d xy orbital [26,54,55]. More advanced theoretical simulations, such as DFT + DMFT [56] and QSGW + DMFT [51], have had some success in capturing the global electronic structure on the eV scale [56,57], finding strongly incoherent spectral weight at 1-3 eV below E F and sharp quasiparticles only in the near vicinity of E F . However ab-initio efforts still usually overestimate the size of the hole and electron Fermi surfaces, which are much smaller in experiment [26,56]. Most DFT-based simulations additionally predict that the d xy hole band also crosses the Fermi level, suggesting a three hole pocket and two electron pocket Fermi surface [26,50]. Finally, typical DFT-based calculations also suggest that a stripe or staggered-stripe antiferromagnetic ground state is the most stable configuration [54,58], when in reality FeSe remains paramagnetic (albeit with strong antiferromagnetic fluctuations [59][60][61]). Current research is attempting to resolve this discrepancy from a pure ab-initio perspective. Wang. et. al. [61] were able to reproduce the band structure around the Gamma point using a a polymorphus network of local structural distortions. The use of hybrid exchange correlation functionals and Hubbard-Hund correlations have recently been shown to also produce a substantial improvement on the tetragonal structure [62].
Due to the current limitations in ab-initio modelling however, a substantial amount of work has gone into developing quantitatively accurate tight binding models of FeSe [8,48,50,63,64]. These models bypass the limitations in our current ab-initio theories, allowing for an accurate, albeit phenomenological, description of the single-particle electronic structure to be defined, which we can compare with experimental measurements. Several hopping parameters sets have been developed, which have been obtained by directly comparing the numerical band dispersion with experimental ARPES data in the tetragonal state [18,48,63,64]. These models have been shown to reproduce the single-particle electronic properties of tetragonal FeSe much better than conventional DFT-based approaches [48,50,63]. In particular these models accurately capture the small Fermi energy of FeSe, which has been shown to lead to strong chemical potential renormalising effects as a function of temperature and nematic ordering [30,48,[65][66][67]. By construction, such models allow for a quantitative description of the band positions of the hole and electron bands such that a comparison of the electronic structure in the nematic state can take place.
III. EXPERIMENTAL EVIDENCE FOR A MISSING ELECTRON POCKET IN THE NEMATIC STATE
We now focus on the electronic structure in the nematic state. Here experimental measurements encounter a major challenge. The nematic state is accompanied by a tetragonal to orthorhombic structural transition, at which point multiple orthorhombic domains form in the crystal. It has been identified that these domains are typically on the order of 1-5 µm in size [38,[68][69][70], which is much smaller than the cross section of the photon beam used in most high resolution synchrotronbased ARPES measurements (> 50µm [71]), as sketched in Fig. 3(a). Most of the initial photoemission data of FeSe in the nematic phase was collected on "twinned" crystals. In such measurements, the band dispersion measured along the experimental k x axis contains contributions from domains with the orthorhombic a axis both along, or perpendicular to, this direction, i.e. one measures a superposition of the spectral function arising from both domains. This creates an apparent C 4 symmetry in the measurements even at low temperatures (in the sense that the measured spectra are invariant under 90 degree rotation of the sample; the as-measured spectra are not generally fourfold-symmetric due to the ARPES matrix elements [72,73]). This can lead to ambiguity about which band arises from which domain.
ARPES measurements on twinned crystals
Multiple ARPES measurements on twinned crystals of FeSe have been reported [16, 17, 23-29, 52, 65, 74] and have been extensively reviewed [1,10,30,75]. We present a representative Fermi surface obtained from a twinned crystal in Fig. 3(b) from Ref. [29]. The hole pockets appear as two overlapping ellipses. Meanwhile, at the corner of the Brillouin zone, measurements reveal two electron pockets, which have been pinched in to produce what looks like two overlapping "peanuts".
The challenge now lies in identifying which of these pockets, comes from which domain. The two hole pockets can be easily understood as one ellipse from each orthorhombic domain. Measurements of the band dispersion around the hole pocket reveal that the inner hole band (h 2 ) undergoes a Lifshitz transition as a function of temperature and resides below the Fermi level at 10 K, whilst the outer hole band (h 1 ) elongates into an elliptical shape. As all three hole bands can be tracked as a function of temperature from the tetragonal to nematic state, there is little ambiguity about the shape of the hole pocket Fermi surface at low temperatures. However, it is not possible to identify the orientation of the elliptical hole pocket from a single domain, i.e to identify whether it elon-gates along the orthorhombic a or b axis simply from these twinned measurements.
For the electron pocket, however, the understanding was less clear, and historically several distinct band structures have been interpreted from nearly identical data sets [16,17,26,27,29]. As can be seen in Fig. 3(c), two electron pockets can be observed which look like overlapping "peanuts" in the twinned data. As the tetragonal state also exhibits two electron pockets, this may not appear that surprising. Indeed one interpretation was that the two oval shaped electron pockets in the tetragonal state simply pinched in at the sides, due to raising the binding energy of vH 1 [23,29]. In other words, the electron pockets could retain approximate fourfold symmetry around the M point, and the pockets from each domain simply overlapped in twinned data sets [17]. However, other interpretations, particularly those attempting to understand the nematic band shifts from theoretical grounds, believed that the nematic state should have two differently shaped electron pockets [26,27]. It was also equally plausible, experimentally at least, that only one electron pocket existed per domain [16,25,56]. Distinguishing between these scenarios was particularly challenging due to the broadness of the spectral weight around the M point in the tetragonal state (see Fig. 2(d)), which made a precise interpretation of the temperature evolution of the two van-Hove singularities ambiguous.
ARPES measurements on detwinned crystals
Compared to the measurements on twinned data, a much more preferable method to study the Fermi surface of FeSe would be to experimentally overcome the limitation imposed by these orthorhombic domains, and directly measure the electronic structure from a single crystallographic orientation. There are two strategies to overcome the twinning issue faced by ARPES measurements. Either 1) generate a sample with macroscopic ordering of the orthorhombic domains on length scales larger than the photon beam cross section, or 2) make the photon beam much smaller than the size of an orthorhombic domain. It has been known from earlier work on the 122 family of Fe-based superconductors that upon the application of "uniaxial" strain along the Fe-Fe direction, it becomes energetically favourable for a majority of the orthorhombic domains to align along that axis [76]. While the resulting domain population is unlikely to be 100% pure, measurements on strained, or "detwinned", samples, as sketched in Fig. 3(d), allows one to distinguish between the intense spectral weight arising from the majority domain and the weak spectral weight arising from the 90 degree rotated minority domain.
The first ARPES measurements on uniaxial strained samples of FeSe were performed in 2014 by Shimojima et. al. [25], where it was shown that the single hole pocket was elongated along the k y axis. Later, in 2017, Watson et. al. [31] was additionally able to resolve the detail of the electron pockets, as shown in Fig. 3(e). These measurements on detwinned crystals confirmed that the Fermi surface consisted of one elliptical hole-pocket, as expected from interpretation of the twinned measurements, but additionally revealed only one electron-pocket around the M point. This is shown in Fig. 3(f), where the majority of the spectral weight intensity now comes from one domain, and only a weak residual intensity comes from the minority domain. Unlike in the tetragonal state, at low temperatures, the electronic band structure around the M point produces sharp quasiparticle bands, a saddle point can be observed at −5 meV, which is electron like along the minor length of the electron pocket (as shown in Fig 3(g)), but hole-like when rotated by 90 degrees (Fig. 3(j)). Additionally, along the major length of the electron pocket, a deeper electron band and saddle point at ∼ −60 meV can be observed that is dominated by d xy orbital weight. This gap between the upper and lower saddle points, is approximately 50 meV, and has been previously quoted as a "nematic energy scale" [26,32,36]. However, as we will discuss in the theoretical section below, the exact energy scale of nematic shifts and splittings is slightly more complex and requires a linear combination of order parameters of different energy scales [64].
This finding of only a single electron pocket at the Fermi level was not the expected theoretical result [63], but nevertheless has now been reproduced by Yi et. al [32] (Fig. 3(h,k)) and Huh et. al. [33] (Fig. 3(i,l)). Further measurements on sulphur doped FeSe 1−x S x crystals under uniaxial strain by Cai et. al. [34,35] have also reported very similar Fermi surfaces.
Temperature dependent detwinned ARPES measurements
A natural question when studying the evolution of the electronic structure of FeSe from the tetragonal to nematic state is to ask how do ARPES measurements evolve as a function of temperature. Many data sets on twinned samples exist (as discussed e.g by Coldea and Watson [30]), and recently Yi et. al. and [32], Huh et. al. [33] have presented temperature dependent measurements on detwinned samples of FeSe. Similarly Cai et. al. [35] have reported temperature dependent measurements on detwinned samples of 9% doped FeSe 1−x S x .
In twinnned crystals, the temperature dependence of the hole bands leaves little room for ambiguity, and can be neatly tracked as a function of temperature [56]. However the bands around the M point are a bit more ambiguous, due to the broad spectral feature of the M point in the tetragonal state (as shown in Fig. 2(d)). Whilst this broad spectral feature is observed to split as a function of temperature, precisely tracking the vHs from high to low temperatures requires a degree of interpretation and peak fitting, with multiple papers suggesting different evolution of the spectral weight [23,27,29,55,77].
Unfortunately, the additional complication of uniaxial strain in detwinned measurements makes temperature dependent analysis technically even more challenging. By changing the temperature of your system, you inevitably alter the amount of strain applied to the sample due to thermal expansion of the rig, which in turn may alter the relative pop- (D-F) Equivalent sketch and measurement for a detwinned crystal of FeSe, which probes a majority of orthorhombic domains aligned in one direction. (G) Band dispersion of a detwinned crystal centered at the electron pocket. The insert shows the band path, from Watson et. al. [31]. h) Second derivative band dispersions of a detwinned crystal along the same path as (G) but extended from Z to A, taken from Yi. et. al. [32]. i) Band dispersion of a detwinned crystal along the same path as (G) from Huh. et. al. [33]. ulation of orthorhombic domains you are probing, and moreover the energetics of domain formation may be temperaturedependent.
-0.5 0 0.5 1 1st domain 2nd domain 1 1.2 1.4 −0.2 −0.1 0 0.1 0.2 a strain i k Y (Å −1 ) k X (Å −1 ) 1st domain 2nd domain 1 1.2 1.4 −0.2 −0.1 0 0.1 0.2 i k Y (Å −1 ) k X (Å −1 ) Watson et. al. (2017) Huh et. al. (2020) A B C D E F G H I J K L Yi et. al. (2019) k Y (Å −1 ) k Y (Å −1 ) 1 0.5 0 -0.5 -1 k X (Å −1 ) -0.5 0 0.5 1 1 0.5 0 -0.5 -1 k X (Å
Cai et. al. [35], observed that as a function of decreasing temperature, the spectral weight of the second electron pocket simply decreases, which could be explained as a change in orthorhombic domain populations originating from a Fermi surface consisting of just one electron pocket per domain. However, Cai et. al. argue that this is not the case and that the spectral weight loss is intrinsic to the nematic state [35].
On the other hand, Yi et. al. [32] and Huh. et. al. [33] argue that they observe a band shifting above the Fermi level in their temperature dependent measurements. A temperature band shift would be independent of orthorhombic domain population, however it should also have been detected within twinned ARPES measurements, which so far has not been reported.
We conclude by noting that temperature-dependent ARPES measurements of the electron pocket are very challenging to perform, firstly because of the issue with orthorhombic domains, and secondly due to the intrinsic broadness of the van hove singularities measured by ARPES. One approach to overcome this limitation would be to systematically study the evolution of the low temperature electronic structure across the series of isoelectronic sulphur substituted FeSe 1−x S x crystals, extending the existing measurements on twinned samples [52,74]. This will require future experimental investigation.
NanoARPES
There are experimental complications with performing ARPES measurements on uniaxially strained crystals, which may leave doubt as to the validity of the conclusions presented above. First, it is hard to fully exclude if the application of uniaxial strain has actually perturbed the underlying electronic structure of the crystal you are measuring. For example in the tetragonal material Sr 2 RuO 4 , uniaxial strain on the order of 1% shifts the position of the vHs by nearly 20 meV [79]. In order to fully support the conclusions from these ARPES measurements on detwinned crystals, complementary techniques must be employed and their results compared. To this end, nanoARPES has also been performed on crystals of FeSe. In these technically demanding measurements, the photon beam is focused to sub-micrometer spatial resolution using a focusing optic close to the sample [80]. The reduction of the spot size comes at the cost of dramatically reducing the photon flux, and thus the energy and angular resolutions are typically relaxed (compared to the earlier high-resolution results presented) in order to have a reasonable signal of photoelectrons. Nevertheless, the technique has been improved over the past 10 years to allow for energy resolution better than 20 meV [81]. This sub-micrometer spot size is smaller than a single orthorhombic domain, allowing for a spatial map of the sample from which the two orthorhombic domains can be distinguished by analysing their differing ARPES spectra, shown as red and blue stripes in Fig. 4(a,b). Measurements of the Fermi surface and band dispersion around the electron pocket in both domains (Fig. 4(c-f)) reveal an electronic structure totally consistent with that extracted from the ARPES measurements under uniaxial strain. In summary, the nanoARPES results fully support the conclusion of a Fermi surface in the nematic state consisting of a single hole pocket and a single electron pocket.
STM measurements
An entirely independent method to study the momentum resolved electronic structure within a single domain is to use scanning tunneling microscopy (STM). STM utilises quantum tunnelling, between the surface of a material and an atomically sharp tip, to study the electronic structure on the subnanometer scale. Information about the electronic structure can then be extracted in two ways. The first is by studying the differential conductance (dI/dV ) to obtain a quantity proportional to the local density of states of the system. The second is to measure quasiparticle interference (QPI), to measure the perturbations to the local density of states generated by the presence of defects such as impurities or atomic vacancies. The wavelength associated with this perturbation contains direct information about the allowed momentum dependent scattering vectors associated with an electronic structure at a constant energy via q = k − k .
Multiple STM measurements have been reported for FeSe, and information regarding the nematic [12,39,41] and superconducting state [12,39,40,82] have been determined, tetragonal state information has also been obtained from studies of isoelectronic sulphur doped crystals [40]. These measurements all contain a plethora of information regarding the local structure of the surface of FeSe, as well as information on defects [83,84]. Here, however, we focus on what the STM measurements can tell us about the low energy electronic structure in the nematic state, and whether this is consistent with the ARPES measurements discussed above. Although measuring QPI is an indirect method to measuring the electronic structure of a material, it is particularly powerful in determining band minimas and maximas, especially above the Fermi level, as well identifying whether bands have hole or electron scattering characteristics within a certain energy range.
The scattering vector vs energy dispersion along the q x and q y directions, taken from Ref. [40], are presented in Fig. 4(h,i). In agreement with other data sets [39,41], several holelike scattering vectors can be observed predominately along the q x axis, with a narrower hole-like dispersion along the q y direction. Also along the q y axis, one very clear electronlike scattering vector can be detected, which has a minima at ∼ −5 meV, and has been identified as a scattering vector that connects the d yz parts of the electron pocket in FeSe ( Fig. 4(g) [39][40][41]78]. No corresponding electron-like dispersion can be observed along the q x direction, which should be the [38]. (E,F) Fermi surface around the electron pocket and Ay − Z cut taken in an adjacent orthorhombic domain [38]. (G) Sketch of the Fermi surface scattering vectors inferred from STM measurements, as suggested by Ref. [40].
A b -0.5 0 0.5 1 -0.5 0 0.5 1 0.2 0 -0.2 1 1.2 1.4 1 1.2 1.4 0.2 0 -0.2 k (Å -1 ) X k (Å -1 ) Y k (Å -1 ) X k (Å -1 ) Y k (Å -1 ) Y k (Å -1 ) Y F E-E (eV) -0.15 -0.05 E-E (eV) -0.15 -0.05 F F 0 -0.1 -0.1 0 B C D E F k -40 -
(H,I) STM measurements of the QPI scattering dispersions as a function of energy along the qx and qy high symmetry axes respectively, reproduced from Ref. [40]. (J,K) Simulated QPI scattering dispersion from a model of FeSe which described the band structure shown in Fig. 3, from Ref. [78]. case in a two electron pocket scenario where all bands scatter equally. This was therefore interpreted as further evidence, from an independent technique to ARPES, that the Fermi surface of FeSe only consists of one hole pocket and one electron pocket, as sketched in Fig. 4(g).
We note that due to the indirect nature of QPI measurements, there is a degree of interpretation and uncertainty about the assignment of the electronic states and often it is necessary to directly simulate the QPI dispersion from a theoretical assumption of the electronic structure and compare the agreement. Due to the intrinsic broadness of the experimentally measured scattering vectors, this can lead to differing conclusions based on initial assumptions. For example, Kostin et. al. [41], assuming that two electron pockets must be present at the Fermi level, interpreted a weak spectral feature as evidence for a second electron pocket, with a greatly reduced scattering intensity. Whereas Rhodes et. al. [78], assuming that only one electron pocket was present at the Fermi level, interpreted this weak feature as an artifact of the Feenstra function, used in the experimental processing [85]. Importantly however, both theoretical simulations agree that a Fermi surface consisting of one hole pocket and two electron pockets can not independently reproduce the observed data without some additional form of anisotropy, which implies that ARPES and STM are probing the same underlying electronic structure. We present the numerical simulations from Ref. [78] in Fig. 4(j,k).
As an aside, it is interesting to note that the hole band maxima in 4(h) extends to +25 meV [40]. It is known from ARPES that only one hole-like scattering vector at this energy can exist, and specifically must be generated by the k z = π states [30]. This reveals that QPI measurements are sensitive to states with different k z . From arguments about the group velocity of electrons scattering off of defects [86,87], and the short range nature of quantum tunneling, it actually implies that QPI measurements will exhibit a k z -selectivity rule [78], such that all stationary points along the k z axis will contribute to scattering vectors that will be detected by STM measurements, this has recently been realised in the fully 3D system, PbS [88].
Points of contention
While we have so far presented a unified picture of the electronic structure of FeSe and have focused on points where broad agreement is found in the recent literature, historically there have been many points of disagreement surrounding the identification of bands and the nature of the Fermi surface, and there remain some points of contention.
Regarding the hole pockets, an outlying report is a recent claim from laser-ARPES measurements that there is additional splitting, most prominently resulting in two hole pockets at the Fermi level instead of one [89]. The implication is that the Kramers degeneracy of the bands is lifted, i.e. that either time-reversal or inversion symmetry is broken. However, it is worth noting that at low photon energies used the k z is not well-defined as the final states are not free electron-like, and the two Fermi contours identified appear to be fairly close to the known Fermi contours at k z = 0 and k z = π. Moreover, synchrotron-ARPES measurements with equally high energy resolution and better angular resolution (due to better definition of k z ) do not identify any additional splitting either in the Γ or Z planes [16], and neither has any comparable splitting been observed for the electron pockets. Finally, there is no supporting evidence for time-reversal symmetry breaking from other techniques. Thus it remains our view that the Kramers degeneracy holds for all states and that there is only one hole pocket crossing EF, which is significantly warped along the k z axis.
Regarding the electron pockets, while several groups have now coalesced around the one electron pocket scenario, it has previously been claimed that the ARPES data on twinned crystals is consistent with four features in the EDC at the M point [23] such that there are two electron pockets per domain, with each domain contributing a pair of crossed peanuts with slightly differing shapes [17]. This scenario is perhaps the most natural, as it is based on DFT predictions, and comes down to somewhat technical questions of whether asymmetric lineshapes at the M point contain one or two peaks, and whether the proposed small splittings can be resolved. Some of this groups data on twinned samples does indeed seem to show a splitting, which at face value would support their scenario. However, neither our group nor other groups have observed these claimed features and peak splittings in comparable data on twinned samples. Moreover, the detwinned data shows a complete absence of any spectral weight aside from the peanut along the a direction, in multiple experimental geometries, which cannot easily be explained away by matrix element effects in ARPES (and similarly in QPI). We encourage all groups to continue to push for higher resolution data which could finally settle the controversy, especially on detwinned samples.
IV.THEORETICAL EXPLANATIONS FOR THE MISSING ELECTRON POCKET
As we have discussed, the low energy electronic structure of the tetragonal state of FeSe can be qualitatively understood just from symmetry based arguments regarding the crystal structure and the d xz , d yz and d xy orbitals of the Fe atoms. This band structure can be explained both from the framework of tight-binding modelling [8,48,50,63] as well as DFTbased simulations. All of this implies that, although a true quantitative explanation describing the renormalisation of the band structure from correlation effects may be missing, our understanding of the single-particle physics is complete.
Within the nematic state, however, this is not the case. Following the previous logic, it would be assumed that the orthorhombic distortion produces a negligible change to the electronic structure, such that two hole pockets and two electron pockets should be present in the nematic state, which as the experimental data has revealed is clearly not the case. It is for this reason that the nematic state is believed to be of electronic or magnetic origin, yet the microscopic details still remain unclear. To address this, there has been a great deal of focus on trying to model how the nematic state evolves the electronic structure of a tetragonal-based model of FeSe, such as that shown in Fig. 5(a-c) originally presented in Ref. [64]. Specifically, theoretical research has attempted to develop a nematic order parameter which • Lowers the symmetry from C 4 to C 2 whilst still preserving mirror symmetry.
• Generates an elliptical hole pocket dominated by d xz orbital weight.
• Removes one of the two electron pockets from the Fermi surface.
Historically, the first attempt to describe such a mechanism assumed that the C 4 symmetry breaking was governed by a lifting of the energy degeneracy of the d xz and d yz orbitals [90].
Φ 1 (n xz − n yz ),(1)
where n xz/yz = c † A,xz/yz c A,xz/yz + c † B,xz/yz c B,xz/yz is the number operator for the xz or yz orbital respectively on atom A and B in a two atom unit cell model of FeSe, and Φ 1 is a scalar value used to describe the magnitude of the nematic order, which can in principle be fit to experiment.
This term, referred to in the literature as ferro-orbital ordering, is the simplest form of C 4 symmetry breaking possible in this system. It acts in a momentum independent fashion to raise the binding energy of the d xz bands and lower the binding energy of the d yz band, similar to a Jahn-teller distortion [91]. In this scenario, the electronic structure would evolve to produce a Fermi surface as shown in Fig 5(d), which despite producing the correct elliptical hole pocket, does not generate the one-electron-pocket Fermi surface determined from experiment.
Following the train of thought that the phenomenology of the nematic state may be captured by a degeneracy breaking of the d xz and d yz states, it was also noted that there are two additional B 1g symmetry breaking terms that can be defined and are equally valid in the nematic state [53,63] Φ 2 (n xz + n yz )(cos(k x ) − cos(k y ))
Φ 3 (n xz − n yz )(cos(k x ) + cos(k y ))(2)
Here, n xz/yz = c † A,xz/yz c B,xz/yz + c † B,xz/yz c A,xz/yz describes a hopping from an xz or yz orbital on atom A (B) to a xz or yz orbital on atom B (A). These two terms, referred to as d-wave nematic bond order (Φ 2 ) and extended-s wave bond order (Φ 3 ) respectively, in combination with the ferro orbital order (Φ 1 ) are the only possible nematic order parameters that can be defined for the d xz and d yz orbitals up to nearest neighbour hopping [53], and have been extensively used in previous theoretical descriptions of the nematic state of FeSe [12,15,16,20,21,29,41,77,78,[92][93][94][95][96][97][98][99]. The individual consequences of these order parameters are shown in Fig. 5(e) and 5(f). However, despite this vast amount of literature assuming these three d xz /d yz nematic order parameters as the starting point for theoretical analysis, there lies one big problem. No matter what values of Φ 1 , Φ 2 and Φ 3 are chosen, a Fermi surface consisting of one hole pocket and a single electron pocket can not be produced, at least not starting from a quantitatively accurate ARPES-based model of FeSe in the tetragonal state [64]. The best attempts to describe the ARPES data within this limitation result in a Fermi surface consisting of the correct elliptical hole pocket, a first electron pocket, of correct shape and size, and a second large electron pocket, dominated by d xy orbital character, as shown in Fig 5(g-i).
There is no experimental evidence for this large second electron pocket in the nematic state, and this discrepancy between theory and experiment has posed a major challenge for our theoretical understanding of nematicity. This is the central origin of the missing electron pocket problem. It has now become clear that a theory of nematicity only involving the physics captured in Eq. (1)-(3), i.e nematicity derived solely from d xz and d yz orbital ordering, is insufficient to reproduce our experimental measurements, and additional explanations for this discrepancy have had to be developed.
Orbital selective quasiparticle weights
The earliest attempt to explain this discrepancy came from attempts to understand local spin fluctuations in tetragonal FeSe, such as those incorporated by DFT + dynamic mean field theory (DMFT). Within this framework it has been shown that the self-consistently determined quasiparticle weight (Z) of the d xy orbital was significantly smaller than the quasiparticle weight of the d xz/yz orbitals [51,54,100], approximately half. As the spectral function intensity measured by ARPES is directly proportional to the quasiparticle weight, the contribution of d xy dominated bands should be significantly reduced, compared to the d xz and d yz dominated bands in ARPES measurements. It was thus argued that ARPES measurements may not be able to observe the d xy orbital, and thus would not detect the second d xy dominated electron pocket in the nematic state, e.g in Ref. [77], shown in Fig 6(a,b).
This argument however has not been supported by experimental measurements. Both in the tetragonal and nematic state, bands of d xy orbital character have been identified, particularly around the M point [30]. And although it is true that the d xy orbital appears to exhibit a larger effective mass renormalisation than the d xz and d yz orbitals [26], this extra renormalisation appears to not be enough to mask d xy spectral weight from ARPES-based measurements.
A similar, more phenomenological, approach was later employed by Kreisel et. al. [18] and popularised by Sprau et. al. [12]. Here the values of the nematic order parameters (Φ 1 − Φ 3 ) were adjusted such that two similar shaped electron pockets were generated (Fig. 6(d)), one dominated by d xz orbital weight and one dominated by d yz orbital weight, with the tips retaining significant d xy orbital character. Specifically, starting from an ARPES-based tetragonal model of FeSe [18] values of Φ 1 = 9.6 meV, Φ 2 = −8.9 meV and Φ 3 = 0 meV were used. It was then assumed that the nematic state could exhibit a significant reduction in the d xz quasiparticle weight compared to the d yz weight and, following the same argument as before, hidden from ARPES measurements of the spectral function. This is shown in Fig. 6(e). Following this logic, Sprau et. al. attempted to determine which values of Z by fitting them to experimental measurements of the angular dependence of the superconducting gap (discussed in Section 5) and the quasiparticle weight values chosen were Z xy = 0.1 Z xz =0.2 and Z yz =0.8, which in a later study was refined to Z xy = 0.073, Z xz = 0.16 and Z yz = 0.86 [41]. In order to reproduce experimental data, it was also necessary to strongly suppress the quasiparticle weight of the d xy orbital, which as a consequence effectively fully suppressed one of the two electron pockets at the Fermi level. Slave-spin calculations, starting from a DFT-based tight binding model and varying the contributions of Φ 1 −Φ 3 have also been performed and found that similar anisotropic ratios of the quasiparticle weights can be obtained [21], as shown in Fig. 6(c). A review of the slave-spin approach can be found in Ref. [94].
This formalism of "orbital selective quasiparticle weights", i.e suppressing the contribution of electronic states with d xz and d xy orbital character in the nematic state, has received the most traction out of the potential theories of the missing electron pocket of FeSe. It has been claimed to be in agreement with STM and QPI measurements of the electronic structure [41], the superconducting gap properties [12], the spin susceptibility measured by inelastic neutron scattering [60], µSR measurements of spin relaxations [97] and thermodynamic based-measurements [95]. A recent review on the topic can be found in Ref. [1].
In our view, however, the success of this approach is due to accurately generating a Fermi surface of FeSe that has the correct one hole pocket and one electron pocket structure, and not necessarily due to the underlying assumptions behind the ansatz of highly anisotropic quasiparticle weights. Indeed, a change in spectral weight, on the order of magnitude as proposed by this theory, is something that should be directly observable with ARPES based measurements. In the tetragonal state, the quasiparticle weight of the d xz and d yz orbitals must be equivalent by symmetry, and thus, under this assumption, there would be a strong sudden suppression of the d xz dominated bands upon entering the nematic state. This is not what is observed in experimental measurements, bands of d xz dominated weight are detected at all temperatures within the nematic state, with no obvious reduction to the spectral intensity [14-17, 27, 37]. Additionally, alternate explanations of the STM data and superconducting gap data, that do not rely on the assumption of orbital-selective quasiparticle weights, have been presented [16,19,40,78].
E-type order parameters
More recent attempts to explain the missing electron pocket have gone back to studying the single-particle physics of FeSe. A recent DFT + U calculation by Long et. al. [101], involving symmetry preconditioned wavefunctions, found a lower energy configuration of FeSe by breaking the E symmetry via a multipole nematic order, as shown in Fig 6(f). This has been further studied by Yamada et. al. [102]. This symmetry breaking essentially generates a tetragonal to monolclinic distortion by generating an overlap between a d xy orbital and either d xz or d yz orbital, which as a bi-product also breaks C 4 symmetry. This consequentially generates a hybridisation between the d xy dominated electron band and either the d xz or d yz dominated electron band and was shown to produce a one-electron pocket Fermi surface within a certain parameter regime.
A stable E-type nematic order parameter was equally identified, within a tight-binding framework using parameters extracted from LDA-based calculations, by Steffensen et. al. [103]. Here it was shown that including nearest-neighbour Coloumbic repulsion, the self consistently calculated meanfield nematic order parameter that had the largest magnitude was an inter-orbital term hybridising the d xz and d xy orbitals (or d yz and d xy ). This order parameter was equally able to generate a one-electron pocket Fermi surface, via a similar hybridisation mechanism as the DFT-based calculation as shown in Fig. 6(g,h).
This appears to suggest that long-range Coulomb repulsion can stabilise a C 4 symmetry breaking ground state in FeSe. However, in this scenario, the E-type order parameter would also reduce the crystal symmetry of FeSe from tetragonal to monoclinic. Currently, the experimental evidence suggesting a tetragonal to monoclinic structural distortion in FeSe is lacking. However, upon > 85% Te doping of the Se sites, a tetragonal to monoclinic transition has been realised [104]. This could hint that the known monoclinic structure of FeTe is actually stabilised by electron interactions [105], however whether this mechanism can describe the physics of FeSe will require further experimental investigation.
non-local dxy nematic order parameter
When considering the relevant d xz , d yz and d xy orbitals of tetragonal FeSe within a tight binding framework, there are only four order parameters that can be defined which break the B 1g rotational symmetry of the material within a single unit cell. The first three, described in Eq. (1) -(3), involve breaking the degeneracy of the d xz and dyz orbitals. However, a fourth equally valid order parameter involving the dxy orbital can also be defined as, Φ 4 (n xy )(cos(k x ) − cos(k y )).
This term acts as a hopping anisotropy for nearest neighbour d xy orbitals, in a similar manner as (2) for the d xz and d yz orbitals. It was initially defined by Fernandes et. al. [53], however in subsequent works it was assumed that this d xy nematic term would be much smaller, or negligible, compared to Eq. (1) -(3) [53]. Renormalisation group theory [106][107][108] additionally found, that whilst Eq. (4) was symmetry allowed, nematic symmetry breaking only had stable RG flow in either the d xz /d yz channel or the d xy channel, implying that finite Φ 1 − Φ 3 and Φ 4 would not both be present simultaneously [106]. However a weakly unstable trajectory suggested that this may not be the case [107].
In Ref. [64] Rhodes et. al. looked at the qualitative effect Φ 4 has on the electronic structure. They showed that a one-electron pocket Fermi surface could be generated from a ARPES-based tight binding model of FeSe solely using the Φ 4 term, as shown in Fig. 6(i). It was shown that Φ 4 has the effect of breaking the degeneracy of the d xy vHs (vH 2 in Fig. 2(a)), which if made large enough (∼50 meV) would induce a Lifshitz transition of the d xy band, and thus reduce the total number of electron pockets crossing the Fermi level to one. This is shown in Fig. 6(k). in combination with Φ 1 to Φ 3 , the addition of Φ 4 made it possible to generate a Fermi surface in agreement with the ARPES measurements, as shown in Fig. 6(j). A recent study has also found this order to be consistent with specific heat measurements [109].
However, in order to get quantitative agreement with the Fermi surface and low-energy electronic structure using Eq.
(1) -(4), it was observed that the the splitting of the d xy van-Hove singularity must be asymmetric. Specifically, ARPES measurements as a function of temperature find that the lower part of the d xy vHs around the M point remains approximately at the same energy [27,29,32]. This is not captured by the Φ 4 term that assumes a symmetric splitting of the bands. To account for this, Rhodes et. al. [64] included a d xy -specific Hartree shift, a constant energy shift of the d xy orbital at the M point, that although allowed by symmetry, did not have an obvious origin. Additionally, in order to generate a Lifshitz transition of the electron pocket, and obtain quantitative agreement with experimental data as a function of temperature both d xy terms, Φ 4 and the Hartree shift, had to be significantly larger than the the d xz /d yz terms (Φ 1 − Φ 3 ). Specifically, in order to reproduce the ARPES measurements Φ 1 +Φ 3 = 15 meV, Φ 1 +Φ 2 = −26 meV and Φ 4 = ∆ Hartree = 45 meV [64]. It is also worth noting that the mean-field analysis by Steffensen et. al. [103] equally found that the Φ 4 nematic order parameter should be finite, but found it to be of approximately equal magnitude as Φ 1 -Φ 3 rather than twice as large, as suggested by Rhodes et. al. [64].
Importance of the dxy orbital in theories of nematicity Each theory proposed to describe the low-energy electronic structure of the nematic state of FeSe has it's relative strengths and weaknesses. Nevertheless a common theme in these dif- ferent attempts has begun to emerge. In all methods used to theoretically remove an electron pocket from the Fermi level, it has been necessary to modify the d xy orbital in some way. Whether that's suppressing its contribution via quasiparticle weights, gapping out the d xy band via hybridisation, or rigidly shifting the d xy band above the Fermi level. What we can gleam from this analysis therefore, is that we should view the nematic state in a new light, not originating from a specific orbital ordering mechanism of d xz and d yz states, but rather as a symmetry breaking phenomena which couples to every or-bital at the Fermi level. Further theoretical investigations are required in order to elucidate the origin of the nematic state. The importance of the d xy orbital has also been recently noted from NMR measurements [110] and angular dependent magnetoresistance [111].
V. CONSEQUENCES FOR THE SUPERCONDUCTING GAP SYMMETRY
One of the most striking properties of FeSe is it's highly tuneable superconducting transition temperature, ranging from 8 K in bulk crystals [42], 36.7 K under pressure [112], and up to 65 K when a monolayer is placed on SrTiO 3 [113], and hence the nature of superconductivity in FeSe is an important question that attracted a lot of attention.
From an experimental point of view, the momentum dependence of the superconducting gap of bulk FeSe, has been extensively determined from ARPES [13][14][15][16][17], STM [12,40,82], Spectific heat [114,115] and muSR measurements [97], with surprisingly near unanimous agreement as to the angular dependence of the gap structure around both the hole and electron pocket. This achievement provided the perfect opportunity to directly compare theories of superconductivity with experimental measurements.
In this section, we will review the experimental data of the momentum dependence of the superconducting gap, particularly from ARPES measurements, and discuss the theoretical consequence the updated Fermi surface topology has on the theoretical understanding of superconductivity in FeSe.
Experimental measurements of the superconducting gap
The key findings from the multiple ARPES and QPI measurements are presented in Fig. 7. For the gap situated on the hole pocket, a highly two-fold anisotropic momentum dependence of the gap was measured, as shown from QPI analysis by Sprau et. al. in Fig. 7(a). The angular dependence of the hole pocket using ARPES was first reported in 2016 by Xu et. al. [13] on 7% sulphur doped FeSe measured at 6.3 K, as shown in Fig. 7(f). It was found that the angular dependence at both k z = 0 (using a photon energy of hν = 37 eV) and k z = π (hν = 21 eV) produced near identical momentum distributions. This sulphur doped system has a very similar electronic structure to undoped FeSe, albeit with a slightly reduced nematic transition temperature [116] and slightly higher superconducting transition temperature (9.8 K [13]). Later, in 2018, Liu et. al. [15] and Hashimoto et. al. [14] used laser ARPES, with hν = 6.994 eV, on FeSe at 1.6 K and observed the same highly anistropic angular dependence of the gap at the hole pocket, as shown in Fig. 7(b,c). By using such a low photon energy and temperature these authors ensured the greatest possible energy resolution for resolving the gap of the hole pocket. However the trade-off here is that information about states with large angular momentum, e.g the electron pockets, as well as the k z -dependence of the hole pocket, can not be obtained. Kushnirenko et. al. [17], as well as Rhodes et. al. [16], were able to resolve the three dimensional gap structure of both the hole and electron pockets using synchrotron radiation, as shown in Fig. 7(d,e). In these manuscripts, it was again confirmed that the gap structure of the hole pocket at both k z = 0 and k z = π exhibited the same highly anisotropic two-fold angular dependence of the gap as determined in the Sulphur doped sample of Xu. et. al. [13]. Kushnirenko et. al. claimed that the superconducting gap that was larger at k z = π and smaller at k z = 0, however Rhodes et. al. suggested the opposite: the gap was observed to be larger at k z = 0 and smaller at k z = π. We note that in order to reach the k z = 0 hole pocket, a higher photon energy of 37 eV is required, which makes the measurement of the gap at the Γ point exceedingly challenging, and the measurements are at the cutting edge of what is currently achievable by synchrotron-based ARPES measurements.
Hashimoto et. al. additionally claimed that the gap structure produced a different behaviour with and without the presence of uniaxial strain. Without strain, they observed a cos(8θ) behaviour [14], which when accidentally detwinned via uniaxial strain, yielded a gap structure that is consistent with the other measurements. So far this cos(8θ) dependence of the gap has not been reproduced.
As for the electron pocket, the angular dependence of the gap from QPI measurements is presented in Fig. 7(g). Revealing a particularly constant gap magnitude across the length of the ellipse, which quickly decays towards zero at the tips of the pocket. This is where the orbital character of the pocket transforms from predominantly d yz weight to d xy weight. ARPES measurements by Kushnirenko et. al [17], and Rhodes et. al. [16], were also able to resolve the angular dependence of the superconducting gap at the electron pocket. ARPES measurements along the minor length of the electron pocket, above and below T c , are shown in Fig. 7(h,i). Thanks to the orbital sensitivity of ARPES-based measurements, Rhodes et. al. found a direct correlation between the intensity of d yz orbital weight and the size of the superconducting gap, establishing a direct link between orbital character and gap magnitude. Kushnirenko et. al. [17] also observed that the rate that the gap decreased as a function of momentum was slightly different for intermediate k z values (Fig. 7(m).
This extremely aniostropic gap structure for both the hole and electron pocket raises a question as to whether FeSe is a nodal or nodeless superconductor, which could have a profound effect on our understanding of the gap symmetry in this system. For example, neglecting the electron pocket, it was argued by Hashimoto et. al. that a nodal gap structure of the hole pocket would be consistent with p-wave superconductivity [14] (This is not consistent once the gap structure of the electron pocket is additionally taken into account). It is not possible to clearly distinguish between a nodal gap or a very small gap in ARPES measurements, due to the limitations of energy resolution arising from thermal broadening and the choice of photon energy. Alternate techniques, [13], showing equivalent momentum dependence as the undoped sample. (G) Angular dependence of the gap around the electron pocket as extracted from BQPI measurements from Sprau et. al. [12]. (H,I) Band dispersion along the minor length of the electron pocket above and below Tc, along the high symmetry axis from Rhodes et. al. [16]. (J) Comparison of the gap magnitude (Leading Edge Gap -LEG) and the intensity of the spectral weight from Linear Vertical polarised light as a function of kx, which is directly correlated to the amplitude of dyz orbital weight. The gap is observed to decrease with decreasing dyz weight. Taken from Rhodes et. al. [16]. (K) Sketch of the angular dependence of the electron pocket at kz = 0 (bottom) kz = π 2 (middle) and kz = π (top) from Kushirenko et. al. [17]. (A,G) Reproduced from Ref. [12] with permission from the AAAS. (B) Reproduced from Ref [15] under the Creative Commons Attribution 4.0 International License. (C) b) Reproduced from Ref [14] under the Creative Commons Attribution 4.0 International License. (F) Reproduced from Ref. [13] with permission from the American Physical Society. (E,K) Reproduced from Ref. [17] with permission from the American Physical Society. such as STM and specific heat measurements, do have sufficient energy and thermal resolution to tackle this issue, but here STM measurements of the density of states by Sprau et. al. [12] suggest a fully gapped, nodeless, superconducting ground state, whereas specific heat measurements have argued that the measured data is consistent with a nodal superconducting gap [114]. It is still unclear whether FeSe exhibits nodes or very small superconducting gaps, however as we will discuss below, theoretical arguments appear to suggest that if any nodes do exist, they would be accidental in nature.
Theoretical understanding of the superconducting gap
The most striking result from the experimentally determined gap structure of FeSe, is the clear realisation that the size of the superconducting gap at the Fermi level is correlated with the magnitude of d yz orbital weight. This tells us that the superconducting pairing mechanism is sensitive to orbital character, and is evidence for superconductivity mediated by Coulomb interactions, such as via a spin-fluctuation mechanism of superconductivity.
Although the idea that spin fluctuations govern the Cooper pairing in the iron-based superconductors, was originally proposed back when superconductivity in these materials were first discovered [7], the evidence for this has often been inferred from gap symmetry arguments, such as a sign-changing s ± order parameter [12], or from the general argument that FeSe is close to a magnetic instability. FeSe, being such a clean system, has enabled a direct comparison between theoretical simulations and experimental data.
Indeed many theoretical simulations of the angular dependence of the superconducting gap in FeSe have been performed [12, 16, 18-21, 64, 103]. However, as the formation of Cooper pairs are directly sensitive to the states at the Fermi level, the starting model used to describe FeSe is very important. Numerical simulations have shown that models of FeSe which do not account for the missing electron pocket of the nematic state, i.e a model Fermi surface which describes two electron pockets around the M point, can not reproduce the experimentally observed gap structure [12,16,19,20].
Initially, this was a confusing result, but with hindsight it is not that surprising. The presence of an extra electron pocket in the simulations would naturally influence the superconducting pairing. Due to the local nature of Coulomb repulsion, the pairing between electrons in real space will be largest for electrons located on the same atom in the same orbital. It follows from this argument, that the pairing of electrons in momentum space would be favoured if a spin scattering process occurs which couples electronic states of the same orbital character. In the nematic state of FeSe, spin-fluctuations are strongest when connecting the hole and electron pocket [60,117,118]. In a one-electron pocket scenario, the only common orbital content between the two pockets are the d yz orbital weight, as shown in Fig. 8(a-d), and thus this would dominate the superconducting gap magnitude. This would not be the case in a two-electron pocket scenario, where scattering with d xz electrons between the hole and electron pocket would also contribute.
It has now been shown that irregardless of the theoretical mechanism employed to remove this second electron pocket from the superconducting calculation, whether that's orbital selective quasiparticle weights [12,18,21], orbital selective spin fluctuations [19], E-type nematic ordering [103], a nonlocal d xy nematic order parameter [64] or simply ignoring it from simulations of the superconducting pairing outright [16] (as shown in Fig. 8), the correct momentum dependence of the gap structure can be naturally captured assuming weakcoupling spin fluctuation mediated pairing. This is a remarkable finding, not only does it further support the theory of spin-fluctuation mediated superconductivity in the iron-based superconductors, but it provides another independent piece of evidence for a single electron pocket around the M point in the nematic state of FeSe. This result highlights the incredible importance of correctly accounting for the missing electron pocket in the nematic state, as without it we can not begin to understand the superconducting properties of this material.
DISCUSSION
This review has been wholly focused on what at first glance might appear to be an esoteric point of discussion, namely, the characterisation and modelling of the Fermi surface of FeSe in the nematic state. However, we propose that after the hundreds of papers and many years of debate and controversy on the subject, that there are very important conclusions to be drawn, which have wider implications for our understanding of both nematic ordering and superconductivity across the wider family of Fe-based superconductors.
The first conclusions surround nematic ordering, where the results establish • That nematic ordering affects all bands at the Fermi level, with the d xy derived bands playing as significant a role as the d xz and d yz derived bands.
• That nematic order manifests in the band structure through a combination of all allowed symmetrybreaking terms, primarily anisotropic hopping terms, and cannot be exclusively treated by on-site orbital ordering.
• That nematic ordering does not cause a minor perturbation of the electronic structure, but can lift an entire electron pocket away from the Fermi level.
We believe that these conclusions should be widely applicable across other Fe-based superconductors. While these conclusions do not yet constitute a self-consistent microscopic mechanism of nematic order, they do present strong constraints to any proposed microscopic models. The second set of conclusions relate to the superconductivity:
• The superconducting gap of FeSe is remarkably anisotropic.
• The fact that the gap follows the d yz orbital character is strong experimental evidence that the pairing mechanism is sensitive to local orbital degrees of freedom, i.e. for spin-fluctuation pairing.
• The superconducting gap of FeSe can be naturally reproduced by spin-fluctuation calculations assuming only one electron pocket at the Fermi level.
There has long been a consensus that the superconductivity in the Fe-based systems is mediated by spin-fluctuation pairing, but we argue that FeSe provides some of the most direct experimental support for this. As long as one starts with the one-electron pocket Fermi surface, the further details of the calculation are not critical, because in this scenario the only orbital component which is present on both the hole and electron pockets is the d yz character, and so this channel dominates the structure of the gap. The success of this result justifies the use of similar spin-fluctuation pairing calculations on other Fe-based superconductors, although we emphasize the importance of starting with an experimentally accurate Fermi surface.
Importantly this insight has only been unlocked once we understand that the true Fermi surface of FeSe consists of one hole pocket and a single electron pocket, rather than one hole pocket and two electron pockets as was initially believed. However, despite us emphasizing how the one electron pocket scenario is key to the understanding of the unusual properties of FeSe, we believe it is still an open question as to what mechanism really drives this modification of the electronic structure. The models of describing the electronic structure in the nematic state have grown more accurate and more sophisticated, yet there is a lack of intuition about what is the real driving force for the evolution of the electronic structure that we observe. In our opinion it remains a delicate and important open question, but solving it in the case of FeSe could unlock a wider understanding of nematicity in the iron-based superconductors.
Additionally, whilst the experimental challenge imposed by measuring the electronic structure of orthorhombic crystals has always been present, the focus on an answer to the origin of nematicity in FeSe has particularly emphasised the continued development of detwinning methods in ARPES [25, 31-37, 81, 119], as well as showcasing the potential of NanoARPES for strongly correlated materials with local domain structures [64,81].
OUTLOOK AND CONCLUSION
With an outlook to the future, there are still multiple open questions regarding the missing electron pocket problem, nematicity and superconductivity in FeSe. Firstly, can we experimentally identify the exact conditions when one of the electron pockets in the tetragaonal state appears or disappears from the Fermi level? So far, this has remained slightly ambiguous, with some experiments claiming a gradual disappearance of the electron pocket [34] and others claiming a Lifshitz transition around 70 K [32,33,64].
Another open question is how the missing electron pocket scenario can be reconciled with the QPI measurements as a function of sulphur doping [40] or Tellurium doping [2], each providing an isoelectronic tuning parameter to control the evolution of the Fermi surface. The systematic evolution of the Fermi surface has been studied by Quantum Oscillations [120], however due to the tiny size of the Fermi energy in this system, the unambiguous assignment of the quantum oscillation frequencies is challenging [64]. Equally, twinned ARPES measurements on sulphur doped FeSe have already been performed [52,74], as well as several studies on detwinned crystals for 9% sulphur doping [34,35]. So far it is unclear when the missing electron pocket reappears, and so further measurements of detwinned FeSe 1−x S x are desirable, although by 18% the system is tetragonal once more and two electron pockets are certainly observed [74].
Finally, an important avenue of research is how does the momentum-dependence of the superconducting gap change as nematicity is supressed, e.g as a function of sulphur doping. The momentum dependence of the superconducting gap for undoped FeSe has now been extensively characterised, and theoretical predictions of how the gap should evolve as nematicity is suppressed have been proposed [64]. This much needed experimental data would again place important constraints on our theories of nematicity and superconductivity in these systems.
As the study of the Fe-based superconductors has matured since they exploded onto the scene in 2008, the emphasis has shifted from basic characterisation of a wide variety of superconducting families, to detailed examination of particular cases. FeSe has been the subject of particularly focused attention, and the effort has been worthwhile, with two remarkable results emerging: the one electron pocket Fermi surface, and the highly anisotropic superconducting gap structure. We have argued that these two, taken together, provide strong evidence for spin-fluctuation pairing in FeSe, which is presumably applicable to the wider family of Fe-based superconductors. However, the extent to which the one electron pocket phenomenology may be applicable to the nematic phase of other material systems is a large open question; as well as FeSe 1−x S x and FeSe 1−x Te x , we propose NaFeAs [119] as a candidate worthy of re-examination. Thus as this review of FeSe concludes, we propose it is time to take the experimental and theoretical tools developed for case of FeSe, and apply them with renewed vigour to the wider field of Fe-based superconductors.
FIG. 1 .
1Summary of the Fermi surface of FeSe and photoemission measurements of a single electron pocket. (A) Crystal structure of FeSe, Fe (black), Se (grey). (B)
(C) Sketch of the experimentally determined Fermi surface of FeSe in the tetragonal state and (D) in the nematic state. (E-J) Summary of the Fermi surface of FeSe measured around the M/A point via different photoemission techniques. (E) Measurement in the tetragonal state (100 K, LV hv = 56 eV [29]) showing two electron pockets, (F) ARPES Measurement in the nematic state (10 K, LV, hv = 56 eV [29]) arising from a superposition of two orthorhombic domain orientations (red and blue regions), referred to as a "twinned" measurement. (G,H) ARPES measurement of a detwinned crystal in the nematic state (10 K, hv = 56 eV [31], where strain is applied either along the a or b crystallographic axis and predominately probes a single domain orientation. (I,J) NanoARPES measurement using a photon beamspot of < 1 µm (30 K, hv = 56 eV [38]) in individual orthorhombic domains.
FIG. 2 .
2Electronic structure in the tetragonal state of FeSe. (A) Sketch of the low-energy band structure of a typical P 4/nmm Febased superconductor along the Γ − M (kz = 0) or Z − A (kz = π) high symmetry points. The colours indicate the dominant orbital character of the bands. (B) Fermi surface in the tetragonal state measured at 100 K close to kz = π (hν = 56 eV). (C) Cut along the Z − A direction (equivalent to Γ − M but at kz = π) direction for the hole bands around the Z point for the same photon energy (hν = 56 eV). (D) Cut along the Z − A direction for the electron bands around the A point. Figures adapted from[26,29].
FIG. 3 .
3Summary of ARPES measurements on detwinned crystals of FeSe (A) sketch of a photoemission setup on a twinned crystal, showing equal coverage of both red and blue orthorhombic domains. (B) Fermi surface measured from a measurement on a twinned crystal (hν=56 eV). Taken from Watson et. al. [31] (C) Close up of the electron pocket near the A point from Watson et. al. [29] (hν=56 eV).
(J-L) Equivalent measurements but taken along the length of the electron pocket. (H,K) reproduced from Ref. [32] under the Creative Commons Attribution 4.0 International License. (I,J) reproduced from Ref. [33] under the Creative Commons Attribution 4.0 International License.
FIG. 4 .
4Electronic structure within a single domain without the application of uniaxial strain. (A) Sketch of a NanoARPES measurement, where the beam is focused to have a cross section < 1µm. (B) Experimental spatial map of FeSe where the colour corresponds to the orthorhombic domains of FeSe, reproduced from Ref. [38]. (C,D) Fermi surface around the electron pocket (hν = 56 eV, 30 K) and Ay − Z cut taken within a single orthorhombic domain
(G-I) are reproduced under the Creative Commons Attribution 4.0 International License.
FIG. 5 .
5Limitations of dxz/dyz nematic ordering and origin of the missing electron pocket problem. (A,B,C) Fermi surface and band dispersions around the Z and A point, for a tetragonal state model of the electronic structure from ref. [64] in quantitative agreement with ARPES measurements. (D,E,F) The individual effect of the three symmetry breaking d xz/yz nematic order terms on the Fermi surface of the tetragonal state model. (D) Ferro orbital order (Φ1 = 26 meV) (E) d-wave bond order (Φ2 = −26 meV) (F) Extended s-wave bond order (Φ3=15 meV). (G,H,I) Fermi surface and band dispersions around the Z and A point, using a combination of Φ1, Φ2 and Φ3 as is often used in the literature. No matter what linear combination of these order parameters are used, a Fermi surface in agreement with the experimental data can not be produced.
FIG. 6 .
6Theoretical attempts to resolve the missing electron pocket problem. (A) Fermi surface of the electron pockets in the nematic state proposed by Christensen et. al.[77]. The spectral function is argued to have an increased decoherence of dxy weight, which is argued would not be observed by experiment and is simulated in (B). (C) Slave-spin calculations from Yu et. al.[21], revealing the possibility of highly anisotropic quasiparticle weights with local Coulomb repulsion. (D,E) Spectral function of the 1-Fe unit cell tight binding model from Kreisel et. al.[18], with and without orbital-selective quasiparticle weights, highlighting the possible suppression of the second electron pocket via incoherent dxz and dxy spectral weight. f) Band dispersion of FeSe obtained from a DFT+U calculation with symmetry preconditioned wavefunctions from Long et. al.[101], highlighting the band hybridisation obtained if an E-type nematic order parameter is considered.(G) Fermi surface of the 1-Fe unit cell model from Steffensen et. al. [103] taking into account a self consistently obtained E-type nematic order parameter. (H) Band dispersion from the model used by Steffensen et. al.[103] showing a band hybridisation of the dxz (red) and dxy (blue) bands around the Y point (1-Fe unit cell), gapping out the second electron pocket. (I) Fermi surface obtained from the 2-Fe unit cell tetragonal model from Fig. 5(a) assuming dominant dxy nematic ordering, as suggested by Rhodes et. al.[64]. (J) Equivalent Fermi surface including all four symmetry allowed nematic order parameters of FeSe and a symmetry allowed Hartree shift. (L) Mean-field temperature evolution of the electronic states at the high symmetry M point, highlighting a Lifshitz transition of the dxy band and removal of the second electron pocket as proposed by Rhodes. et. al. [64]. (A,B) Reproduced from Ref. [77] under the Creative Commons Attribution 4.0 International License. (C) Reproduced from Ref. [21] with permission from the American Physical Society. (D,E) Reproduced from Ref. [18] with permission from the American Physical Society. (F) Reproduced from Ref. [101] under the Creative Commons Attribution 4.0 International License.
FIG. 7 .
7Experimental measurements of the superconducting gap of FeSe. (A) Angular dependence of the gap around the hole pocket as extracted from BQPI measurements from Sprau et. al.[12]. (B) Angular dependence of the gap around the hole pocket as extracted from Laser ARPES measurements from Liu et. al.[15]. (C) Angular dependence of the gap around the hole pocket from Hashimoto et. al.[14]. Red dots are data from a twinned sample, whereas green data was measured on an accidentally strained sample. (D) Band dispersion of the kz = 0 hole band (hν = 37 eV) from Rhodes et. al.[16] taken along the direction where the hole band gap is largest, above and below Tc. (E) Equivalent band dispersion of the kz = π hole band (hν = 21 eV) below Tc from Kushnirenko et. al.[17]. (F) Angular dependence of the hole band of FeSe0.93S0.07 from Xu et. al.
FIG. 8 .
8Theoretical simulation of the momentum dependence of the superconducting gap from Rhodes et. al. [16]. Here, a Fermi surface consisting of one hole pocket a single electron pocket were considered and a spin fluctuation pairing mechanism was assumed. (A,B) Fermi surface of the hole pocket and angular dependence of the orbital content of the hole pocket. (C,D) Fermi surface of the one electron pocket and angular dependence of the orbital content of the electron pocket. The colour labels are red -dxz, green dyz and blue dxy. (E) Simulated angular dependence of the superconducting gap for the hole pocket (red) and electron pocket (blue), revealing a direct correlation with the dyz weight shown in (B) and (D). The crosses and dots are experimental data extracted from STM[12] and ARPES[16] measurements respectively.
CONFLICT OF INTEREST STATEMENTThe authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.AUTHOR CONTRIBUTIONSAll authors contributed to the development of this review article.FUNDING LCR acknowledges funding from the royal commission for the exhibition for the 1851.
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[
"SEMIGROUP GRADED ALGEBRAS AND CODIMENSION GROWTH OF GRADED POLYNOMIAL IDENTITIES",
"SEMIGROUP GRADED ALGEBRAS AND CODIMENSION GROWTH OF GRADED POLYNOMIAL IDENTITIES"
]
| [
"A S Gordienko "
]
| []
| [
"Mathematics Subject Classification"
]
| We show that if T is any of four semigroups of two elements that are not groups, there exists a finite dimensional associative T -graded algebra over a field of characteristic 0 such that the codimensions of its graded polynomial identities have a fractional exponent of growth. In particular, we provide an example of a finite dimensional graded-simple semigroup graded algebra over an algebraically closed field of characteristic 0 with a fractional graded PI-exponent, which is strictly less than the dimension of the algebra. However, if T is a left or right zero band and the T -graded algebra is unital, or T is a cancellative semigroup, then the T -graded algebra satisfies the graded analog of Amitsur's conjecture, i.e. there exists an integer graded PI-exponent. Moreover, in the first case it turns out that the ordinary and the graded PI-exponents coincide. In addition, we consider related problems on the structure of semigroup graded algebras. | 10.1016/j.jalgebra.2015.04.027 | [
"https://arxiv.org/pdf/1409.0151v3.pdf"
]
| 117,649,321 | 1409.0151 | 98241199221f9cf8bd00ab38e495bbb97559547b |
SEMIGROUP GRADED ALGEBRAS AND CODIMENSION GROWTH OF GRADED POLYNOMIAL IDENTITIES
2010
A S Gordienko
SEMIGROUP GRADED ALGEBRAS AND CODIMENSION GROWTH OF GRADED POLYNOMIAL IDENTITIES
Mathematics Subject Classification
2010and phrases Associative algebraJacobson radicalpolynomial identitygradingsemigroupzero bandH-(co)module algebrabialgebracodimensionAmitsur's conjecture Supported by Fonds Wetenschappelijk Onderzoek -Vlaanderen Pegasus Marie Curie post doctoral fel- lowship (Belgium) and RFBR grant 13-01-00234a (Russia)
We show that if T is any of four semigroups of two elements that are not groups, there exists a finite dimensional associative T -graded algebra over a field of characteristic 0 such that the codimensions of its graded polynomial identities have a fractional exponent of growth. In particular, we provide an example of a finite dimensional graded-simple semigroup graded algebra over an algebraically closed field of characteristic 0 with a fractional graded PI-exponent, which is strictly less than the dimension of the algebra. However, if T is a left or right zero band and the T -graded algebra is unital, or T is a cancellative semigroup, then the T -graded algebra satisfies the graded analog of Amitsur's conjecture, i.e. there exists an integer graded PI-exponent. Moreover, in the first case it turns out that the ordinary and the graded PI-exponents coincide. In addition, we consider related problems on the structure of semigroup graded algebras.
The notion of a semigroup graded algebra is a natural generalization of the notion of a group graded algebra, however the first notion is much less restricting: e.g. if an algebra is the direct sum of its left ideals or if an algebra is the direct sum of a subalgebra and an ideal, this can be expressed in the language of semigroup gradings.
In 2010-2011 E. Aljadeff, A. Giambruno, and D. La Mattina [1,2,5] proved that if an associative PI-algebra is graded by a finite group, then there exists an integer exponent of codimensions of its graded polynomial identities, i.e. the graded analog of Amitsur's conjecture holds. In [7,Theorem 1] and [9,Theorem 3] the author proved the same for finite dimensional associative and Lie algebras graded by any groups. In [10] A. V. Kelarev studied semigroup graded PI-algebras.
The next question that naturally arises in this investigation is as to whether the results on codimension growth of graded polynomial identities hold for semigroup graded associative algebras.
In the associative case the main properties that we use in order to prove the graded analog of Amisur's conjecture (Theorem 2) are the gradedness (or homogeneity) of the Jacobson radical and the graded version of the Wedderburn -Artin theorem. We consider these properties in Sections 2 and 3 and obtain the graded analog of Amitsur's conjecture for algebras graded by cancellative semigroups (Theorem 6) and unital algebras graded by left or right zero bands (Theorem 7). In the first case we use Kelarev and Plant's result on gradedness of Jacobson radicals in algebras graded by cancellative groupoids [11,Corollary 4.1].
Until now, there were no examples known of an associative algebra with a fractional PIexponent of any kind (graded, Hopf, etc.). In 1999 S. P. Mishchenko and M. V. Zaicev gave an example of an infinite dimensional Lie algebra with a fractional PI-exponent [12] (see the proof in [13]). Here we use their ideas to present a finite dimensional semigroup graded associative algebra with a fractional exponent of codimension growth of graded polynomial identities (Theorems 3-5) for each of four semigroups of two elements that are not groups.
The PI-exponent of a finite dimensional graded-simple group graded Lie or associative algebra over an algebraically closed field of characteristic 0 equals the dimension of the algebra [7,Example 12] and [9,Theorem 4]. In Theorem 5 we provide an example of a finite dimensional graded-simple semigroup graded algebra over an algebraically closed field of characteristic 0 with a fractional graded PI-exponent, which is strictly less than the dimension of the algebra.
Semigroups of two elements
First we describe all the five non-isomorphic semigroups of two elements. Let T 1 = {0, 1} be the multiplicative semigroup of the field Z 2 . Let T 2 = {0, v} be the semigroup defined by relations v 2 = 0 2 = 0 · v = v · 0 = 0. Recall that a semigroup T is a right zero band if t 1 t 2 = t 1 for every t 1 , t 2 ∈ T and a left zero band if t 1 t 2 = t 2 for every t 1 , t 2 ∈ T .
Let T 3 be the right zero of two elements.
Proposition 1. Let T be a semigroup that consists of two elements. Then T is isomorphic to one of semigroups from the list {T 1 , T 2 , T 3 , T op 3 , (Z 2 , +)} and each two semigroups from this list are non-isomorphic. (Here T op 3 is anti-isomorphic to T 3 .)
Proof. First, consider the case when T = {a, a 2 } for some a ∈ T . Then if a 3 = a, we have a 4 = a 2 , a 2 is the identity element of T , and T ∼ = (Z 2 , +). If a 3 = a 2 , then a 4 = a 3 = a 2 , a 2 is the zero element of T , and T ∼ = T 2 . Now consider the case when T = {a, a 2 } for all a ∈ T . Then T = {a, b}, a 2 = a, b 2 = b. If ab = ba, then T ∼ = T 1 . If ab = ba, then T ∼ = T 3 for ab = b, ba = a, and T ∼ = T op 3 for ab = a, ba = b.
Gradedness of the Jacobson radical
Let T be a semigroup. An algebra A is T -graded if A = ⊕ t∈T A (t) (direct sum of subspaces) and
A (h) A (t) ⊆ A (ht) . A subspace V of A is graded (or homogeneous) if V = t∈T V ∩ A (t) .
It is known [11,Example 4.2], that the Jacobson radical is not necessarily graded. (See also the survey of positive results in [11,Section 4.4].) Here we provide examples and results related to semigroups of two elements and left and right zero bands.
Denote by M k (F ) is the full k × k matrix algebra over a field F and UT k (F ) is the algebra of upper triangular k ×k matrices. In M k (F ) we fix the basis of matrix units e i , 1 i, k.
Example 1. Let A = M k (F ) ⊕ UT k (F ) (direct sum of ideals) where F is a field, k 2. Define a T 1 -grading on A by A (0) = (M k (F ), 0), A (1) = {(ϕ(a), a) | a ∈ UT k (F )} where ϕ : UT k (F ) → M k (F ) is the natural embedding. Then J(A) = {(0, e ij ) | 1 i < j k} ⊂ (0, UT k (F )), J(A) ∩ A (0) = J(A) ∩ A (1) = 0, and J(A) is not a graded ideal. Example 2. Let A = M k (F ) ⊕ V (direct sum of ideals) where V ∼ = M k (F ) as a vector space, k ∈ N, V 2 = 0, and F is a field. Denote by ϕ : V → M k (F ) the corresponding isomorphism. Define a T 2 -grading on A by A (0) = (M k (F ), 0), A (v) = {(ϕ(a), a) | a ∈ V }. Then J(A) = (0, V ), J(A) ∩ A (0) = J(A) ∩ A (v) = 0,
and J(A) is not a graded ideal.
Example 3. Let A = M k (F ) ⊕ V (direct sum of ideals) where V ∼ = M k (F ) as a left M k (F )- module, k ∈ N, V 2 = V M k (F ) = 0, and F is a field. Denote by ϕ : V → M k (F ) the corresponding isomorphism. Define a T 3 -grading on A by A (e 1 ) = (M k (F ), 0), A (e 2 ) = {(ϕ(a), a) | a ∈ V }. Then J(A) = (0, V ), J(A) ∩ A (0) = J(A) ∩ A (v) = 0,
and J(A) is not a graded ideal.
Remark. One can use the opposite example to show that the Jacobson radical is not necessarily T op 3 -graded. However, if an algebra is unital and T 3 -or T op 3 -graded, then the Jacobson radical is graded. In fact, a more general result holds.
Proposition 2. Let A be a T -graded associative algebra with 1 over a field F for some left or right zero band T . Then every ideal of A is graded.
Proof. Consider the case when t 1 t 2 = t 2 for every t 1 , t 2 ∈ T . (Another case is considered analogously.) Then all A (t) , t ∈ T , are left ideals of A and 1 = t∈T e t for some e t ∈ A (t) . Let I be an ideal. Then for every a ∈ I we have a = t∈T ae t where ae t ∈ I ∩ A (t) for every t ∈ T . Hence I = t∈T I ∩ A (t) is a graded ideal.
Graded analogs of the Wedderburn theorems and T -graded simplicity
Now we study whether the graded analogs of the Wedderburn theorems hold for T -graded algebras where T is a semigroup. Recall that a T -graded algebra A is a graded-simple algebra if A has no graded ideals other than A and 0.
Example 4. Let B = M k (F ) ⊕ M k (F ) (direct sum of ideals), k ∈ N, where F is a field. Define a T 1 -grading on A by A (0) = (M k (F ), 0), A (1) = {(a, a) | a ∈ M k (F )}.
Then B cannot be presented as the direct sum of T 1 -graded ideals that are T 1 -graded-simple algebras, i.e. the T 1 -graded analog of the Wedderburn -Artin theorem does not hold.
Proof. Note that the semisimple algebra B has only four ideals: 0, B, (0, M k (F )), and (M k (F ), 0). Three of them are T 1 -graded, namely, 0, B, and (M k (F ), 0), and only (M k (F ), 0) is a T 1 -graded-simple algebra.
Remark. Since B (0) is always a graded ideal, every T 1 -graded-simple algebra B has the trivial grading, i.e. B = B (0) . Therefore, every T 1 -graded-simple algebra is simple as an ordinary algebra.
Proposition 3. Let B be a finite dimensional associative T 2 -graded semisimple algebra over a field F . Then B = B (0) , and by the ordinary Wedderburn -Artin theorem, B is the direct sum of T 2 -graded ideals that are simple algebras (with the trivial grading). Moreover, every simple T 2 -graded algebra is simple as an ordinary algebra. In particular, the T 2 -graded analog of the Wedderburn -Artin theorem holds.
Proof. Suppose B = B (0) . Note that B (0) is an ideal and, by the ordinary Wedderburn -Artin theorem, B = B (0) ⊕ I for some semisimple ideal I of B. However, (B/B (0) ) 2 = 0 since (B (1) ) 2 ⊆ B (0) , and I ∼ = B/B (0) cannot be semisimple. Hence B = B (0) , the algebra B has the trivial T 2 -grading, and we can apply to B the ordinary Wedderburn -Artin theorem.
Proposition 4. Let B be a finite dimensional associative T -graded semisimple algebra over a field F for some left or right zero band T . Then B is the direct sum of T -graded ideals that are simple algebras. In particular, the T -graded analog of the Wedderburn -Artin theorem holds and every finite dimensional semisimple T -graded-simple algebra is simple as an ordinary algebra.
Proof. Note that any subset of T is again a left or right zero band. Therefore, reducing T to the support of the grading, we may assume that T is finite. Consider the case when T is a right zero band. The other case is considered analogously.
By the ordinary Wedderburn -Artin theorem,
B = B 1 ⊕ B 2 ⊕ . . . ⊕ B s (direct sum of ideals)
for some simple algebras B i . If I is a left ideal in B, then
I = 1 B 1 I ⊕ 1 B 2 I ⊕ . . . ⊕ 1 Bs I (direct sum of ideals), and 1 B i I = I1 B i = I ∩ B i . Hence if B = B (t 1 ) ⊕ . . . ⊕ B (ts)
is the T -grading, then for every i, we have
B i = 1 B i B = (1 B i B (t 1 ) ) ⊕ . . . ⊕ (1 B i B (es) ) = (B i ∩ B (t 1 ) ) ⊕ . . . ⊕ (B i ∩ B (ts) ).
Therefore, each B i is a T -graded ideal. Now the theorem follows.
However there exist non-semisimple T 3 -graded-simple algebras. (See Proposition 6 below.) By Proposition 2, if a T -graded algebra, where T is a left or right zero band, contains unity, the its Jacobson radical is graded (as well as all the other ideals). Therefore, one may ask whether the T -graded analog of the Wedderburn -Mal'cev theorem holds for such algebras. In fact, the answer is true. Proof. Without lost of generality, we may assume that T is a right zero band. First we consider the case J 2 = 0.
Note that 1 A = t∈T e t for some e t ∈ A (t) . Moreover, since 1 A e t = r∈T e r e t and e r e t ∈ A (t) for every t ∈ T , we have e 2 t = e t and e r e t = 0 for all r = t. Using the ordinary Wedderburn -Mal'cev theorem we choose a maximal semisimple subalgebra B such that A = B⊕J (direct sum of subspaces). Let π : A → A/J be the natural projection which is a graded map since J is graded. Let ϕ : A/J → A be a homomorphic embedding such that ϕ(A/J) = B and πϕ = id A/J . Note that π(1 A ) = t∈T π(e t ) is the unity of A/J. Therefore 1 A = 1 B = ϕπ(1 A ).
Let T = {t 1 , . . . , t s }. If ϕπ(e t i ) = e t i for all 1 i s, then Be t i ⊆ B for all 1 i s and B = ⊕ t∈T Be t is a graded subalgebra and the theorem is proved. Choose 1 k s such that ϕπ(e t i ) = e t i for all 1 i k and ϕπ(e t k+1 ) = e t k+1 . Note that π(ϕπ(e t k+1 ) − e t k+1 ) = 0 and ϕπ(e t k+1 ) = e t k+1 + j for some j ∈ J. In addition, je t i = (ϕπ(e t k+1 ) − e t k+1 )e t i = ϕπ(e t k+1 e t i ) − e t k+1 e t i = 0 for all 1 i k. Analogously, e t i j = 0 for all 1 i k. Moreover, since (e t k+1 + j) 2 = e t k+1 + j, we have j = e t k+1 j + je t k+1 and e t k+1 je t k+1 = 0.
Letφ : A/J → A be the homomorphism defined bỹ
ϕ(a) = (1 A + e t k+1 j − je t k+1 )ϕ(a)(1 A + e t k+1 j − je t k+1 ) −1 = (1 A + e t k+1 j − je t k+1 )ϕ(a)(1 A − e t k+1 j + je t k+1 ).
Note that
ϕπ(e t i ) = (1 A + e t k+1 j − je t k+1 )e t i (1 A − e t k+1 j + je t k+1 ) = e t i for all 1 i k. Moreoverφ π(e t k+1 ) = (1 A + e t k+1 j − je t k+1 )ϕπ(e t k+1 )(1 A − e t k+1 j + je t k+1 ) = t∈T ϕπ(e t ) + e t k+1 j − je t k+1 ϕπ(e t k+1 ) t∈T ϕπ(e t ) − e t k+1 j + je t k+1 = (ϕπ(e t k+1 ) + e t k+1 j − je t k+1 )ϕπ(e t k+1 )(ϕπ(e t k+1 ) − e t k+1 j + je t k+1 ) = (e t k+1 + j + e t k+1 j − je t k+1 )(e t k+1 + j)(e t k+1 + j − e t k+1 j + je t k+1 ) = e t k+1 .
ThereforeB =φ(A/J) is a maximal semisimple subalgebra such that A =B ⊕J (direct sum of subspaces) andφπ(e t i ) = e t i for all 1 i k + 1. Thus, using the induction argument, we may assume that e t =φπ(e t ) ∈B for all t ∈ T . HenceB = ⊕ t∈TB e t is a graded subalgebra of A.
We have proved the theorem for the case J 2 = 0. The general case is proved by induction on dim A. Suppose J 2 = 0. Then A/J 2 = B 0 ⊕ J/J 2 (direct sum of graded subspaces) for some graded maximal semisimple subalgebra B 0 of A/J 2 . Consider the preimage
B 1 of B 0 in A under the natural map π 1 : A → A/J 2 . Since B 0 ∼ = A/J is semisimple, J(B 1 ) = J 2 .
Moreover dim B 1 < dim A and, by the induction assumption, we have B 1 = B ⊕ J 2 (direct sum of graded subspaces) for some graded maximal semisimple subalgebra B in A. Hence A = B ⊕ J (direct sum of graded subspaces) and the theorem is proved.
Recall that a semigroup T is cancellative if for every a, b, c ∈ T each of the conditions ac = bc and ca = cb implies a = b.
Proposition 5. Let B be a finite dimensional associative T -graded semisimple algebra over a field F for some cancellative semigroup T . Then B is the direct sum of T -graded ideals that are T -graded-simple algebras. In particular, the T -graded analog of the Wedderburn -Artin theorem holds.
Proof. By the ordinary Wedderburn -Artin theorem, B = B 1 ⊕ . . . ⊕ B s (direct sum of ideals) for some simple algebras B i . Let I be a minimal T -graded ideal in B. Then
I = B i 1 ⊕ . . . ⊕ B i k for some i 1 , . . . , i k . Define N := i∈{1,...,s}\{i 1 ,...,i k } B i . Since all B i are semisimple, we have N = {b ∈ B | ba = 0 for all a ∈ I}.
Since T is cancellative and I is graded, the ideal N is graded too. Hence B = I ⊕ N (direct sum of graded ideals) where I is a T -graded-simple algebra. Applying to N the inductive argument, we get the proposition.
Graded polynomial identities, their codimensions and cocharacters
Let T be a semigroup and let F be a field. Denote by F X T -gr the free T -graded associative algebra over F on the countable set
X T -gr := t∈T X (t) , X (t) = {x (t) 1 , x (t) 2
, . . .}, i.e. the algebra of polynomials in non-commuting variables from X T -gr . The indeterminates from X (t) are said to be homogeneous of degree t. The T -degree of a monomial x (t 1 )
i 1 . . . x (tt)
is ∈ F X T -gr is defined to be t 1 t 2 . . . t s , as opposed to its total degree, which is defined to be s. Denote by F X T -gr (t) the subspace of the algebra F X T -gr spanned by all the monomials having T -degree t. Notice that
F X T -gr (t) F X T -gr (h) ⊆ F X T -gr (th) ,
for every t, h ∈ G. It follows that
F X T -gr = t∈G F X T -gr (t) is a T -grading. Let f = f (x (t 1 ) i 1 , . . . , x (ts) is ) ∈ F X T -gr . We say that f is a graded polynomial identity of a T -graded algebra A = t∈T A (t) and write f ≡ 0 if f (a (t 1 ) i 1 , . . . , a (ts) is ) = 0 for all a (t j ) i j ∈ A (t j ) , 1 j s. The set Id T -gr (A) of graded polynomial identities of A is a graded ideal of F X T -gr . Example 5. Let T = (Z 2 , +) = {0,1}, M 2 (F ) = M 2 (F ) (0) ⊕ M 2 (F ) (1) where M 2 (F ) (0) = F 0 0 F and M 2 (F ) (1) = 0 F F 0 . Then x (0) y (0) − y (0) x (0) ∈ Id T -gr (M 2 (F )). Let P T -gr n := x (t 1 ) σ(1) x (t 2 ) σ(2) . . . x (tn) σ(n) | t i ∈ T, σ ∈ S n F ⊂ F X T -gr , n ∈ N.
Then the number
c T -gr n (A) := dim P T -gr n P T -gr n ∩ Id T -gr (A)
is called the nth codimension of graded polynomial identities or the nth graded codimension of A.
The analog of Amitsur's conjecture for graded codimensions can be formulated as follows.
Conjecture. There exists
PIexp T -gr (A) := lim n→∞ n c T -gr n (A) ∈ Z + .
If T is the trivial (semi)group of one element, we get the notion of ordinary polynomial identities, ordinary codimensions c n (A), and the ordinary PI-exponent PIexp(A).
As we shall see in Theorems 3-5 below, the analog of Amitsur's conjecture fails for all semigroups T of two elements that are not groups.
However, in Theorem 2 below we provide sufficient conditions for a graded algebra to satisfy the analog of Amitsur's conjecture. As a consequence, we prove that if T is a cancellative semigroup or T is a left or right zero band, and a finite dimensional T -graded algebra A contains 1, then A satisfies the graded analog of Amitsur's conjecture (Theorems 6 and 7).
Polynomial H-identities of associative algebras with a generalized H-action
In our case, instead of working with graded codimensions directly, it is more convenient to replace the grading with the corresponding dual structure and study the asymptotic behaviour of polynomial H-identities.
Let H be an arbitrary associative algebra with 1 over a field F . We say that an associative algebra A is an algebra with a generalized H-action if A is endowed with a homomorphism
H → End F (A) and for every h ∈ H there exist h i , h i , h i , h i ∈ H such that h(ab) = i (h i a)(h i b) + (h i b)(h i a) for all a, b ∈ A.
(1)
Remark. We use the term "generalized H-action" in order to distinguish from the case when an algebra is an H-module algebra for some Hopf algebra H which is a particular case of the generalized H-action.
Let F X be the free associative algebra without 1 on the set X :
= {x 1 , x 2 , x 3 , . . .}. Then F X = ∞ n=1 F X (n) where F X (n)
is the linear span of all monomials of total degree n.
Let H be an arbitrary associative algebra with 1 over F . Consider the algebra
F X|H := ∞ n=1 H ⊗n ⊗ F X (n) with the multiplication (u 1 ⊗ w 1 )(u 2 ⊗ w 2 ) := (u 1 ⊗ u 2 ) ⊗ w 1 w 2 for all u 1 ∈ H ⊗j , u 2 ∈ H ⊗k , w 1 ∈ F X (j)
, w 2 ∈ F X (k) . We use the notation
x h 1 i 1 x h 2 i 2 . . . x hn in := (h 1 ⊗ h 2 ⊗ . . . ⊗ h n ) ⊗ x i 1 x i 2 . . . x in . Here h 1 ⊗ h 2 ⊗ . . . ⊗ h n ∈ H ⊗n , x i 1 x i 2 . . . x in ∈ F X (n) .
Note that if (γ β ) β∈Λ is a basis in H, then F X|H is isomorphic to the free associative algebra over F with free formal generators x
γ β i , β ∈ Λ, i ∈ N.
We refer to the elements of F X|H as associative H-polynomials. Note that here we do not consider any H-action on F X|H .
Let A be an associative algebra with a generalized H-action. Any map ψ : X → A has a unique homomorphic extensionψ :
F X|H → A such thatψ(x h i ) = hψ(x i ) for all i ∈ N and h ∈ H. An H-polynomial f ∈ F X|H is an H-identity of A ifψ(f ) = 0 for all maps ψ : X → A. In other words, f (x 1 , x 2 , . . . , x n ) is an H-identity of A if and only if f (a 1 , a 2 , . . . , a n ) = 0 for any a i ∈ A. In this case we write f ≡ 0. The set Id H (A) of all H-identities of A is an ideal of F X|H .
We denote by P H n the space of all multilinear H-polynomials in
x 1 , . . . , x n , n ∈ N, i.e. P H n = x h 1 σ(1) x h 2 σ(2) . . . x hn σ(n) | h i ∈ H, σ ∈ S n F ⊂ F X|H . Then the number c H n (A) := dim P H n P H n ∩Id H (A)
is called the nth codimension of polynomial H-identities or the nth H-codimension of A.
One of the main tools in the investigation of polynomial identities is provided by the representation theory of symmetric groups. The symmetric group S n acts on the space
of irreducible characters χ(λ). Let e T λ = a T λ b T λ and e * T λ = b T λ a T λ where a T λ = π∈R T λ π and b T λ = σ∈C T λ
(sign σ)σ, be Young symmetrizers corresponding to a Young tableau T λ . Then
M (λ) = F Se T λ ∼ = F Se * T λ is an irreducible F S n -module
corresponding to a partition λ n. We refer the reader to [3,4,6] for an account of S n -representations and their applications to polynomial identities.
Generalized (F T ) * -action on T -graded algebras
In this section we show that every finite dimensional semigroup graded algebra is an algebra with a generalized H-action for a suitable associative algebra H.
For an arbitrary semigroup T one can consider the semigroup algebra F T over a field F which is the vector space with the formal basis (t) t∈T and the multiplication induced by the one in T .
Consider the space (F T ) * dual to F T . Then (F T ) * is an algebra with the multiplication defined by (hw)(t) = h(t)w(t) for h, w ∈ (F T ) * and t ∈ T . The identity element is defined by 1 (F T ) * (t) = 1 for all t ∈ T . In other words, (F T ) * is the algebra dual to the coalgebra F T .
Let Γ : A = ⊕ t∈T A (t) be a grading on an algebra A. We have the following natural (F T ) *action on A:
ha (t) = h(t)a (t) for all h ∈ (F T ) * , a (t) ∈ A (t) and t ∈ T .
Remark. If T is a finite group, then A is an F T -comodule algebra for the Hopf algebra F T and an (F T ) * -module algebra for the Hopf algebra (F T ) * .
For every t ∈ T define h t ∈ (F T ) * by h t (g) = 0 if g = t, 1 if g = t for g ∈ T .
If A is finite dimensional, the set supp Γ := {t ∈ T | A (t) = 0} is finite and
h t (ab) = g,w∈supp Γ, gw=t h g (a)h w (b) for all a, b ∈ A.
By linearity, we get (1). Therefore, A is an algebra with a generalized (F T ) * -action.
Note that ha = t∈supp Γ h(t)h t a for all a ∈ A and x h − t∈supp Γ h(t)x ht ∈ Id (F T ) * (A)(2)
for all h ∈ (F T ) * . Proof. Denote the grading
A = ⊕ t∈T A (t) by Γ. Let ξ : F X | (F T ) * → F X T -gr be the homomorphism of algebras defined by ξ(x h i ) = t∈supp Γ h(t)x (t) i , i ∈ N, h ∈ (F T ) * .
Suppose f ∈ Id (F T ) * (A). Consider an arbitrary graded homomorphism ψ : F X T -gr → A.
Then the homomorphism of algebras ψξ : F X | (F T ) * → A satisfies the condition
ψξ(x h i ) = t∈supp Γ h(t)ψ x (t) i = h t∈supp Γ ψ x (t) i = h ψξ(x i ).
Thus ψξ(f ) = 0 and ξ(f ) ∈ Id T -gr (A). Hence ξ Id (F T ) * (A) ⊆ Id T -gr (A). Denote bỹ
ξ : F X | (F T ) * / Id (F T ) * (A) → F X T -gr / Id T -gr (A) the homomorphism induced by ξ. Let η : F X T -gr → F X | (F T ) * be the homomorphism defined by η x (t) i = x ht i for all i ∈ N and t ∈ T .
Consider an arbitrary graded polynomial identity f ∈ F X T -gr . Let ψ : F X | (F T ) * → A be a homomorphism satisfying the condition ψ(x h i ) = hψ(x i ) for every i ∈ N and h ∈ (F T ) * . Then for any i ∈ N and g, t ∈ T we have
h g ψη x (t) i = h g ψ(x ht i ) = h g h t ψ(x i ) = 0 if g = t, ψη x (t) i if g = t
Thus ψη x
f + Id (F T ) * (A) ∈ F X | (F T ) * / Id (F T ) * (A) for f ∈ F X | (F T ) * andf = f + Id T -gr (A) ∈ F X T -gr / Id T -gr (A) for f ∈ F X T -gr . We havẽ ηξ x h i =η t∈supp Γ h i (t)x (t) i = t∈supp Γ h i (t)x ht i =x h i for every h ∈ (F T ) * and i ∈ N. (Here we use (2).) Thusηξ = id F X|(F T ) * / Id (F T ) * (A) . Moreover ξη x (t) i =ξ x ht i =x (t)
i for every t ∈ T and i ∈ N. Therefore,ξη = id F X T -gr / Id T -gr (A) and
F X T -gr / Id T -gr (A) ∼ = F X | (F T ) * / Id (F T ) * (A)c (F T ) * n (A) = dim P (F T ) * n P (F T ) * n ∩ Id (F T ) * (A) = dim P T -gr n P T -gr n ∩ Id T -gr (A) = c T -gr n (A).
Now we can provide a sufficient condition for a graded algebra to satisfy the graded analog of Amitsur's conjecture.
Theorem 2. Let A be a finite dimensional non-nilpotent T -graded associative algebra over an algebraically closed field F of characteristic 0 for some semigroup T . Suppose that the Jacobson radical J := J(A) is a graded ideal. Let
A/J = B 1 ⊕ . . . ⊕ B q (direct sum of graded ideals)
where B i are graded-simple algebras and let κ : A/J → A be any homomorphism of algebras (not necessarily graded) such that πκ = id A/J where π : A → A/J is the natural projection. Then there exist constants
C 1 , C 2 > 0, r 1 , r 2 ∈ R such that C 1 n r 1 d n c T -gr n (A) C 2 n r 2 d n for all n ∈ N where d = max dim B i 1 ⊕ B i 2 ⊕ . . . ⊕ B ir r 1, ((F T ) * κ(B i 1 ))A + ((F T ) * κ(B i 2 ))A + . . . ((F T ) * κ(B i r−1 ))A + ((F T ) * κ(B ir )) = 0 and A + := A + F · 1.
Proof. The theorem is an immediate consequence of Lemma 1 and [8, Theorem 1].
Remark. The existence of the map κ follows from the ordinary Wedderburn -Mal'cev theorem.
Remark. If A is nilpotent, i.e. x 1 . . . x p ≡ 0 for some p ∈ N, then P T -gr n ⊆ Id T -gr (A) and c T -gr n (A) = 0 for all n p. Corollary. The above analog of Amitsur's conjecture holds for such codimensions.
Partitions restricted to convex polytopes
Here we apply ideas from [13] and prove auxiliary results that we use in the construction of algebras with fractional graded PI-exponents.
In this section we show that if all the partitions λ n that correspond to irreducible F S n -modules with nonzero multiplicities m(A, H, λ) belong to a convex polyhedron, then lim n→∞ n c H n (A) is bounded by the maximum of a particular function F on the "continuous" version of the polyhedron.
Fix q ∈ N. Let F (α 1 , . . . , α q ) = Suppose we have some numbers γ ij ∈ R for 1 i m, 0 j q and θ k ∈ R for q < k r where m, r ∈ Z + , r q. Define Ω = (α 1 , . . . , α q ) ∈ R q i=1 α i = 1, α 1 α 2 . . . α q 0, q j=1 γ ij α j 0 for 1 i m .
For every n ∈ N we define Ω n = λ n q j=1 γ ij λ j + γ i0 0 for 1 i m, λ i θ i for q < i r, λ r+1 = 0 .
We treat Ω and Ω n as the "continuous" and the "discrete" version of the same polyhedron. Denote by d the maximum of F on the compact set Ω. (We assume Ω to be non-empty.)
for some C 1 , C 2 > 0 and r 1 , r 2 ∈ R that do not depend on λ i . Let ε > 0. Since F is continuous, there exists δ > 0 such that for every x from the domain of F such that the distance between x and Ω is less than δ, we have F (x) < d + ε.
Therefore, by (3), there exists n 0 ∈ N such that for all n n 0 and λ n such that m(A, H, λ) = 0 we have dim M (λ) C 2 n r 2 (d + ε) n .
By [8,Theorem 5], there exist C 3 > 0, r 3 ∈ Z + such that λ n m(A, H, λ) C 3 n r 3 for all n ∈ N.
Hence c H n (A) = λ n m(A, H, λ) dim M (λ) C 2 C 3 n r 2 +r 3 (d + ε) n and lim n→∞ n c H n (A) d + ε.
Since ε > 0 is arbitrary, we get the lemma. Lemma 3. Let q ∈ N, q 4,
Ω = (α 1 , . . . , α q ) ∈ R q q i=1 α i = 1, α 1 α 2 . . . α q 0, α q + α q−1 α 1 . Then d := max x∈Ω F (x) = (q − 3) + 2 √ 2 = q − 0.1716 . . .
Proof. We express α 1 in terms of α 2 , . . . , α q and consider F 0 (α 2 , . . . , α q ) := F 1 − q i=2 α i , α 2 , . . . , α q = 1
(1 − q i=2 α i ) (1− q i=2 α i) α α 2 2 . . . α αq q
on the segment Ω 0 = (α 2 , . . . , α q ) α 2 0, . . . , α q 0, α 2 + . . . + α q−2 + 2α q−1 + 2α q 1 .
Note that we have weakened the restrictions on α i . However, we will see that max x∈Ω F (x) = max x∈Ω 0 F 0 (x).
We have F 0 (α 2 , . . . , α q ) = e −(1− q i=2 α i) ln(1− q i=2 α i) − q i=2 (α i ln α i ) and ∂F 0 ∂α k (α 2 , α 3 , . . . , α q ) = ln 1 − q i=2 α i − ln α k e −(1− q i=2 α i) ln(1− q i=2 α i) − q i=2 (α i ln α i ) .
Hence the only critical point is (α 2 , α 3 , . . . , α q ) = 1 q , . . . , 1 q / ∈ Ω 0 . Therefore, F 0 takes its maximal values on the border ∂Ω 0 of Ω 0 . Note that ∂Ω 0 = Υ ∪ q i=2 Ω i where Ω i = {(α 2 , . . . , α q ) ∈ Ω 0 | α i = 0} and Υ = {(α 2 , . . . , α q ) ∈ Ω 0 | α 2 + . . . + α q−2 + 2α q−1 + 2α q = 1}. Determining the critical points once again, we get F 0 (x) q − 1 for all x ∈ q i=2 Ω i . Consider F 0 on Υ. We express α 2 in terms of α 3 , . . . , α q and define
F 1 (α 3 , . . . , α q ) = F 0 (1 − α 3 − . . . − α q−2 − 2α q−1 − 2α q , α 3 , . . . , α q ) = 1 (α q−1 + α q ) α q−1 +αq (1 − α 3 − . . . − α q−2 − 2α q−1 − 2α q ) 1−α 3 −...−α q−2 −2α q−1 −2αq α α 3 3 . . . α αq q = e −(α q−1 +αq) ln(α q−1 +αq)−(1−α 3 −...−α q−2 −2α q−1 −2αq) ln(1−α 3 −...−α q−2 −2α q−1 −2αq)−α 3 ln α 3 −...−αq ln αq on Υ 1 = {(α 3 , . . . , α q ) | 1 − α 3 − . . . − α q−2 − 2α q−1 − 2α q 0, α 3 0, . . . , α q 0} . Then ∂F 1 ∂α i (α 3 , . . . , α q ) = (ln(1 − α 3 − . . . − α q−2 − 2α q−1 − 2α q ) − ln α i )·− α 3 − . . . − α q−2 − 2α q−1 − 2α q ) − ln(α q−1 + α q ) − ln α i )· e −(α q−1 +αq) ln(α q−1 +αq)−(1−α 3 −...−α q−2 −2α q−1 −2αq) ln(1−α 3 −...−α q−2 −2α q−1 −2αq)−α 3 ln α 3 −...−αq ln αq , for i = q − 1, q. Therefore, (α 3 , . . . ,α q ) whereα 3 = . . . =α q−2 = √ 2 4+(q−3) √ 2 andα q−1 =α q = 1 4+(q−3) √ 2 , is the critical point of F 1 . Hence F 1 (α 3 , . . . ,α q ) = (q − 3) + 2 √ 2 = q − 0.1716 . . . is the maximum of F 1 on Υ 1 .
Now we return to the original variables. Sinceα 1 =α q−1 +α q = 2 4+(q−3) √ 2 and (1) M 2 (F )W = 0 and ϕ is a homomorphism of algebras;
α 2 = 1 − α 3 − . . . − α q−2 − 2α q−1 − 2α q = √ 2 4 + (q − 3) √ 2 , we haveα 1 α 2 . . . α q . Therefore d = F (α 1 , . . . ,α q ) = (q − 3) + 2 √ 2 = q − 0.1716 . . . is the maximum of F on Ω.
(2) M 2 (F )W = 0 and W 2 = 0;
(3) W 2 = 0 and ϕ is a homomorphism of left M 2 (F )-modules. Then if m(A, (F T ) * , λ) = 0 for some λ n, n ∈ N, we have λ q+1 = 0 and λ q−1 + λ q λ 1 + 1 where q := dim A.
Proof. In order to prove that m(A, (F T ) * , λ) = 0 for some λ n, it is sufficient to show that e * T λ f ≡ 0 for every f ∈ P (F T ) * n and a Young tableau T λ . Note that e * T λ f is alternating in the variables of each column of T λ . Since f is multilinear, it is sufficient to substitute only basis elements. However dim A = q and if λ q+1 > 0, then at least two of the basis elements corresponding to the variables of the first column coincide and e * T λ f vanish. Hence if λ q+1 > 0, we have e T λ f ≡ 0.
Consider and e * T λ f does not vanish under substitution of some basis elements a 1 , . . . , a n . Again, e * T λ f is alternating in the variables of each column. In each of the first λ q columns we have q boxes and in each of the next (λ q−1 − λ q ) columns we have (q − 1) boxes. Therefore, the impact in n i=1 θ(a i ) of the basis elements corresponding to the first λ q−1 columns is at least λ q . Since θ(a) = −1 only for a = (e 21 , 0) and we cannot substitute more than one such element for the variables of the same column, by (4) there must be at least (λ q − 1) other columns. Hence λ 1 − λ q−1 λ q − 1 and we get the lemma.
Proof.
Let by (α 1 , . . . , α q ) ∈ Ω such a point that F (α 1 , . . . , α q ) = d. For every n ∈ N define µ n by µ i = [α i n] for 2 i q and µ 1 = n − q i=1 µ i . For every ε > 0 there exists n 0 ∈ N such that for every n n 0 we have F µ 1 n , . . . , µq n > d − ε. By the assumptions of the lemma, m(A, (F T ) * , µ) = 0 and by the hook and the Stirling formulas, there exist C 1 > 0 and r 1 ∈ R such that we have To prove Theorem 3, we need the following lemma. We omit ϕ for shortness and write (e ij , e ij ) instead of (e ij , ϕ −1 (e ij )). Lemma 6. Let λ n, n ∈ N, λ 8 = 0, and λ 6 + λ 7 λ 1 . Then m(A, (F T 1 ) * , λ) = 0.
Ω = (α 1 , . . . , α q ) ∈ R q q i=1 α i = 1, α 1 α 2 . . . α q 0, α q + α q−1 α 1 . By Lemma 3, d := max x∈Ω F (x) = (q − 3) + 2 √ 2 = q − 0.1716 . . . Denotec (F T ) * n (A) dim M (µ) = n! i,j h ij n! (µ 1 + q − 1)! . . . (µ q + q − 1)! n! n q(q−1) µ 1 ! . . . µ q !
Proof. It is sufficient to show that for some f ∈ P n and some T λ we have e T λ f ≡ 0 on A.
Let β 2 = λ 6 − λ 7 . Fix numbers β 3 , . . . , β 12 0 such that β 3 + β 5 + β 7 + β 9 + β 11 = λ 7 , β 3 + β 4 = λ 5 − λ 6 , β 5 + β 6 = λ 4 − λ 5 , β 7 + β 8 = λ 3 − λ 4 , β 9 + β 10 = λ 2 − λ 3 , and β 11 + β 12 = λ 1 − λ 2 . In other words, we have We fix some Young tableau T λ of the shape λ filled in with the numbers from 1 to n. For each column of T λ we define a multilinear alternating polynomial depending on the variables with the indexes from the column. For shortness, we denote the polynomials corresponding to different columns in the ith block by the same letter f i . By (i 1 , . . . , i ) we denote the -tuple of numbers from a column (from up to down). By S{i 1 , . . . , i } we denote the symmetric group on i 1 , . . . , i . We define 10 10 f β 12 12 ∈ P n . As we have already mentioned, here different copies of f i depend on different variables.
f 1 := σ∈S{i 1 ,...,i 7 } (sign σ)x h 0 σ(i 3 ) x h 1 σ(i 2 ) x h 0 σ(i 6 ) x h 1 σ(i 4 ) x h 0 σ(i 5 ) x h 0 σ(i 1 ) x h 1 σ(i 7 ) , f 2 := σ∈S{i 1 ,...,i 6 } (sign σ)x h 0 σ(i 3 ) x h 1 σ(i 2 ) x h 0 σ(i 6 ) x h 1 σ(i 4 ) x h 0 σ(i 5 ) x h 0 σ(i 1 ) , f 3 := σ∈S{i 1 ,...,i 5 } (sign σ)x h 0 σ(i 5 ) x h 1 σ(i 4 ) x h 0 σ(i 1 ) x h 1 σ(i 2 ) x h 0 σ(i 3 ) , f 4 := σ∈S{i 1 ,...,i 5 } (sign σ)x h 1 σ(i 2 ) x h 0 σ(i 3 ) x h 1 σ(i 5 ) x h 1 σ(i 4 ) x h 0 σ(i 1 ) , f 5 := σ∈S{i 1 ,...,i 4 } (sign σ)x h 1 σ(i 4 ) x h 0 σ(i 1 ) x h 1 σ(i 2 ) x h 0 σ(i 3 ) , f 6 := σ∈S{i 1 ,...,i 4 } (sign σ)x h 1 σ(i 2 ) x h 1 σ(i 3 ) x h 1 σ(i 4 ) x h 0 σ(i 1 ) , f 7 := σ∈S{i 1 ,i 2 ,i 3 } (sign σ)x h 0 σ(i 1 ) x h 1 σ(i 2 ) x h 0 σ(i 3 ) , f 8 := σ∈S{i 1 ,i 2 ,i 3 } (sign σ)x h 1 σ(i 2 ) x h 1 σ(i 3 ) x h 0 σ(i 1 ) , f 9 := σ∈S{i 1 ,i 2 } (sign σ)x h 0 σ(i 1 ) x h 1 σ(i 2 ) , f 10 := σ∈S{i 1 ,i 2 } (sign σ)x h 0 σ(i 2 ) x h 0 σ(i 1 ) , f 11 := x h 0 i 1 , f 12 := x h 1 i 1 . Define the polynomial f = (f 1 f 3 ) β 3 (f 1 f 5 ) β 5 (f 1 f 7 ) β 7 (f 1 f 9 ) β 9 (f 1 f 11 ) β 11 f β 2 2 f β 4 4 f β 6 6 f β 8 8 f β
The copies of f 1 are alternating polynomials of degree 7 corresponding to the first λ 7 columns of height 7.
The copies of f 2 are alternating polynomials of degree 6 corresponding to the next β 2 columns of height 6.
. . .
The copies of f 12 are polynomials of degree 1 with the indexes from the last β 12 columns of height 1.
We claim that e T λ f ≡ 0. In order to verify this, we fill D λ with the homogeneous elements and denote the tableau obtained by τ . (See Figure 1.) Now for each variable we substitute the element from the corresponding box in τ . Note that f does not vanish under this substitution.
Figure 1. τ = λ 7 β 2 β 3 β 4 β 5 β 6 β 7 β 8 β 9 β 10 β 11 β 12 (e 21 ,
Recall that e T λ = a T λ b T λ where a T λ is the symmetrization in the variables of each row and b T λ is the alternation in the variables of each column. Since all f i are alternating polynomials, b T λ f is a nonzero multiple of f . Two sets of variables correspond to the second row of T λ . For the variables of the first group we substitute (e 11 , e 11 ) ∈ A (1) , for the second one, we substitute (e 12 , 0) ∈ A (0) . Thus if an item in a T λ mixes variables from these two groups, at least one variable from the second group, i.e. in f 11 , is replaced with a variable form the first one. However, f 10 vanishes if at least one variable of it is replaced with an element of A (1) since h 0 is applied for both variables of f 10 . Thus all items in a T λ b T λ f where variables from these two groups are mixed, vanish.
Therefore, if an item in a T λ replaces a variable from the first two columns with a variable with a different value from the tableau τ , we will have too many elements from A (1) substituted for the variables of f 1 and f 2 and the result is zero in virtue of the action of h 0 . Therefore all items in a T λ b T λ f where variables from the first two columns having different values are mixed, vanish. We continue this procedure and finally show that if an item in a T λ does not stabilize the sets of variables with the same values from the tableau τ , the corresponding item in a T λ b T λ f vanishes. Hence the value of a T λ b T λ f is a nonzero multiple of the value of b T λ f , i.e. is nonzero. The lemma is proved.
Proof of Theorem 3. We use Lemmas 5 and 6. 9. A T 2 -graded algebra with a fractional graded PI-exponent
Theorem 4. Let A 2 = M 2 (F ) ⊕ F j 11 ⊕ F j 12 ⊕ F j 22 (direct sum of ideals) where F isi ) = x (v) i . We claim that Θ(Id T 1 -gr (A)) = Id T 2 -gr (A 2 ).(6)
Since F is of characteristic 0, both Id T 1 -gr (A) and Id T 2 -gr (A 2 ) are generated by multilinear polynomials. (The proof is completely analogous to [6,Theorem 1.3.8].) In other words, in order to prove (6), it is sufficient to show that if f ∈ F X T 1 -gr is multilinear as an ordinary polynomial in variables x
(t 1 ) 1 , x (t 2 ) 2 , . . . , x (tn) n where t i ∈ T 1 , then f ∈ Id T 1 -gr (A) if and only if Θ(f ) ∈ Id T 2 -gr (A 2 )
. We substitute only homogeneous elements. Note that πψ(a) = π(a) where π is the projection on the first component: π(a, b) = a for all (a, b) ∈ A and (a, b) ∈ A 2 . Moreover, if a ∈ A (0) ∪ A (1) , then π(a) = 0 if and only if a = 0. Since T 1 and T 2 are commutative, the value of f under the substitution of homogeneous elements is again a homogeneous element. Applying π, we show that f ∈ P T 1 -gr n vanishes under the substitution of homogeneous elements a Remark. Note that A does not contain unity. If a T 3 -graded algebra is unital, its graded PI-exponent is integer. (See Theorem 7 below.)
Remark. The algebra A is T 3 -graded-simple (see Proposition 6 below), however 3 + 2 √ 2 = PIexp T 3 -gr (A) < dim A = 6 even if F is algebraically closed.
Proposition 6. The algebra A is T 3 -graded-simple.
Proof. First, we notice that J(A) = I. Suppose W = 0 is a graded ideal of A. Then there exists an nonzero homogeneous element (a 1 , b 1 ) ∈ W where a 1 ∈ M 2 (F ) and b 1 ∈ I. Since I does not contain homogeneous elements, we have a 1 = 0. However (a 1 , b 1 )(E, 0) = (a 1 , 0) ∈ (M 2 (F ), 0) ∩ W . Since W is a two sided ideal and M 2 (F ) is simple, we get (M 2 (F ), 0) ⊆ W . Furthermore (0, I) = (M k (F ), 0)(0, I) ⊆ W . Thus W = A, and A is T 3 -graded-simple.
To prove Theorem 5, we need Lemma 7 which is an analog of Lemma 6. We omit ϕ for shortness and write (e ij , e ij ) instead of (e ij , ϕ −1 (e ij )).
Lemma 7. Let λ n, n ∈ N, λ 7 = 0, and λ 5 + λ 6 λ 1 . Then m(A, (F T 3 ) * , λ) = 0.
Proof. It is sufficient to show that for some f ∈ P n and some T λ we have e T λ f ≡ 0 on A.
Let β 2 = λ 5 − λ 6 . Fix numbers β 3 , . . . , β 10 0 such that β 3 + β 5 + β 7 + β 9 = λ 6 , β 3 + β 4 = λ 4 − λ 5 , β 5 + β 6 = λ 3 − λ 4 , β 7 + β 8 = λ 2 − λ 3 , and β 9 + β 10 = λ 1 − λ 2 . In other words, we have We fix some Young tableau T λ of the shape λ filled in with the numbers from 1 to n. Like in Lemma 6, for each column of T λ we define a multilinear alternating polynomial depending on the variables with the indexes from the column. For shortness, we denote the polynomials corresponding to different columns in the ith block by the same letter f i . By (i 1 , . . . , i ) we denote the -tuple of numbers from a column (from up to down). By S{i 1 , . . . , i } we denote the symmetric group on i 1 , . . . , i . We define 10 10 ∈ P n . As we have already mentioned, here different copies of f i depend on different variables.
f = (f 3 f 1 ) β 3 (f 5 f 1 ) β 5 (f 7 f 1 ) β 7 (f 9 f 1 ) β 9 f β 2 2 f β 4 4 f β 6 6 f β 8 8 f β
The copies of f 1 are alternating polynomials of degree 6 corresponding to the first λ 6 columns of height 6.
The copies of f 2 are alternating polynomials of degree 5 corresponding to the next β 2 columns of height 5.
. . .
The copies of f 10 are polynomials of degree 1 with the indexes from the last β 10 columns of height 1.
We claim that e T λ f ≡ 0. In order to verify this, we fill D λ with the homogeneous elements and denote the tableau obtained by τ . (See Figure 2.) Now for each variable we substitute the element from the corresponding box in τ . Note that f does not vanish under this substitution.
Recall that e T λ = a T λ b T λ where a T λ is the symmetrization in the variables of each row and b T λ is the alternation in the variables of each column. Since all f i are alternating polynomials, b T λ f is a nonzero multiple of f . Two sets of variables correspond to the second row of T λ . For the variables of the first group we substitute (e 22 , e 22 ) ∈ A (e 2 ) , for the second one, we substitute (e 12 , 0) ∈ A (e 1 ) . Thus if an item in a T λ mixes variables from these two groups, at least one variable from the second group, i.e. in f 8 , is replaced with a variable form the first one. However, f 8 vanishes if at least one variable of it is replaced with an element of A (e 2 ) since h e 1 is applied for both variables of f 8 . Thus all items in a T λ b T λ f where variables from these two groups are mixed, vanish.
Therefore, if an item in a T λ replaces a variable from the first three columns with a variable with a different value from the tableau τ , we have too many elements from A (e 2 ) substituted for the variables of f 1 , f 2 , and f 3 and the result is zero in virtue of the action of h e 1 . Therefore all items in a T λ b T λ f where variables from the first three columns having different values are mixed, vanish. We continue this procedure and finally show that if an item in a T λ does not stabilize the sets of variables with the same values from the tableau τ , the corresponding item in a T λ b T λ f vanishes. Hence the value of a T λ b T λ f is a nonzero multiple of the value of b T λ f , i.e. is nonzero. The lemma is proved.
Proof of Theorem 5. We use Lemmas 5 and 7.
11. Positive results on the analog of Amitsur's conjecture for polynomial T -graded identities Theorem 6. Let A be a finite dimensional non-nilpotent T -graded associative algebra with 1 over a field F of characteristic 0 for some cancellative semigroup T . Then there exist constants C 1 , C 2 > 0, r 1 , r 2 ∈ R, d ∈ N, such that C 1 n r 1 d n c T -gr n (A) C 2 n r 2 d n for all n ∈ N.
Corollary. The graded analog of Amitsur's conjecture holds for such codimensions.
Proof of Theorem 6. Note that graded codimensions do not change upon an extension of the base field. The proof is analogous to the case of ordinary codimensions [6, Theorem 4.1.9]. Hence we may assume F to be algebraically closed. By [11,Corollary 4.1], J(A) is a graded ideal. By Proposition 5, A/J(A) is the sum of graded ideals that are T -graded-simple algebras. Now we apply Theorem 2.
Theorem 7. Let A be a finite dimensional non-nilpotent T -graded associative algebra with 1 over a field F of characteristic 0 for some left or right zero band T . Then there exist constants C 1 , C 2 > 0, r 1 , r 2 ∈ R, such that Corollary. The graded analog of Amitsur's conjecture holds for such codimensions.
Proof of Theorem 7. Since A is finite dimensional, we may assume that T is finite. Again, without lost of generality, we may assume F to be algebraically closed.
By Propositions 2 and 4, the Jacobson radical of A is a graded ideal and A/J(A) is the sum of graded ideals that are simple algebras. Therefore, by Theorem 2 there exists an integer PIexp T -gr (A). By Theorem 1, we can choose a graded maximal semisimple subalgebra B such that A = B ⊕ J(A) (direct sum of graded subspaces). Then we may define an embedding κ : A/J → B (see Theorem 2) to be graded. Let B = B 1 ⊕ . . . ⊕ B s (direct sum of ideals) for some simple algebras B i . Then by Proposition 2 the ideals B i are graded, A/J = κ −1 (B 1 ) ⊕ . . . ⊕ κ −1 (B s ) (direct sum of graded ideals), and
PIexp T -gr (A) = max dim B i 1 ⊕ B i 2 ⊕ . . . ⊕ B ir r 1,
((F T ) * B i 1 )A + ((F T ) * B i 2 )A + . . . ((F T ) * B i r−1 )A + ((F T ) * B ir ) = 0 = max dim B i 1 ⊕ B i 2 ⊕ . . . ⊕ B ir r 1, B i 1 A + B i 2 A + . . . B i r−1 A + B ir = 0 = max dim B i 1 ⊕ B i 2 ⊕ . . . ⊕ B
Theorem 1 .
1Let A be a finite dimensional associative T -graded algebra with unity over a field F where T is a left or right zero band and A/J(A) is a separable algebra. (E.g., F is a perfect field.) Then there exists a graded maximal semisimple subalgebra B such that A = B ⊕ J (direct sum of graded spaces) where J := J(A).
H (A) by permuting the variables. Irreducible F S n -modules are described by partitions λ = (λ 1 , . . . , λ s ) n and their Young diagrams D λ . The character χ H n (A) of the F S n -module H (A) is called the nth cocharacter of polynomial H-identities of A.
Lemma 1 .
1Let A be a finite dimensional algebra over a field F graded by a semigroup T . Then c T -gr n (A) = c (F T ) * n (A) for all n ∈ N.
∈
A (t) and ψη is a graded homomorphism. Therefore, ψη(f ) = 0 and η(Id T -gr (A)) ⊆ Id (F T ) * (A). Denote byη : F X T -gr / Id T -gr (A) → F X | (F T ) * / Id (F T ) * (A) the induced homomorphism. Now we use the notationf =
T -gr (A) . Hence
.
Define 0 0 := 1. Then F is continuous on the segment {(α 1 , . . . , α q ) | α i 0}.
Lemma 2 .
2Let A be an algebra with a generalized H-action where H is an associative algebra with unity over a field F of characteristic 0. Suppose m(A, H, λ) = 0 for all λ n, λ / ∈ Ω n , n ∈ N. Then lim n→∞ n c H n (A) d. Proof. Let λ n such that m(A, H, λ) = 0. By the hook formula, dim M (λ) = n! i,j h ij where h ij is the length of the hook with the edge in (i, j) in the Young diagram D λ . Hence dim M (λ) n! λ 1 !...λs! . Note that (x x ) = e x ln x = (ln x + 1)e x ln x and x x is decreasing for x 1 e . By the Stirling formula, for all sufficiently large n we have dim M (λ)
e
−(α q−1 +αq) ln(α q−1 +αq)−(1−α 3 −...−α q−2 −2α q−1 −2αq) ln(1−α 3 −...−α q−2 −2α q−1 −2αq)−α 3 ln α 3 −...−αq ln αq for i = 3, . . . , q − 2 and ∂F 1 ∂α i (α 3 , . . . , α q ) = (2 ln(1
Lemma 4 .
4Let B 0 ⊆ {e11 , e 12 , e 22 } be some subset where e 12 ∈ B 0 and let W be an ideal such that there exists a linear isomorphism ϕ :W → B 0 F ⊂ M 2 (F ). Suppose A = M 2 (F ) ⊕ W (directsum of subspaces) is a T -graded algebra over a field of characteristic 0 where T = {t 1 , t 2 } is a semigroup of two elements, A (t 1 ) = (M 2 (F ), 0) and A (t 2 ) = {(ϕ(a), a) | a ∈ W }. Suppose W M 2 (F ) = 0 and one of the following three conditions holds:
in A the homogeneous basis B = {(e 11 , 0), (e 12 , 0), (e 21 , 0), (e 22 , 0)} ∪ {(a, ϕ −1 (a) | a ∈ B 0 }. Note that the product of any two elements of B is either 0 or again an element of B. Define the function θ : B → Z by θ(e ij , ϕ −1 (e ij )) = θ(e ij , 0) = j − i. Let a 1 , . . . , a k ∈ B. If a 1 . . . a k a i ) = θ(a 1 . . . a k ) 1. (4) Note that b∈B θ(b) = 1 and q−1 i=1 θ(a i ) 0 for any different a i ∈ B. If m(A, (F T ) * , λ) = 0, then e * T λ f ≡ 0 for some f ∈ P (F T ) * n
Lemma 5 .
5Let A be an algebra from Lemma 4. Suppose that for every λ n, n ∈ N, such that λ q−1 + λ q λ 1 , we have m(A, (F T ) * , λ) = 0. Then there exists lim n→∞ n c T -gr n (A) = (q − 3) + 2 √ 2 = q − 0.1716 . . .
n r 1 (d − ε) n .
.
Let A = M 2 (F ) ⊕ UT 2 (F ) (direct sum of ideals) where F is a field of characteristic 0. Define a T 1 -grading on A by A (0) = (M 2 (F ), 0), A (1) = {(ϕ(a), a) | a ∈ UT 2 (F )} where ϕ : UT 2 (F ) → M 2 (F )is the natural embedding. In other words, A is an algebra from Example 1 for k = 2. Then there exists lim
β 2 β 3 β 4 β 5 β 6 β 7 β 8 β 9 β 10 β 11 β 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (Here β i denotes the number of columns in each block of columns.)
21 , 0) (e 21 , 0) (e 21 , 0) (e 11 , e 11 ) (e 11 , e 11 ) (e 11 , e 11 ) (e 11 , e 11 ) (e 11 , e 11 ) (e 11 , e 11 ) (e 11 , e 11 ) (e 11 , e 11 ) (e 11 , e 11 ) (e 11 , e 11 ) (e 12 , 0) (e11 , 0) (e 11 , 0) (e 11 , 0) (e 11 , 0) (e 11 , 0) (e 12 , e 12 ) (e 11 , 0) (e 12 , e 12 ) (e 22 , e 22 ) (e 22 , e 22 ) (e 22 , e 22 ) (e 22 , e 22 ) (e 22 , e 22 ) (e 22 , e 22 ) (e 22 , 0) (e 22 , 0) (e 22 , 0) (e 12 , e 12 ) (e 12 , 0) (e 12 , 0) (e 12 , e 12 ) (Here in the ith block we have β i columns with the same values in all cells of a row. For shortness, we depict each value for each block only once. The tableau τ is still of the shape λ.)
= 0 .
0Define a T 2 -grading on A 2 by A 11 , j 11 ), (e 12 , j 12 ), (e 22 , j 22 ) . Then there exists lim n→∞ n c T 2 8284 . . ..Proof. Let A be the algebra from Theorem 3. Define a linear isomorphism ψ : A → A 2 by ψ(e ij , e k ) = (e ij , j k ) and ψ(e ij , 0) = (e ij , 0). Then ψ(A (0) isomorphism Θ : F X T 1 -gr → F X T 2 -gr of algebras by Θ
∈
A (t i ) , t i ∈ T 1 , if and only if Θ(f ) vanishes under the substitution of ψ a (t i ) i ∈ A (t i ) 2 . (Here0 = 0 and1 = v.) Hence (6) holds and c T 1 -gr n (A) = c T 2 -gr n (A 2 ) for all n ∈ N. Now we apply Theorem 4.
.
Let F be a field of characteristic 0. Denote by I the irreducible left M 2 (F )module isomorphic to the minimal left ideal e 12 , e 22 F ⊂ M 2 (F ). Let A = M 2 (F ) ⊕ I (direct sum of left ideals) where IM 2 (F ) := 0 and I 2 := 0. Define a T 3 -grading on A by A (e 1 ) = (M 2 (F ), 0), A (e 2 ) = {(ϕ(a), a) | a ∈ I} where ϕ : I → M 2 (F ) is the natural embedding which is a homomorphism of M 2 (F )-modules. Then there exists lim
β 2 β 3 β 4 β 5 β 6 β 7 β 8 β 9 β 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (Here β i denotes the number of columns in each block of columns.)
Figure 2
221 , 0) (e 21 , 0) (e 21 , 0) (e 22 , e 22 ) (e 22 , e 22 ) (e 22 , e 22 ) (e 22 , e 22 ) (e 22 , e 22 ) (e 22 , e 22 ) (e 22 , e 22 ) (e 22 , e 22 ) (e 12 , 0) (e 11 , 0) (e 11 , 0) (e 11 , 0) (e 11 , 0) (e 11 , 0) (e 12 , e 12 ) (e 22 , 0) (e 22 , 0) (e 22 , 0) (e 12 , e 12 ) (e 12 , e 12 ) (e 12 , e 12 ) (e 12 , 0) (Here in the ith block we have β i columns with the same values in all cells of a row. For shortness, we depict each value for each block only once. The tableau τ is still of the shape λ.)
C
1 n r 1 d n c T -gr n (A) C 2 n r 2 d n for all n ∈ N where d = PIexp(A) is the ordinary PI-exponent of A.
ir r 1 ,
1B i 1 J(A) B i 2 J(A) . . . B i r−1 J(A) B ir = 0 = PIexp(A) since B i are simple as ordinary algebras. Remark. The equality PIexp T -gr (A) = PIexp(A) does not imply the equality of codimensions. Indeed, let k ∈ N, k 2, A = M k (F ), T = {t 1 , . . . , t k } where t i t = t for all 1 i, k, A (t i ) = e 1i , . . . , e ki F . Then x (t i ) 1 are linearly independent modulo Id T -gr (A). In order to check this, it is sufficient to substitute x (t i ) 1 = e ii . Thus c 1 (A) = 1 < c T -gr 1 (A) = k.
f 1 := σ∈S{i 1 ,...,i 6 } (sign σ)x he 1 σ(i 3 ) x he 2 σ(i 5 ) x he 1 σ(i 4 ) x he 2 σ(i 2 ) x he 1 σ(i 1 ) x he 1 σ(i 6 ) ,
A. S. GORDIENKO f 2 := σ∈S{i 1 ,...,i 5 } (sign σ)xhe 1 σ(i 1 ) x he 1 σ(i 3 ) x he 2 σ(i 5 ) x he 2 σ(i 2 ) x he 1 σ(i 4 ) , f 3 := σ∈S{i 1 ,...,i 4 } (sign σ)x he 1 σ(i 4 ) x he 2 σ(i 2 ) x he 1 σ(i 1 ) x he 1 σ(i 3 ) , f 4 := σ∈S{i 1 ,...,i 4 } (sign σ)x he 1 σ(i 1 ) x he 1 σ(i 3 ) x he 2 σ(i 4 ) x he 2 σ(i 2 ) , f 5 := σ∈S{i 1 ,i 2 ,i 3 } (sign σ)x he 2 σ(i 2 ) x he 1 σ(i 1 ) x he 1 σ(i 3 ) , f 6 := σ∈S{i 1 ,i 2 ,i 3 } (sign σ)x he 2 σ(i 2 ) x he 1 σ(i 1 ) x he 2 σ(i 3 ) , f 7 := σ∈S{i 1 ,i 2 } (sign σ)x he 2 σ(i 2 ) x he 1 σ(i 1 ) , f 8 := σ∈S{i 1 ,i 2 } (sign σ)x he 1 σ(i 1 ) x he 1 σ(i 2 ) , f 9 := xhe 1 i 1 , f 10 := x he 2 i 1 . Define the polynomial
AcknowledgementsI am grateful to E. Jespers, M. V. Zaicev, and E. Iwaki for helpful discussions.
Multialternating graded polynomials and growth of polynomial identities. E Aljadeff, A Giambruno, Proc. Amer. Math. Soc. 141Aljadeff, E., Giambruno, A. Multialternating graded polynomials and growth of polynomial identities. Proc. Amer. Math. Soc. 141:9 (2013), 3055-3065.
Graded polynomial identities and exponential growth. E Aljadeff, A Giambruno, D La Mattina, J. reine angew. Math. 650Aljadeff, E., Giambruno, A., La Mattina, D. Graded polynomial identities and exponential growth. J. reine angew. Math., 650 (2011), 83-100.
Identical relations in Lie algebras. Yu A Bakhturin, VNU Science PressUtrechtBakhturin, Yu. A. Identical relations in Lie algebras. VNU Science Press, Utrecht, 1987.
Free algebras and PI-algebras: graduate course in algebra. V S Drensky, Springer-VerlagSingaporeDrensky, V. S. Free algebras and PI-algebras: graduate course in algebra. Singapore, Springer-Verlag, 2000.
Graded polynomial identities and codimensions: computing the exponential growth. A Giambruno, D La Mattina, Adv. Math. 225Giambruno, A., La Mattina, D. Graded polynomial identities and codimensions: computing the expo- nential growth. Adv. Math., 225 (2010), 859-881.
Polynomial identities and asymptotic methods. A Giambruno, M V Zaicev, AMS Mathematical Surveys and Monographs. 122Giambruno, A., Zaicev, M. V. Polynomial identities and asymptotic methods. AMS Mathematical Sur- veys and Monographs Vol. 122, Providence, R.I., 2005.
Amitsur's conjecture for polynomial H-identities of H-module Lie algebras. A S Gordienko, Tran. Amer. Math. Socto appearGordienko, A. S. Amitsur's conjecture for polynomial H-identities of H-module Lie algebras. Tran. Amer. Math. Soc. (to appear).
Asymptotics of H-identities for associative algebras with an H-invariant radical. A S Gordienko, J. Algebra. 393Gordienko, A. S. Asymptotics of H-identities for associative algebras with an H-invariant radical. J. Algebra, 393 (2013), 92-101.
Co-stability of radicals and its applications to PI-theory. A S Gordienko, Algebra Colloqium (to appearGordienko, A. S. Co-stability of radicals and its applications to PI-theory. Algebra Colloqium (to appear).
On semigroup graded PI-algebras. A V Kelarev, Semigroup Forum. 47Kelarev, A. V. On semigroup graded PI-algebras. Semigroup Forum, 47:1 (1993), 294-298.
Ring constructions and applications. Series in Algebra 9. A V Kelarev, World ScientificSingaporeKelarev, A. V. Ring constructions and applications. Series in Algebra 9, World Scientific, Singapore, 2002.
An example of a variety of Lie algebras with a fractional exponent. S P Mishchenko, M V Zaicev, J. Math. Sci. 936Mishchenko, S. P., Zaicev M. V. An example of a variety of Lie algebras with a fractional exponent. J. Math. Sci. (New York), 93:6 (1999), 977-982.
A sufficient condition for coincidence of lower and upper exponents of the variety of linear algebras. S P Mishchenko, A B Verevkin, M V Zaitsev, Mosc. Univ. Math. Bull. 662Mishchenko, S.P., Verevkin, A.B., Zaitsev, M.V. A sufficient condition for coincidence of lower and upper exponents of the variety of linear algebras. Mosc. Univ. Math. Bull., 66:2 (2011), 86-89.
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| In this paper, local well-posedness is shown for the one dimensional cubic nonlinear Schrödinger equation in L p -spaces for 2 < p < 4, which generalizes a classical result for p = 2 by Y. Tsutsumi and recent work for 1 < p < 2 by Y. Zhou. As a consequence, a local theory of solutions is developed for a class of data which decay more slowly than square integrable functions. Regularity properties of the local solutions in the L p -based Sobolev spaces and Stricharz spaces are also proved. | null | [
"https://arxiv.org/pdf/2204.06202v2.pdf"
]
| 248,157,456 | 2204.06202 | 6487a76668b9cb4893434bfc6955fa9cb04e63ab |
18 May 2022
18 May 2022arXiv:2204.06202v2 [math.AP] WELL-POSEDNESS FOR THE 1D CUBIC NONLINEAR SCHRÖDINGER EQUATION IN L p , p > 2 RYOSUKE HYAKUNA
In this paper, local well-posedness is shown for the one dimensional cubic nonlinear Schrödinger equation in L p -spaces for 2 < p < 4, which generalizes a classical result for p = 2 by Y. Tsutsumi and recent work for 1 < p < 2 by Y. Zhou. As a consequence, a local theory of solutions is developed for a class of data which decay more slowly than square integrable functions. Regularity properties of the local solutions in the L p -based Sobolev spaces and Stricharz spaces are also proved.
Introduction
We consider the Cauchy problem for the one dimensional cubic nonlinear Schrödinger equation:
(1.1) iu t + u xx + |u| 2 u = 0, u| t=0 = φ.
It is well known that (1.1) is locally well posed in L 2 (R). Here the "well-posedness" means the existence of a local solution u : [0, T ]×R → C, T T ( φ L 2 ) > 0 for any φ ∈ L 2 , uniqueness (in a suitable solution space), continuous dependence on data, and the persistence property of the solution-i.e., u ∈ C([0, T ]; L 2 ). This is a classical result by Y. Tsutsumi [17]. Our interest in this paper is to extend this standard well-posedness in L 2 into L p , p = 2 in a natural manner. As far as the author knows, there are not many studies on (1.1) in this direction. One reason for that is it is widely believed that (1.1) is not locally well posed in L p if p = 2. In fact, as early as the 1960s, it was alreadly proved that the initial value problem for the corresponding linear equation (1.2) iu t + u xx = 0, u| t=0 = φ,
is not well posed in L p (R) unless p = 2 (see [12]). In particular, if φ ∈ L p , p = 2 we cannot expect the solution u(t) of (1.2) to belong to L p . Recently, though, there are several works that treat (1.1) in data spaces whose norm are characterized by some kind of p-th integrability with p = 2. For example, Grünrock [10] proved that (1.1) is locally well posed in L p for 1 < p < ∞, where L p {φ ∈ S ′ (R) |φ ∈ L p ′ }. Notice that by the Hausdorff-Young inequality
L p ⊂ L p , if p ≤ 2, L p ⊂ L p , if p ≥ 2.
Thus (1.1) can be well posed in spaces which are similar to the L p -spaces even if p = 2. Moreover, one remarkable work was done by Zhou. In [18] he consider the corresponding integral equation where U (t) denotes the free Schrödinger group, and introduced the twisted variable v(t) = U (−t)u(t) to rewrite (1.3) as
(1.4) v(t) = φ + i t 0 U (−s) |U (s)v(s)| 2 (U (s)v(s)) ds.
Then he discovered that (1.3) is locally well posed in L p , 1 < p < 2 in the sense that a local solution v ∈ C([0, T ( φ L p )]; L p ) of (1.4) exists for any data φ ∈ L p with uniqueness and continuous depedence on data. This result suggests that (1.1) can be locally well posed in L p , p = 2 if one consider the "twisted" persistence property U (−t)u(t) ∈ C([0, T ]; L p ) of the solution in place of the usual persistency. Note that by the unitarity property u(t) ∈ C(I; L 2 ) if and only if U (−t)u(t) ∈ C(I; L 2 ) for any I ⊂ R. Thus this kind of well-posedness in L p can be regarded as a natural extension of the one in L 2 in the usual sense. Now one question arises: does the similar well-posedness hold true for p > 2? As far as the author knows there is no previous work that addresses this problem. But L p -spaces for p > 2 are most typical spaces whose functions decay at |x| → ∞ more slowly than square integrable fuctions, and we believe it is very interesting to establish a theory of solutions to (1.1) for such a class of data. As a study in a similar direction, we refer to the most recent work [5,15,16] where the authors consider Bessel potential spaces H s,p for p > 2 with s > 0. Here in this paper we consider mere L p -spaces -without any smoothness or other additional assumptions. Throughout the paper we use the following notations. For r ∈ [1, ∞], r ′ denotes the conjugate of r: 1/r + 1/r ′ = 1. F and F −1 denote the Fourier transform and the inverse Fourier transform respectively. We also use φ to denote the Fourier transform of φ. c, C are positive constants that may vary line to line. In particular, we use C A,B,··· when we want to emphasize that the constant depends on the parameters A, B, · · · . Let I ⊂ R be an interval and let X be a Banach space of complex valued fuctions on R. The Bessel potential space is denoted by H s,p . We define the space C S (I; X) by
C S (I; X) {u : I × R → C | U (−t)u(t) ∈ C(I; X) }.
Now we give the definition of local well-posedness introduced by Zhou [18].
Definition 1.1. Let X be a Banach space of C-valued functions on R. We say that (1.1) is locally well posed in X if, for any φ ∈ X there are T > 0 and a unique solution u : [0, T ]×R → C of (1.1) such that
u ∈ C S ([0, T ]; X) ∩ Z T S T ,
where Z T is some auxillary space. Moreover, the map φ → u is locally Lipschitz from L p to S T .
The main result of the present paper is a local well-posedness in L p with p > 2 in the above sense. It is obvious that if X has the unitarity property i.e. U (t)φ X = φ X , ∀t ∈ I, then C S (I; X) = C(I; X). In such cases, local well-posedness in the sense of Definition 1.1 is equivalent to the usual one equipped with the persistence property of solution. In particular, the classical result [17] says (1.1) is locally well posed in L 2 in the sense of Definition 1.1.
We prove Theorem 1.2 by constructing a local solution in a space Y p q,θ (T ) of functions defined
on [0, T ] × R such that Y p q,θ (T ) ֒→ C S ([0, T ]; L p ).
The definition of Y p q,θ is given at the end of this section. Below we state our local well-posedness result in a more precise manner. For M > 0 and 1 ≤ p ≤ ∞ we set
(1.5) B p M {φ ∈ L p (R) | φ L p ≤ M }. Proposition 1.4. Let 2 ≤ p < 4. For any M > 0 there is a T M > 0 with lim M →∞ T M = ∞ such that: for any φ ∈ B p M there is a unique solution u ∈ Y p q,θ (T M ) of (1.1). Moreover, the map φ → u is Lipschitz from B p M to Y p q,θ (T M ).
We also present miscellaneous results on the solution given by Proposition 1.4. It would be of interest to pursue the regularity of the solution at the L p -level. Unfortunately, we cannot expect the solution to be in L p in general. This is not unexpected from the well known ill-posed result for the corresponding linear equation. For the regularity in the L p -framework we have the following results: let 2 ≤ p < 4 and M > 0. For φ ∈ B p M we let S M φ ∈ Y p q,θ (T M ) be the (unique) solution to (1.1) given by Proposition 1.4.
(i) S M φ ∈ C([0, T M ] ; H s,p (R)) for every φ ∈ B p M . (ii) There exists C M > 0 such that sup t∈[0,T M ] S M φ 1 − S M φ 2 H s,p (R) ≤ C M φ 1 − φ 2 L p for any φ 1 , φ 2 ∈ B p M .
On the other hand, if s > −(1 − 2/p) the Cauchy problem (1.1) is not locally well posed from L p to H s,p in the above sense. So in general, from a standpoint of the well-posedness with the L p -based regularity, there is a loss of smoothness of order 1 − 2/p. More precisely, we have the following result: Remark 1.7. The case 3 < p < 4 is excluded from the statement of Corollary 1.6. This is due to the availability of key Strichartz type estimates which lead to the ill-posedness results.
Our next interest is the regularity in the so-called Strichartz space. It is well known that the solution u :
I × R → C of (1.1) for φ ∈ L 2 belongs to the space L ρ (I; L r (R)) for ρ, r ∈ [2, ∞] satisfying 2 ρ + 1 r = 1 2 .
Recall that such a pair of exponents (ρ, r) is called admissible. Our next regularity result shows that this kind of property for the L 2 -solution can be extended to the L p -setting for p > 2 in a very natural manner. For 1 ≤ ρ, r < ∞, α ∈ R and I ⊂ [0, ∞) we define the space of functions L ρ (I, t α dt ; L r (R)) by
L ρ (I, t α dt ; L r (R)) {u : I × R → C | u L ρ (I, t α dt ;L r (R)) < ∞}, where u L ρ (I, t α dt ;L r (R)) I R |u(t, x)| r dx ρ r t α dt 1 ρ .
We have the following regularity property for the solution u to (1.1) given by Theorem 1.2:
Corollary 1.8. Let M > 0 and let 2 ≤ p < 4 and 2 < ρ, r < ∞. Assume that 2 ρ + 1 r + 1 p = 1.
and q, r satisfy either of (i),(ii) below:
(i) 0 ≤ 1 ρ < min 1 4 , 1 2 − 1 r , (ii) 4 < r ≤ ∞ and ρ = 1 4 .
Then
(1.6) S M φ ∈ L ρ ([0, T M ], t 1 p − 1 2 dt ; L r (R)) for any φ ∈ B p
M . Now we give the definition of the space Y p q,θ along with related spaces. These spaces were first introduced by Zhou in [18] to show the well-posedness of (1.1) in L p , 1 < p < 2. Definition 1.9. Let T > 0 and let 1 ≤ p, q < ∞ and θ ∈ R.
(i) The spaceX p q,θ (T ) is defined bỹ
X p q,θ (T ) {v : [0, T ] × R → C | v X p q,θ (T ) < ∞}, where v X p q,θ (T ) T 0 s θ (∂ s v)(s, ·) L p q ds 1 q , and X p q,θ (T ) by X p q,θ (T ) {v ∈X p q,θ (T )|v(0) ∈ L p } equipped with the norm v X p q,θ (T ) v(0) L p + v X p q,θ (T ) . (ii) The spaceỸ p q,θ (T ) is defined bỹ Y p q,θ (T ) {u : [0, T ] × R → C | U (−t)u(t) ∈X p q,θ (T ) } with u Ỹ p q,θ (T ) U (−t)u(t) X p q,θ (T ) . The space Y p q,θ (T ) is defined by Y p q,θ (T ) {u : [0, T ] × R → C |U (−t)u(t) ∈ X p q,θ (T )}, equipped with the norm u Y p q,θ (T ) U (−t)u(t) X p q,θ (T )
. We easily get the following embedding result:
Lemma 1.10. (See e.g. [13, Lemma 2.1]) Let T > 0 and let 1 ≤ p, q < ∞ and θ ∈ R. Suppose that −∞ < q ′ θ < 1. Then X p q,θ (T ) ֒→ C([0, T ]; L p (R)). In particular, the embedding Y p q,θ (T ) ֒→ C S ([0, T ]; L p (R)) holds.
Key Lemmata
Generalized Strichartz type inequality.
The key to our local well-posedness results is generalized Strichartz type estimates:
Lemma 2.1. Let 2 ≤ p < 4. Then the estimate (2.1) U (t)φ L 3p xt (R 2 ) ≤ C φ L p (R) holds true.
In this paper we use estimate (2.1) in the following form:
Corollary 2.2. Let 2 ≤ p < 4. Then the estimate (2.2) t − 2 3p ′ U (1/4t)φ L 3p ′ (R + ;L 3p ′ (R)) ≤ C φ L p (R)
holds true.
Remark 2.3.
Here are some comments on the above estimates.
• The estimates can be regarded as a generalization of the well-known Strichartz estimate for φ ∈ L 2 and can be traced back to [6]. See also [4] and introduction in [9]. The proof can be found in e.g. [14]. • It is well known that the local in time Strichartz estimate
U (t)φ L 6 ([0,T ];L 6 (R)) ≤ C φ L 2 (R)
is exploited to prove the existence of an L 2 -solution to (1.1) for a sufficiently small time T > 0. In our analysis we essentially use (2.1) of the following form to establish a local solution on [0, T ]:
(2.3) U (t)φ L 3p xt ([T −1 ,∞)×R) ≤ C φ L p (R)
. This implies that one needs a global in time version of Strichartz type estimates, even in proving the existence of a "local" solution for data φ ∈ L p , p > 2.
2.2.
Factorization of U (t) and cubic nonlinearity. In order to construct a solution u of (1.1) with twisted persistence property U (−t)u(t) ∈ L p we rewrite the cubic nonlinearity in terms of v(t) U (−t)u(t). This is done via factorization of the free Schrödinger group U (−t). This kind of expresssion is known and has been used in [18] and in much earlier studies. See e.g. [11]. Here we present the details for completeness and for convenience of the reader. To this end we introduce several operators. The phase modulation operator M t is defined by
M t : w → e i x 2 4t w.
The dilation operator D t is defined by
(D t w)(x) (4πit) − 1 2 w x 4πit .
R is the reflection operator: (Rw)(x) w(−x). Using these operators we get a factorization of U (t) and U (−t) as follows(see [3,Chapter 4]):
U (t) = M t D t FM t , U (−t) = M −1 t F −1 D −1 t M −1 t
. This leads to our next key lemma: Lemma 2.4. For t = 0 the following equality holds:
(2.4) U (−t)[u 1 (t)u 2 (t)u 3 (t)] = ct −1 M −1 t (M t U (−t)u 1 (t)) * (RM t U (−t)u 2 (t)) * (M t U (−t)u 3 (t))
, where * denotes the convolution with respect to the space variable and c is an absolute constant.
Proof. By the factorization of U (−t), we have 3. Proof of the well-posedness results 3.1. Non-linear estimates. We first prove a key trilinear estimate in X p q,θ -spaces from which we deduce the desired local well-posedness result via the fixed point theorem. We introduce the trilinear form
D(v 1 , v 2 , v 3 ) by D(v 1 , v 2 , v 3 ) t 0 s −1 M −1 s [(M s v 1 (s)) * (RM s v 2 (s)) * (M s v 3 (s))]ds.
Proposition 3.1. Let T > 0. Assume that 2 ≤ p < 4. Then
(3.1) D(v 1 , v 2 , v 3 ) X p q,θ (T ) ≤ C 3 j=1 v j X p 1,0 (T ) with (3.2) q = p p − 1 (= p ′ ), θ = −(1 − 2 p ).
Proof. By the Hausdorff-Young and Hölder inequalities, we have
t θ ∂ t D(v 1 , v 2 , v 3 ) L p = t θ−1 (M t v 1 (t)) * (RM t v 2 (t)) * (M t v 3 (t)) L p ≤ C 3 j=1 t − 1−θ 3 F −1 M t v j (t) L 3p ′ ,
where we have also used the identity F −1 Rf = F −1 f . Taking L q ([0, T ])-norm of both sides and using Hölder's inequality in the time variable, and the fact that U (−1/4t) = F −1 M t F, t = 0, we get
D(v 1 , v 2 , v 3 ) X p q,θ (T ) ≤ C 3 j=1 t − 1−θ 3 U (−1/4t)F −1 v j (t) L 3q ([0,T ];L 3p ′ ) .
We estimate the right hand side. Observe that for each j = 1, 2, 3
t − 1−θ 3 U (−1/4t)F −1 v j (t) L 3q ([0,T ];L 3p ′ ) = t − 1−θ 3 RU (−1/4t)F −1 v j (t) L 3q ([0,T ];L 3p ′ ) = t − 1−θ 3 U (1/4t)F −1 v j (t) L 3q ([0,T ];L 3p ′ ) . Now we write v j (t) = v j (0) + t 0 (∂ s v j )(s)ds. Using the symbol U (t) t − 1−θ 3 U (1/4t)F −1 we have U (t)v j (t) L 3q ([0,T ];L 3p ′ ) ≤ U (t)v j (0) L 3q ([0,T ];L 3p ′ ) + t 0 U (t)(∂ s v j )(s)ds L 3q ([0,T ];L 3p ′ ) ≤ U (t)v j (0) L 3q ([0,T ];L 3p ′ ) + t 0 U (t)(∂ s v j )(s) L 3p ′ (R) ds L 3q ([0,T ]) ≤ U (t)v j (0) L 3q ([0,T ];L 3p ′ ) + T 0 U (t)(∂ s v j )(s) L 3p ′ (R) ds L 3q ([0,T ]) ≤ U (t)v j (0) L 3q ([0,T ];L 3p ′ ) + T 0 U (t)(∂ s v j )(s) L 3q ([0,T ];L 3p ′ ) ds.
Now we take q, θ as in (3.2). Then by (2.2) the right hand side of the above inequalities is smaller than
C v j (0) L p + T 0 ∂ s v j (s) L p ds = C v j (0) L p + T 0 ∂ s v j (s) L p ds = C v j X p 1,0 (T )
. This proves the nonlinear estimate in question. Let q, θ be as in (3.2). Using the nonlinear estimate (3.1), we want to find a fixed point of the operator
(Φu)(t) U (t)φ + i t 0 U (t − s)|u(s)| 2 u(s)ds
in a closed subset of Y p q,θ (T ) for a suitable T > 0. Throughout the proof we use the convention that v(t) = U (−t)u(t), v j (t) = U (−t)u j (t). Using these notations and Lemma 2.4, we have
(3.4) U (−t)Φu(t) = φ + cD(v, v, v).
For a > 0 we define V (a) by
V (a) {u ∈ Y p q,θ (T ) | u(0) = φ, u Ỹ p q,θ (T ) ≤ a} equipped with the distance d(u 1 , u 2 ) u 1 − u 2 Ỹ p q,θ (T )
. We first estimate Φu for u ∈ V (a). By (3.4) and (3.1), we have
Φu Ỹ p q,θ (T ) = U (−t)Φu X p q,θ (T ) = c D(v, v, v) X p q,θ (T ) ≤ C v 3 X p 1,0 (T ) = C u 3 Y p 1,0 (T ) .
By Hölder's inequality we get
u 3 Y p 1,0 (T ) ≤ φ L p + T 1− 1 p u Ỹ p q,θ (T ) 3 ≤ 8 φ 3 L p + 8T 3(1− 1 p ) u 3 Y p q,θ (T ) . Therefore, Φ : V (a) → V (a) is well defined if we choose a, T so that (3.5) 8C φ 3 L p ≤ a 2 , 8CT 3(1− 1 p ) a 3 ≤ a 2 .
Similarly, for u 1 , u 2 ∈ V (a) we have
Φu 1 − Φu 2 Ỹ p q,θ (T ) = D(v 1 , v 1 , v 1 ) − D(v 2 , v 2 , v 2 ) X p q,θ (T ) ≤ D(v 1 − v 2 , v 1 , v 1 ) X p q,θ (T ) + D(v 2 , v 1 − v 2 , v 1 ) X p q,θ (T ) + D(v 2 , v 2 , v 1 − v 2 ) − D(v 2 , v 2 , v 2 ) X p q,θ (T ) ≤ CT 1− 1 p v 1 − v 2 X p q,θ (T ) × 1≤j,k≤2 ( φ L p + T 1− 1 p v j X p q,θ (T ) )( φ L p + T 1− 1 p v k X p q,θ (T ) ) ≤ 8CT 1− 1 p ( φ 2 L p + T 2(1− 1 p ) a 2 ) u 1 − u 2 Ỹ p q,θ (T ) . Thus Φ is a contraction mapping if (3.6) 8CT 1− 1 p ( φ 2 L p + T 2(1− 1 p ) a 2 ) < 1 2 .
Now we prove the existence of a local solution by the fixed point argument. Let M > 0. For any φ ∈ B p M we put
(3.7) a = 16CM 3 , T = εM − 2p p−1 ( T M ),
where ε > 0 is a constant independent of a, M . Then it is easy to see that a and T defined by (3.7) satisfy (3.5) and (3.6) if ε is sufficiently small. We choose such an ε. Then Φ : V (a) → V (a) is well defined and is a contraction mapping. By the fixed point theorem, there exists a solution u ∈ Y p q,θ (T M ) of the integral equation (3.3). Moreover, the uniqueness in Y p q,θ (T M ) and continuous dependence on data follows from a similar difference estimate as above. Consequently, the desired local well-poedness result has been proved. Proof. We prove the continuity. Take ε > 0 arbitrarily and fix it. For t, t ′ ∈ [0, T ] we write
U (t)φ H s,p (R) ≤ C T φ L p ,(4.2) U (t)φ − U (t ′ )φ H s,p ≤ U (t)(φ −φ) H s,p + U (t)φ − U (t ′ )φ H s,p + U (t ′ )(φ −φ) H s,p
for someφ ∈ C ∞ 0 (R). By Lemma 4.1 and density, we may chooseφ ∈ C ∞ 0 (R) so that max
U (t)(φ −φ) H s,p , U (t ′ )(φ −φ) H s,p ≤ C T φ −φ L p ≤ ε 3 .
For the second term in the right hand side of (4.2), we have
U (t)φ − U (t ′ )φ H s,p ≤ U (t)φ − U (t ′ )φ L p ≤ C (e it|·| 2 − e it ′ |·| 2 )Fφ L p ′ ≤ C|t − t ′ | | · | 2 Fφ L p ′ . Therefore, there is δ = δ(ε) > 0 such that U (t)φ − U (t ′ )φ H s,p ≤ ε 3 for any t, t ′ ∈ [0, T ] with |t − t ′ | < δ(ε)
. Now the desired continuity assertion follows from the elementary ε/3-argument.
Proof of Corollary 1.5. It is enough to show the embedding
(4.3)
Y p q,θ (T ) ֒→ C([0, T ] ; H s,p (R)). To prove this we first check that
(4.4) Y p q,θ (T ) ֒→ L ∞ ([0, T ] ; H s,p (R)). Let u ∈ Y p q,θ (T ) and write u(t) = U (t)v(t), v(t) = U (−t)u(t). We have U (t)v(t) = U (t)v(0) + t 0 U (t)(∂ s v)(τ )dτ.
Taking H s,p -norm of both sides and using Lemma 4.1, we have
u(t) H s,p ≤ U (t)v(0) H s,p + t 0 U (t)(∂ τ v)(τ ) H s,p dτ ≤ C T v(0) L p + C T T 0 (∂ τ v)(τ ) L p dτ ≤ C T v X p 1,0 (T ) ≤ C T v X p q,θ (T ) = u Y p q,θ (T )
for any t ∈ [0, T ]. This proves (4.4). Now it is enough to check the continuity of the map t → u(t) from [0, T ] to H s,p to show (4.3).
For t, t ′ ∈ [0, T ] we write
u(t) − u(t ′ ) = U (t)v(0) + t 0 U (t)(∂ τ v)(s)ds − U (t ′ )v(0) − t ′ 0 U (t ′ )(∂ τ v)(τ )dτ = U (t)v(0) − U (t ′ )v(0) + t 0 U (t)(∂ τ v)(τ ) − U (t ′ )(∂ τ v)(τ ) dτ + t ′ t U (t ′ )(∂ τ v)(τ )dτ = I 1 + I 2 + I 3 .
Now taking H s,p -norm and letting t ′ tend to t, we see that I 1 H s,p converges to 0 by Lemma 4.2. Similarly, I 2 also tends to 0 in H s,p since
I 2 H s,p ≤ T 0 U (t)(∂ τ v)(τ ) − U (t ′ )(∂ τ v)(τ ) H s,p dτ and U (t)(∂ τ v)(τ ) L 1 τ ([0,T ];H s,p ) ≤ C T u Y p q,θ , ∀t ∈ [0, T ]
by Lemma 4.1. For I 3 we have
I 3 H s,p ≤ t ′ t U (t ′ )(∂ τ v)(τ ) H s,p dτ ≤ C T t ′ t ∂ τ v(τ ) L p dτ ≤ C T |t − t ′ | q ′ θ+1 v X p q,θ (T )
, from which it follows that I 3 converges to 0 in H s,p as t ′ → t. Consequently, we see that (t → u(t)) ∈ C([0, T ]; H s,p ).
4.2.
Proof of Corollary 1.6. We need an off-diagonal generalization of the generalized Strichartz estimate (2.1). Lemma 4.3. ([14]) Let 2 ≤ p < 4 and let q, r be such that 2 q
+ 1 r = 1 p ′ .
Moreover, assume either of (i),(ii) below:
(i) 0 ≤ 1 q < min 1 4 , 1 2 − 1 r .
(ii) 4 < r ≤ ∞ and q = 1 4 .
Then the estimate
(4.5) U (t)φ L q (R;L r (R) ≤ C φ L p .
holds true. In particular, the estimate
(4.6) t − 2 q U (1/4t)φ L q (R + ;L r (R)) ≤ C φ L p .
holds true.
The key to the ill-posedness result is an "L 2 -smoothing" for the Duhamel contribution of the solution.
Otherwise one may establish estimate (4.1) by density and the Banach-Steinhaus theorem. Now we write U (t)φ n = S M φ n + (U (t)φ n − S M φ n ) and apply Lemma 4.4 to obtain
sup t∈[0,T M ] U (t)φ n H s,p ≤ sup t∈[0,T M ] S M φ n H s,p + sup t∈[0,T M ] S M φ n − U (t)φ n H s,p ≤ sup t∈[0,T M ] S M φ n H s,p + sup t∈[0,T M ] S M φ n − U (t)φ n L 2 ≤ sup t∈[0,T M ] S M φ n H s,p + CM 1 p−1 .
Letting n → ∞ we see that
lim n→∞ sup t∈[0,T M ] S M φ n H s,p = ∞.
This implies that property (ii) in Corollay 1.5 does not hold.
Second proof of Corollary 1.6. Let M > 0. In the second proof we show that assertion (i) in the statement of Corollary 1.5 fails by showing the existence of data φ ∈ B p M such that (S M φ)(t 0 ) / ∈ H s,p for some t 0 ∈ (0, T M ) if 2 < p ≤ 4 and s > 2/p − 1. We first recall the following result on the L p -regularity for the homogeneous data:
Lemma 4.5. [3, Theorem 2.6.1] Let 0 < a < 1 and ψ a (x) = |x| −a , x ∈ R. Then U (t)ψ a ∈ L p (R)
for all t > 0 and for any p such that
(4.7) p > max 1 a , 1 1 − a .
Observe that when p > 2 we may choose a such that
(4.8) 1 p < a < 1 − 1 p .
We take such an a and for t 0 ∈ (0, T M ) and we set
φ a cU (−t 0 )ψ a = cU (t 0 )ψ a ,
which belongs to L p for any p satisfying (4.7). The contant c can be varied depending on the size of M . Here we assume c = 1 for simplicity. Clearly, U (t 0 )φ = ψ a = |x| −a . We show that there is an a such that (S M φ a )(t 0 ) / ∈ H s,p if s > 2/p − 1 and 2 < p ≤ 3. As in the first proof, we may assume that s < 1/p − 1/2. Then by the L 2 -smoothing, (S M φ a )(t 0 ) − U (t 0 )φ a ∈ L 2 ⊂ H s,p and it is enough to check that U (t 0 )φ a / ∈ H s,p to conclude that (S M φ a )(t 0 ) / ∈ H s,p . We estimate D s U (t 0 )φ a F −1 · s U (t 0 )φ a , where ξ s (1 + |ξ| 2 ) 1/2 . It is known ([8, Proposition 1.2.5]) that F −1 · s is strictly positive and satisfies
[F −1 · s ](x) ≥ C|x| −s−1 + O(|x| −s+1 )
for |x| ≤ 2. In particular, for a sufficiently small δ > 0 one has where we have used some basic facts (see e.g. [7]) on the convolution and the Fourier transform of the homogeneous functions. Clearly, | · | −s−a = H 1 + H 2 / ∈ L p loc (R) if a ≥ 1/p − s. In view of (4.8), such an a exists if 1/p − s < 1 − 1/p, which is equivalent to s > 2/p − 1. On the other hand, we see that |x| < δ/4 and |x − y| > δ implies |y| ≥ (3/4)δ and (2/3)|y| ≤ |x − y|. Thus we have holds. To prove this inclusion relation it is enough to show the following Strichartz type estimate:
[F −1 · s ](x) ≥ c|x| −
(4.10) U (t)φ L q (R,t 1/p−1/2 dt;L r (R)) ≤ C φ L p .
Indeed, once (4.10) is verified, the desired embedding follows arguing as in the proof of (4.4). Essentially, the estimate is equivalent to (4.5). We prove (4.10) by showing this. Recall the factorization of U (t) in Section 2. For f ∈ S(R) we have
|U (t)f | = |D t FM t F −1 Ff | = |D t FM t F −1 Ff | = |D t F −1 M t FF −1 f | = |D t U (1/4t)F −1 f |.
Now we substitute φ = F −1 f (i.e. f = F −1 φ) into the above equality and apply (4.5) to obtain D t U (1/4t)φ L q (R;L r (R)) = U (t)F −1 φ L q (R;L r (R)) ≤ C φ L p = C φ L p .
Finally, a suitable change of the space and time variables in the left hand side of the above inequality yields (4.10).
( 1 .
13) u(t) = U (t)φ + i t 0 U (t − s)|u(s)| 2 u(s)ds,2000 Mathematics Subject Classification. 35Q55. Key words and phrases.
Theorem 1. 2 .
2For 2 ≤ p < 4 Cauchy problem (1.1) is locally well posed in L p (R) in the sense of Definition 1.1.
Corollary 1 . 5 .
15Let 2 ≤ p < 4 and s < −(1 − 2/p). Then (1.1) is locally well posed from L p to H s,p in the following sense: for any M > 0 one has
Corollary 1 . 6 .
16Let 2 < p ≤ 3 and s > −(1 − 2/p). Then Cauchy problem (1.1) is not locally well posed from L p to H s,p in the sense of Corollary 1.5.
3. 2 .U
2Proof of Proposition 1.4. Now we prove the main local well-posedness result. (t − s)|u(s)| 2 u(s)ds.
4 .
4Proof of the regularity results 4.1. Proof of Corollary 1.5. We recall some classical results on the L p -regularity for the solution to the linear Schrödinger equation. Denote B s p,r by the Besov space of order s. For the definition of the Besov space see e.g. [1].
]
Let 1 ≤ p ≤ ∞ and T > 0. The estimate (4.1) sup t∈[0,T ]U (t)φ L p (R) ≤ C T φ B s p,1 (R) , ∀φ ∈ B s p,1 (R)holds true for some C T > 0 if and only if s ≥ 2|1/p − 1/2|.
∀φ ∈ L p holds true if s < −(1 − 2/p) and fails if s > −(1 − 2/p).
Lemma 4. 2 .
2Let T > 0. Let 2 < p < ∞ and s < −(1 − 2/p). Then U (t)φ ∈ C([0, T ]; H s,p (R))for any φ ∈ L p (R).
y| −1−s |y| −a dy.Now we can easily verify that the function in the right hand side is not in L p ([−δ/4, δ/4]) as follows. We may writeC|x| −s−a = (| · | −1−s * | · | −a )
dy = C δ < ∞ for any x ∈ [−δ/4, δ/4]. Hence H 1 / ∈ L p (R) and consequently, we see that U (t 0 )φ a / ∈ H s,p . 4.3. Proof of Corollary 1.8. Finally, we prove the result on the Strichartz regularity for the local solution. It suffices to prove that the embedding (4.9) Y p q,θ (T M ) ֒→ L ρ ([0, T M ],
M t U (−t)u 1 (t)u 2 (t)u 3 (t) = F −1 D −1 t M −1 t u 1 (t)u 2 (t)u 3 (t) = F −1 D −1 t (M −1 t u 1 (t))(M −1 t u 2 (t))(M −1 t u 3 (t)) = ct −1 F −1 (D −1 t M −1 t u 1 (t))(D −1 t M −1 t u 2 (t))(D −1 t M −1 t u 3 (t)) = ct −1 (M t U (−t)u 1 (t)) * (RM t U (−t)u 2 (t)) * (M t U (−t)u 3 (t)),where we have also used the following trivial equalitiesD −1 t (f gh) = (4πit) −1 (D −1 t f )(D −1 t g)(D −1 t h), (F −1 f )(x) = (RF −1 f )(x).
Proof. Let q 0 ≥ 1. Observe first thatby Plancherel's identity. So we estimate Y 2 q 0 ,0 -norm of the Duhamel terms of the solution. Arguing as in the proof of Proposition 3.1 we haveNow we put q 0 = 4p 5p−6 . Then (q, r) = (3q 0 , 6) satisfies either (i) or (ii) in the statement of Lemma 4.3 as long as 2 ≤ p ≤ 3. Thus arguing as in the proof of Proposition 3.1 we haveConsequently, for T M and a as in (3.7) we haveWe present two proofs of Corollary 1.6. First proof of Corollary 1.6. Assume s > −(2/p − 1). Let M > 0. We show that there is a sequence of data (φ n ) n ⊂ B p M (R) ∩ S(R) such that lim n→∞ sup t∈[0,T M ] S M φ n L p = ∞. We may assume that s is sufficiently close to −(1 − 2/p), say s < −(1/2 − 1/p), so that Sobolev's embedding U (t)φ n H s,p > n.
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| []
|
[
"Fine-grained complexity of graph homomorphism problem for bounded-treewidth graphs *",
"Fine-grained complexity of graph homomorphism problem for bounded-treewidth graphs *"
]
| [
"Karolina Okrasa [email protected] \nFaculty of Mathematics and Information Science\nFaculty of Mathematics and Information Science, Warsaw\nWarsaw University of Technology\nPoland\n\nUniversity of Technology\nPoland\n",
"Paweł Rzążewski [email protected] "
]
| [
"Faculty of Mathematics and Information Science\nFaculty of Mathematics and Information Science, Warsaw\nWarsaw University of Technology\nPoland",
"University of Technology\nPoland"
]
| []
| For graphs G and H, a homomorphism from G to H is an edge-preserving mapping from the vertex set of G to the vertex set of H. For a fixed graph H, by H (H) we denote the computational problem which asks whether a given graph G admits a homomorphism to H. If H is a complete graph with k vertices, then H (H) is equivalent to the k-C problem, so graph homomorphisms can be seen as generalizations of colorings. It is known that H (H) is polynomial-time solvable if H is bipartite or has a vertex with a loop, and NP-complete otherwise [Hell and Nešetřil, JCTB 1990].In this paper we are interested in the complexity of the problem, parameterized by the treewidth of the input graph G. If G has n vertices and is given along with its tree decomposition of width tw(G), then the problem can be solved in time |V (H)| tw(G) · n O(1) , using a straightforward dynamic programming. We explore whether this bound can be improved. We show that if H is a projective core, then the existence of such a faster algorithm is unlikely: assuming the Strong Exponential Time Hypothesis (SETH), the H (H) problem cannot be solved in time (|V (H)| − ε) tw(G) · n O(1) , for any ε > 0. This result provides a full complexity characterization for a large class of graphs H, as almost all graphs are projective cores.We also notice that the naive algorithm can be improved for some graphs H, and show a complexity classification for all graphs H, assuming two conjectures from algebraic graph theory. In particular, there are no known graphs H which are not covered by our result.In order to prove our results, we bring together some tools and techniques from algebra and from fine-grained complexity. | 10.1137/1.9781611975994.97 | [
"https://arxiv.org/pdf/1906.08371v3.pdf"
]
| 195,218,496 | 1906.08371 | 8e662ddb86103de8e32211465bd892f8fdab312c |
Fine-grained complexity of graph homomorphism problem for bounded-treewidth graphs *
19 Jun 2019 June 21, 2019
Karolina Okrasa [email protected]
Faculty of Mathematics and Information Science
Faculty of Mathematics and Information Science, Warsaw
Warsaw University of Technology
Poland
University of Technology
Poland
Paweł Rzążewski [email protected]
Fine-grained complexity of graph homomorphism problem for bounded-treewidth graphs *
19 Jun 2019 June 21, 2019* This work is supported by Polish National Science Centre grant no. 2018/31/D/ST6/00062. § Corresponding author.
For graphs G and H, a homomorphism from G to H is an edge-preserving mapping from the vertex set of G to the vertex set of H. For a fixed graph H, by H (H) we denote the computational problem which asks whether a given graph G admits a homomorphism to H. If H is a complete graph with k vertices, then H (H) is equivalent to the k-C problem, so graph homomorphisms can be seen as generalizations of colorings. It is known that H (H) is polynomial-time solvable if H is bipartite or has a vertex with a loop, and NP-complete otherwise [Hell and Nešetřil, JCTB 1990].In this paper we are interested in the complexity of the problem, parameterized by the treewidth of the input graph G. If G has n vertices and is given along with its tree decomposition of width tw(G), then the problem can be solved in time |V (H)| tw(G) · n O(1) , using a straightforward dynamic programming. We explore whether this bound can be improved. We show that if H is a projective core, then the existence of such a faster algorithm is unlikely: assuming the Strong Exponential Time Hypothesis (SETH), the H (H) problem cannot be solved in time (|V (H)| − ε) tw(G) · n O(1) , for any ε > 0. This result provides a full complexity characterization for a large class of graphs H, as almost all graphs are projective cores.We also notice that the naive algorithm can be improved for some graphs H, and show a complexity classification for all graphs H, assuming two conjectures from algebraic graph theory. In particular, there are no known graphs H which are not covered by our result.In order to prove our results, we bring together some tools and techniques from algebra and from fine-grained complexity.
Introduction
Many problems that are intractable for general graphs become significantly easier if the structure of the input instance is "simple". One of the most successful measures of such a structural simplicity is the treewidth of a graph, whose notion was rediscovered by many authors in different contexts [3,22,39,1]. Most classic NP-hard problems, including I S , D S , H C , or C , can be solved in time f (tw(G)) · n O(1) , where tw(G) is the treewidth of the input graph G and n is the number of its vertices [2,7,11,13]. In other words, many problems become polynomially solvable for graphs with bounded treewidth.
In past few years the notion of fine-grained complexity gained popularity, and the researchers became interested in understanding what is the optimal dependence on the treewidth, i.e., the function f in the complexity of algorithms solving particular problems. This lead to many interesting algorithmic results and lower bounds [43,6,29,33,38,13]. Note that the usual assumption that P = NP is not strong enough to obtain tight bounds for the running times of algorithms. In the negative results we usually assume the Exponential Time Hypothesis (ETH), or the Strong Exponential Time Hypothesis (SETH) [27,28]. Informally speaking, the ETH asserts that 3-S with n variables and m clauses cannot be solved in time 2 o(n+m) , while the SETH implies that CNF S with n variables and m clauses cannot be solved in time (2 − ε) n · m O(1) , for any ε > 0.
For example, it is known that for every fixed k, the k-C problem can be solved in time k tw(G) · n O(1) , if a tree decomposition of G of width tw(G) is given. On the other hand, Lokshtanov, Marx, and Saurabh showed that this results is essentially optimal, assuming the SETH.
Theorem 1 (Lokshtanov, Marx, Saurabh [34]). Let k 3 be a fixed integer. Assuming the SETH, the k-C problem on a graph G with n vertices cannot be solved in time (k − ε) tw(G) · c · n d for any ε > 0 and any constants c, d.
Homomorphisms. For two graphs G and H, a homomorphism is an edge-preserving mapping from V (G) to V (H). The graph H is called the target of the homomorphism. The existence of a homomorphism from any graph G to the complete graph K k is equivalent to the existence of a k-coloring of G. Because of that we often refer to a homomorphism to H as an H-coloring and think of vertices of H as colors. We also say that a graph G is H-colorable if it admits a homomorphism to H. For a fixed graph H, by H (H) we denote the computational problem which asks whether a given instance graph G admits a homomorphism to H. Clearly H (K k ) is equivalent to k-C . Since k-C is one of the best studied computational problems, it is interesting to investigate how these results generalize to H (H) for non-complete targets H. For example, it is known that k-C is polynomial-time solvable for k 2, and NP-complete otherwise. A celebrated result by Hell and Nešetřil [24] states that H (H) is polynomially solvable if H is bipartite or has a vertex with a loop, and otherwise is NP-complete. The polynomial part of the theorem is straightforward and the main contribution was to prove hardness for all non-bipartite graphs H. The difficulty comes from the fact that the local structure of the graph H is not very helpful, but we need to consider H as a whole. This is the reason why the proof of Hell and Nešetřil uses a combination of combinatorial and algebraic arguments. Several alternative proofs of the result have appeared [10,41], but none of them is purely combinatorial.
If it comes the to the running times of algorithms for k-C , it is well-known that the trivial k n · n O(1) algorithm for k-C can be improved to c n · n O(1) for a constant c, which does not depend on k (currently best algorithm of this type has running time 2 n · n O(1) [4]). Analogously, we can ask whether the trivial |H| n · n O(1) algorithm for H (H) can be improved. There were several algorithms with running times c(H) n · n O(1) , where c(H) is some structural parameter of H, that could be much smaller than |H| [20,44,40]. However, the question whether there exists an absolute constant c, such that for every H the H (H) problem can be solved in time c n · n O (1) , remained open. Finally, it was answered in the negative by Cygan et al. [12], who proved that the |H| n · n O(1) algorithm is essentially optimal, assuming the ETH.
Using a standard dynamic programming approach, H (H) can be solved in time |H| t · n O(1) , if an input graph is given along with its tree decomposition of width t [5,13]. Theorem 1 asserts that this algorithm is optimal if H is a complete graph with at least 3 vertices, unless the ETH fails. A natural extension of this result would be to provide analogous tight bounds for non-complete targets H.
Egri, Marx, and Rzążewski [15] considered this problem in the setting list homomorphisms. Let H be a fixed graph. The input of the LH (H) problem consists of a graph G, whose every vertex is equipped with a list of vertices of the target H. We ask if G has a homomorphism to H, respecting the lists. Egri et al. provided a full complexity classification for the case if H is reflexive, i.e., every vertex has a loop. It is perhaps worth mentioning that the P / NP-complete dichotomy for LH (H) was first proved for reflexive graphs as well: If H is a reflexive graph, then the LH (H) problem is polynomial time-solvable if H is an interval graph, and NP-complete otherwise [17]. Egri et al. defined a new graph invariant i * (H), based on incomparable sets of vertices and a new graph decomposition, and proved the following.
Theorem 2 (Egri, Marx, Rzążewski [15]). Let H be a fixed non-interval reflexive graph with i * (H) = k. Let n and t be, respectively, the number of vertices and the treewidth of an instance graph G. (a) Assuming a tree decomposition of G of width t is given, the LH (H) problem can be solved in time k t · c · n d , for some constants c, d. (b) There is no algorithm solving LH (H) in time (k − ε) t · c · n d for any ε > 0, and any constants c, d, unless the SETH fails.
In this paper we are interested in showing tight complexity bounds for the complexity of the non-list variant of the problem. Let us point out that despite the obvious similarity of H (H) and LH (H) problems, they behave very differently when it comes to showing hardness results. Note that if H ′ is an induced subgraph of H, then any instance of LH (H ′ ) is also an instance of LH (H), where the vertices of V (H) \ V (H ′ ) do not appear in any list. Thus in order to prove hardness of LH (H), it is sufficient to find a "hard part" H ′ of H, and perform a reduction for the LH (H ′ ) problem. The complexity dichotomy for LH (H) was proven exactly along these lines [17,18,19]. Also the proof of Theorem 2 (b) heavily uses the fact that we can work with some local subgraphs of H and ignore the rest of vertices. In particular, all these proofs are purely combinatorial.
On the other hand, in H (H) problem we need to capture the structure of the whole graph H, which is difficult using only combinatorial tools. This is why typical tools used in this area come from abstract algebra and algebraic graph theory.
For more information about graph homomorphisms we refer the reader to the comprehensive monograph by Hell and Nešetřil [26].
Our contribution. It is well known that in the study of graph homomorphisms the crucial role is played by the graphs that are cores, i.e., they do not have a homomorphism to any of its proper subgraphs. In particular, in order to provide a complete complexity classification of H (H), it is sufficient to consider the case that H is a core (we explain this in more detail in Section 2.1). Also, the complexity dichotomy by Hell and Nešetřil [24] implies that H (H) is polynomial-time solvable if H is a core on at most two vertices. So from now on let us assume that H is a fixed core, which is non-trivial, i.e., has at least three vertices.
We split the analysis into two cases, depending on the structure of H. First, in Section 4.1, we consider targets H, that are projective (the definition of this class is rather technical, so we postpone it to Section 2.2). We show that for projective cores the straightforward dynamic programming on a tree decomposition is optimal, assuming the SETH. Theorem 3. Let H be a fixed non-trivial projective core on k vertices, and let n and t be, respectively, the number of vertices and the treewidth of an instance graph G. (a) Assuming a tree decomposition of G of width t is given, the H (H) problem can be solved in time k t ·c·n d , for some constants c, d. (b) There is no algorithm solving H (H) in time (k − ε) t · c · n d for any ε > 0, and any constants c, d, unless the SETH fails.
The proof brings together some tools and ideas from algebra and fine-grained complexity theory. The main technical ingredient is the construction of a so-called edge gadget, i.e., a graph F with two specified vertices u * and v * , such that: (a) for any distinct vertices x, y of H, there is a homomorphism from F to H, which maps u * to x and v * to y, and (b) in any homomorphism from F to H, the vertices u * and v * are mapped to distinct vertices of H.
Using this gadget, we can perform a simple and elegant reduction from k-C . If G is an instance of k-C , we construct an instance G * of H (H) by taking a copy of G and replacing each edge xy with a copy of the edge gadget, whose u * -vertex is identified with x, and v * -vertex is identified with y. By the properties of the edge gadget it is straightforward to observe that G * is H-colorable if and only if G is k-colorable. Since the size of F depends only on H, we observe that the treewidth of G * differs from the treewidth of G by an additive constant, which is sufficient to obtain the desired lower bound.
Although the statement of Theorem 3 might seem quite specific, it actually covers a large class of graphs. Indeed, Hell and Nešetřil observed that almost all graphs are cores [25], see also [26,Corollary 3.28]. Moreover, Łuczak and Nešetřil proved that almost all graphs are projective [35]. From these two results, we can easily obtain that almost all graphs are projective cores. This, combined by Theorem 3, implies the following.
Corollary 4. For almost all graphs H, the H (H) problem on instance graphs with n vertices and treewidth t cannot be solved in time (|H| − ε) t · c · n d for any ε > 0, and any constants c, d, unless the SETH fails.
In Section 4.2 we consider the case that H is a non-projective core. First, we show that the approach that we used for projective cores cannot work in this case: it appears that one can construct the edge gadget for a core H with the properties listed above if and only if H is projective. What makes studying non-projective cores difficult is that we do not understand their structure well. In particular, we know that a graph H = H 1 × H 2 , where H 1 and H 2 are non-trivial and × denotes the direct product of graphs (see Section 2.2 for a formal definition), is non-projective, and by choosing H 1 and H 2 appropriately, we can ensure that H is a core. However, we do not know whether there are any non-projective non-trivial connected cores that are indecomposable, i.e., they cannot be constructed using direct products. This problem was studied in a slightly more general setting by Larose and Tardif [32,Problem 2], and it remains wide open. We restate it here, only for restricted case that H is a core, which sufficient for our purpose. Since we do not know any counterexample to Conjecture 1, in the remainder we consider cores H that are built using the direct product. We show a lower complexity bound for H (H 1 × . . . × H m ), where each H i is indecomposable, under an additional assumption that one of the factors of H is truly projective.
The definition of truly projective graphs is rather technical and we present it in Section 4.2. Graphs with such a property (actually, a slightly more restrictive one) were studied by Larose [30,Problems 1b. and 1b'.] in connection with some problems related to unique colorings, considered by Greenwell and Lovász [21]. Larose [30,31] defined and investigated even more restricted class of graphs, called strongly projective (see Section 5 for the definition). We know that every strongly projective graph is truly projective, and every truly projective graph is projective. Larose [30,31] proved that all known projective graphs are in fact strongly projective. This raises a natural question whether projectivity and strong projectivity are in fact equivalent [30,31]. Of course, an affirmative answer to this question would in particular mean that all projective cores are truly projective. Again, we state the problem in this weaker form, which is sufficient for our application.
Conjecture 2. Every projective core is truly projective.
Actually, it appears that if we assume both Conjecture 1 and Conjecture 2, we are able to provide a full complexity classification for the H (H) problem, parameterized by the treewidth of the input graph. Let us point out that despite some work on both conjectures by members of graph homomorphisms community [32,30,31], we know no graph H, for which the bounds from Theorem 5 do not hold.
Notation and preliminaries
For n ∈ N , by [n] we denote the set {1, 2, . . . , n}. All graphs considered in this paper are finite, undirected and with no multiple edges. For a graph G, by V (G) and E(G) we denote the set of vertices and the set of edges of G, respectively, and we write |G| for the number of vertices of G. Let K * 1 be the single-vertex graph with a loop. A graph is ramified if it has no two distinct vertices u and v such that the neighborhood of u is contained in the neighborhood of v. For a graph G, denote by ω(G), χ(G), and og(G), respectively, the size of the largest clique contained in G, the chromatic number of G, and the odd girth of G.
A tree decomposition of a graph G is a pair T , {X a } a∈V (T ) , in which T is a tree, whose vertices are called nodes and {X a } a∈V (T ) is the family of subsets (called bags) of V (G), such that
1. every v ∈ V (G) belongs to at least one bag X a , 2. for every uv ∈ E(G) there is at least one bag X a such that u, v ∈ X a , 3. for every v ∈ V (G) the set T v := {a ∈ V (T ) | v ∈ X a } induces a connected subgraph of T . The width of a tree decomposition T , {X a } a∈V (T ) is the number max a∈V (T ) |X a | − 1.
The minimum possible width of a tree decomposition of G is called the treewidth of G and denoted by tw(G).
Graph homomorphisms and cores
For graphs G and H, a function f : Figure 1). If G admits a homomorphism to H, we denote this fact by G → H and we write f :
V (G) → V (H) is a homomorphism, if it preserves edges, i.e., for every uv ∈ E(G) it holds that f (u)f (v) ∈ E(H) (seeG → H if f is a homomorphism from G to H. If there is no homomorphism from G to H, we write G → H. Graphs G and H are homomorphically equivalent if G → H and H → G, and incomparable if G → H and H → G.
Observe that homomorphic equivalence is an equivalence relation on the class of all graphs. An endomorphism of G is any homomorphism from f : G → G. A graph G is a core if G → H for every proper subgraph H of G. Equivalently, we can say G is a core if and only if every endomorphism of G is an automorphism. Note that a core is always ramified. If H is a subgraph of G such that G → H and H is a core, we say that H is a core of G. Notice that if H is a subgraph of G, then it always holds that H → G, so every graph is homomorphically equivalent to its core. Moreover, if H is a core of G, then H is always an induced subgraph of G, because every endomorphism f : G → H restricted to H must be an automorphism. It was observed by Hell and Nešetřil that every graph has a unique core (up to an isomorphism) [25]. Note that if f : G → H is a homomorphism from G to its core H, then it must be surjective.
We say that a core is trivial if it is isomorphic to K 1 , K * 1 , or K 2 . It is easy to observe that these three graphs are the only cores with fewer than 3 vertices. In general, finding a core of a given graph is computationally hard; in particular, deciding if a graph is a core is coNP-complete [25]. However, the graphs whose cores are trivial are simple to describe. Observation 6. Let G be a graph, whose core H is trivial. In particular, there are no non-trivial cores with loops. The following conditions are necessary for G to have a homomorphism into H.
Observation 7 ([26]). Assume that G → H and G and H have no loops. Then ω(G) ω(H), χ(G) χ(H), and og(G) og(H).
We denote by H 1 + . . . + H m a disconnected graph with connected components H 1 , . . . , H m . Observe that is f is a homomorphism from G = G 1 + . . . + G ℓ to H = H 1 + . . . + H m , then it maps every connected component of G into some connected component of H. Also note that a graph does not have to be connected to be a core, in particular the following characterization follows directly from the definition of a core.
Observation 8. A disconnected graph H is a core if and only if its connected components are pairwise incomparable cores.
An example of a pair of incomparable cores is shown in Figure 2: it is the Grötzsch graph, denoted by G G , and the clique K 3 . Clearly, og(G G ) > og(K 3 ) and χ(G G ) > χ(K 3 ), so by Observation 7, they are incomparable. Therefore, by Observation 8, the graph G G + K 3 is a core. Finally, let us observe that we can construct arbitrarily large families of pairwise incomparable cores. Let us start the construction with an arbitrary non-trivial core H 0 . Now suppose we have constructed pairwise incomparable cores H 0 , H 1 , . . . , H k and we want to construct H k+1 . Let ℓ = max i∈{0,...,k} og(H i ) and r = max i∈{0,...,k} χ(H i ). By the classic result of Erdős [16], there is a graph H with og(H) > ℓ and χ(H) > r. We set H k+1 to be the core of H. Observe that og(H k+1 ) = og(H) > ℓ and χ(H k+1 ) = χ(H) > r, so, by Observation 7, we have that for every i ∈ {0, . . . , k} the core H k+1 is incomparable with H i .
Graph products
Define the direct product of graphs H 1 and H 2 , denoted by H 1 ×H 2 , as follows:
V (H 1 ×H 2 ) = {(x, y) | x ∈ V (H 1 ) and y ∈ V (H 2 )} and E(H 1 × H 2 ) = {(x 1 , y 1 )(x 2 , y 2 ) | x 1 x 2 ∈ E(H 1 ) and y 1 y 2 ∈ E(H 2 )}. If H = H 1 ×H 2 , then H 1 ×H 2 is a factorization of H, and H 1 and H 2 are its factors. Note that H 1 ×H 2 ≃ H if and only if H 1 ≃ K * 1 or H 2 ≃ K * 1 .
Clearly, the binary operation × is commutative, so will identify H 1 ×H 2 and H 2 × H 1 . Since × is also associative, we can extend the definition for more than two factors:
H 1 × · · · × H m−1 × H m := (H 1 × · · · × H m−1 ) × H m .
Moreover, in the next sections, we will sometimes consider products of graphs, that are products themselves. Formally, the vertices of such graphs are tuples of tuples. If it does not lead to confusion, forx := (x 1 , . . . , x k 1 ) andȳ := (y 1 , . . . , y k 2 ), we will treat tuples (x,ȳ), (x 1 , . . . , x k 1 , y 1 , . . . , y k 2 ), (x, y 1 , . . . , y k 2 ), and (x 1 , . . . , x k 1 ,ȳ) as equivalent. This notation is generalized to more factors in a natural way. We denote by H m the product of m copies of H.
The direct product appears in the literature under different names: tensor product, cardinal product, Kronecker product, relational product. It is also called categorical product, because it is the product in the category of graphs (see [23,37] for details).
We say that a graph H is directly indecomposable (or indecomposable for short) if the fact that H =
H 1 × H 2 implies that either H 1 ≃ K * 1 or H 2 ≃ K * 1 . A graph that is not indecomposable, is decomposable. A factorization,
where each factor is directly indecomposable and not isomorphic to K * 1 , is called a prime factorization. Clearly, K * 1 does not have a prime factorization. The following property will be very useful (see also Theorem 8.17 in [23]).
Theorem 9 (McKenzie [36]). Any connected non-bipartite graph with more than one vertex has a unique prime factorization into directly indecomposable factors (with possible loops).
Let i ∈ [m] and let H 1 × . . . × H m be some factorization of H (not necessary prime). A function π i : V (H) → V (H i ) such that for every (x 1 , . . . , x m ) ∈ V (H) it holds that π i (x 1 , . . . , x m ) = x i is a projection on the i-th coordinate.
It follows from the definition of the direct product that every projection π i is a homomorphism from H to H i .
Below we summarize some basic properties of direct products.
Observation 10. Let H be a graph on k vertices. Then (a) H × K 1 consists of k isolated vertices, in particular its core is K 1 , (b) if H has at least one edge, then the core of H × K 2 is K 2 , (c) the graph H m contains a subgraph isomorphic to H, induced by the set {(x, . . . , x) | x ∈ V (H)}; in particular, if m 2, then H m is never a core, (d) if H = H 1 × . . . × H m , then for every G it holds that G → H if and only if G → H i for all i ∈ [m]. Proof. Items (a), (b), (c) are straightforward to observe. To prove (d), consider a homomorphism f : G → H. Clearly, H → H i for every i ∈ [m] because each projection π i : H → H i is a homomorphism. So π i • f is a homomorphism from G to H i . On the other hand, if we have some f i : G → H i for every i ∈ [m], then we can define a homomorphism f : G → H by f (x) := (f 1 (x), . . . , f m (x)). A homomorphism f : H m → H is idempotent, if for every x ∈ V (H) it holds that f (x, x, . . . , x) = x.
One of the main characters of the paper is the class of projective graphs, considered e.g. in [30,31,32]. A graph H is projective (or idempotent trivial), if for every m 2, every idempotent homomorphism from H m to H is a projection.
Observation 11. If H is a projective core and f : H m → H is a homomorphism, then f ≡ g • π i for some i ∈ [m] and some automorphism g of H.
Proof. If f is idempotent, then it is a projection and we are done. Assume f is not idempotent and define g : f (x, . . . , x). The function g is an endomorphism of H and H is a core, so g is in fact an automorphism of H. Observe that g −1 • f is an idempotent homomorphism, so it is equal to π i for some i ∈ [m], because H is projective. From this we get that f ≡ g • π i .
V (H) → V (H) by g(x) =
It is known that projective graphs are always connected [32]. Observe that the definition of projective graphs does not imply that their recognition is decidable. However, an algorithm to recognize these graph follows from the following, useful characterization.
Theorem 12 (Larose, Tardif [32]). A connected graph H with at least three vertices is projective if and only if every idempotent homomorphism from H 2 to H is a projection.
Recall from the introduction that almost all graphs are projective cores [26,35]. It appears that the properties of projectivity and being a core are independent. In particular, the graph in Figure 3 is not a core, as it can be mapped to a triangle. However, Larose [30] proved that all non-bipartite, connected, ramified graphs which do not contain C 4 as a (non-necessarily induced) subgraph, are projective (this will be discussed in more detail in Section 5, see Theorem 25). On the other hand, there are also non-projective cores, an example is G G × K 3 , see Figure 2. We discuss such graphs in detail in Section 4.2. Figure 3: An example of a projective graph which is not a core.
Complexity of finding graph homomorphisms
Note that if two graphs H 1 and H 2 are homomorphically equivalent, then the H (H 1 ) and H (H 2 ) problems are also equivalent. So in particular, because every graph is homomorphically equivalent to its core, we may restrict our attention to graphs H which are cores. Also, recall from Observation 6 that H (H) can be solved in polynomial time if H is isomorphic to K * 1 , K 1 , or K 2 . So we will be interested only in non-trivial cores H. In particular, we will assume that H is non-bipartite and has no loops.
We are interested in understanding the complexity bound of the H (H) problem, parameterized by the treewidth of the input graph. The standard dynamic programming approach (see for example Cygan et al. [13]) gives us the following upper bound.
Theorem 13 (Folklore). Let H be a fixed graph on k vertices. Assuming a tree decomposition of width t of the instance graph on n vertices is given, the H (H) problem can be solved in time k t · c · n d for some constants c, d.
m · α t · c · n d = α t · c ′ · n d for c ′ = c · m. This proves (a).
To see (b), consider an instance G of H (H i ) on n vertices and treewidth t. Let V (H i ) = {z 1 , . . . , z k } and let u be some fixed vertex of G. We construct an instance G * of H (H) as follows. We take a copy G ′ of G and a copy H k i of H k i , and identify the vertex corresponding to u in G ′ with the vertex corresponding to (z 1 , . . . , z k ) in H k i . Denote this vertex of G * byz. We claim that G → H i if and only if G * → H. Indeed, if f : G → H i , then there exists j ∈ [k] such that f (z) = z j , so we can define a homomorphism g : G * → H i (which is also a homomorphism from G * to H) by
g(x) = f (x) if x ∈ G ′ , π j (x) otherwise.
Clearly, both f and π j are homomorphisms andz is a cutvertex in G * for which f (z) = π j (z), so g is a homomorphism from G * to H. Conversely, if we have g : G * → H, we know that g maps G * to a connected component H j of H, for some j ∈ [m], because G * is connected. But G * contains an induced copy H k i of H k i , so also an induced copy of H i , say H i (recall Observation 10 (c)). So g| V ( H i ) is in fact a homomorphism from H i to H j . Recall from Observation 7 that since H 1 + . . . + H m is a core, its connected components are pairwise incomparable cores -so j must be equal to i. It means that g| V (G ′ ) is a homomorphism from G ′ to H i , so we conclude that G → H i .
Note that the number of vertices of G * is n + |H k i | − 1 |H k i | · n. Now let T , {X a } a∈V (T ) be a tree decomposition of G of width t, and let b be a node of T , such that u ∈ X b . Define
X b ′ := X b ∪ V (H k i ) and let V (T * ) = V (T )∪{b ′ } and E(T * ) = E(T )∪{bb ′ }. Clearly, T * , {X a } a∈V (T * ) is a tree decomposition of G * . This means that tw(G * ) t + |H k i |.
The graph H i is fixed, so the number of vertices of H k i is a constant. By our assumption we can decide if
G * → H in time α tw(G * ) · c · |G * | d , so we can decide if G → H i in time α tw(G * ) · c · (|H k i |n) d α t α |H k i | · c · |H k i | d n d = α t · c ′ · n d , where c ′ = c · α |H k i | · |H k i | d .
Theorem 15 implies that for our purpose it is sufficient to consider connected cores.
Lower bounds
In this section we will investigate the lower bounds for the complexity of H (H). The section is split into two main parts. In Section 4.1 we consider projective cores. Then, in Section 4.2, we consider nonprojective cores.
Projective cores
The main result of this section is Theorem 3.
Theorem 3.
Let H be a fixed non-trivial projective core on k vertices, and let n and t be, respectively, the number of vertices and the treewidth of an instance graph G. (a) Assuming a tree decomposition of G of width t is given, the H (H) problem can be solved in time k t ·c·n d , for some constants c, d. (b) There is no algorithm solving H (H) in time (k − ε) t · c · n d for any ε > 0, and any constants c, d, unless the SETH fails.
Observe that Theorem 3 (a) follows from Theorem 13, so we need to show the hardness counterpart, i.e., the statement (b). A crucial building block in our reduction will the graph called the edge gadget, whose construction is described in the following lemma.
Lemma 16. For every non-trivial projective core H, there exists a graph F with two specified vertices u * and v * , satisfying the following: (a) for every x, y ∈ V (H) such that x = y, there exists a homomorphism f : . . . , z i−1 , z i+1 , . . . z k ). We claim that F := H (k−1)k and vertices To see that (a) holds, observe that if x and y are distinct vertices from V (H), then there always exists i ∈ [k(k − 1)] such that π i (u * ) = x and π i (v * ) = y. This means that π i is a homomorphism from F = H k(k−1) to H satisfying π i (u * ) = x and π i (v * ) = y.
F → H such that f (u * ) = x and f (v * ) = y, (b) for every f : F → H it holds that f (u * ) = f (v * ). Proof. Let V (H) = {z 1 , . . . , z k }. For i ∈ [k] denote by z k−1 i the (k − 1)-tuple (z i , . . . , z i ) and by z i the (k − 1)-tuple (z 1 ,
To prove (b), recall that since H is projective, by Observation 11, the homomorphism f is a composition of some automorphism g of H and π i for some i ∈ [k(k − 1)]. Observe that u * and v * are defined in a way such that π j (u * ) = π j (v * ) for every j ∈ [k(k − 1)]. As g is an automorphism, it is injective, which gives us f (u * ) = g(π i (u * )) = g(π i (v * )) = f (v * ).
Finally, we are ready to prove Theorem 3 (b).
Proof of Theorem 3 (b).
Note that since H is non-trivial, we have k 3. Since H is projective, it is also connected. We reduce from k-C , let G be an instance with n vertices and treewidth t. We construct an instance G * of H (H) as follows. First, for every z ∈ V (G) we introduce a vertex z ′ of V (G * ). Let V ′ denote the set of these vertices. Now, for every edge xy of G, we introduce to G * a copy of the edge gadget, constructed in Lemma 16, and denote it by F xy . We identify the vertices u * and v * of F xy with vertices x ′ and y ′ , respectively. This completes the construction of G * .
We claim that G is k-colorable if and only if G * → H. Indeed, let ϕ be a k-coloring of G. For simplicity of notation, we label the colors used by ϕ in the same way as the vertices of H, i.e., z 1 , z 2 , . . . , z k . Define g : V (V ′ ) → V (H) by setting g(v ′ ) := ϕ(v ′ ) Now consider an edge xy of G and the edge gadget F xy . Since c is a proper coloring, we have g(x ′ ) = g(y ′ ). So by Lemma 16 (a), we can find a homomorphism f xy : F xy → H, such that f xy (x ′ ) = g(x ′ ) and f xy (y ′ ) = g(y ′ ). Repeating this for every edge gadget, we can extend g to a homomorphism from G * to H.
Conversely, from Lemma 16 (b), we know that for any f : G * → H and every edge xy of G it holds that f (x ′ ) = f (y ′ ), so any homomorphism from G * induces a k-coloring of G.
The number of vertices of G * is at most |F |n 2 . Now let T , {X a } a∈V (T ) be a tree decomposition of G of width t. Let us extend it to a tree decomposition of G * . For each edge xy of G there exists a bag X b such that x, y ∈ X b . We add to T the node b ′ with X b ′ := X b ∪ V (F xy ), and an edge bb ′ . It is straightforward to observe that by repeating this step for every edge of G, we obtain a tree decomposition of G * of width
at most t + |F |. Recall that H is fixed, so |F | is a constant. So if we could decide if G * → H in time (k − ε) tw(G * ) · c · |G * | d (k − ε) t+|F | · c · |F | d · n 2d , then we would be able to decide if G is k-colorable in time (k − ε) t · c ′ · n d ′ for constants c ′ = c · (k − ε) |F | · |F | d and d ′ = 2d
. By Theorem 1 this contradicts the SETH.
Non-projective cores
Now we will focus on non-trivial connected cores, which are additionally non-projective, i.e., they do not satisfy the assumptions of Theorem 3. First, let us argue that the approach from Section 4.1 cannot work in this case. In particular, we will show that an edge gadget with properties listed in Lemma 16 cannot be constructed for non-projective graphs H.
We will need the definition of constructible sets, see Larose and Tardif [32]. For a graph H, a set
C ⊆ V (H) is constructible if there exists a graph K, vertices x 0 , . . . , x ℓ ∈ V (F ) and y 1 , . . . , y ℓ ∈ V (H) such that {f (x 0 ) ∈ V (H) | f : K → H such that f (x i ) = y i for every i ∈ [ℓ]} = C.
We might think of C as the set of colors that might appear on the vertex x 0 , when we precolor each x i with the color y i and try to extend this partial mapping to a homomorphism to H. The tuple (K, x 0 , . . . , x ℓ , y 1 , . . . , y ℓ ) is called a construction of C. It appears that the notion of constructible sets is closely related to projectivity. Take k − ℓ copies of F , say F 1 , . . . , F k−ℓ and denote the vertices u * and v * of i-th copy F i by u * i and v * i , respectively. Identify the vertices u * i of all these copies, denote the obtained vertex by u * , and the obtained graph by K. Now set x 0 := u * and for each i ∈ [k − ℓ] set x i := v * i . It is easy to verify that this is a construction of the set C. Indeed, observe that if x ∈ C, then, from Lemma 16 (a), for each copy F i there exists a homomorphism f i :
F i → H such that f i (v * i ) = f i (x i ) = y i and f i (u * ) = f i (x 0 ) = x.
Combining these homomorphisms yields a homomorphism f : K → H. On the other hand, if x ∈ C, then x = y i for some i ∈ [k − ℓ]. But from Lemma 16 (b) we know that for every homomorphism f :
F → H it holds that x = y i = f (v * i ) = f (u * ) = f (x 0 )
, so x 0 cannot be mapped to x by any homomorphism from K to H.
Observe that if H is projective, then it must be indecomposable. Indeed, assume that for some nontrivial H it holds that H = H 1 ×H 2 , H ≃ K * 1 and H 2 ≃ K * 1 . Consider a homomorphism f : (H 1 ×H 2 ) 2 → H 1 × H 2 , defined as f ((x, y), (x ′ , y ′ )) = (x, y ′ ). Note that it is idempotent, but not a projection, so H is not projective.
In the light of the observation above, it is natural to ask whether indecomposability implies projectivity. This problem was already stated e.g. by Larose and Tardif [32,Problem 2] and, to the best of our knowledge, no significant progress in this direction was made. Let us recall it here.
Conjecture 1.
Let H be a connected non-trivial core. Then H is projective if and only if it is indecomposable.
Since we know no connected non-trivial non-projective cores that are indecomposable, in the remainder of the section we will assume that H is a decomposable, non-trivial connected core. By Theorem 9 we know that H has a unique prime factorization H 1 × . . . × H m for some m 2. To simplify the notation, for any given homomorphism f :
G → H 1 × . . . × H m and i ∈ [m], we define f i ≡ π i • f . Then for each vertex x of G it holds that f (x) = (f 1 (x), . . . , f m (x)),
and f i is a homomorphism from G to H i . The following observation follows from Observation 10. · c · n d , where n and t are the number of vertices and the treewidth of the input graph, respectively, and c, d are constants. We believe that this bound is actually tight, and prove a matching lower bound under some additional assumption.
We say that a graph H is truly projective if it has at least three vertices and for every s 2 and every connected core W incomparable with H, it holds that the only homomorphisms f : H s × W → H which satisfy f (x, x, . . . , x, y) = x for any x ∈ V (H), y ∈ V (W ), are projections. It is easy to verify that truly projective graphs are projective. Indeed, by Theorem 12, we need to show that any idempotent homomorphism g : H 2 → H is a projection. Consider a core W , which is incomparable with H, and a homomorphism f : H 2 × W → H, defined by f (x 1 , x 2 , y) := g(x 1 , x 2 ). Since H is truly projective, f is a projection, and so is g.
We show the following lower bound. The proof of Theorem 20 is similar to the proof of Theorem 3 (b). We start with constructing an appropriate edge gadget. We will use the following result (to avoid introducing new definitions, we state the theorem in a sightly weaker form, using the terminology used in this paper, see also [23,Theorem 8.18]).
H (H) in time (|H i | − ε) t · c · n d ,
Theorem 21 (Dörfler,[14]). Let ϕ be an automorphism of a connected, non-bipartite, ramified graph H, with the prime factorization H 1 × . . . × H m . Then for each i ∈ [m] there exists an automorphism ϕ (i) of H i such that ϕ i (t 1 , . . . , t m ) ≡ ϕ (i) (t i ).
In particular, it implies the following.
Corollary 22.
Let µ be an automorphism of a connected, non-trivial core H = H 1 × R, where H 1 is indecomposable and R ≃ K * 1 . Then there exist automorphisms µ (1) : H 1 → H 1 and µ (2) : R → R such that µ(t, t ′ ) ≡ (µ (1) (t), µ (2) (t ′ )).
Proof. By Observation 19, R is a non-trivial core, so it admits a unique prime factorization, say R = H 2 × . . . × H m . Therefore H 1 × H 2 × . . . × H m is the unique prime factorization of H. From Theorem 21 we know that for each i ∈ [m] there exists an automorphism ϕ (i) of H i such that µ(t 1 , . . . , t m ) ≡ (ϕ (1) (t 1 ), . . . , ϕ (m) (t m )). Define µ (1) by setting µ (1) (t) := ϕ (1) (t) for every vertex t ∈ V (H 1 ). Analogously, we define µ (2) by setting µ (2)
(t 2 , . . . , t m ) := (ϕ (2) (t 2 ), . . . , ϕ (m) (t m )) for every vertex (t 2 , . . . , t m ) of R (for each i ∈ [m] \ {1} we have t i ∈ V (H i )).
It is straightforward to verify that µ (1) and µ (2) satisfy the statement of the corollary.
In the following lemma we construct an edge gadget, that will be used in the hardness reduction. The construction is similar to the one in Lemma 16, but more technically complicated.
Lemma 23.
Let H = H 1 × R be a connected, non-trivial core, such that H 1 is truly projective and R ≃ K * 1 . Let w be a fixed vertex of R. Then there exists a graph F and vertices u * , v * of F , satisfying the following conditions: (a) for every xy ∈ E(H 1 ) there exists f : ) and v * := (v, w). We will treat vertices u and v as 2s-tuples, and vertices u * and v * as (2s + 1)-tuples.
F → H such that f (u * ) = (x, w) and f (v * ) = (y, w), (b) for any f : F → H it holds that f 1 (u * )f 1 (v * ) ∈ E(H 1 ).
Observe that, if xy ∈ E(H 1 ), then, by the definition of u * and v * , there exists i ∈ [2s] such that x = π i (u) and y = π i (v). Define a function f : V (F ) → V (H) as f (x 1 , . . . , x 2s , w) := (π i (x 1 , . . . , x 2s ), w). Observe that this is a homomorphism, for which f (u * ) = f (u, w) = (x, w) and f (v * ) = f (v, w) = (y, w), which is exactly the condition (a) in the statement of Lemma 23.
We prove (b) in two steps. First, we observe the following.
Claim. Let ϕ : F → H. If for every z ∈ V (H 1 ) and r ∈ V (R) it holds that ϕ 1 (z, . . . , z, r) = z then
ϕ 1 (u * )ϕ 1 (v * ) ∈ E(H 1 ).
Proof of Claim. Recall that R is a connected core incomparable with H 1 , and H 1 is truly projective. It means that if ϕ 1 : H 2s 1 × R → H 1 satisfies the assumption of the claim, then it is equal to π i for some i ∈ [2s]. From the definition of u * and v * we have that π i (u * )π i (v * ) ∈ E(H 1 ).
Note that the set {(z, . . . , z, r) ∈ F | z ∈ V (H 1 ), r ∈ V (R)} induces in F a subgraph isomorphic to H, let us call it H. Let σ be an isomorphism from H to H defined as σ(z, . . . , z, r) := (z, r).
Consider any homomorphism f : F → H. We observe that f | V ( H) is an isomorphism from H to H, because H is a core. If f | V ( H) ≡ σ then for every z ∈ V (H) and r ∈ V (R) it holds that f 1 (z, . . . , z, r) = σ 1 (z, . . . , z, r) = z, so, by the Claim above, we are done. If not, observe that there exists the inverse isomorphism g : H → H such that g • f | V ( H) is the identity function on V ( H). Define µ := σ • g. Observe that µ is an endomorphism of H 1 × R, so an automorphism, since H 1 × R is a core. Also note that (µ • f ) : F → H 1 × R is a homomorphism such that for every (z, . . . , z, r) ∈ V ( H) it holds that . . . , z, r) = σ(z, . . . , z, r) = (z, r),
(µ • f )(z, . . . , z, r) = (σ • g • f )(z, . . . , z, r) = (σ • id)(z,so (µ • f ) 1 (z, .
. . , z, z ′ ) = z. This means that µ • f satisfies the assumption of the Claim, so
(µ • f ) 1 (u * )(µ • f ) 1 (v * ) ∈ E(H 1 ).(1)
Clearly, for every vertexz of F it holds that
(µ • f ) (z) = µ f 1 (z), f 2 (z) = µ 1 f 1 (z), f 2 (z) , µ 2 f 1 (z), f 2 (z) .(2)
Note that Corollary 22 implies that there exist automorphisms µ (1) and µ (2) of H 1 and R, respectively, such that for everyz ∈ V (F ) it holds that
µ 1 f 1 (z), f 2 (z) =µ (1) (f 1 (z)) µ 2 f 1 (z), f 2 (z) =µ (2) (f 2 (z)),(3)
In particular, (2) and (3) imply that (µ • f ) 1 = µ (1) • f 1 . Combining this with (1) we get that
µ (1) • f 1 (u * ) µ (1) • f 1 (v * ) ∈ E(H 1 ).(4)
Since µ (1) is the automorphism of H 1 , there exists the inverse automorphism µ (1) −1 of H 1 . Because µ (1) −1 is an automorphism, (4) implies that f 1 (u * )f 1 (v * ) ∈ E(H 1 ), which completes the proof. Now we can proceed to the proof of Theorem 20.
Proof of Theorem 20. Since × is commutative, without loss of generality we can assume that H 1 is truly projective. Define R := H 2 × . . . × H m , so H = H 1 × R. Since H 1 is truly projective, it is projective, so Theorem 3 can be applied here. Hence we known that assuming the SETH, there is no algorithm which solves instances of H (H 1 ) with n vertices and treewidth t in time (|H 1 | − ε) t · c ′ · n d ′ , for any ε > 0 and constants c ′ , d ′ .
Let G be an instance of H (H 1 ) with n vertices and treewidth t. The construction of the instance G * of H (H) is analogous as in the proof of Theorem 3 (b). Let w be a fixed vertex of R and let F be a graph obtained by calling Lemma 23 for H and w. For every vertex z of G, we introduce to G * a vertex z ′ . Then we add a copy F xy of F for every pair of vertices x ′ , y ′ , which corresponds to an edge xy in G, and identify vertices x ′ and y ′ with vertices u * and v * of F xy , respectively.
As in the proof of Theorem 3, we observe that G * is a yes-instance of H (H) if and only if G is a yes-instance of H (H 1 ). Moreover, |G * | |F | · n 2 and tw(G * ) t + |F |. Thus, if we could decide if G * → H in time (|H 1 | − ε) tw(G * ) · |G * | d · c, then we would be able to decide if G → H 1 in time (|H 1 | − ε) t · n d ′ · c ′ for constants c ′ , d ′ . By Theorem 3 (b), such an algorithm contradicts the SETH.
Note that combining the results from Theorem 14 and Theorem 20 we obtain a tight complexity bound for graphs H, whose largest factor is truly projective.
Corollary 24.
Let H be a non-trivial, connected core with prime factorization H 1 × . . . × H m and let H i be the factor with the largest number of the vertices. Assume that H i is truly projective. Let n and t be, respectively, the number of vertices and the treewidth of an instance graph G. (a) If a tree decomposition of G of width t is given, the H (H) problem can be solved in time |H i | t · c · n d , for some constants c, d > 0. (b) There is no algorithm solving H (H) in time (|H i | − ε) t · c · n d for any ε > 0 and any constants c, d, unless the SETH fails.
Conclusion
Recall that in Theorem 20 and Corollary 24 we presented lower complexity bounds for H (H) in the case that one of factors of H is truly projective. In the light of Conjecture 1, we would like to weaken this assumption by substituting "truly projective" with "projective". Let us discuss the possibility of obtaining such a result. As mentioned in the introduction, a class of graphs very close to truly projective graphs was considered by Larose [30]. In the same paper, he defined and studied the so-called strongly projective graphs. A graph H on at least three vertices is strongly projective, if for every connected graph W on at least two vertices and every s 2, the only homomorphisms f : H s × W → H satisfying f (x, . . . , x, y) = x for all x ∈ V (H) and y ∈ V (W ), are projections. Note that this definition is very similar, but more restrictive than the definition of truly projective graphs. Indeed, for truly projective graphs H we restricted the homomorphisms from H s × W to H only for connected cores W , that are incomparable with H. Thus it is clear that every strongly projective graph is truly projective, and, as observed before, every truly projective graph is projective. Among other properties of strongly projective graphs, Larose [30,31] shows that their recognition is decidable -note that this does not follow directly from the definition.
Let us recall some results on strongly projective graphs, as they show that many natural graphs satisfy the assumptions of Theorem 20 and Corollary 24. We say that graph is square-free if it does not contain a copy of C 4 as a (not necessarily induced) subgraph. Larose proved the following.
Theorem 25 (Larose [30]). If H is a square-free, connected, non-bipartite core, then it is strongly projective.
Example. Consider the graph G B on 21 vertices, shown on Figure 4 (left), it is called the Brinkmann graph [9]. It is connected, its chromatic number of 4 and its girth is 5. In particular, it is square-free. Thus by Theorem 25 we know that G B is strongly projective. By exhaustive computer search we verified that K 3 × G B is a core. Let us consider the complexity of H (K 3 × G B ) for input graphs with n vertices and treewidth t. The straightforward dynamic programming approach from Theorem 13 results in the running time 63 t · c · n d , where c and d are constants. However, Theorem 14 gives us a faster algorithm, whose running time is 21 t · c · n d . Moreover, by Corollary 24 we know that this algorithm is likely to be asymptotically optimal, i.e., there is no algorithm with running time (21 − ε) t · c · n d for any ε > 0 and any constants c, d, unless the SETH fails.
A graph is said to be primitive if there is no non-trivial partition of its vertices which is invariant under all automorphisms of this graph (see e.g. [42]).
Theorem 26 (Larose [30]). If H is a directly indecomposable primitive core, then it is strongly projective. In particular, Theorem 26 implies that Kneser graphs are strongly projective. Note that Kneser graphs might have 4-cycles, so this statement does not follow from Theorem 25.
Interestingly, Larose [30,31] proved that all known projective graphs are in fact strongly projective (and thus of course truly projective). He also asked whether the same holds for all projective graphs. We recall this problem in a weaker form, which would be sufficient in our setting.
Conjecture 2. Every projective core is truly projective.
Clearly, if both Conjecture 1 and Conjecture 2 are true, there is another characterization of non-trivial indecomposable connected cores.
Observation 27. Assume that Conjecture 1 and Conjecture 2 hold. Let H be a connected non-bipartite core. Then H is indecomposable if and only if it is truly projective.
Note that Theorem 3, Corollary 24, and Observation 27 imply the following result.
Theorem 5. Assume that Conjecture 1 and Conjecture 2 hold. Let H be a fixed non-trivial connected core with prime decomposition H 1 × . . . × H m , and define k := max i∈[m] |H i |. Let n and t be, respectively, the number of vertices and the treewidth of an instance graph G. (a) Assuming a tree decomposition of G of width t is given, the H (H) problem can be solved in time k t ·c·n d , where c, d are constants. (b) There is no algorithm solving H (H) in time (k − ε) t · c · n d for any ε > 0 and any constants c, d, unless the SETH fails.
We conjecture that the bounds from Theorem 5 are tight for all connected cores. Finally, let us point out one more problem, related to the ones discussed in this paper. Recall that if H = H 1 × H 2 is a connected, non-trivial core and H 1 ≃ K * 1 , H 2 ≃ K * 1 , then H 1 and H 2 must incomparable cores. We believe it would be interesting to know if the opposite implication holds as well.
To motivate the study on this problem, we state the following conjecture. Conjecture 3. Let H 1 and H 2 be connected, indecomposable, incomparable cores. Then H 1 × H 2 is a core.
We confirmed this conjecture by exhaustive computer search for some small graphs. In particular, the conjecture is true for graphs K 3 × H, where H is any 4-vertex-critical, triangle-free graph with at most 14 vertices [8], the Grötzsch graph (see Figure 2), the Brinkmann graph (see Figure 4 (left)), or the Chvátal graph (see Figure 4 (right)).
Conjecture 1 .
1Let H be a connected non-trivial core. Then H is projective if and only if it is indecomposable.
Theorem 5 .
5Assume that Conjecture 1 and Conjecture 2 hold. Let H be a fixed non-trivial connected core with prime decomposition H 1 × . . . × H m , and define k := max i∈[m] |H i |. Let n and t be, respectively, the number of vertices and the treewidth of an instance graph G. (a) Assuming a tree decomposition of G of width t is given, the H (H) problem can be solved in time k t ·c·n d , where c, d are constants. (b) There is no algorithm solving H (H) in time (k − ε) t · c · n d for any ε > 0 and any constants c, d, unless the SETH fails.
Figure 1 :
1An example of a homomorphism from G (left) to H (right). Colors of the vertices indicate the mapping.
(a) H ≃ K 1 if and only if χ(G) = 1, i.e., G has no edges, (b) H ≃ K 2 if and only if χ(G) = 2, i.e., G is bipartite and has at least one edge, (c) H ≃ K * 1 if and only if G has vertex with a loop.
Figure 2 :
2An example of incomparable cores, the Grötzsch graph (left) and K 3 (right).
)
and v * := (z 1 , . . . , z k ) satisfy the statement of the lemma. Note that the vertices of F are k(k − 1)-tuples.
Theorem 17 (
17Larose, Tardif [32]). A graph H on at least three vertices is projective if and only if every subset of its vertices is constructible. Now we show that Lemma 16 cannot work for non-projective graphs H. Proposition 18. Let H be a fixed non-trivial connected core. Then an edge gadget F with properties listed in Lemma 16 exists if and only if H is projective. Proof. The 'if' statement follows from Lemma 16. Let k := |H| and suppose that there exists a graph F with properties given in Lemma 16. Consider a set C ⊆ V (H) and define ℓ := |C|. Let {y 1 , . . . , y k−ℓ } be the complement of C in V (H).
for any ε > 0 and any constants c, d, where n and t are, respectively, the number vertices and the treewidth of an instance graph.
Proof. Let E(H 1 ) = {e 1 , . . . , e s } and let e i = u i v i for every i ∈ [s] (clearly, one vertex can appear many times as some u i or v j ). Consider the vertices u :=(u 1 , . . . , u s , v 1 , . . . , v s ) v :=(v 1 , . . . , v s , u 1 , . . . , u s ) of H 2s 1 . Let F := H 2s 1 × R, and let u * := (u, w
Figure 4 :
4The Brinkmann graph (left) and the Chvátal graph (right).
By Theorem 1, this bound is tight if H is a complete graph with at least three vertices, unless the SETH fails. We are interested in extending this result for other graphs H.First, let us observe that there are cores, for which the bound from Theorem 13 can be improved. Indeed, let H be a decomposable core, isomorphic to H 1 × . . . × H m (see discussion in Section 4.2 for more about cores that are products.). Recall from Observation 10 (d) that for every graph G it holds thatG → H if and only if G → H i for every i ∈ [m]. So, given an instance G of H (H), we can call the algorithm from Theorem 13 to solve H (H i ) for each i ∈ [m] and return a positive answer if and only if we get a positive answer in each of the calls. This way we obtain the following result.Theorem 14. Let H be a fixed core with prime factorization H 1 × . . . × H m . Assuming a tree decomposition of width t of the instance graph with n vertices is given, the H (H) problem can be solved in time max j∈[m] |H j | t · c · n d for some constants c, d. Let us conclude this section with a simple observation about the complexity of H (H) for disconnected cores H.Theorem 15. Consider a fixed disconnected core H = H 1 + . . . + H m . Consider an instance G with n vertices, given along with its tree decomposition of width t. (a) Assume that for every i ∈ [m] the H (H i ) problem can be solved in time α t · c · n d , where α, c, d are constants. Then H (H) can be solved in time α t · c ′ · n d for some constant c ′ . (b) Assume that H (H) can be solved in time α t · c · n d , where α, c, d are constants. Then for every i ∈ [m] the H (H i ) problem can be solved in time α t · c ′ · n d for some constant c ′ , d. Proof. First, observe that if G is disconnected, say G = G 1 + . . . + G ℓ , then G → H if and only if G i → H for every i ∈ [ℓ]. Also, tw(G) = max i∈[ℓ] tw(G i ). It means that if the instance graph is disconnected, we can just consider the problem separately for each of its connected components. So we assume that G is connected. First, recall from Observation 8 that G → H if and only if G → H i for some i ∈ [m]. Again, we can solve H (H i ) for each i ∈ [m] (for the same instance G) and return a positive answer for H (H) if and only if we get a positive answer for at least one i ∈ [m]. The total complexity of this algorithm is
Observation 19. Let H be a connected, non-trivial core with factorization H = H 1 × . . . × H m , such that H i ≃ K * 1 for all i ∈ [m]. Then for i ∈ [m] the graph H i is a connected non-trivial core, incomparable with H j for j ∈ [m] \ {i}. Now let us consider the complexity of H (H), where H has a prime factorization H 1 × H 2 × . . . H m for m 2. By Theorem 14, the problem can be solved in time max i∈[m] |H i |t
Theorem 20 .
20Let H be a fixed non-trivial connected core, with prime factorization H 1 × . . . × H m . Assume there exists i ∈ [m] such that H i is truly projective. Unless the SETH fails, there is no algorithm solving
Acknowledgment. The authors are grateful to D. Marx for introducing us to the problem, and to B. Larose, C. Tardif, B. Martin, and Mi. Pilipczuk for useful comments.
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| []
|
[
"Superconducting pairing symmetry in the kagome-lattice Hubbard model",
"Superconducting pairing symmetry in the kagome-lattice Hubbard model"
]
| [
"Chenyue Wen \nSchool of Physics\nBeihang University\n100191BeijingChina\n",
"Xingchuan Zhu \nCenter for Basic Teaching and Experiment\nNanjing University of Science and Technology\n214443JiangyinPeople's Republic of China\n",
"Zhisong Xiao \nSchool of Physics\nBeihang University\n100191BeijingChina\n",
"Ning Hao \nAnhui Key Laboratory of Condensed Matter Physics at Extreme Conditions\nHigh Magnetic Field Laboratory\nHFIPS\nChinese Academy of Sciences\n230031HefeiAnhuiChina\n",
"Rubem Mondaini \nBeijing Computational Science Research Center\n100084BeijingChina\n",
"Huaiming Guo \nSchool of Physics\nBeihang University\n100191BeijingChina\n\nBeijing Computational Science Research Center\n100084BeijingChina\n",
"Shiping Feng \nDepartment of Physics\nBeijing Normal University\n100875BeijingChina\n"
]
| [
"School of Physics\nBeihang University\n100191BeijingChina",
"Center for Basic Teaching and Experiment\nNanjing University of Science and Technology\n214443JiangyinPeople's Republic of China",
"School of Physics\nBeihang University\n100191BeijingChina",
"Anhui Key Laboratory of Condensed Matter Physics at Extreme Conditions\nHigh Magnetic Field Laboratory\nHFIPS\nChinese Academy of Sciences\n230031HefeiAnhuiChina",
"Beijing Computational Science Research Center\n100084BeijingChina",
"School of Physics\nBeihang University\n100191BeijingChina",
"Beijing Computational Science Research Center\n100084BeijingChina",
"Department of Physics\nBeijing Normal University\n100875BeijingChina"
]
| []
| The dominating superconducting pairing symmetry of the kagome-lattice Hubbard model is investigated using the determinant quantum Monte Carlo method. The superconducting instability may occur when doping the correlated insulators formed by the Hubbard interaction near the Dirac filling, and the possible superconducting state exhibits an electron-hole asymmetry. Among the pairing symmetries allowed, we demonstrate that the dominating channel is d-wave in the hole-doped case. This opens the possibility of condensation into an unconventional d x 2 −y 2 + idxy phase, which is characterized by an integer topological invariant and gapless edge states. In contrast, the s * -wave channel, which has no sign change in the pairing function, is favored by the electron doping. We further find the dominating s * -wave pairing persists up to the Van Hove singularity. The results are closely related to the recent experimental observations in kagome compounds AV3Sb5(A: K, Rb,Cs), and provide insight into the pairing mechanism of their superconducting states. arXiv:2109.12582v2 [cond-mat.str-el] | 10.1103/physrevb.105.075118 | [
"https://arxiv.org/pdf/2109.12582v2.pdf"
]
| 237,940,462 | 2109.12582 | f71023b1a373fdfa070c6f8c289ab3f140adea91 |
Superconducting pairing symmetry in the kagome-lattice Hubbard model
Chenyue Wen
School of Physics
Beihang University
100191BeijingChina
Xingchuan Zhu
Center for Basic Teaching and Experiment
Nanjing University of Science and Technology
214443JiangyinPeople's Republic of China
Zhisong Xiao
School of Physics
Beihang University
100191BeijingChina
Ning Hao
Anhui Key Laboratory of Condensed Matter Physics at Extreme Conditions
High Magnetic Field Laboratory
HFIPS
Chinese Academy of Sciences
230031HefeiAnhuiChina
Rubem Mondaini
Beijing Computational Science Research Center
100084BeijingChina
Huaiming Guo
School of Physics
Beihang University
100191BeijingChina
Beijing Computational Science Research Center
100084BeijingChina
Shiping Feng
Department of Physics
Beijing Normal University
100875BeijingChina
Superconducting pairing symmetry in the kagome-lattice Hubbard model
The dominating superconducting pairing symmetry of the kagome-lattice Hubbard model is investigated using the determinant quantum Monte Carlo method. The superconducting instability may occur when doping the correlated insulators formed by the Hubbard interaction near the Dirac filling, and the possible superconducting state exhibits an electron-hole asymmetry. Among the pairing symmetries allowed, we demonstrate that the dominating channel is d-wave in the hole-doped case. This opens the possibility of condensation into an unconventional d x 2 −y 2 + idxy phase, which is characterized by an integer topological invariant and gapless edge states. In contrast, the s * -wave channel, which has no sign change in the pairing function, is favored by the electron doping. We further find the dominating s * -wave pairing persists up to the Van Hove singularity. The results are closely related to the recent experimental observations in kagome compounds AV3Sb5(A: K, Rb,Cs), and provide insight into the pairing mechanism of their superconducting states. arXiv:2109.12582v2 [cond-mat.str-el]
I. INTRODUCTION
The kagome lattice, composed of corner-sharing triangles whose lattice points each have four nearest neighbors 1 , combines the intriguing physics of geometry frustration, flat band, Van Hove singularity (VHS) and Dirac fermion, setting an ideal platform for novel quantum phases [2][3][4][5][6][7][8][9][10][11][12][13] . Recently, a new family of kagome materials AV 3 Sb 5 (A: K, Rb, Cs) was discovered 14 . They are composed of a layered structure, with an ideal kagome network of vanadium layers separated by alkali metal ions. The common properties of these compounds include: Z 2 topological metal state, charge density wave (CDW) order occurring below T CDW c ≈ 80 − 110K, and unconventional superconductivity with critical temperature T c ≈ 0.9 − 2.7K [15][16][17][18] .
Despite intense investigation on these compounds, debate over the nature of CDW and superconductivity persists [19][20][21][22][23] . In particular, the controversy regarding the superconducting (SC) pairing and its mechanism abounds. While a significant residual linear term in the thermal conductivity, and the V -shaped spectral gap opening in the differential conductance indicate the unconventional nodal superconductivity 24,25 , the magnetic penetration depth and nuclear magnetic resonance measurements suggest a nodeless s-wave superconductor [26][27][28] . Furthermore, a recent scanning tunneling microscopy study in CsV 3 Sb 5 finds evidence of the existence of gap nodes but the absence of sign change in the SC order parameter 29 . The disagreement among the dif- * These authors contributed equally to this work ferent experimental setups is certainly influenced by the complexity induced by the multiband nature of the superconductivity in these compounds [30][31][32][33][34] . Since existing theoretical and experimental studies indicate that strong electron correlations play an essential role in the appearance of superconductivity 35,36 , it is highly desirable to gain insight on the dominating pairing symmetry. For that, a good starting point is the Hubbard Hamiltonian on a kagome lattice.
In this paper, we employ the determinant quantum Monte Carlo method (DQMC) [37][38][39][40] , combined with cues coming from mean-field (MF) theory, to study the SC pairing symmetry in the kagome-lattice Hubbard model. The quasiparticle spectra of the pairing symmetries allowed on the kagome lattice are first analyzed to demonstrate properties of the different types of superconducting states. Then DQMC calculations reveal that doping around the Dirac point, the dominating pairing channel is d (s * )-wave in the hole (electron)-doped case, and the s *wave channel remains dominating even when the system is doped to the upper VHS. These results are consistent with some of recent experimental observations, and provide a theoretical understanding of the superconducting states in the newly discovered kagome materials.
II. THE MODEL AND METHOD
We start from the kagome-lattice Hubbard model, where c † iσ and c iσ are the creation and annihilation operators, respectively, at site i with spin σ =↑, ↓; ij denotes nearest neighbors; n iσ = c † iσ c iσ is the number of electrons of spin σ on site i, and U is the on-site repulsion. Throughout the paper, the hopping amplitude is set to t = 1 as the unit of energy.
H = −t ij σ c † iσ c jσ + U i n i↑ − 1 2 n i↓ − 1 2 ,(1)
The kagome lattice has a three-site unit cell [ Fig. 1(a)]. In momentum space, the U = 0 Hamiltonian is given by 41
H 0 (k) = −2t 0 cos k 1 cos k 3 cos k 1 0 cos k 2 cos k 3 cos k 2 0 ,(2)
where k n = k · a n (the sublattice index n = 1, 2, 3) with a 1 = (1, 0), a 2 = (−1, √ 3)/2, and a 3 = −(a 1 + a 2 ). The spectrum of H 0 (k) has one flat band E 3 (k) = 2t and two dispersive ones,
E 1,2 (k) = t[−1 ± 4f (k) − 3],(3)
with f (k) = cos 2 k 1 + cos 2 k 2 + cos 2 k 3 . Bands 1 and 2 touch at two inequivalent Dirac points K ± = (±2π/3, 0) at energy −t, see Figs. 1(b) and 1(c). For 1 3 filling, the lowest band is filled, and the low-energy excitations resemble those of graphene, which are linear, 1,2 = ± √ 3t| q|, with q = (q x , q y ) a small displacement away from the Dirac points.
For the dispersive bands, three momenta M at the centers of the edges of the Brillouin zone (BZ) are saddle points, resulting in VHSs at fillings ρ = 1/4 and 5/12, respectively. The corresponding energies are E M /t = −2 and 0, and the Dirac points are exactly located at the middle point between them.
Conversely, at finite interactions, Eq.(1) is solved numerically via DQMC, where one decouples the on-site interaction term through the introduction of an auxiliary Hubbard-Stratonovich field, which is integrated out stochastically. The only errors are those associated with the statistical sampling, the finite spatial lattice size, and the inverse temperature discretization. These errors are well controlled in the sense that they can be systematically reduced as needed, and further eliminated by appropriate extrapolations. Since the kagome lattice is nonbipartite, the infamous sign problem exists at all densities, and can become severe upon lowering the temperature and increasing the interaction strength [42][43][44] . Nevertheless, the sign problem is found to be significantly reduced at some specific fillings, where the simulations can be carried out at relatively low temperatures. Generally, we access the temperatures with the average sign higher than 0.5 to obtain reliable results. In the following, we use the inverse temperature discretization ∆τ = 0.1, and lowest temperature accessed is T /t = 1/25. The lattice has N = 3 × L × L sites with L up to 9.
III. PAIRING SYMMETRIES AND PROPERTIES OF THE SC STATE
The on-site repulsive interactions tend to drive nonlocal pairings, and hence nearest-neighbor SC pairings are considered. Their symmetries should be compatible with the underlying geometry of the lattice. As the kagome lattice is described by C 6v point group symmetry, the possible pairing states can be classified by the irreducible representations of C 6v . These include the singlet pairing symmetries: s * -wave, d x 2 −y 2 -wave, and d xy -wave; and triplet pairing symmetries: p x -wave, p ywave, and f -wave, all of which are schematically represented in Fig. 2. While s * -and f -wave correspond to one-dimensional representations, d x 2 −y 2 and d xy belong to two-dimensional representation E 2 , and thus are degenerate, which also holds for any linear combination of them. The value of the pairing susceptibility is the same for all the degenerate combinations, thus can not distinguish them from one another. A qualitative argument for the dominance of a chiral combination, i.e., d + id pairing, is that it opens a gap everywhere in the spectrum. Compared to the individual d x 2 −y 2 or d xy pairing which has nodes in the spectrum, the gap opening enable an overall energy lowering, hence makes the d + id pairing energetically favored. Similarly, p x and p y belong to two-dimensional representation E 1 , and the above argument is also applicable to the formation of chiral p + ip superconductivity if the p-wave channel dominates.
We then explore the properties of the SC state with the above allowed pairing symmetries based on the BCS Hamiltonian, which writes as
H SC = k Ψ † k H k Ψ k ,(4)with Ψ k = c 1,k↑ , c 2,k↑ , c 3,k↑ , c † 1,−k↓ , c † 2,−k↓ , c † 3,−k↓ T and H k = H 0 (k) − µ ∆ † k ∆ k −H 0 (k) + µ , ∆ k = 0 η 1 (k) η 3 (k) η 1 (k) 0 η 2 (k) η 3 (k) η 2 (k) 0 .(5)
Here µ is the chemical potential. η n (k) = −∆ n (e −ikn − ζe ikn ) with the pairing amplitude ∆ n , which can be read from the real space arrangement in Fig. 2; ζ = −1(+1) for singlet (triplet) pairing.
To demonstrate the symmetry of each pairing channel, the SC Hamiltonian H k should be transformed into the band basis, under which
H 0 (k) becomes diagonal, i.e., V † k H 0 (k)V k = diag[E 1 (k), E 2 (k), E 3 (k)], where V k is the eigenvector matrix of H 0 (k). In turn, the pair- ing matrix becomes∆ k = V † k ∆ k V k ,
and is composed of inter-and intra-band pairings. As shown in Fig. 2, the symmetries of the different pairing channels are clearly reflected in the intra-band pairings.
In the presence of the complex inter-band pairings, the quasiparticle spectrum does not follow the standard BCS form, and it is not straightforward to identify whether there are zero-energy quasiparticles. Instead, the Hamiltonian in Eq. (4) should be investigated via numerical diagonalization. We first focus on the case of the electron doping on the Dirac cones. It is found that the s * and f -wave states are fully gapped. Although the d x 2 −y 2 -and d xy -wave pairings are gapless, a chiral lin-ear combination of them, i.e., d x 2 −y 2 + id xy , is gapped. The chiral d-wave state is a topological superconductor characterized by an integer Chern number C = −2, corresponding to which two pairs of gapless states traversing the gap appear in the presence of sawtooth edges [see Fig. 3(b)]. The p-wave pairings are similar, and the chiral p x + ip y -wave state is also a topological superconductor with C = −2.
IV. DQMC STUDY OF THE DOMINATING PAIRING SYMMETRY
The low energy physics at density ρ = 2/3 should be compared to the one of the honeycomb Hubbard model at half-filling (ρ = 1), which undergoes a Dirac semimetal to insulator transition at sufficiently large interactions. As we are interested in the SC instabilities, the best candidates are when doping away from this regime. We investigate densities symmetrically around the Dirac point, ρ = 2/3 ± δρ (with δρ 0.046). To determine the dominating pairing symmetry, we evaluate the uniform pairing susceptibility [45][46][47] ,
χ α = 1 N β 0 dτ ij ∆ α i (τ )∆ α † j (0) ,(6)
where ∆ α i (τ ) = j f α ij e τ H c i↑ c j↓ e −τ H is the timedependent pairing operator with form-factors f α ij = 0, ±1 or ± 2 for the bond connecting sites i and j, depending on the pairing symmetry α (see Fig. 2). The effective susceptibility, χ α eff ≡ χ α − χ α 0 , subtracts the uncorrelated part χ α 0 from χ α , thereby directly capturing the interaction effects, and can be further used to evaluate the pairing vertex. Figure 4 shows χ eff versus temperature for the pairing channels, d-wave at ρ 1 = 0.62 and s * -wave at ρ 2 = 0.713, for several values of U . χ d eff in (a) and χ s * eff in (b) are positive, and tend to increase rapidly for large U at low temperatures. In contrast, the values for triplet pwave and f -wave pairings are increasingly negative with decreasing the temperature and increasing the interaction, suggesting these symmetry channels are suppressed (see Appendix D). These results demonstrate that the possible SC states on the upper and lower sides of the Dirac points are asymmetric: While the dominating pairing symmetry is d-wave in the hole-doped case, the s *wave channel, which has no sign change in the pairing function, is favored by the electron doping on the Dirac points. As the fillings approach the VHSs, the sign problem gets continuously worse, and the simulations are limited to relatively high temperatures. Although the values of χ d,s * eff still dominate at both VHSs, Fig. 4(a) shows that χ d eff drops quickly, and becomes negative at low temperatures, implying the possible SC instability is destructed in the heavily hole-doped case. In comparison, the s * -wave pairing is successively enhanced by the electron doping, and remains dominating even at the upper VHS [ Fig. 4(b)]. Here the temperatures accessed by DQMC is still much higher than the SC transition temperature. Since the long-range SC order is still under development, the pairing susceptibility does not show significant finitesize effect.
To reveal the microscopic origin of the SC pairing interaction, the charge and spin correlations are calculated at both densities ρ = 0.62 and ρ = 0.713. No clear difference in the density-density correlations between the two fillings is observed under the same conditions (see Appendix D). The spin-spin correlations, on the other hand, are quite distinct. Specifically, at both fillings, the nearest-neighbor and next-nearest-neighbor spin correlations are antiferromagnetic and ferromagnetic, respectively. However the next-next-nearest-neighbor ones are antiferromagnetic (ferromagnetic) for ρ = 0.62 (ρ = 0.713). This is consistent with the asymmetry of the SC states below and above the Dirac points, which suggests superconductivity may be stimulated (and coupled) by magnetic fluctuations.
V. CONCLUSIONS
We have studied the SC pairing symmetry of the kagome-lattice Hubbard model using the DQMC method. From the high temperature trends of the pairing susceptibility, we find the possible superconducting states are asymmetric on the hole-and electron-doped sides of the Dirac points. While the dominating channel is d-wave for the hole doping, the s * -wave pairing is favored in the electron doped regime. Besides, the s * -wave symmetry channel remains dominating even when the system is doped to the upper VHS. We find the spin-spin correlation exhibit a similar electron-hole asymmetry, suggestive that the SC pairing interaction may be connected to magnetic fluctuations.
Experimentally, angle-resolved photoemission spectrum and other techniques have revealed that the Fermi surface contains pockets formed by different kinds of bands, and the superconductivity is of multi-band nature. Although a description of the electronic structure using a simple one-band model is often insufficient, the predicted s * -wave pairing symmetry near the VHS is consistent with the experimental results [26][27][28][29] . Thus our results imply that the kagome-lattice Hubbard Hamiltonian includes some of the key ingredients that may ex-plain the superconductivity in the family of kagome materials AV 3 Sb 5 (A: K, Rb,Cs), and further suggest the role of spin excitations in influencing the pairing interactions.
Although the on-site Hubbard repulsion can produce results consistent with some experimental observations, long-range interactions may not be omitted in real kagome superconductors. A recent theoretical study finds that the non-local Coulomb repulsion is a critical parameter to determine the pairing symmetry 22 . Besides, an exotic SC state, known as pair density wave, is revealed in recent STM experiments 25 . Hence simulating long-range interactions with the aim to explain the intriguing phenomena in kagome materials will be a significant direction of future DQMC research. The pairing function in the band basis is a 3 × 3 matrix, in which the diagonal (off-diagonal) elements correspond to intra-band (inter-band) pairings. Figure A1 plots the momentum dependence of the d x 2 −y 2 -wave pairing function. Since the pairing matrix is symmetric, only nonequivalent elements are presented. As shown in Fig. A1, the symmetry of the pairing function is clearly reflected in the intra-band pairings. The pairing functions for the d xy -wave and s-wave are shown in Fig. A2 and Fig. A3, respectively. The triplet pairing functions can be similarly obtained, and are not plotted here. The quasiparticle spectrum evolves with the pairing strength ∆. Figure A4 plots the quasiparticle energy gap as a function of ∆ for d + id and p + ip superconductors. For both pairing channels, the gap closes two times in the calculated range 0 < ∆ < 0.5, accompanied by a change of the Chern number. Between two gapless points, the gap value exhibits a dome-like shape. In practice, ∆ should be a small fraction of the energy scale t. Thus the gap should be within the first dome, and both the d + id and p + ip states are nodeless superconductors. Figure A5(a) plots the quasiparticle energy gap and the Chern number as a function of the chemical potential for the p + ip state. The gap of the triplet p + ip state closes at both the Dirac points and VHSs, and the Chern number always changes its value and sign across the gapless point. In stark contrast to the s * -wave and d + id cases presented in the main text [ Fig. 3], the energy gap here is symmetric to the Dirac points, and the Chern number is inversion symmetric. To demonstrate the topological properties of the chiral p x + ip y state, the quasiparticle spectrum on open lattice at µ = −0.95 is plotted in Fig. A5(b). Since it is a topological superconductor with C = −2, two pairs of gapless states traverse the gap. The kagome lattice is nonbipartite, and has no particle-hole symmetry, thus the DQMC calculations are limited by the sign problem at all densities. Generally, the mean value of the sign is s ∝ e −βLδf 42,43 , which decreases exponentially with increasing the lattice size L and the inverse temperature β. We first fix the lattice size, and investigate the evolution of the sign problem with the interaction strength and the inverse temperature. Figure A6(a) presents the landscape of the average sign for the interaction magnitudes ranging from U/t = 0 to 9, at all densities in our single-band model. As expected, the sign problem only becomes severe at large interactions and low temperatures. Besides, the average sign is s = 1 for both empty and fully-filled bands. At a fixed interaction, s goes down slowly away from ρ = 0, but it falls steeply near ρ = 2 when the Fermi level is in the flat band. As shown in Fig.A6(b), the average sign becomes almost zero in a finite range near each VHS filling at low enough temperatures. Interestingly, s takes a maximum value between the two VHS points, and the optimal filling approximately corresponds to the location of the Dirac points. Thus DQMC can be performed to larger interactions and lower temperatures near the Dirac points, providing a unique opportunity for DQMC to explore the physics of the correlated Dirac fermions. Figure A7 shows the effective susceptibility for pand f -wave triplet pairings; χ p,f eff is increasingly negative as the temperature is lowered. Besides, the values of The finite-temperature DQMC method works in the grant canonical ensemble, with the average density being controlled by the chemical potential. Usually µ ρ (the average density is ρ at µ) is a function of the temperature and the interaction, i.e., µ ρ (T, U ). Thus the chemical potential corresponding to a fixed density has to be found for each set of parameters (T, U ). In practice, we perform DQMC calculations with equally-spaced chemical potentials near the desired density. Then from a ρ − µ curve fitting, the wanted µ is determined from interpolation. Figures A8 (a) and A8 (b) plot the average density corresponding to the chemical potentials found using the above procedure for various U at different T . The chemical potentials for small interactions and relatively high temperatures can be determined very accurately. However ρ at large U and low T has a relatively large error bar, owing to the severe sign problem in this regime. As shown in Figures A8 (c) and A8 (d), the average sign begins to drop quickly from a critical temperature. For larger U , a clear drop of the average sign happens at higher temperature. These results are consistent with the general rule of the sign problem, i.e., becoming worse at large U and low T 42,44 . In our calculations, we have probed the lowest possible temperature for each U , average sign permitting. drops quickly and becomes negative at low temperature, χ s * eff at ρ = 5 6 has a substantial enhancement. It implies that the d-wave pairing is destructed at the lower VHS, and the s * -wave channel persists to the upper VHS. The corresponding average signs are shown in Fig. A9 (c) and A9 (d), which begin to drop quickly from a higher temperature than that in a more lightly doped case. For the on-site Hubbard interaction U/t = 6 used in the figure, the temperature accessed is relatively high, which is T > t/5 (t/4) at ρ = 1 2 ( 5 6 ). We also plot the average densities corresponding to the manually determined chemical potentials targeting the VHS densities at various temperatures. As shown in Figs.A9 (e) and A9(f), the chemical potentials are well controlled to obtain the desired average densities. The charge correlations along the paths shown in (a). Here the Hubbard interaction is U/t = 4, and the inverse temperature is βt = 6. Figure A10 plots the density-density correlations at ρ = 0.62 and 0.713, which is defined as
1.0 <sign> (b) β =5 β =6 β =7 β =8 β =9 β =10C ij = n i n j − n i n j ,(D1)
with n i = n i↑ + n i↓ the total density on site i. In Fig.A10 (b), only the charge correlations for nonequivalent pairs of sites are shown. At both fillings, the values of C ij are pretty small, and decrease quickly to zero as i and j depart away from each other. Besides, the values of C ij at the two different fillings differ little from each other. These results suggest the charge fluctuations do not cause the asymmetry of the superconductivity, thus may not account for the superconducting pairing.
FIG. 1 :
1(a) The kagome lattice which is a triangular Bravais lattice with a three-site unit cell. The sublattice is labeled as A,B,C. (b) The first Brillouin zone with the high symmetry points marked. (c) The band structure along the high symmetry directions in the Brillouin zone (left) and the density of states (right).
FIG. 2 :
2The pairing symmetries allowed by the point group of the kagome lattice: (a) s * -and f -wave, (b) d x 2 −y 2 -and px-wave, and (d) dxy-and py-wave. For triplet pairing, there is an additional sign when the pairing is along the opposite direction of the arrow. The momentum dependence of the intra-band pairing function for (d) s * , (e) d x 2 −y 2 , (f) dxy, (g) f , (h) px, and (i) py pairing channels.
FIG. 3 :
3(a), The quasiparticle energy gap as a function of the chemical potential for the s * -wave and d + id superconductors. (b), The quasiparticle spectrum of the d + id chiral superconducting state on kagome nanoribbons with a pair of sawtooth edges. The spectrum is symmetric to kx = π, and we only show the low-energy part near the left valley. The gap parameter is ∆ = 0.01. In (b), µ = −0.95 corresponds to an electron doping on the Dirac cones.
Figure 3 (
3a) plots the quasiparticle energy gap as a function of the chemical potential for the s * -wave and d + id superconductors. The gap of the d + id state closes at the Dirac points (µ = −1), where the Chern number changes its sign. For both channels, the gap is asymmetric to the Dirac points, and continuously decreases as µ increases. Above the Dirac points, both pairing channels have almost identical gap values, and become nodal superconductors at the upper VHS. In contrast, the gap of the triplet p+ip state closes at both the Dirac points and the two VHSs, and the Chern number always changes the value and sign across the gapless points (see Appendix B).
FIG. 4 :
4The effective susceptibility of the dominating pairing channel as a function of temperature for several values of U : (a) d-wave at ρ 1 = 0.62; (b) s * -wave at ρ 2 = 0.713. Here ρ 1 , ρ 2 are symmetric to the Dirac points, and correspond to hole and electron dopings, respectively. χ d eff (χ s * eff ) at the lower (upper) VHS is also plotted in the left (right) panel to demonstrate the evolution with doping.
FIG. 5 :
5The spin-spin correlation at: (a) ρ = 0.62; (b) ρ = 0.713. The star symbol marks the reference site, and here a site in Sublattice A is chosen. The magnitude of the correlation is represented by the radii of the solid circle. The blue (red) color corresponds to the negative (positive) sign. The interaction strength is U/t = 4, and the inverse temperature is βt = 6.
Acknowledgments
The authors thank Fan Yang and Wen Yang for helpful discussions. J.S and H.G. acknowledge support from the National Natural Science Foundation of China (NSFC) grant Nos. 11774019 and 12074022, the NSAF grant in NSFC with grant No. U1930402, the Fundamental Research Funds for the Central Universities and the HPC resources at Beihang University. Z. X. acknowledges support from NSFC grant No. 61975005. R.M. acknowledges support from NSFC grants No. U1930402, 12050410263, 12111530010 and No. 11974039. N.H. acknowledges support from NSFC Grants No. 12022413, No. 11674331, the "Strategic Priority Research Program (B)" of the Chinese Academy of Sciences, Grant No. XDB33030100, the '100 Talents Project'of the Chinese Academy of Sciences, the Collaborative Innovation Program of Hefei Science Center, CAS (Grants No. 2020HSC-CIP002), the CASHIPS Director's Fund (BJPY2019B03). S.F. is supported by the National Key Research and Development Program of China, and NSFC under Grant Nos. 11974051 and 11734002. Appendix A: The momentum dependence of the pairing functions
FIG. A1 :
A1The momentum dependence of the d x 2 −y 2 -wave intraband pairing function for the component (a) , j) represent the band index. FIG. A2: The momentum dependence of the dxy-wave intraband pairing function for the component (a) (1, 1), (b) (2, 2), (c) (3, 3). The interband pairing function for the component (d) (1, 2), (e) (1, 3), (f) (2, 3). Appendix B: More results on the quasiparticle spectrum
FIG. A3 :
A3Momentum dependence of the s-wave intraband pairing function for the component (a) (2, 2), (b) (3, 3). All the other elements vanish for the s-wave channel.
A4: (a), The quasiparticle energy gap and the Chern number as a function of the pairing strength ∆ for: (a) d + id and (b) p + ip superconductors. Here µ = −0.95 corresponds to an electron doping on the Dirac cones.
FIG. A5: (a), The quasiparticle energy gap and the Chern number as a function of the chemical potential for the p + ip superconductor. (b), The quasiparticle spectrum of the p + ip chiral superconducting state on kagome nanoribbons with a pair of sawtooth edges. This plot is similar to Fig. 3 of the main text and the same parameters are used.
FIG. A6 :
A6(a), The contour plot of the average sign in the (ρ, U ) plane at β = 6 on a lattice with N = 54 sites. (b), The average sign as a function of ρ for U/t = 6 at various inverse temperatures on a L = 9 kagome lattice. The dotted (dashed-dotted) vertical line marks the position of the Dirac point (the VHSs).Appendix D: More DQMC results In the main text we have ostensibly focused on the s * (d)-wave pairing symmetry in electron(hole) doping the Dirac point fillings. Here we provide further justification, by investigating other symmetry channels, starting with Fig. A7, in complement to Fig. 4 in the main text.
FIG. A7 :
A7The effective pairing susceptibility of other pairing channels: (a) s * -wave, (c) p-wave, (e) f -wave at ρ 1 = 0.62;(b) d-wave, (d) p-wave, (f) f -wave at ρ 2 = 0.713.χ p,f eff decreases as U is increased, suggesting these symmetries are suppressed. The singlet s * (d)-wave pairing in the hole(electron)-doped case exhibits similar trend with the temperature. Although χ s * (d) eff increases with U , the values are much smaller than those of the dominating pairing at low temperatures, implying these latter pairing channels are not favored by the interaction.
FIG. A8 :
A8The average density at the manually determined chemical potential targetting a fixed density: (a) ρ = 0.62 and (b) ρ = 0.713. (c) and (d) are the corresponding average signs of (a) and (b), respectively.
FIG
. A9: The effective pairing susceptibility of all allowed pairing channels: (a) ρ = 0.5; (b) ρ = 0.833. (c) and (d) are the corresponding average signs of (a) and (b), respectively. The average sign at ρ = 0.62 (ρ = 0.713) is also plotted in (c) [(d)] for comparison. The average density at the manually determined chemical potential targeting a fixed density: (e) ρ = 0.5 and (f) ρ = 0.833. Here the Hubbard interaction is U/t = 6.
Figure A9 (
A9a) and A9 (b) plot the effective susceptibilities for all allowed pairing channels at the VHSs ρ = 1 2 and ρ = 5 6 , respectively. The values of the d(s * )-wave channel at ρ = 1 2 ( 5 6 ) still dominate. While χ d eff at ρ = 1 2
FIG
. A10: (a) The L = 6 kagome lattice on which the DQMC simulations are performed. The star symbol represents the reference site. The colored lines with arrows mark the sites having nonequivalent distances with the reference site. (b)
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[
"ON THE SEMI-SIMPLICITY CONJECTURE FOR Q ab",
"ON THE SEMI-SIMPLICITY CONJECTURE FOR Q ab"
]
| [
"Marco D ' Addezio "
]
| []
| []
| We show that the semi-simplicity conjecture for finitely generated fields follows from the conjunction of the semi-simplicity conjecture for finite fields and for the maximal abelian extension of the field of rational numbers. | null | [
"https://arxiv.org/pdf/1805.11071v1.pdf"
]
| 119,149,883 | 1805.11071 | 62f2df2052eb528789806b52130a80757f5622ce |
ON THE SEMI-SIMPLICITY CONJECTURE FOR Q ab
Marco D ' Addezio
ON THE SEMI-SIMPLICITY CONJECTURE FOR Q ab
We show that the semi-simplicity conjecture for finitely generated fields follows from the conjunction of the semi-simplicity conjecture for finite fields and for the maximal abelian extension of the field of rational numbers.
. For every field k we choose an algebraic closure k and we denote by Γ k the group Gal(k/k). We denote by Q ab the maximal abelian extension of Q in Q.
1.2. Let k be a field. A variety over k will be for us a separated and reduced scheme of finite type over k. If X is a scheme over k, we denote by X k the scheme X ⊗ k k, over k. Let be a prime different from the characteristic of k. We denote by Rep Q (Γ k ) the neutral Tannakian category of finite-dimensional -adic Q -linear representations of Γ k . We will refer to objects in Rep Q (Γ k ) simply as -adic representations of Γ k . Let C(k, ) be the smallest neutral Tannakian subcategory of Rep Q (Γ k ), closed under subquotients, which contains all the -adic representations of Γ k of the form H í et (X k , Q ), where i is an integer, X is a smooth and projective variety over k and X k := X ⊗ k k. We will say that an object in C(k, ) is an -adic representation of Γ k coming from geometry.
1.3. For a field K of characteristic 0 and V a finite-dimensional K-vector space, we will say that a linear endomorphism ϕ of K is semi-simple if it is diagonalizable after a finite extension of K. Let V ρ be an -adic representation of Γ Q and p = a prime where V ρ is unramified, we will say that ρ is semi-simple at p if one (or equivalently any) Frobenius element at p acts via a semi-simple automorphism.
Introduction
Let k be a field, we consider the following statement: S(k): For every prime different from the characteristic of k, an -adic representation of Γ k coming from geometry is semi-simple. Grothendieck and Serre have conjectured that for a finitely generated field k, then S(k) holds (see [Tat65]). This conjecture is commonly known as the semi-simplicity conjecture. Notice that the conjecture predicts that S(k) is true even for fields k which are infinite Galois extensions of a finitely generated field. Indeed, if k /k is a Galois extension, then S(k) implies S(k ), because the restriction of a semisimple representation to a normal subgroup is semi-simple. At the same time, we recall that S(k) is false for arbitrary fields, for example it is false for Q p and C((t)).
In this article we prove that, if we assume S(F p ) for every prime p, we can deduce S(Q) from S(Q ab ). This yields the following result.
Theorem 2.1 (Theorem 4.7). Let k be a Galois extension of a finitely generated field. The conjunction of S(F p ) for every prime p and S(Q ab ) implies S(k).
Let us make a brief summary on what is already known about the conjecture. The first result has been obtained by Weil, who has proven in 1948, thanks to the positivity of the Rosati involution, the conjecture for abelian varieties (and hence for curves) over finite fields. In 1983, Faltings has proven the semi-simplicity conjecture for abelian varieties over number fields as an intermediate step for his proof of the Mordell conjecture. By the work of Deligne, both these results can be extended to K3 surfaces, thanks to the Kuga-Satake construction.
In 1980, Deligne has also obtained in positive characteristic a weak form of the general conjecture, as a consequence of his theory on weights.
Theorem 2.2 ([Del80, Théorème 3.4.1.(iv)]). Let X be a normal scheme of finite type over F p . The inverse image of a pure lisse Q -sheaf on X to X Fp is semi-simple.
This result suggests that in positive characteristic the semi-simplicity conjecture reduces to the conjecture over F p . This has been indeed proven by Lei Fu via a monodromy argument.
Theorem 2.3 ([Fu90])
. Let X be a normal connected scheme of finite type over F p and let F be a ι-pure lisse Q -sheaf on X. If there exists a closed point x of X such that the Frobenius automorphism at x is semi-simple, then F is a semi-simple lisse sheaf on X.
Corollary 2.4. Let k be a Galois extension of a finitely generated field of positive characteristic p, then S(F p ) implies S(k).
In characteristic 0, thanks to Serre's specialization argument, via Hilbert's irreducibility theorem, the semi-simplicity conjecture reduces to S(Q).
Theorem 2.5 ([Ser00]). Let k be a Galois extension of a finitely generated field of characteristic 0, then S(Q) implies S(k).
Some analogies
We intend to investigate in this article the following mixed-characteristic analogue of Theorem 2.3.
Conjecture A. If an -adic representation of Γ Q coming from geometry is semi-simple at some unramified prime p different from , then it is semi-simple as a representation of Γ Q .
We will show in the next section how to adapt Fu's proof of Theorem 2.3 to prove that S(Q ab ) implies Conjecture A.
Remark 3.1. We think S(Q ab ) as the analogue of Theorem 2.2. We do not know whether is it possible to prove S(Q ab ) via a suitable theory of weights for Γ Q ab .
More concretely, let V ρ be an -adic representation coming from geometry of Γ Q of weight 0. Let N be a positive multiple of every prime where V ρ is ramified. We consider the vector space
H ρ := H 1 et (Spec(Z[ζ ∞ , N −1 ]), V ρ )
. By transport of structure, H ρ is endowed with an action of the monoid
End(Z[ζ ∞ , N −1 ]) = ( Z, ×),
where ( Z, ×) is the multiplicative monoid of the ring Z. Let ϕ p be the endomorphism on H ρ induced by p ∈ ( Z, ×). If we know that for every eigenvalue α of ϕ p , there exists ι : Q → C such that |ι(α)| < 1, then we also know that ϕ p acts without fixed points. Thus we prove that there are no non-trivial extensions of Q by
Our main result
We adapt Fu's proof in [Fu90] in our situation. To do this we first introduce the notion of the Weil group of Q associated to a Frobenius element at a prime p.
We take the natural exact sequence
1 → Γ Q ab → Γ Q δ − → Z × → 1,
where Z × = Gal(Q ab /Q). We choose a closed embedding Γ Qp ⊆ Γ Q induced by a field inclusion Q → Q p and a Frobenius lift F p ∈ Γ Qp ⊆ Γ Q .
Definition 4.1. We define the Weil group of Q with respect to F p as the semi-direct product W Q,Fp := Γ Q ab Z, where Z is endowed with the discrete topology and the action of 1 ∈ Z on Γ Q ab is the adjoint action of F p ∈ Γ Q on Γ Q ab .
We have a canonical injective continuous map W Q,Fp → Γ Q which sends 1 ∈ Z to F p . Contrary to the Weil group of a connected scheme over a finite field, in this case for every choice of a Frobenius element F p , the subgroup W Q,Fp is not dense in Γ Q . Its closure is not even of finite index in Γ Q . We will see in Proposition 4.5 how to overcome this problem.
We start by recalling a well-known fact on the quotients of Z × . It is worth mentioning that this lemma is also one of the main ingredients of Moonen's recent result on the semi-simplicity conjecture [Moo17].
Lemma 4.2. The profinite group Z × admits a unique continuous quotient isomorphic to Z .
Proof. The subgroup Z × ⊆ Z × is dense, hence it is enough to prove the result on Z × . We know that Hom( Z × , Z ) = Hom(Z × , Z ) and by an explicit computation
Hom(Z × , Z ) = 1 for = Z for = .
This shows that Hom( Z × , Z ) = Z , which implies the result.
We fix for every prime a quotient δ : Γ Q Z . Proof. Let K be the closure of the group generated by δ (F p ) in Z . It is endowed with a natural Z -module structure, hence it is an ideal of Z . We need to show that K is non-trivial. Let Q ur p be the maximal unramified extension of Q p in Q p . The map Γ Qp → Gal(Q(ζ ∞ )/Q) induced by the inclusion Γ Qp ⊆ Γ Q we have chosen, factors through Gal(Q ur p /Q p ), because the extension Q(ζ ∞ )/Q is unramified at p. By definition, F p is sent to the unique Frobenius element in Gal(Q ur p /Q p ), thus its image in Gal(Q(ζ ∞ )/Q) is the unique automorphism of Q(ζ ∞ ) which raises each root of unit to its p-th power. This automorphism has infinite order in Gal(Q(ζ ∞ )/Q), thus its image δ (F p ) Z -quotient of Gal(Q(ζ ∞ )/Q) is non-trivial. . In particular, is non-trivial, as we wanted.
Let be a prime and ρ an -adic representation of Γ Q . We denote by Π the image of Γ Q , by Π 0 the image of Γ Q ab , by Π the quotient Π/Π 0 and by π the natural projection π : Π Π. We obtain the following commutative diagram with exact rows of profinite groups
1 Γ Q ab Γ Q Z × 1 1 Π 0 Π Π 1. ρ δ ρ π
Lemma 4.4. The group Π is either a finite abelian group or it admits a surjective morphism π : Π Z with finite Kernel such that δ = π • π • ρ.
Proof. The group Π is a closed subgroup of the topological group GL(V ρ ), thus by [DSMS91, Theorem 9.6], it can be endowed with the structure of an -adic analytic group. By [ibid. , Theorem 8.32], it contains an open topologically finitely generated pro--subgroup. As π is surjective, the same is true for Π.
The group Π, being a quotient of Z × , is an abelian group. Let Π ⊆ Π be the maximal pro--subgroup of Π. By the previous argument, Π is open in Π and it is topologically finitely generated. Hence Π is a finitely generated Z -module with respect to its natural Z -module structure.
At the same time, the group Π is a quotient of Z × , thus by Lemma 4.2, it is a Z -module of rank at most one. This means precisely that either Π is finite or it is isomorphic to Λ × Z , with Λ a finite group.
Notice that again, by the uniqueness of the Z -quotients of Z × , we can choose a projection π : Π Z with finite Kernel such that δ = π • π • ρ.
We can prove now the main technical result needed for the final theorem.
Proposition 4.5. An -adic representation ρ of Γ Q is semi-simple if for some prime p = and for some choice of F p , its restriction to W Q,Fp is semi-simple.
Proof. Let Π p be the closure of the image of W Q,Fp in Π. In light of [Fu90, Lemma 1], it is enough to show that Π p has finite index in Π. We notice that as Π 0 ⊆ Π p , if we set Π p := π(Π p ), then [Π : Π p ] = [Π : Π p ], thus we are reduced to show that Π p has finite index in Π.
By Lemma 4.4, we have two cases. If Π is finite the result holds trivially. If Π is infinite, we take π : Π Z as in the lemma. As π has finite Kernel, it is enough to show that [Z : π (Π p )] is finite. The profinite group Π p is topologically generated by ρ(F p ), hence the group π (Π p ) is topologically generated by δ (F p ). We conclude in virtue of Lemma 4.3.
Theorem 4.6. Let ρ be an -adic representation of Γ Q which is semi-simple when restricted to Γ Q ab . If there exists a prime p = and F p ∈ Γ Q , a Frobenius element at p, such that ρ(F p ) is semi-simple, then ρ is a semi-simple representation of Γ Q . In particular, S(Q ab ) implies Conjecture A.
Proof. By Proposition 4.5, it is enough to check that ρ is semi-simple when restricted to W Q,Fp → Γ Q .
Let G 0 be the Zariski closure of Π 0 in GL(V ρ ) and let G Fp be the semi-direct product G 0 Z, where Z is endowed with the discrete topology and 1 ∈ Z acts on G 0 as ρ(F p ) acts on G 0 by conjugation.
We define ρ : W Q,Fp → G Fp as the only morphism making the following diagram commuting
1 Γ Q ab W Q,Fp Z 1 1 G 0 G Fp Z 1. ρ ρ
At the same time, let σ : G Fp → GL(V ρ ) be the representation which extends the tautological representation G 0 → GL(V ρ ) by sending 1 ∈ Z to ρ(F p ). The composition σ • ρ : W Q,Fp → GL(V ρ ) is equal to ρ. As ρ has Zariski-dense image, to show that the restriction of ρ to W Q,Fp is semi-simple it is enough to show that σ is semi-simple. The group G 0 is reductive because, by assumption, ρ is semi-simple when restricted to Γ Q ab . In virtue of [Del80, Lemme 1.3.10], there exists a Q -point g in the center of G Fp of the form (g , d) where g ∈ G 0 (Q ) and d = 0. As ρ(F p ) is semi-simple, we can apply [Fu90, Lemma 2] to the representation σ, obtaining thereby the desired result.
Theorem 4.7. Let k be a Galois extension of a finitely generated field. The conjunction of S(F p ) for every prime p and S(Q ab ) implies S(k).
Proof. In positive characteristic this is Fu's result (Corollary 2.4). In characteristic zero, in virtue of S(Q ab ), Theorem 4.6 implies Conjecture A. As we are assuming S(F p ) for every prime p, Conjecture A implies S(Q). Finally Theorem 2.5 yields the desired result.
V ρ over Spec(Z[ζ ∞ , N −1 ]) which descend to Spec(Z[N −1 ]). For this strategy, the ring Z[ζ ∞ , N −1 ] can be also replaced by the Dedekind domain Z[ζ ∞ , N −1 , −1 ].
Lemma 4. 3 .
3If = p, the closure of the group generated by δ (F p ) in Z is an open subgroup.
1 .
1Notation 1.1. If R is a ring, for every n ∈ Z >0 we denote by R[ζ n ] the quotient R[t]/(t n − 1) and by R[ζ ∞] the
ring lim
− →n
R[ζ n ]. At the same time, if is a prime number we denote by R[ζ ∞ ] the ring lim
− →n
R[ζ n ]
Acknowledgments I thank my advisor Hélène Esnault and Matteo Tamiozzo for all the valuable corrections and remarks on a first draft of this article. I also thank Ben Moonen and Kazuya Kato for some enlightening discussions on the topic.
. P Deligne, La Conjecture De, Weil, Publ. Math. IHES. IIP. Deligne, La Conjecture de Weil, II, Publ. Math. IHES 52 (1980), 137-252.
Analytic pro-p-groups. J D Dixon, M P F Sautoy, A Mann, D Segal, Cambridge Studies in Advanced Math. 61Cambridge University PressJ. D. Dixon, M. P. F. du Sautoy, A. Mann, D. Segal, Analytic pro-p-groups, Cambridge Studies in Advanced Math. 61, Cambridge University Press, Cambridge, 1991.
B Moonen, arXiv:1709.04489A remark on the Tate conjecture. to appear in the J. of Alg. GeomB. Moonen, A remark on the Tate conjecture, arXiv: 1709.04489 (2017), to appear in the J. of Alg. Geom.
On the semisimplicity of pure sheaves. L Fu, Proc. Amer. Math. Soc. 127L. Fu, On the semisimplicity of pure sheaves, Proc. Amer. Math. Soc. 127 (1999), 2529-2533.
Lettresà Ken Ribet du 1/1/1981 et du 29/1/1981, OEuvres -Collected Papers IV. J.-P Serre, Springer-VerlagHeidelbergJ.-P. Serre, Lettresà Ken Ribet du 1/1/1981 et du 29/1/1981, OEuvres -Collected Papers IV, Springer-Verlag, Heidelberg, 2000, 1-12.
Algebraic cycles and poles of zeta functions, Arithmetical Algebraic Geometry. J Tate, Proc. Conf. Purdue Univ. Conf. Purdue UnivNew YorkHarper and RowJ. Tate, Algebraic cycles and poles of zeta functions, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper and Row, New York, 1965, 93-110.
| []
|
[
"A BOOTSTRAPPING APPROACH TO JUMP INEQUALITIES AND THEIR APPLICATIONS",
"A BOOTSTRAPPING APPROACH TO JUMP INEQUALITIES AND THEIR APPLICATIONS"
]
| [
"Mariusz Mirek ",
"ANDElias M Stein ",
"Pavel Zorin-Kranich "
]
| []
| []
| The aim of this paper is to present an abstract and general approach to jump inequalities in harmonic analysis. Our principal conclusion is the refinement of r-variational estimates, previously known for r > 2, to end-point results for the jump quasi-seminorm corresponding to r = 2. This will be applied to the dimension-free results in [Bou+18a] and [Bou+18b] and operators of Radon type treated in [JSW08]. | 10.2140/apde.2020.13.527 | [
"https://arxiv.org/pdf/1808.09048v2.pdf"
]
| 54,629,567 | 1808.09048 | cfdcca386a1258bf5db34d540e17eac17475b55c |
A BOOTSTRAPPING APPROACH TO JUMP INEQUALITIES AND THEIR APPLICATIONS
Mariusz Mirek
ANDElias M Stein
Pavel Zorin-Kranich
A BOOTSTRAPPING APPROACH TO JUMP INEQUALITIES AND THEIR APPLICATIONS
The aim of this paper is to present an abstract and general approach to jump inequalities in harmonic analysis. Our principal conclusion is the refinement of r-variational estimates, previously known for r > 2, to end-point results for the jump quasi-seminorm corresponding to r = 2. This will be applied to the dimension-free results in [Bou+18a] and [Bou+18b] and operators of Radon type treated in [JSW08].
Introduction
Variational and jump inequalities in harmonic analysis, probability, and ergodic theory have been studied extensively since [Bou89], where a variational version of the Hardy-Littlewood maximal function was introduced. The purpose of this paper is to formulate general sufficient conditions that allow us to deal with variational and jump inequalities for a wide class of operators. Our approach will be based on certain bootstrap arguments. As an application we extend the known L p estimates for r-variations for r > 2 (see definition (1.2)) to end-point assertions for the jump quasi-seminorm J p 2 (see definition (1.3)), which corresponds to r = 2. In this way our results will extend previously recently obtained assertions in [Bou+18a] and [Bou+18b] for dimension-free estimates given for r > 2, as well as a number of results in [JSW08] for operators of Radon type.
We recall the notation for jump quasi-seminorms from [MSZ18a]. For any λ > 0 and I ⊂ R the λ-jump counting function of a function f : I → C is defined by
:= sup J∈N sup t 0 <···<t J t j ∈I J j=1 |f (t j ) − f (t j−1 )| r 1/r , 0 < r < ∞, sup t 0 <t 1 t j ∈I |f (t 1 ) − f (t 0 )|, r = ∞, (1.2)
where the former supremum is taken over all finite increasing sequences in I. Throughout the article (X, B, m) denotes a σ-finite measure space. For a function f : X × I → C the jump quasi-seminorm on L p (X) for 1 < p < ∞ is defined by J p 2 (f ) := J p 2 (f : X × I → C) := J p 2 ((f (·, t)) t∈I ) := J p 2 ((f (·, t)) t∈I : X → C) := sup λ>0 λN λ (f (·, t) : t ∈ I) 1/2 L p (X) .
(1.3)
In this connection by [MSZ18a, Lemma 2.12] we note that (1.4) V r (f ) L p,∞ (X) p,r J p 2 (f ) ≤ V 2 (f ) L p (X) for r > 2, and the first inequality fails for r = 2.
We now briefly list our main results. (1) The extension to the jump quasi-seminorm J p 2 of dimension-free estimates for maximal averages over convex sets, as given by Theorem 1.9, Theorem 1.11 and Theorem 1.14 below.
(2) The corresponding extension to J p 2 of the previous dimension-free estimates for cubes in the discrete setting, see Theorem 1.18.
(3) The general J p 2 results for operators of Radon type (both averages and singular integrals) in Theorem 1.22 and Theorem 1.30, related to the previous results in [JSW08]. Underlying the proofs of all these results will be the basic facts about the jump quantity J p 2 obtained in our recent paper [MSZ18a], and the bootstrap arguments in Section 2 of the present paper. The reader might compare the methods in Section 2 with related arguments in [Bou+18a, Section 2.2] as well as [NSW78], [DR86], [Car86], and Christ's observation included in [Car88]. The techniques in Section 2 will be carried out in the following framework. We assume that we are given a measure space (X, B, m) which is endowed with a sequence of linear operators (S j ) j∈Z acting on L 1 (X) + L ∞ (X) that play the role of the Littlewood-Paley operators. Namely, the following conditions are satisfied:
(1) The family (S j ) j∈Z is a resolution of the identity on L 2 (X), i.e. the identity j∈Z S j = Id (1.5) holds in the strong operator topology on L 2 (X).
(2) For every 1 < p < ∞ we have
j∈Z |S j f | 2 1/2 L p f L p , f ∈ L p (X). (1.6)
Suppose now we have a family of linear operators (T t ) t∈I acting on L 1 (X)+L ∞ (X), where the index set I is a countable subset of (0, ∞). We assume that I ⊆ (0, ∞) to make our exposition consistent with the results in the literature. One of our aims is to understand what kind of conditions have to be imposed on the family (T t ) t∈I , in terms of its interactions with the Littlewood-Paley operators (S j ) j∈Z to obtain the inequality
J p 2 ((T t f ) t∈I : X → C) f L p (1.7)
in some range of p's. We accomplish this task in Section 2 by proving Theorem 2.14 and Theorem 2.39 for positive operators 1 by certain bootstrap arguments, and Theorem 2.28 for general operators. Our approach will be based on extension of ideas from [DR86] and [Bou+18a] to a more abstract setting.
As mentioned above it has been very well known since Bourgain's article [Bou89] that r-variational estimates (and consequently maximal estimates, see (1.2)) can be deduced from jump inequalities. Namely, a priori jump estimates (1.7) in an open range of p ∈ (1, ∞) imply V r (T t f : t ∈ I) L p p,r f L p in the same range of p's and for all r ∈ (2, ∞]. This follows from (1.4) and interpolation. Therefore, it is natural to say that the jump inequality in (2.2) is an endpoint for r-variations at r = 2. On the other hand, we also know that the range of r ∈ (2, ∞] in r-variational estimates, for many operators in harmonic analysis, is sharp due to the sharp estimates in Lépingle's inequality for martingales, see [MSZ18a] and the references therein.
Here and later we write a b if a ≤ Cb, where the constant 0 < C < ∞ is allowed to depend on p, but not on the underlying abstract measure space X or function f . If C is allowed to depend on some additional parameters this will be indicated by adding a subscript to the symbol .
1.1. Applications to dimension-free estimates. An important application of the results from Section 2 will be bounds independent of the dimension in jump inequalities associated with the Hardy-Littlewood averaging operators. Let G ⊂ R d be a symmetric convex body, that is, a non-empty symmetric convex open bounded subset of R d . Define for t > 0 and x ∈ R d the averaging operator
(1.8) A G t f (x) := |G| −1 G f (x − ty)dy, f ∈ L 1 loc (R d ).
It follows from the spherical maximal theorem that in the case that G is the Euclidean ball the maximal operator Ste82]. This result was extended to arbitrary symmetric convex bodies G ⊂ R d in [Bou86a] (for p = 2) and [Bou86b; Car86] (for p > 3/2). For unit balls G = B q induced by q norms in R d the full range p > 1 of dimension-free estimates was established in [Mül90] (for 1 ≤ q < ∞) and [Bou14] (for cubes q = ∞) with constants depending on q. In the latter case the product structure of the cubes is important; this result was recently extended to products of Euclidean balls of arbitrary dimensions [Som17].
A G f := sup t>0 |A G t f | corresponding to (1.8) is bounded on L p (R d ) for all p > 1, uniformly in d ∈ N [
Variational versions of most of the aforementioned dimension-free estimates were obtained in [Bou+18a] for r > 2. In this article we give a shorter and more selfcontained proof of the main results of [Bou+18a] and extend them to the endpoint r = 2 by appealing to Theorem 2.14 and Theorem 2.39. A notable simplification is that we do not use the maximal estimates as a black box. In particular, we reprove all dimension-free estimates for the maximal function A G .
In view of (1.4) and by real interpolation, Theorem 1.9 below extends [Bou+18a, Theorem 1.2]. Theorem 1.9. Let d ∈ N and G ⊂ R d be a symmetric convex body. Then for every
1 < p < ∞ and f ∈ L p (R d ) we have (1.10) J p 2 ((A G 2 k f ) k∈Z : R d → C) f L p (R d ) ,
where the implicit constant is independent of d and G.
As a consequence of Theorem 1.9 and the decomposition into long and short jumps, see (2.2), Theorems 1.11 and 1.14 below extend [Bou+18a, Theorem 1.1] and [Bou+18a, Theorem 1.3], respectively. Hence Theorem 1.9 can be thought of as the main result of this paper, since inequalities (1.12) and (1.15) were obtained in [Bou+18a]. However, we shall present a different approach to establish the estimates in (1.12) and (1.15).
Theorem 1.11. Let G be as in Theorem 1.9. Then for every 3/2 < p < 4 and f ∈ L p (R d ) we have
(1.12) k∈Z V 2 (A G t f : t ∈ [2 k , 2 k+1 ]) 2 1/2 L p (R d ) f L p (R d ) .
In particular,
(1.13) J p 2 ((A G t f ) t>0 : R d → C) f L p (R d ) ,
where the implicit constants in (1.12) and (1.13) are independent of d and G.
Theorem 1.14. Let d ∈ N and G ⊂ R d be the unit ball induced by the q norm in R d for some 1 ≤ q ≤ ∞. Then for every 1 < p < ∞ and f ∈ L p (R d ) we have
(1.15) k∈Z V 2 (A G t f : t ∈ [2 k , 2 k+1 ]) 2 1/2 L p (R d ) q f L p (R d ) .
In particular
(1.16) J p 2 ((A G t f ) t>0 : R d → C) q f L p (R d ) ,
where the implicit constants in (1.15) and (1.16) are independent of d.
The method of the present paper also allows us to provide estimates independent of the dimension in jump inequalities associated with the discrete averaging operator along cubes in Z d . For every x ∈ Z d and N ∈ N let
A N f (x) := 1 |Q N ∩ Z d | y∈Q N ∩Z d f (x − y), f ∈ 1 (Z d ), (1.17) be the discrete Hardy-Littlewood averaging operator, where Q N = [−N, N ] d .
Theorem 1.18. For every 3/2 < p < 4 and f ∈ p (Z d ) we have
J p 2 ((A N f ) N ∈N : Z d → C) f p (Z d ) . (1.19)
Moreover, if we consider only lacunary parameters, then (1.19) remains true for all 1 < p < ∞ and we have
(1.20) J p 2 ((A 2 k f ) k≥0 : Z d → C) f p (Z d ) ,
where the implicit constants in (1.19) and (1.20) are independent of d.
Theorem 1.18 provides the endpoint estimate at r = 2 for the recent dimension-free estimates [Bou+18b] for r-variations corresponding to operator (1.17).
The dimension-free results are proved in Section 3.1 by combining the results from Section 2 (Theorem 2.14 and Theorem 2.39) with the jump estimates for the Poisson semigroup from [MSZ18a] and Fourier multiplier estimates from [Bou86a] and [Mül90;Bou14].
1.2. Applications to operators of Radon type. Another important class of operators which was extensively studied in [JSW08] in the context of jump inequalities are operators of Radon type modeled on polynomial mappings.
Let P = (P 1 , . . . , P d ) : R k → R d be a polynomial mapping, where each component P j : R k → R is a polynomial with k variables and real coefficients. We fix Ω ⊂ R k a convex open bounded set containing the origin (not necessarily symmetric), and for every x ∈ R d and t > 0 we define the Radon averaging operators
M P t f (x) := 1 |Ω t | Ωt f (x − P (y))dy, (1.21)
where Ω t = {x ∈ R k : t −1 x ∈ Ω}. Using Theorem 2.14 and Theorem 2.39 we easily deduce Theorem 1.22, see Section 3.3.
Theorem 1.22. For every 1 < p < ∞ and f ∈ L p (R d ) we have
(1.23) J p 2 ((M P t f ) t>0 : R d → C) d,p f L p (R d ) ,
where the implicit constant is independent of the coefficients of P .
Before we formulate a corresponding result for truncated singular integrals we need to fix some definitions and notation. A modulus of continuity is a function ω : [0, ∞) → [0, ∞) with ω(0) = 0 that is subadditive in the sense that
u ≤ t + s =⇒ ω(u) ≤ ω(t) + ω(s).
Substituting s = 0 one sees that ω(u) ≤ ω(t) for all 0 ≤ u ≤ t. The basic example is ω(t) = t θ , with θ ∈ (0, 1). Note that the composition and sum of two moduli of continuity is again a modulus of continuity. In particular, if ω(t) is a modulus of continuity and θ ∈ (0, 1), then ω(t) θ and ω(t θ ) are also moduli of continuity.
The Dini norm and the log-Dini norm of a modulus of continuity are defined respectively by setting For any c > 0 the integral can be equivalently (up to a c-dependent multiplicative constant) replaced by the sum over 2 −j/c with j ∈ N.
Finally, for every x ∈ R d and t > 0 we will consider the truncated singular Radon transform
H P t f (x) := R k \Ωt
f (x − P (y))K(y)dy, (1.25) defined for every Schwartz function f in R d , where K : R k \ {0} → C is a kernel satisfying the following conditions:
(1) the size condition, i.e. there exists a constant C K > 0 such that
(1.26) |K(x)| ≤ C K |x| −k , for all x ∈ R k ;
(2) the cancellation condition
Ω R \Ωr K(y)dy = 0, for 0 < r < R < ∞; (1.27) (3) the smoothness condition (1.28) sup R>0 sup |y|≤Rt/2 R≤|x|≤2R |K(x) − K(x + y)|dx ≤ ω K (t),
for every t ∈ (0, 1) with some modulus of continuity ω K .
In many applications it is easy to verify the somewhat stronger pointwise version of the smoothness estimate from (1.28). Namely,
(1.29) |K(x) − K(x + y)| ≤ ω K (|y|/|x|)|x| −k , provided that |y| ≤ |x|/2,
for some modulus of continuity ω K . One can immediately see that condition (1.29) implies condition (1.28). Our next result establishes an analogue of the inequality (1.23) for the operators in (1.25).
Theorem 1.30. Suppose that ω θ logDini + ω θ/2 Dini < ∞ for some θ ∈ (0, 1]. Then for every p ∈ {1 + θ, (1 + θ) } and f ∈ L p (R d ) we have
(1.31) J p 2 ((H P t f ) t>0 : R d → C) d,p f L p (R d ) ,
where the implicit constant is independent of the coefficients of P . More precisely,
(1) if ω θ logDini < ∞, then J p 2 ((H 2 k f ) k∈Z : R d → C) f L p ; (1.32) (2) if ω θ/2 Dini < ∞, then (1.33) k∈Z V 2 (H t f : t ∈ [2 k , 2 k+1 ]) 2 1/2 L p f L p .
The inequality (1.23) was proved in [JSW08] for the averages M P t over Euclidean balls. The inequality (1.31) was proved in [JSW08] for monomial curves, i.e. in the case k = 1, d = 2, K(y) = y −1 and P (x) = (x, x a ), where a > 1. General polynomials were considered in [MST15] (although jump estimates are not explicitly stated in that article they can also be obtained with minor modifications of the proofs). Multi-dimensional variants of H P t were also studied in [MST15] under stronger regularity conditions imposed on the kernel K. Inequalities (1.23) and (1.31) will be used to establish jump inequalities for the discrete analogues of (1.21) and (1.25) in [MSZ18b].
Finally we provide van der Corput integral estimates in Lemma B.1 and Proposition B.2, which have the feature that permit to handle the oscillatory integrals with non-smooth amplitudes. Its broader scope will be needed in the proof of Theorem 1.30.
2. An abstract approach to jump inequalities 2.1. Preliminaries. Let (X, B, m) be a σ-finite measure space endowed with a sequence of linear Littlewood-Paley operators (S j ) j∈Z satisfying (1.5), (1.6). Assume that (T t ) t∈I is a family of linear operators acting on L 1 (X) + L ∞ (X), where the index set I is a subset of (0, ∞). Under suitable conditions imposed on the family (T t ) t∈I in terms of its interactions with the Littlewood-Paley operators (S j ) j∈Z as in the introduction, we will study strong uniform jump inequalities
J p 2 ((T t f ) t∈I : X → C) f L p (2.1)
in various ranges of p's, see Theorem 2.14, Theorem 2.28 and Theorem 2.39.
To avoid further problems with measurability we will always assume that I is countable. Usually I = D := {2 n : n ∈ Z} the set of all dyadic numbers or I = U := n∈Z N/2 n the set of non-negative rational numbers whose denominators in reduced form are powers of 2. In practice, the countability assumption may be removed if for every f ∈ L 1 (X) + L ∞ (X) the function I t → T t f (x) is continuous for m-almost every x ∈ X. In our applications this will be always our case.
We recall the decomposition into long and short jumps from [JSW08, Lemma 1.3], which tells that for every λ > 0 we have
λN λ (T t f (x) : t ∈ I) 1/2 λN λ/3 (T t f (x) : t ∈ D) 1/2 + k∈Z λN λ (T t f (x) : t ∈ [2 k , 2 k+1 ) ∩ I) 1/2 2 1/2 . (2.2)
In other words the λ-jump counting function can be dominated by the long jumps (the first term in (2.2) with t ∈ D) and the short jumps (the square function in (2.2)). Similar inequalities hold for the maximal function and for r-variations.
We deal with L p bounds for the long jump counting function corresponding to T t with t ∈ D in two ways, similarly to [DR86]. The first approach is to find an approximating family of operators (see the family (P k ) k∈Z in Theorem 2.14) for which the bound in question is known and control a square function that dominates the error term, see (2.15) in Theorem 2.14. In our case this method works for positive operators with martingales or related operators as the approximating family. The second approach is to express T 2 k as a telescoping sum
T 2 k f = j≥k T 2 j f − T 2 j+1 f = j≥m B j f (2.3)
and try to deduce bounds in question from the behavior of B j = T 2 j −T 2 j+1 . This approach is needed if T t is a truncated singular integral type operator, see Theorem 2.28. Similar strategies also yield L p bounds for maximal functions
sup k∈Z |T 2 k f (x)| or r- variations V r (T 2 k f (x) : k ∈ Z).
In order to deal with short jumps we note that the square function on the righthand side of (2.2) is dominated by the square function associated with 2-variations, which in turn is controlled by a series of square functions
k∈Z V 2 (T t f (x) : t ∈ [2 k , 2 k+1 ) ∩ I) 2 1/2 ≤ √ 2 l≥0 k∈Z 2 l −1 m=0 |(T 2 k +2 k−l (m+1) − T 2 k +2 k−l m )f (x)| 2 1/2 . (2.4)
The square function on the right-hand side of (2.4) gives rise to assumption (2.40). Inequality (2.4) follows from the next lemma with g(t) = T 2 k +t f (x) and r = 2.
Lemma 2.5. Let r ∈ [1, ∞), k ∈ Z, and a function g : [0, 2 k ] ∩ U → C be given. Then
V r g(t) : t ∈ [0, 2 k ] ∩ U ≤ 2 r−1 r l≥0 2 l −1 m=0 |g(2 k−l (m + 1)) − g(2 k−l m)| r 1/r . (2.6)
The variation norm on the left-hand side of (2.6) can be extended to all t ∈ [0, 2 k ] if g : [0, 2 k ] → C is continuous. Lemma 2.5 originates in the paper of Lewko and Lewko [LL12], where it was observed that the 2-variation norm of a sequence of length N can be controlled by the sum of log N square functions and this observation was used to obtain a variational version of the Rademacher-Menshov theorem. Inequality (2.6), essentially in this form, was independently proved by the first author and Trojan in [MT16] and used to estimate r-variations for discrete Radon transforms. Lemma 2.5 has been used in several recent articles on r-variations, including [Bou+18a]. For completeness we include a proof, which is shorter than the previous proofs.
Proof of Lemma 2.5. Due to monotonicity of r-variations it suffices to prove (2.6)
with U N = {u/2 N : u ∈ N and 0 ≤ u ≤ 2 k+N } in place of [0, 2 k ] ∩ U. Observe that V r g(t) : t ∈ U N = V r g(t/2 N ) : t ∈ [0, 2 k+N ] ∩ Z .
The proof will be completed if we show that
V r g(t) : t ∈ [0, 2 n ] ∩ Z ≤ 2 1−1/r n l=0 2 n−l −1 m=0 |g(2 l (m + 1)) − g(2 l m)| r 1/r . (2.7)
Once (2.7) is established we apply it with g(t/2 N ) in place of g(t) and n = k + N and obtain (2.6). We prove (2.7) by induction on n. The case n = 0 is easy to verify. Let n ≥ 1 and suppose that the claim is known for n − 1. Let 0 ≤ t 0 < · · · < t J < 2 n be an increasing sequence of integers. For j ∈ {0, . . . , J} let s j ≤ t j ≤ u j be the closest smaller and larger even integer, respectively. Then
J j=1 |g(t j ) − g(t j−1 )| r 1/r = J j=1 |(g(t j ) − g(s j )) + (g(s j ) − g(u j−1 )) + (g(u j−1 ) − g(t j−1 ))| r 1/r ≤ J j=1 |g(s j ) − g(u j−1 )| r 1/r + J j=1 |(g(t j ) − g(s j )) + (g(u j−1 ) − g(t j−1 ))| r 1/r .
In the first term we notice that the sequence u 0 ≤ s 1 ≤ u 1 ≤ · · · is monotonically increasing and takes values in 2N, so we can apply the induction hypothesis to the function g(2·). In the second term we use the elementary inequality (a + b) r ≤ 2 r−1 (a r + b r ) and observe |t j − s j | ≤ 1, |t j−1 − u j−1 | ≤ 1, and s j ≥ u j−1 , so that this is bounded by the l = 0 summand in (2.7).
Preparatory estimates.
We recall Lemma 2.8 that deduces a vector-valued inequality from a maximal one. Then we apply it to obtain Lemma 2.9.
M * ,J f := sup k∈J sup |g|≤|f | |M k g|,
where the supremum is taken in the lattice sense. Let q 0 , q 1 ∈ [1, ∞] and 0 ≤ θ ≤ 1 with 1 2 = 1−θ q 0 and q 0 ≤ q 1 . Let q θ ∈ [q 0 , q 1 ] be given by 1
q θ = 1−θ q 0 + θ q 1 = 1 2 + 1−q 0 /2 q 1 . Then k∈J |M k g k | 2 1/2 L q θ ≤ (sup k∈J M k L q 0 →L q 0 ) 1−θ M * ,J θ L q 1 →L q 1 k∈J |g k | 2 1/2 L q θ .
Proof. Consider the operatorM g := (M k g k ) k∈J acting on sequences of functions g = (g k ) k∈J in L 1 (X) + L ∞ (X). By Fubini's theorem
M g L q 0 ( q 0 ) = M k g k L q 0 q 0 ≤ (sup k∈J M k L q 0 →L q 0 ) g k L q 0 q 0 = (sup k∈J M k L q 0 →L q 0 ) g L q 0 ( q 0 ) .
By definition of the maximal operator
M g L q 1 ( ∞ ) = sup k∈J |M k g k | L q 1 ≤ M * ,J (sup k∈J |g k |) L q 1 ≤ M * ,J L q 1 →L q 1 sup k∈J |g k | L q 1 = M * ,J L q 1 →L q 1 g L q 1 ( ∞ ) .
The claim for q θ ∈ [q 0 , q 1 ] follows by complex interpolation between L q 0 (X; q 0 (J)) and L q 1 (X; ∞ (J)).
Lemma 2.9. Suppose that (X, B, m) is a σ-finite measure space with a sequence of operators (S k ) k∈Z that satisfy the Littlewood-Paley inequality (1.6). Let 1 ≤ q 0 ≤ q 1 ≤ 2 and L ∈ N be a positive integer and let
V L = {(k, l) ∈ Z 2 : 0 ≤ l ≤ L − 1}. Let (M k,l ) (k,l)∈V L be a sequence of operators bounded on L q 1 (X) such that (2.10) k∈Z L−1 l=0 |M k,l S k+j f | 2 1/2 L 2 ≤ a j f L 2 , f ∈ L 2 (X)
for some positive numbers (a j ) j∈Z . Then for p = q 1 and for all f ∈ L p (X) we have
k∈Z L−1 l=0 |M k,l S k+j f | 2 1/2 L p L 2−q 1 2−q 0 1 2 sup (k,l)∈V L M k,l 2−q 1 2−q 0 q 0 2 L q 0 →L q 0 M * ,V L 2−q 1 2 L q 1 →L q 1 a q 1 −q 0 2−q 0 j f L p .
(2.11)
If M k are convolution operators on an abelian group G, then (2.11) also holds for q 1 ≤ p ≤ q 1 . The implicit constants in the conclusion do not depend on the qualitative bounds that we assume for the operators M k,l on L q 1 (X).
Proof. First we show (2.11). In the case q 1 = 2 this is identical to the hypothesis (2.10), so suppose q 1 < 2. Let θ and q θ ∈ [q 0 , q 1 ] be as in Lemma 2.8, then by that lemma and Littlewood-Paley inequality (1.6) we obtain
k∈Z L−1 l=0 |M k,l S k+j f | 2 1/2 L q θ sup (k,l)∈V L M k,l 1−θ L q 0 →L q 0 M * ,V L θ L q 1 →L q 1 k∈Z L−1 l=0 |S k+j f | 2 1/2 L q θ L 1/2 sup (k,l)∈V L M k,l 1−θ L q 0 →L q 0 M * ,V L θ L q 1 →L q 1 f L q θ .
(2.12)
Since q θ ≤ q 1 < 2, there is a unique ν ∈ (0, 1] such that 1 q 1 = ν q θ + 1−ν 2 . Substituting the definition of q θ we obtain 1 q 1 = νθ q 1 + 1 2 . It follows that
1 − θ = q 0 2 , θ = 2 − q 0 2 , νθ = 2 − q 1 2 , ν = 2 − q 1 2 − q 0 , ν(1 − θ) = 2 − q 1 2 − q 0 q 0 2 , 1 − ν = q 1 − q 0 2 − q 0 .
Interpolating (2.12) with the hypothesis (2.10) gives the claim (2.11) for p = q 1 .
If M k are convolution operators, then by duality the first inequality in (2.12) also holds with q θ replaced by q θ . Also, 1 q 1 = ν q θ + 1−ν 2 , so the same argument as before also works for p = q 1 . The conclusion for q 1 < p < q 1 follows by complex interpolation.
2.3. Long jumps for positive operators. Suppose now we have a sequence of positive linear operators (A k ) k∈Z and an approximating family of linear operators (P k ) k∈Z both acting on L 1 (X) + L ∞ (X) such that for every 1 < p < ∞ the maximal lattice operator
P * f := sup k∈Z sup |g|≤|f | |P k g|,
satisfies the maximal estimate P * L p →L p 1.
(2.13) Theorem 2.14 will be based on a variant of bootstrap argument discussed in the context of differentiation in lacunary directions in [NSW78]. These ideas were also used to provide L p bounds for maximal Radon transforms in [DR86]. It was observed by Christ that the argument from [NSW78] can be formulated as an abstract principle, which was useful in many situations [Car88] and also in the context of dimension-free estimates [Car86].
Theorem 2.14. Assume that (X, B, m) is a σ-finite measure space endowed with a sequence of linear operators (S j ) j∈Z satisfying (1.5) and (1.6). Given parameters 1 ≤ q 0 < q 1 ≤ 2, let (A k ) k∈Z be a sequence of positive linear operators such that sup k∈Z A k L q 0 →L q 0 1. Suppose that the maximal function P * satisfies (2.13) with p = q 1 and
k∈Z |(A k − P k )S k+j f | 2 1/2 L 2 ≤ a j f L 2 , f ∈ L 2 (X) (2.15)
for some positive numbers (a j ) j∈Z satisfying a := j∈Z a
q 1 −q 0 2−q 0 j < ∞. Then for all f ∈ L p (X) with p = q 1 we have k∈Z |(A k − P k )f | 2 1/2 L p (1 + a 2/q 1 ) f L p . (2.16)
In particular
(2.17) A * L p →L p 1 + a 2/q 1 .
If in addition we have the jump inequality
(2.18) J p 2 ((P k f ) k∈Z : X → C) f L p ,
then also
(2.19) J p 2 ((A k f ) k∈Z : X → C) (1 + a 2/q 1 ) f L p .
In the case of convolution operators on an abelian group G all these implications also hold for q 1 ≤ p ≤ q 1 , and we have the vector-valued estimate
(2.20) k∈Z |A k f k | r 1/r L p k∈Z |f k | r 1/r L p in the same range q 1 ≤ p ≤ q 1 for all 1 ≤ r ≤ ∞.
A few remarks concerning the assumptions in Theorem 2.14 are in order. In applications it is usually not difficult to verify the assumption (2.15). For general operators the most reasonable and efficient way is to apply T T * methods. However, for convolution operators on G assumption (2.15) can be verified using Fourier transform methods, which may be simpler than T T * methods. Let us explain the second approach more precisely when G = R d . We first have to fix some terminology.
Let A be a d × d real matrix whose eigenvalues have positive real part. We set
t A := exp(A log t), for t > 0. (2.21)
Let q be a smooth A-homogeneous quasi-norm on R d , that is, q : R d → [0, ∞) is a continuous function, smooth on R d \ {0}, and such that (1) q(x) = 0 ⇐⇒ x = 0;
(2) there is C ≥ 1 such that for all x, y ∈ R d we have q(x + y) ≤ C(q(x) + q(y));
(3) q(t A x) = tq(x) for all t > 0 and x ∈ R d . Let also q * be a smooth (away from 0) A * -homogeneous quasi-norm, where A * is the adjoint matrix to A. We only have to find a sequence of Littlewood-Paley projections associated with the quasi-norm q * . For this purpose let φ 0 : [0, ∞) → [0, ∞) be a smooth function such that 0 ≤ φ 0 ≤ 1 [1/2,2] and its dilates φ j (x) :
= φ 0 (2 j x) satisfy j∈Z φ 2 j = 1 (0,∞) . (2.22)
For each j ∈ Z we define the Littlewood-Paley operatorS j such that S j f = ψ j f corresponds to a smooth function ψ j (ξ) := φ j (q * (ξ)) on R d . By (2.22) we see that (1.5) holds for S j =S 2 j . Moreover, by [Riv71, Theorem II.1.5] we obtain the Littlewood-Paley inequality (1.6) for the operators S j andS j .
If (Φ t : t > 0) is a family of Schwartz functions such that Φ t (ξ) = Φ(tq * (ξ)), where Φ is a non-negative Schwartz function on R d with integral one, then by [JSW08, Theorem 1.1] we know that for every 1 < p < ∞ we have
J p 2 ((Φ 2 k * f ) k∈Z : R d → C) f L p , f ∈ L p (R d ). (2.23)
The maximal version of inequality (2.23) has been known for a long time and follows from the Hardy-Littlewood maximal theorem [Ste93].
Suppose now we have a family (A k ) k∈Z of convolution operators A k f = µ 2 k * f corresponding to a family of probability measures (µ t : t > 0) on R d such that
| µ t (ξ) − µ t (0)| ≤ ω(tq * (ξ)) if tq * (ξ) ≤ 1, (2.24) | µ t (ξ)| ≤ ω((tq * (ξ)) −1 )
if tq * (ξ) ≥ 1, (2.25) for some modulus of continuity ω.
Theorem 2.14, taking into account all the facts mentioned above, yields
J p 2 ((µ 2 k * f ) k∈Z : R d → C) f L p , f ∈ L p (R d ) (2.26)
for p = q 1 and q 0 = 1 as long as a = j∈Z ω(2 −|j| ) q 1 −q 0 2−q 0 < ∞, since (2.15) can be easily verified with a j = ω(2 −|j| ) using (2.24), (2.25) and the properties of S j and Φ.
Proof of Theorem 2.14. We begin with the proof of (2.16). If q 1 = 2 then we use (1.5) and (2.15) and we are done. We now assume that q 1 < 2. By the monotone convergence theorem it suffices to consider only finitely many M k := A k − P k 's in (2.16), let us say those with |k| ≤ K. Restrict all summations and suprema to |k| ≤ K and let B be the smallest implicit constant for which (2.16) holds with p = q 1 . In view of the qualitative boundedness hypothesis we obtain B < ∞, but the bound may depend on K. Our aim is to show that B 1 + a 2/q 1 . There is nothing to do if B 1. Therefore, we will assume that B 1, so by (1.5), (2.13) and (2.11) with L = 1 and M k,0 := M k , we obtain
|k|≤K |M k f | 2 1/2 L p ≤ j∈Z |k|≤K |M k S k+j f | 2 1/2 L p 1 + M * 2−q 1 2 L p →L p a f L p .
By positivity we have |A * f | ≤ sup |k|≤K A k |f | and consequently, we obtain
|A * f | ≤ sup |k|≤K A k |f | ≤M * L p →L p ≤ P * L p →L p + A * L p →L p ≤ 2 P * L p →L p + B 1 + B.
Taking into account these inequalities we have
|k|≤K |(A k − P k )f | 2 1/2 L p = |k|≤K |M k f | 2 1/2 L p 1 + a(1 + B) 2−q 1 2 f L p .
Taking the supremum over f gives
B 1 + a(1 + B) 2−q 1 2
(1 + a)B
J p 2 ((A k f ) k∈Z ) J p 2 ((P k f ) k∈Z ) + J p 2 ((M k f ) k∈Z ) f L p + V 2 (M k f : k ∈ Z) L p f L p + k∈Z |M k f | 2 1/2 L p .
In the case of convolution operators by duality and interpolation we extend (2.16) to L p (G) for q 1 ≤ p ≤ q 1 , and all other inequalities follow as before. Finally, the vector-valued estimate (2.20) with r = ∞ is equivalent to the maximal estimate by positivity, with r = 1 it follows by duality, and with 1 < r < ∞ by complex interpolation.
2.4. Long jumps for non-positive operators. We now drop the positivity assumption and we will be working with general operators (B k ) k∈Z acting on L 1 (X) + L ∞ (X). This will require some knowledge about the maximal lattice operator B * defined in (2.29) and about the sum of B k 's over k ∈ Z. No bootstrap argument seems to be available for non-positive operators and therefore additional assumptions like (2.30) and (2.32) will be indispensable. The proof of Theorem 2.28 is based on the ideas from [DR86].
Theorem 2.28. Assume that (X, B, m) is a σ-finite measure space endowed with a sequence of linear operators (S j ) j∈Z satisfying (1.5) and (1.6). Let 1 ≤ q 0 < q 1 ≤ 2 and let (B k ) k∈Z be a sequence of linear operators commuting with the sequence (S j ) j∈Z such that sup k∈Z B k L q 0 →L q 0 1. Suppose that the maximal lattice operator
B * f := sup k∈Z sup |g|≤|f | |B k g|, (2.29) satisfies B * L q 1 →L q 1 1. (2.30) We also assume k∈Z |B k S k+j f | 2 1/2 L 2 ≤ a j f L 2 , f ∈ L 2 (X) (2.31)
for some positive numbers (a j ) j∈Z .
(1) Suppose that (B k ) k∈Z additionally satisfies k∈Z B k L q 1 →L q 1
(2.32)
Let P k := j>k S j and assume that the jump inequality (2.18) holds for the sequence (P k ) k∈Z with p = q 1 . Then for all f ∈ L p (X) with p = q 1 we have
J p 2 j≥k B j f k∈Z : X → C k∈Z B k L q 1 →L q 1 + sup k∈Z B k 2−q 1 2−q 0 q 0 2 L q 0 →L q 0 B * 2−q 1 2 L q 1 →L q 1ã f L p , (2.33) whereã := j∈Z ja q 1 −q 0 2−q 0 j < ∞.
(2) Suppose that there is a sequence of self-adjoint linear operators (S j ) j∈Z such that S j =S 2 j for every j ∈ Z and satisfying (1.6) and (2.31) withS k+j in place of S k+j . Then for every sequence (ε k ) k∈Z bounded by 1 and for all f ∈ L p (X) with p = q 1 we have
k∈Z ε k B k f L p sup k∈Z B k 2−q 1 2−q 0 q 0 2 L q 0 →L q 0 B * 2−q 1 2 L q 1 →L q 1 a f L p , (2.34)
where a is as in Theorem 2.14.
In the case of convolution operators on an abelian group G all these implications also hold for q 1 ≤ p ≤ q 1 .
In applications in harmonic analysis we will take B k = T 2 k −T 2 k+1 for k ∈ Z, where T t is a truncated singular integral operator of convolution type, see (2.3). This class of operators motivates, to a large extent, the assumptions in Theorem 2.28. In many cases they can be verified if we manage to find positive operators A k such that |B k f | A k |f | for every k ∈ Z and f ∈ L 1 (X) + L ∞ (X). In practice, A k is an averaging operator. We shall illustrate this more precisely by appealing to the discussion after Theorem 2.14.
Suppose that (B k ) k∈Z is a family of convolution operators B k f = σ 2 k * f corresponding to a family of finite measures (σ t : t > 0) on R d such that sup t>0 σ t < ∞ and for every k ∈ Z and t ∈ [2 k , 2 k+1 ] we have
| σ t (ξ)| ≤ ω(2 k q * (ξ)) if 2 k q * (ξ) ≤ 1, (2.35) | σ t (ξ)| ≤ ω((2 k q * (ξ)) −1 ) if 2 k q * (ξ) ≥ 1, (2.36)
for some modulus of continuity ω. Additionally, we assume that |σ 2 k | µ 2 k for some family of finite positive measures (µ t : t > 0) on R d such that sup t>0 µ t < ∞ and satisfying (2.24) and (2.25). In view of these assumptions and Theorem 2.14 we see that condition (2.30) holds, since
|B k f | A k |f |, where A k f = µ 2 k * f . Therefore, k∈Z B k f L p a f L p ,
implies (2.32) with p = q 1 and q 0 = 1, provided that a = j∈Z ω(2 −|j| ) q 1 −q 0 2−q 0 < ∞, since (2.31) can be verified with a j = ω(2 −|j| ) using (2.35), (2.36) and the properties ofS j associated with (2.22). Having proven (2.30) and (2.32) we see that (2.33) holds for the operators B k f = σ 2 k * f with p = q 1 and q 0 = 1 as long asã = j∈Z jω(2 −|j| )
q 1 −q 0 2−q 0 < ∞.
Proof of Theorem 2.28. In order to prove inequality (2.33) we employ the following decomposition
j≥k B j = P k j∈Z B j − l>0 j<0 S k+l B k+j + l≤0 j≥0
S k+l B k+j (2.37) (cf. [DR86, p. 548]). The jump inequality corresponding to the first term on the right-hand side in (2.37) is bounded on L p (X) with p = q 1 , due to (2.18), and (2.32), which ensures boundedness of the operator j∈Z B j .
The estimates for the second and the third term are similar and we only consider the last term. We take the 2 norm with respect to the parameter k and estimate
J p 2 l≤0 j≥0 B k+j S k+l f k∈Z : X → C ≤ k∈Z | l≤0 j≥0 B k+j S k+l f | 2 1/2 L p = k∈Z | m≥0 k n=k−m B n+m S n f | 2 1/2 L p ≤ m≥0 k∈Z | k n=k−m B n+m S n f | 2 1/2 L p by triangle inequality ≤ m≥1 m 1/2 k∈Z k n=k−m |B n+m S n f | 2 1/2 L p by Hölder's inequality = m≥1 m n∈Z |B n+m S n f | 2 1/2 L p .
By (2.11), with L = 1 and M k,0 := B k , we obtain
j∈Z j k∈Z |B k S k+j f | 2 1/2 L p sup k∈Z B k 2−q 1 2−q 0 q 0 2 L q 0 →L q 0 B * 2−q 1 2 L q 1 →L q 1ã f L p .
To prove the second part observe that for a sequence of functions (f j ) j∈Z in L p (X; 2 (Z)) we have the following inequality
j∈ZS j f j L p j∈Z |f j | 2 1/2 L p , (2.38)
which is the dual version of inequality (1.6) for the sequence (S j ) j∈Z . To prove (2.34) we will use (1.5) and (2.38). Indeed,
k∈Z ε k B k f L p ≤ j∈Z k∈Z ε k B k S k+j f L p by (1.5) = j∈Z k∈ZS k+j (ε k B kSk+j f ) L p since S j =S 2 j j∈Z k∈Z |B kSk+j f | 2 1/2 L p by (2.38) sup k∈Z B k 2−q 1 2−q 0 q 0 2 L q 0 →L q 0 B * 2−q 1 2 L q 1 →L q 1 a f L p ,
where in the last step we have used Lemma 2.9, with L = 1 and M k,0 := B k .
2.5. Short variations. We will work with a sequence of linear operators (A k ) k∈Z (not necessarily positive) acting on L 1 (X) + L ∞ (X). However, positive operators will be distinguished in our proof and in this case we can also proceed as before using some bootstrap arguments. For every k ∈ Z and t ∈ [2 k , 2 k+1 ] we will use the following notation
∆((A s ) s∈I ) t f := ∆(A t )f := A t f − A 2 k f.
Theorem 2.39. Assume that (X, B, m) is a σ-finite measure space endowed with a sequence of linear operators (S j ) j∈Z satisfying (1.5) and (1.6). Let (A t ) t∈U be a family of linear operators such that the square function estimate
(2.40) k∈Z 2 l −1 m=0 |(A 2 k +2 k−l (m+1) − A 2 k +2 k−l m )S j+k f | 2 1/2 L 2 ≤ 2 − l 2 a j,l f L 2
holds for all j ∈ Z and l ∈ N with some numbers a j,l ≥ 0 such that for every 0 < ε < ρ we have l≥0 j∈Z 2 −εl a ρ j,l < ∞. (2.41)
(1) Let 1 < q 0 < 2 and 4 < q ∞ < ∞, and suppose that for each q 0 < p < q ∞ the vector-valued estimate
(2.42) k∈Z |A 2 k (1+t) f k | 2 1/2 L p k∈Z |f k | 2 1/2 L p holds uniformly in t ∈ U ∩ [0, 1]. Then for each 3 1+1/q 0 < p < 4 1+2/q∞ we have (2.43) k∈Z V 2 (A t f : t ∈ [2 k , 2 k+1 ] ∩ U) 2 1/2 L p f L p ,
and for each 4 ≤ p < q ∞ and r > p 2 q∞−2 q∞−p we have
(2.44) k∈Z V r (A t f : t ∈ [2 k , 2 k+1 ] ∩ U) r 1/r L p f L p for all f ∈ L p (X).
(2) Let q 0 ∈ [1, 2) and α ∈ [0, 1] be such that αq 0 ≤ 1. Suppose that we have the operator norm Hölder type condition
(2.45) A t+h − A t L q 0 →L q 0 h t α , h, t ∈ U, and h ∈ (0, 1].
Then for every exponent q 1 satisfying
(2.46) q 0 ≤ 2 − 2 − q 0 2 − αq 0 < q 1 ≤ 2,
and such that (3) Moreover, if (A t ) t∈U is a family of positive linear operators, then the condition (2.47) may be replaced by a weaker condition (2.48) A * ,D L q 1 →L q 1 1 and the estimate (2.43) holds as well with the implicit constant which is a constant multiple of 1 + a 2/q 1 . In the case of convolution operators on an abelian group G the implication from (2.48) to (2.43) also holds with p replaced by p .
Theorem 2.39 combined with the results formulated in the previous two paragraphs for dyadic scales will allow us to control, in view of (2.2), the cases for general scales. The first part of Theorem 2.39 gives (2.43) in a restricted range of p's. If one asks for a larger range, a smoothness condition like in (2.45) must be assumed. Inequality (2.45) combined with maximal estimate (2.47) gives larger range of p's in (2.43). If we work with a family of positive operators the condition (2.47) may be relaxed to (2.48) by some bootstrap argument. In the context of discussion after Theorem 2.14 and Theorem 2.28 let us look at a particular situation of (2) and prove (2.43).
Suppose that (A t ) t>0 is a family of convolution operators A t f = σ t * f corresponding to a family of finite measures (σ t : t > 0) on R d such that sup t>0 σ t < ∞ and satisfying (2.35) and (2.36). We assume that |σ t | µ t for some family of finite positive measures (µ t : t > 0) on R d such that sup t>0 µ t < ∞ and satisfying (2.24) and (2.25) to make sure that condition (2.47) holds. Additionally, let us assume that (2.45) holds with α = 1 and q 0 = 1, 2. By Plancherel's theorem, (2.35) and (2.36) we obtain
(A 2 k +2 k−l (m+1) − A 2 k +2 k−l m )S j+k f L 2 ω(2 −|j| ) S j+k f L 2 .
(2.49) Thus (2.45) with q 0 = 2, t = 2 k + 2 k−l m, h = 2 k−l combined with (2.49) imply
(A 2 k +2 k−l (m+1) − A 2 k +2 k−l m )S j+k f L 2 min(2 −l , ω(2 −|j| )) S j+k f L 2 . (2.50)
Consequently (2.40) holds with a j,l = min{1, 2 l ω(2 −|j| )} and Theorem 2.39 gives the desired conclusion as long as a = l≥0 j∈Z 2 − (q 1 −1)l 2 (min{1, 2 l ω(2 −|j| )}) q 1 −1 < ∞.
Proof of Theorem 2.39: case (1). By Minkowski's inequality for 2 ≤ s ≤ q ∞ < ∞ we have
k∈Z 2 l −1 m=0 |(A 2 k +2 k−l (m+1) − A 2 k +2 k−l m )f k | s 1/s s L q∞ = 2 l −1 m=0 k∈Z |(A 2 k +2 k−l (m+1) − A 2 k +2 k−l m )f k | s L q∞/s ≤ 2 l −1 m=0 k∈Z |(A 2 k +2 k−l (m+1) − A 2 k +2 k−l m )f k | s L q∞/s ≤ 2 l sup 0≤m<2 l k∈Z |(A 2 k +2 k−l (m+1) − A 2 k +2 k−l m )f k | s 1/s s L q∞ ≤ 2 l+2 sup 0≤m≤2 l k∈Z |A 2 k +2 k−l m f k | 2 1/2 s L q∞ 2 l k∈Z |f k | 2 1/2 s L q∞ ,
where we have applied (2.42) in the last step. Using this with f k = S j+k f and applying (1.6) we obtain
k∈Z 2 l −1 m=0 |(A 2 k +2 k−l (m+1) − A 2 k +2 k−l m )S j+k f | s 1/s L q∞ 2 l/s f L q∞ for all 2 ≤ s ≤ q ∞ < ∞. By interpolation with (2.40) we obtain k∈Z 2 l −1 m=0 |(A 2 k +2 k−l (m+1) − A 2 k +2 k−l m )S j+k f | r 1/r L p 2 − θl 2 + (1−θ)l s a θ j,l f L p , (2.51)
where 0 < θ ≤ 1 and 1 r = θ 2 + 1−θ s and 1 p = θ 2 + 1−θ q∞ , so θ = 2 p q∞−p q∞−2 . By Lemma 2.5 or more precisely by an analogue of inequality (2.4) with r norm in place of 2 norm and by (2.51) we obtain
(2.52) k∈Z V r (A t f : t ∈ [2 k , 2 k+1 ] ∩ U) r 1/r L p 2 − θl 2 + (1−θ)l s a θ j,l f L p .
In view of (2.41) with ε = θ 2 − (1−θ) s and ρ = θ this estimate is summable in l and j, provided that −θ/2 + (1 − θ)/s < 0. In particular, for 2 ≤ p < 4 1+2/q∞ we use s = 2. For 4 ≤ p < q ∞ we use s > q∞(p−2) q∞−p and then r > p 2 q∞−2 q∞−p .
For q 0 ∈ (1, 2) by Minkowski's inequality we have
k∈Z 2 l −1 m=0 |(A 2 k +2 k−l (m+1) − A 2 k +2 k−l m )f k | 2 1/2 L q 0 ≤ 2 l −1 m=0 k∈Z |(A 2 k +2 k−l (m+1) − A 2 k +2 k−l m )f k | 2 1/2 L q 0 ≤ 2 l+1 sup 0≤m≤2 l k∈Z |A 2 k +2 k−l m f k | 2 1/2 L q 0 2 l k∈Z |f k | 2 1/2 L q 0 .
Substituting f k = S j+k f , applying (1.6), and interpolating with (2.40) we obtain
k∈Z 2 l −1 m=0 |(A 2 k +2 k−l (m+1) − A 2 k +2 k−l m )S j+k f | 2 1/2 L p 2 − θl 2 +(1−θ)l a θ j,l f L p ,
(2.53)
with 1 p = θ 2 + 1−θ q 0 , for 0 < θ < 1. Hence θ = 2 p p−q 0
2−q 0 and in view of (2.41) with ε = θ 2 − (1 − θ) and ρ = θ this estimate is summable in l and j, provided that −θ/2 + (1 − θ) < 0. The conclusion again follows from Lemma 2.5 and (2.53) like in (2.52) with 3 1+1/q 0 < p ≤ 2.
Proof of Theorem 2.39: case (2) and case (3). By the monotone convergence theorem we may restrict k in (2.43) to |k| ≤ K 0 and t to U k L 0 := {u/2 L 0 : u ∈ N and 2 k+L 0 ≤ u ≤ 2 k+L 0 +1 } for some K 0 ∈ N and L 0 ∈ Z as long as we obtain estimates independent of K 0 and L 0 . Fix K 0 , L 0 and let I := |k|≤K 0 U k L 0 . Let q 1 satisfy (2.46) then invoking (1.5) and (2.11), with L = 2 l , we obtain |k|≤K 0
2 l −1 m=0 |(A 2 k +2 k−l (m+1) − A 2 k +2 k−l m )f | 2 1/2 L p 2 2−q 1 2−q 0 l 2 sup |k|≤K 0 , 0≤m<2 l A 2 k +2 k−l (m+1) − A 2 k +2 k−l m 2−q 1 2−q 0 q 0 2 L q 0 →L q 0 ∆((A s ) s∈U ) * ,I 2−q 1 2 L q 1 →L q 1 · j∈Z (2 − l 2 a j,l ) q 1 −q 0 2−q 0 f L p 2 2−q 1 2−q 0 l 2 (2 −αl ) 2−q 1 2−q 0 q 0 2 ∆((A s ) s∈U ) * ,I 2−q 1 2 L q 1 →L q 1 2 − l 2 q 1 −q 0 2−q 0 j∈Z a q 1 −q 0 2−q 0 j,l f L p .
In order for the right-hand side to be summable in l we need
2 − q 1 2 − q 0 1 2 − α 2 − q 1 2 − q 0 q 0 2 − 1 2 q 1 − q 0 2 − q 0 < 0 ⇐⇒ (2 − q 1 ) − α(2 − q 1 )q 0 − (q 1 − q 0 ) < 0.
It suffices to ensure
(2 − q 1 )(1 − αq 0 ) − (q 1 − q 0 ) < 0 ⇐⇒ q 1 > 2(1 − αq 0 ) + q 0 2 − αq 0 = 2 − 2 − q 0 2 − αq 0 ,
and this is our hypothesis (2.46). Hence under this condition by Lemma 2.5 we conclude for general operators that k∈Z V 2 (A t f : t ∈ I) 2 1/2
L p l≥0 |k|≤K 0 2 l −1 m=0 |(A 2 k +2 k−l (m+1) − A 2 k +2 k−l m )f | 2 1/2 L p ∆((A s ) s∈U ) * ,I 2−q 1 2 L q 1 →L q 1 a f L p .
(2.54)
For positive operators crude estimates and interpolation show that
B := A * ,I L p →L p < ∞ with p = q 1 . Note that sup t∈I |A t f (x)| ≤ sup t∈D |A t f (x)| + k∈Z sup t∈[2 k ,2 k+1 )∩I |(A t − A 2 k )f (x)| 2 1/2 (2.55)
Therefore, appealing to (2.55), (2.48) and (2.54) we obtain by a bootstrap argument that B 1 + B 2−q 1 2 a, since
∆((A s ) s∈U ) * ,I 2−q 1 2 L q 1 →L q 1 B 2−q 1 2 .
Hence, B 1 + a 2/q 1 . In particular, the estimate (2.54) becomes uniform in I ⊂ U, and this simultaneously implies (2.43).
In the case of convolution operators we may replace p = q 1 by p = q 1 in Lemma 2.9 and all subsequent arguments.
Applications
3.1. Dimension-free estimates for jumps in the continuous setting. We begin by providing dimension-free endpoint estimates, for r = 2, in the main results of [Bou+18a]. Let G ⊂ R d be a symmetric convex body. By definition of the averaging operator (1.8)
we have A G tŨ =Ũ A U (G) t ,
whereŨ f := f • U is the composition operator with an invertible linear map U : R d → R d . It follows that all estimates in Section 1 are not affected if G is replaced by U (G).
By [Bou86a], after replacing G by its image under a suitable invertible linear transformation, we may assume that the normalized characteristic function µ :=
|G| −1 1 G satisfies | µ(ξ)| ≤ C|ξ| −1 , (3.1) | µ(ξ) − 1| ≤ C|ξ|, (3.2) | ξ, ∇ µ(ξ) | ≤ C (3.3)
with the constant C independent of the dimension. In [Bou86a] these estimates were proved with |L(G)ξ| in place of |ξ| on the right-hand side, where L(G) is the isotropic constant corresponding to G. The above form is obtained by rescaling.
Then A t := A G t is the convolution operator with µ t and µ t (ξ) = µ(tξ). The Poisson semigroup is defined by
P t f (ξ) := p t (ξ) f (ξ), where p t (ξ) := e −2πt|ξ| .
The associated Littlewood-Paley operators are given by S k := P 2 k − P 2 k+1 . Their Fourier symbols satisfy
(3.4) | S k (ξ)| min{2 k |ξ|, 2 −k |ξ| −1 },
where S k (ξ) is the multiplier associated with the operator S k , i.e. S k f (ξ) = S k (ξ) f (ξ). From now on, for simplicity of notation, we will use this convention. The symbols associated with the Poisson semigroup P k := P 2 k satisfy
(3.5) | P k (ξ) − 1| |2 k ξ|, and | P k (ξ)| 2 −k |ξ| −1 .
Proof of Theorem 1.9. We verify that the sequence (A k ) k∈Z , where A k := A 2 k satisfies the hypotheses of Theorem 2.14 for every 1 = q 0 < q 1 ≤ 2. The maximal inequality (2.13) and the Littlewood-Paley inequality (1.6) for the Poisson semigroup with constants independent of the dimension are well-known [Ste70]. The jump estimate (2.18) was recently established in [MSZ18a,Theorem 1.5].
It remains to verify condition (2.15) for the operators M k := A k − P k . In view of (3.1), (3.2) and (3.5), we have
| M k (ξ)| min{|2 k ξ| −1 , |2 k ξ|}.
For ξ ∈ R d \ {0} let k 0 ∈ Z be such thatξ = 2 k 0 ξ satisfies |ξ| 1. By (3.5) it follows that
k∈Z | M k (ξ) S k+j (ξ)| 2 k∈Z min{|2 k ξ| −1 , |2 k ξ|} 2 min{|2 k+j ξ| −1 , |2 k+j ξ|} 2 = k∈Z min{|2 kξ | −1 , |2 kξ |} 2 min{|2 k+jξ | −1 , |2 k+jξ |} 2 k∈Z min{2 −k , 2 k } 2 min{(2 k+j ) −1 , 2 k+j } 2 2 −δ|j| (3.6)
for δ > 0 with the implicit constant independent of the dimension. By Plancherel's theorem this shows that (2.15) holds with a j 2 −δ|j|/2 .
Proof of Theorem 1.11. We will apply Theorem 2.39 with A t := A t := A G t . By a simple scaling we have A 2 k (1+t) = A (1+t)G 2 k . Hence Theorem 2.14, with A k = A (1+t)G 2 k , applies and we obtain the vector-valued inequality (2.20) for all 1 < p < ∞ and r = 2, which consequently guarantees (2.42). It remains to verify the hypothesis (2.40) of Theorem 2.39. We repeat the estimate [Bou+18a, (4.23)]. By (3.3) for t > 0 and h > 0 we have
(3.7) µ (t + h)ξ − µ tξ ≤ t+h t | ξ, ∇ µ(uξ) |du t+h t du u h t .
By the Plancherel theorem this implies
(3.8) A t+h − A t L 2 →L 2 h t .
This allows us to estimate the square of the left-hand side of (2.40) by
LHS(2.40) 2 = k∈Z 2 l −1 m=0 (A 2 k +2 k−l (m+1) − A 2 k +2 k−l m )S j+k f 2 L 2 k∈Z 2 l −1 m=0 2 −2l S j+k f 2 L 2 = 2 −l k∈Z S j+k f 2 L 2 2 −l f 2 L 2 .
Secondly, by (3.1) and (3.2) for every 0 ≤ m < 2 l we have µ((2 k + 2 k−l (m + 1))ξ) − µ((2 k + 2 k−l m)ξ) min{|2 k ξ|, |2 k ξ| −1 }.
Arguing similarly to (3.6) we obtain
LHS(2.40) 2 2 l 2 −δ|j| f 2 2 .
Hence (2.40) holds with a j,l = min{1, 2 l 2 −δ|j|/2 }.
Proof of Theorem 1.14. By Theorem 1.9 we have the hypothesis (2.48) of Theorem 2.39. The hypothesis (2.40) was verified in the proof of Theorem 1.11. The remaining hypothesis (2.45) is given by [Bou+18a,Lemma 4.2], but we give a more direct proof.
Recall that B q is the unit ball induced by q norm in R d . From [Mül90] (for 1 ≤ q < ∞), and [Bou14] (for q = ∞) we use the multiplier norm estimate m M p p,q,α 1,m = (ξ · ∇) α µ for α ∈ (0, 1) and p ∈ (1, ∞) with implicit constant independent of the dimension. For a Lipschitz function h : (1/2, ∞) → R such that |h(t)| |t| −1 and |h (t)| |t| −1 fractional differentiation can be inverted by fractional integration: [DGM16,Lemma 6.9]. In particular, for t > 1 we obtain
h(t) = 1 Γ(α) +∞ t (u − t) α−1 D α h(u)du, t > 1/2, seeh(t) − h(1) = 1 Γ(α) +∞ 1 ((u − t) α−1 + − (u − 1) α−1 )D α h(u)du,
where u + := max(u, 0) denotes the positive part. In view of (3.1) and (3.3) this result can be applied to the function h(t) = µ(tξ) for any ξ
∈ R d \ {0}. Observing D α h(u) = u −αm (uξ) we obtain µ(tξ) − µ(ξ) = 1 Γ(α) +∞ 1 ((u − t) α−1 + − (u − 1) α−1 + )u −αm (uξ)du.
On the other hand we have
+∞ 1 |(u − t) α−1 + − (u − 1) α−1 + |u −α du α (t − 1) α ,
and for a Schwartz function f ∈ S(R d ) this implies
F −1 ξ (( µ(tξ) − µ(ξ)) f (ξ)) L p ≤ +∞ 1 |(u − t) α−1 + − (u − 1) α−1 + |u −α · F −1 ξ (((uξ · ∇) α µ)(ξ) f (ξ)) L p du α (t − 1) α sup u>0 F −1 ξ (((uξ · ∇) α µ)(uξ) f (ξ)) L p α (t − 1) α ((ξ · ∇) α µ)(ξ) M p f L p ,
where we have used the Fourier inversion formula and Fubini's theorem in the first step and scale invariance of the multiplier norm in the last step. Since the multiplier µ(tξ) − µ(ξ) is (qualitatively) bounded on L p with norm ≤ 2, by density of Schwartz functions this implies
µ(t·) − µ M p α (t − 1) α ,
which by scaling implies the hypothesis (2.45).
Finally we emphasize that once Theorem 1.9 is proved, alternative proofs of Theorem 1.11 an Theorem 1.14 follow by appealing to the short variational estimates given in [Bou+18a].
3.2. Dimension-free estimates for jumps in the discrete setting. We briefly outline the proof of Theorem 1.18. The strategy is much the same as for the proof of Theorem 1.9 and Theorem 1.11. Let
m N (ξ) = 1 (2N + 1) d m∈Q N e 2πim·ξ ,
for ξ ∈ T d be the multiplier corresponding to the operators A N defined in (1.17). Here we remind the reader of the following estimates for m N established recently in [Bou+18b].
Namely there is a constant 0 < C < ∞ independent of the dimension such the for every N, N 1 , N 2 ∈ N and for every ξ ∈ T d ≡ [−1/2, 1/2) d we have
|m N (ξ)| ≤ C(N |ξ|) −1 , |m N (ξ) − 1| ≤ CN |ξ|, |m N 1 (ξ) − m N 2 (ξ)| ≤ C|N 1 − N 2 | max N −1 1 , N −1 2 , (3.9)
where |·| denotes the Euclidean norm restricted to T d . The discrete Poisson semigroup is defined by
P t f (ξ) := p t (ξ) f (ξ), where p t (ξ) := e −2πt|ξ| sin ,
for every ξ ∈ T d and
|ξ| sin := d j=1
(sin(πξ j )) 2 1/2 .
We set P k := P 2 k and the associated Littlewood-Paley operators are given by S k := P 2 k − P 2 k+1 . The maximal inequality (2.13) and the Littlewood-Paley inequality (1.6) for the discrete Poisson semigroup with constants independent of the dimension follow from [Ste70]. The jump estimate (2.18) for discrete Poisson semigroup was recently proved in [MSZ18a, Theorem 1.5]. Moreover, using |ξ| ≤ |ξ| sin ≤ π|ξ| for ξ ∈ T d , we see that the corresponding Fourier symbols S k (ξ) and P k (ξ) satisfy estimates (3.4) and (3.5) as well.
In order to prove (1.20) we have to verify that the sequence (A k ) k∈N , where A k := A 2 k satisfies the hypotheses of Theorem 2.14 for every 1 = q 0 < q 1 ≤ 2. Taking into account (3.9), (3.4) and (3.5) (associated with the discrete Poisson semigroup) it suffices to proceed as in the proof of Theorem 1.9. To prove (1.19) we argue as in the proof of Theorem 1.11.
3.3. Jump inequalities for the operators of Radon type. In this section we prove Theorem 1.22 and Theorem 1.30. By the lifting procedure for the Radon transforms described in [Ste93, Chapter 11, Section 2.4] we can assume without loss of generality that our polynomial mapping P (x) := (x) Γ is the canonical polynomial mapping for some Γ ⊂ N k 0 \ {0} with lexicographical order, given by R k x = (x 1 , . . . , x k ) → (x) Γ := (x γ 1 1 · . . . · x γ k k : γ ∈ Γ) ∈ R Γ , where R Γ := R |Γ| is identified with the space of all vectors whose coordinates are labeled by multi-indices γ = (γ 1 , . . . , γ k ) ∈ Γ.
Throughout what follows A is the diagonal |Γ|×|Γ| matrix such that (Ax) γ = |γ|x γ for every x ∈ R Γ and let q * be the quasi-norm associated with A * = A, given by
q * (ξ) = max γ∈Γ |ξ γ | 1 |γ| , for ξ ∈ R Γ .
We shall later freely appeal, without explicit mention, to the discussions after Theorem 2.14, Theorem 2.28 and Theorem 2.39 with d = |Γ|, A and q * as above.
Proof of Theorem 1.22. Let M t := M P t , where P (x) = (x) Γ . Observe that M t is a convolution operator with a probability measure µ t , whose Fourier transform is defined by
µ t (ξ) := 1 |Ω t | Ωt e −e −2πiξ·(y) Γ K(y)dy, for ξ ∈ R Γ .
For a fixed κ ∈ (0, 1) we claim
|Ψ t (ξ) − Ψ s (ξ)| κ |t A ξ| −1/d ∞ + ω K (|t A ξ| −1/d ∞ ) (tq * (ξ)) −1/d + ω K ((tq * (ξ)) −1/d ), if tq * (ξ) ≥ 1, (3.11)
for all s, t ∈ (0, ∞) such that κt ≤ s ≤ t. Indeed, by Proposition B.2 we obtain
|Ψ t (ξ) − Ψ s (ξ)| = Ωt\Ωs e −2πiξ·(y) Γ K(y)dy sup v∈R k :|v|≤tΛ −1/d |(1 Ωt\Ωs K)(y) − (1 Ωt\Ωs K)(y − v)|dy
with Λ = γ∈Γ t |γ| |ξ γ |. The claim (3.11) clearly holds for Λ ≤ 1. If Λ ≥ 1, then for a fixed v we use (1.28) and the fact that Ω t \ Ω s ⊆ B(0, t) \ B(0, c Ω κt) to estimate the contribution of y such that y, y − v ∈ Ω t \ Ω s . On the set of y such that exactly one of y, y − v is contained in Ω t \ Ω s we use (1.26); the measure of this set is bounded by a multiple of t k−1 |v| due to Lemma A.1. This finishes the proof of (3.11).
Additionally, we have
|Ψ t (ξ) − Ψ s (ξ)| |t A ξ| 1/d ∞ (tq * (ξ)) 1/d + ω K ((tq * (ξ)) 1/d ), if tq * (ξ) ≤ 1 (3.12)
due to the cancellation condition (1.27) and (1.26).
To prove (1.31) we fix θ ∈ (0, 1] and p ∈ {1 + θ, (1 + θ) } and invoking (2.2) it suffices to prove inequalities (1.32) and (1.33). Inequality (1.32) will follow from (2.33) with q 0 = 1, q 1 = 1 + θ and B j := H 2 j − H 2 j+1 upon expressing H 2 k as a telescoping series like in (2.3). Inequality (1.33) will be a consequence of (2.43) with q 0 = 1, q 1 = 1 + θ and A t := H t . Let (σ t : t > 0) be a family of measures defined by
σ t * f (x) = Ωt\Ω 2 k f (x − (y) Γ )K(y)dy, for every t ∈ [2 k , 2 k+1 ], k ∈ Z.
(3.13) Estimates (3.11) and (3.12) allow us to verify (2.35) and (2.36) respectively with ω(t) := t 1/d + ω K (t 1/d ). Moreover |σ 2 k | µ 2 k , where µ t is the measure associated with the averaging operator M t . Hence the discussion after Theorem 2.28 guarantees that inequality (2.33) holds, since B k f = σ 2 k+1 * f . To prove (2.43) it suffices to note that (2.45) holds for all 1 ≤ q 0 < ∞. Moreover inequalities (2.49) and (2.50) remain true for A t = H t . Then Theorem 2.39 completes the proof.
Appendix A. Neighborhoods of boundaries of convex sets
We will show how to control the measure of neighborhoods of the boundaries of convex sets. The proof of the lemma below is based on a simple Vitali covering argument.
Lemma A.1. Let Ω ⊂ R k be a bounded and convex set and let 0 < s diam(Ω). Then |{x ∈ R k : dist(x, ∂Ω) < s}| k s diam(Ω) k−1 .
The implicit constant depends only on the dimension k, but not on the convex set Ω.
Proof. Let r = diam Ω. By translation we may assume Ω ⊆ B(0, r), where B(y, s) denotes an open ball centered at y ∈ R k with radius s > 0. Notice {x ∈ R k : dist(x, ∂Ω) < s} ⊆ y∈∂Ω B(y, s).
By the Vitali covering lemma there exists a finite subset Y ⊂ ∂Ω such that the balls B(y, s) with y ∈ Y , are pairwise disjoint and y∈∂Ω B(y, s) y∈Y B(y, s) .
Consider the nearest-point projection P : R k → clΩ, that is, P (x) = x , where x ∈ clΩ is the unique point such that |x − x | = dist(x, clΩ). It is well-known that P is well-defined and contractive with respect to the Euclidean metric. The restriction of P to the sphere ∂B(0, r) defines a surjection P ∂ : ∂B(0, r) → ∂Ω. This follows from the fact that for every point x ∈ ∂Ω there exists a linear functional φ : R k → R such that φ(y) ≤ φ(x) for every y ∈ clΩ, see e.g. [Roc70, Corollary 11.6.1]). For each y ∈ Y we choose z(y) ∈ ∂B(0, r) such that P ∂ (z(y)) = y. Then the balls B(z(y), s) are pairwise disjoint in view of the contractivity of P and contained in the set {x ∈ R k : r − s < |x| < r + s} that has measure s(r + s) k−1 . But the union of the balls B(z(y), s) has the same measure as y∈Y B(y, s), and the conclusion follows.
Appendix B. Estimates for oscillatory integrals
We present the following variant of van der Corput's oscillatory integral lemma with a rough amplitude function.
Lemma B.1. Given an interval (a, b) ⊂ R suppose that φ : (a, b) → R is a smooth function such that |φ (k) (x)| λ for every x ∈ (a, b) with some λ > 0. Assume additionally that
• either k ≥ 2, • or k = 1 and φ is monotonic.
Then for every locally integrable function ψ : R → C we have Proof. Let η be a smooth positive function with supp η ⊆ [−1, 1] and R η(x)dx = 1. Let ρ(x) := ψ * λ 1/k η(λ 1/k x), and note that |ψ(x) − ρ(x)| ≤ λ 1/k R |ψ(x) − ψ(x − y)||η(λ 1/k y)|dy.
Then we may replace ψ by ρ on the left-hand side of the conclusion. For every x 0 ∈ (a, b) by partial integration and the van der Corput lemma, see for example [Ste93, Section VIII.1.2], we have The latter term is estimated using |ρ (x)| = |(ψ(x) − ψ) * λ 1/k η(λ 1/k ·) (x)| λ 2/k R |ψ(x) − ψ(x − y)||η (λ 1/k y)|dy, and the conclusion follows.
We will also need a multidimensional version of Lemma B.1. As before B(y, s) denotes an open ball centered at y ∈ R k with radius s > 0.
Proposition B.2 ([Zor17]). Given d, k ∈ N, let P (x) = 1≤|α|≤d λ α x α be a polynomial in k variables of degree at most d with real coefficients. Let R > 0 and let ψ : R k → C be an integrable function supported in B(0, R/2). Then where Λ := 1≤|α|≤d R |α| |λ α |.
We include the proof for completeness.
Proof. Changing the variables we have R k e iP (x) ψ(x)dx = R k R k e iP R (x) ψ R (x)dx , where P R (x) = 1≤|α|≤d R |α| λ α x α , ψ R (x) = ψ(Rx) and supp ψ R ⊆ B(0, 1/2). Let us define β = sup
v∈R k :|v|≤Λ −1/d R k |ψ R (x) − ψ R (x − v)|dx,
and observe that ψ R L 1 βΛ 1/d . So there is nothing to prove if Λ 1. We assume that Λ 1. Let η be a non-negative smooth bump function with integral 1, which is supported in the ball B(0, 1/2). Then we define ρ(x) = Λ k/d η(Λ 1/d x) and φ(x) = ψ R * ρ(x) and we note
R k |ψ R (x) − φ(x)|dx ≤ Λ k/d R k R k |ψ R (x) − ψ R (x − y)|dxη(Λ 1/d y)dy β.
The proof will be completed if we show that
R k e iP R (x) φ(x)dx d,k β. (B.3)
Since φ is a smooth function supported in B(0, 1) we invoke [SW01, Lemma 2.2] to get the conclusion. Indeed, [SW01, Lemma 2.2] ensures that there exists a unit vector ξ ∈ R k and an integer m ∈ N such that |(ξ · ∇) m P R | > c k,d Λ on the unit ball B(0, 1) for some c k,d > 0. We may assume, without loss of generality, that ξ = e 1 = (1, 0, . . . , 0) ∈ R k . Then by the van der Corput lemma, see for example [Ste93, Corollary p.334] we obtain since supp φ ⊆ B(0, 1) and φ(1, x ) = 0 for every x ∈ R k−1 ∩ B(0, 1). We now show that ∇φ L 1 Λ 1/d β. Indeed, for every j ∈ N k we have
∂ j φ L 1 = R k R k ψ R (x − y)∂ j ρ(y)dy dx = R k R k ψ R (x) − ψ R (x − y) ∂ j ρ(y)dy dx Λ k/d+1/d R k R k |ψ R (x) − ψ R (x − y)||(∂ j η)(Λ 1/d y)|dxdy Λ 1/d β.
This proves (B.3) and completes the proof of Proposition B.2.
N λ (f ) := N λ (f (t) : t ∈ I):= sup{J ∈ N | ∃ t 0 <···<t J t j ∈I : min 0<j≤J |f (t j ) − f (t j−1 )| ≥ λ}.(1.1) and the r-variation seminorm by V r (f ) :=V r (f (t) : t ∈ I)
Mariusz
Mirek was partially supported by the Schmidt Fellowship and the IAS Found for Math. and by the National Science Center, NCN grant DEC-2015/19/B/ST1/01149. Elias M. Stein was partially supported by NSF grant DMS-1265524. Pavel Zorin-Kranich was partially supported by the Hausdorff Center for Mathematics and DFG SFB-1060.
Lemma 2. 8
8(cf.[DR86, p. 544]). Suppose that (X, B, m) is a σ-finite measure space and (M k ) k∈J is a sequence of linear operators on L 1 (X) + L ∞ (X) indexed by a countable set J. The corresponding maximal operator is defined by
( 2 .
247) ∆((A s ) s∈U ) * ,U L q 1 →L q 1 1we have for all f ∈ L p (X) with p = q 1 that the estimate (2.43) holds with the implicit constant which is a constant multiple of
|∂ 1 φ(x 1 , x )|dx 1 dx Λ −1/d ∇φ L 1 ,
2πiξ·(y) Γ dy, for ξ ∈ R Γ .Condition (2.25) with ω(t) = t 1/d follows from Proposition B.2 and Lemma A.1. It is not difficult to see that (2.24) also holds.In order to prove (1.23) it suffices, in view of (2.2), to show inequality (2.19) with A k := M 2 k and inequality (2.43) with A t := M t for every 1 = q 0 < q 1 ≤ 2. We have already seen that (2.26) holds, hence (2.19) holds and we are done. We now show (2.43). For this purpose note that (2.45) holds for all 1 ≤ q 0 < ∞. This combined with (2.24) and (2.25) permits us to prove (2.49) and (2.50), which imply (2.40) and Theorem 2.39 yields the conclusion.Proof of Theorem 1.30. Let H t := H P t , where P (x) = (x) Γ . Denote the Fourier multiplier corresponding to the truncated singular Radon transform by(3.10)
Ψ t (ξ) :=
R k \Ωt
A linear operator T is positive if T f ≥ 0 for every f ≥ 0.
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Plac Grunwaldzki 2/4, 50-384 Wrocław Poland E-mail address: [email protected]. Usa & Instytut Matematyczny, Uniwersytet Wrocławski, Endenicher Allee. 6053115deMariusz Mirek) Department of Mathematics, Rutgers University ; Department of Mathematics, Princeton University ; Pavel Zorin-Kranich) Mathematical Institute, University of Bonn08544-100 USA E-mail address: [email protected]. Germany E-mail address: [email protected](Mariusz Mirek) Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA & Instytut Matematyczny, Uniwersytet Wrocławski, Plac Grunwaldzki 2/4, 50-384 Wrocław Poland E-mail address: [email protected] (Elias M. Stein) Department of Mathematics, Princeton University, Princeton, NJ 08544-100 USA E-mail address: [email protected] (Pavel Zorin-Kranich) Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany E-mail address: [email protected]
| []
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[
"Alternative formulation of the macroscopic field equations in a linear magneto-dielectric medium: Lagrangian field theory and spacetime setting",
"Alternative formulation of the macroscopic field equations in a linear magneto-dielectric medium: Lagrangian field theory and spacetime setting"
]
| [
"Michael E Crenshaw \nUS Army Combat Capabilities Development Command (DEVCOM) -Aviation and Missile Center\n35898Redstone ArsenalALUSA\n"
]
| [
"US Army Combat Capabilities Development Command (DEVCOM) -Aviation and Missile Center\n35898Redstone ArsenalALUSA"
]
| []
| A transparent linear magneto-dielectric material in free space that is illuminated by a finite quasimonochromatic field is a thermodynamically closed system, definitively, regardless of what field and material subsystems that one defines. The energy-momentum tensor that is formally derived from the Maxwell-Minkowski field equations is inconsistent with both angular and linear momentum conservation in this closed system; this very solid fact is the foundational and continuing issue of the Abraham-Minkowski controversy. The extant resolution of the Abraham-Minkowski dilemma is to treat Maxwellian continuum electrodynamics as being a subsystem and to write the total energymomentum tensor as the sum of a Maxwellian electromagnetic subsystem energy-momentum tensor and a phenomenological material subsystem energy-momentum tensor. We prove that fundamental principles of physics are violated by Maxwellian continuum electrodynamics and that fundamental principles of physics are violated by Maxwellian continuum electrodynamics supplemented by the material subsystem conjecture. We use field theory to derive legitimate equations for macroscopic electromagnetic fields in a transparent linear magneto-dielectric medium. The new field equations are a part of a self-consistent formulation of macroscopic electrodynamics, conservation laws, special relativity, and invariance in a continuous linear medium. In the new formulation, the temporal and spatial coordinates are renormalized by the continuous linear medium instead of the permittivity and permeability being carried as independent material parameters. Then an isotropic, homogeneous, flat, four-dimensional, continuous, linear, non-Minkowski spacetime is the proper setting for the continuum electrodynamics of a simple linear medium in which the effective speed of light is c/n and each medium will be associated with a different spacetime. | null | [
"https://arxiv.org/pdf/1710.00042v4.pdf"
]
| 119,231,465 | 1710.00042 | 6b89df3e6c512daee960831810db36f15045d97b |
Alternative formulation of the macroscopic field equations in a linear magneto-dielectric medium: Lagrangian field theory and spacetime setting
Michael E Crenshaw
US Army Combat Capabilities Development Command (DEVCOM) -Aviation and Missile Center
35898Redstone ArsenalALUSA
Alternative formulation of the macroscopic field equations in a linear magneto-dielectric medium: Lagrangian field theory and spacetime setting
(Dated: February 14, 2022)
A transparent linear magneto-dielectric material in free space that is illuminated by a finite quasimonochromatic field is a thermodynamically closed system, definitively, regardless of what field and material subsystems that one defines. The energy-momentum tensor that is formally derived from the Maxwell-Minkowski field equations is inconsistent with both angular and linear momentum conservation in this closed system; this very solid fact is the foundational and continuing issue of the Abraham-Minkowski controversy. The extant resolution of the Abraham-Minkowski dilemma is to treat Maxwellian continuum electrodynamics as being a subsystem and to write the total energymomentum tensor as the sum of a Maxwellian electromagnetic subsystem energy-momentum tensor and a phenomenological material subsystem energy-momentum tensor. We prove that fundamental principles of physics are violated by Maxwellian continuum electrodynamics and that fundamental principles of physics are violated by Maxwellian continuum electrodynamics supplemented by the material subsystem conjecture. We use field theory to derive legitimate equations for macroscopic electromagnetic fields in a transparent linear magneto-dielectric medium. The new field equations are a part of a self-consistent formulation of macroscopic electrodynamics, conservation laws, special relativity, and invariance in a continuous linear medium. In the new formulation, the temporal and spatial coordinates are renormalized by the continuous linear medium instead of the permittivity and permeability being carried as independent material parameters. Then an isotropic, homogeneous, flat, four-dimensional, continuous, linear, non-Minkowski spacetime is the proper setting for the continuum electrodynamics of a simple linear medium in which the effective speed of light is c/n and each medium will be associated with a different spacetime.
A transparent linear magneto-dielectric material in free space that is illuminated by a finite quasimonochromatic field is a thermodynamically closed system, definitively, regardless of what field and material subsystems that one defines. The energy-momentum tensor that is formally derived from the Maxwell-Minkowski field equations is inconsistent with both angular and linear momentum conservation in this closed system; this very solid fact is the foundational and continuing issue of the Abraham-Minkowski controversy. The extant resolution of the Abraham-Minkowski dilemma is to treat Maxwellian continuum electrodynamics as being a subsystem and to write the total energymomentum tensor as the sum of a Maxwellian electromagnetic subsystem energy-momentum tensor and a phenomenological material subsystem energy-momentum tensor. We prove that fundamental principles of physics are violated by Maxwellian continuum electrodynamics and that fundamental principles of physics are violated by Maxwellian continuum electrodynamics supplemented by the material subsystem conjecture. We use field theory to derive legitimate equations for macroscopic electromagnetic fields in a transparent linear magneto-dielectric medium. The new field equations are a part of a self-consistent formulation of macroscopic electrodynamics, conservation laws, special relativity, and invariance in a continuous linear medium. In the new formulation, the temporal and spatial coordinates are renormalized by the continuous linear medium instead of the permittivity and permeability being carried as independent material parameters. Then an isotropic, homogeneous, flat, four-dimensional, continuous, linear, non-Minkowski spacetime is the proper setting for the continuum electrodynamics of a simple linear medium in which the effective speed of light is c/n and each medium will be associated with a different spacetime.
I. INTRODUCTION
Continuum electrodynamics can be defined as a formal theory whose axioms are the Maxwell-Minkowski equations (macroscopic Maxwell equations), the constitutive relations, and the definitions of the fields in terms of the vector potential. Theorems of continuum electrodynamics are formally generated by the operations of algebra and calculus on the axioms. In particular, the electromagnetic conservation law (see Eq. (2.15)) is a theorem of the Maxwell-Minkowski equations and constitutive relations in the limit that the gradient Minkowski four-force density is negligible. The formal derivation of the electromagnetic conservation law also derives the Minkowski energy-momentum tensor and the Minkowski momentum density.
We address a contradiction between the linear momentum conservation properties of two theorems of formal continuum electrodynamics; the electromagnetic conservation law and the wave equation. Specifically, the vanishing four-divergence of the Minkowski energymomentum tensor for a quasimonochromatic field normally incident on a gradient-index antireflection coated transparent linear magneto-dielectric medium proves, via the (local) electromagnetic conservation law, that the Minkowski linear momentum is conserved [1][2][3][4][5], while global conservation analyses of the same system using the wave equation prove that the Minkowski linear momentum in the medium is greater than the vacuum-incident momentum by a factor of the refractive index n [1, [6][7][8].
The formal theory is a precise axiomatic system and we must acknowledge that the disproof of a theorem of the formal theory, via contradiction between the electromagnetic conservation law and the wave equation, proves that the axioms, the macroscopic Maxwell-Minkowski equations, constitutive relations, and vector potential relations, are false. Instead, scientists assume that the model system is incomplete, ostensibly requiring the inclusion of a material-motion subsystem and a heuristic way to couple the phenomenological material subsystem to the electrodynamic subsystem in order to reconcile Maxwellian electrodynamic theory with the violated physical principles [1,2,6,[9][10][11][12][13][14][15][16][17].
The postulated incompleteness of the Minkowski energy-momentum tensor proves the incompleteness of the electromagnetic conservation law and thereby proves the incompleteness of the axioms from which the electromagnetic conservation law is derived as a theorem. Given that the electromagnetic conservation law, a formal theorem of the Maxwell-Minkowski field equations, results in long-known, scientifically reported [1,2,6,[9][10][11][12][13][14][15][16][17], and easily verified errors in the momentum, the question as to why the incomplete macroscopic Maxwell-Minkowski field equations are regarded as laws of physics, still, and continue to be treated as fundamental equations in textbooks and other scientific publications is a disturbing issue of scientific philosophy.
The foundational issue [18,19] of the century-old Abraham-Minkowski controversy [1,2,6,[9][10][11][12][13][14][15][16][17] is that the non-symmetric Minkowski energy-momentum tensor is not consistent with conservation of angular momen-tum. Originally an issue of angular momentum conservation [18,19], nowadays the Abraham-Minkowski controversy is typically characterized as a question about the physical meaning and usage of two different electromagnetic linear momentum formulas, the Minkowski electromagnetic momentum and the Abraham electromagnetic momentum, and the corresponding material subsystem momentums [1,2,6,[9][10][11][12][13][14][15][16][17]20].
The 'modern' resolution of the Abraham-Minkowski controversy, reviewed in Sec. II, is to posit the existence of a material subsystem with a material subsystem energy-momentum tensor (or a material subsystem momentum) [1,2,6,[9][10][11][12][13][14][15][16][17]. The coupling between the electromagnetic and material subsystems is derived by global conservation of energy, global conservation of linear momentum, and conservation of angular momentum. Then we can write a total (field plus material) energy-momentum tensor (or the total momentum) [1,6,14,15,21]. Except, we prove that using the total (field plus material) energy-momentum tensor [1,21] in the electromagnetic conservation law repairs global linear momentum conservation and angular momentum conservation, by hand, but violates other physical principles, i.e. relativity and the Poynting theorem, that are wellknown to be satisfied by the Maxwell-Minkowski field equations. Then the conservation law is false without the material subsystem conjecture and it is false with the material subsystem conjecture.
Starting with Abraham [19] in 1909, manifold disproofs [1,2,6,[9][10][11][12][13][14][15][16][17] of the electromagnetic conservation law also disprove the Maxwell-Minkowski equations, the axioms from which the conservation law is formally constructed as a theorem that becomes an identity in the limit that a transient is neglected. Then the falsification of the Maxwell-Minkowski equations is a centuryold matter of abstract algebra. Because the literature of the Abraham-Minkowski controversy contains a great amount of deflection about this point, we disprove several hypothetical resolutions of the issue in Sec. II, even though the disproof of the local electromagnetic conservation law is sufficient.
Minkowski spacetime is a model for the vacuum [22]. The spacetime conservation laws, Sec. III, are fundamental physical principles of Minkowski spacetime for the propagation of the unimpeded (no external forces), inviscid, incoherent, incompressible flow of non-interacting particles (nonpolar molecules, dust particles, or photons) in the continuum limit (fluid or light field (light fluid)) through an otherwise empty vacuum [23]. The conservation laws of Minkowski spacetime are fixed and immutable for any countable system obeying the specified conditions, for example, fluid flow in the vacuum or light flow in the vacuum. However, those laws change with physical circumstances. For example, the Navier-Stokes equation is a version of the spacetime conservation law that has been modified for a viscous flow through the otherwise empty vacuum.
Although the vacuum can be modeled as a Minkowski spacetime, we show that the effective medium description of a transparent simple linear magneto-dielectric medium in the continuum limit corresponds to a linear, isotropic, homogeneous, flat, four-dimensional, continuous non-Minkowski 'material' spacetime in which the effective speed of light is c/n. Because the unimpeded, inviscid, incoherent flow of a spatially compressed (compressed and de-compressed at the boundaries of the material, but not otherwise compressible) light field is traveling through a region of space that is not a vacuum, we must develop the physical laws that apply in the 'material' spacetime. The Maxwell-Minkowski equations are readily derived by using Lagrangian field theory in Minkowski spacetime [24,25]. In Secs. IV and V, we develop field theory for a non-empty region of space that responds linearly to electromagnetic radiation with a speed of light c/n. The current author performed a simple application of special relativity using inertial reference frames moving uniformly along the surface of a large transparent linear dielectric in Ref. [26]. It was shown that Einstein's special relativity manifests differently for observers on opposite sides of the interface: i ) An observer on the vacuum side of the interface uses boundary conditions to describe events that occur inside the dielectric using Laue's [27] dielectric special relativity with a vacuum Lorentz factor and a velocity-dependent index of refraction. ii ) An observer on the dielectric side of the interface finds that the application of Einstein's relativity in a dielectric is best described by Rosen's [28] dielectric special relativity in terms of a non-Lorentz (but Lorentz-like) factor that contains the permittivity of the dielectric and does not depend on the velocity of the dielectric. In Sec. VI, the derivation of Ref. [26] is applied to a simple linear magneto-dielectric medium and the result demonstrates that the Lorentz-like factor for the observer inside the continuous medium depends on the permittivity and permeability through the index of refraction. An observer inside an arbitrarily large continuous linear, isotropic, homogeneous, magneto-dielectric medium determines that the velocity of light c/n is independent of the velocity of the source in accordance with the Principle of Relativity.
There are additional optical processes that need to be re-evaluated when they occur in a transparent, isotropic, homogeneous, continuous linear medium, instead of in the vacuum. In Sec. VII, we prove a linear, isotropic, homogeneous, flat, four-dimensional, continuous 'material' spacetime in which the temporal coordinate is normalized by the inverse of the square root of the permittivity and the spatial coordinates are normalized by the square root of the permeability. As has been known [29], Lorentz invariance is not of symmetry of a linear medium. In Sec. VIII, we establish a refractive-index-dependent invariance for the medium-dependent non-Minkowski spacetime of a continuous linear medium. This result implies that Laue's theorem [30,31] and Noether's theorem [32] are re-defined for a simple linear medium by the new invariance principle, however, we focus on the more com-mon electrodynamic principles and we do not derive these two theorems here. In Sec. IX, we construct a tensor formulation of continuum electrodynamics as theorems of the new field equations and note that the energymomentum tensor is diagonally symmetric and that the electromagnetic energy and electromagnetic momentum are locally and globally conserved without the need for a material-motion subsystem. Finally, the new theory of continuum electrodynamics is shown to be consistent with the Balazs [33] thought experiment and the Jones-Richard [34] mirror experiment. It is shown that the uniform velocity of the center of mass-energy theorem depends on a constant mass-energy density. When applied to light propagation, the uniform velocity of the center of mass-energy theorem [34] must be modified to account for the change in the volume occupied by the energy and momentum of the field that is accompanied by the corresponding change in the energy and momentum density, Sec. X.
II. CONTINUUM ELECTRODYNAMICS
The model system consists of a finite quasimonochromatic electromagnetic field and a block of simple linear magneto-dielectric material located in a large finite volume Σ of free space. We define a simple linear medium as a transparent, isotropic, homogeneous, continuous linear medium that has no resonances near the center frequency of the quasimonochromatic field; the material is "effectively dispersionless at frequencies of interest" [10] (There is some ambiguity in the terminology used in the literature because dispersion is actually treated in lowestorder in the 'dispersionless' cases. The refractive index n depends on the frequency of the field, but the frequency is treated as being constant for the duration of the quasimonochromatic field.) The material is initially at rest in the local frame. Unless the radiation is of extraordinary intensity and duration, the velocity of the material in the local frame will be non-relativistic and neglecting the effects of the material motion on the refractive index is an "extremely accurate approximation indeed" [17]. Then dispersion and velocity-dependence can be treated in lowest order such that the permittivity, permeability, and the refractive index can be represented by real constants. The values of these constants depend on the properties of the material and depend on the center frequency of the exciting quasimonochromatic field (dispersion is treated in lowest-order). The model system is the principal model of a simple linear medium that is extensively used in continuum electrodynamics, for example, the derivation of the Fresnel relations [35][36][37]. The stationary 'dispersionless' limit is implicitly and explicitly used in most lowest-order expositions of the Abraham-Minkowski controversy [10].
The electromagnetic theory is developed using vector and tensor formulations. For purposes of illustration, to compare magnitudes, for example, propagation of the field is discussed using the plane-wave limit. The plane-wave limit is a common abstraction with wellknown characteristics that allows paraxial problems to be treated in lowest-order with one spatial dimension, not to be confused with the assumption of uniform plane waves that are nonphysical due to their infinite energy. The plane-wave limit is used in the typical derivation of the Fresnel relations and many other elementary problems of continuum electrodynamics [35][36][37]. The plane-wave limit is explicitly and implicitly used in many lowestorder expositions of the Abraham-Minkowski controversy.
The model quasimonochromatic field is initially in the vacuum and has a constant amplitude except for a short smooth turn-on transition and a short smooth turn-off transition. The field propagates toward and then enters the transparent, isotropic, homogeneous linear medium at normal incidence through a gradient-index antireflection coating. The field re-enters the vacuum through the gradient-index antireflection coating on the opposite side of the medium. The system, as defined, is obviously closed. In particular, any reflected field and whatever material motion that is imparted by the interaction with the field are part of the closed system along with the refracted and transmitted fields. Conservation laws can be applied to the thermodynamically closed system [2].
There is no scientific error in deriving theoretical results for a limiting case in a closed system. Once the theoretical results are derived for the limiting case (quasimonochromatic field, lowest-order dispersion, stationary medium, no sources or sinks, etc), the theory can be extended to more detailed models.
Brevik [2,4] and Wang [5] use the vanishing fourdivergence of the Minkowski energy-momentum tensor (see Eqs. (2.14a) and (2.15)) as a local conservation law [30] to prove that the Minkowski energy
U M = Σ D · E + B · H 2 dv (2.1)
and the Minkowski linear momentum
G M = Σ D × B c dv (2.2)
form a Lorentz four-vector (U M , G M ) in the limit that the Minkowski four-force density f α M that is associated with the gradient-index antireflection coating can be neglected. This is considered to be a resolution of the Abraham-Minkowski controversy because the elements of a Lorentz four-vector are globally conserved [5,30].
Except, that is not the case here. Although the Abraham-Minkowski dilemma was originally about conservation of angular momentum, it was well-known, almost from the outset of the controversy, that the Minkowski linear momentum G M is not globally conserved [1,[6][7][8] thereby contradicting the determination [2,4,5] that (U M , G M ) is a Lorentz four-vector.
That being said, it is common practice to dismiss the global conservation problem with the linear momen-tum by deeming the violation of global momentum conservation to be negligible based on the vanishing fourforce density f α M that appears as the right-hand side of the electromagnetic conservation law (see Eq. (2.14a)). Although the practice is scientifically countenanced by appealing to the electromagnetic conservation law (see Eq. (2.15)), the deduction contradicts the long-known, scientifically reported [1,[6][7][8] and easily verified fact that the Minkowski momentum in the material is the momentum of the field that is incident from the vacuum multiplied by a non-negligible factor of n.
A substantive contradiction exists between the (local) electromagnetic conservation law and the global conservation law. Adopting either the (local) electromagnetic conservation law or global conservation dictates the direction of the analysis and disproves the other. Although both aspects of the contradiction appear in the scientific literature, they typically appear separately thereby avoiding obvious contradictions. In their detailed, comprehensive review article, Pfeifer, Nieminen, Heckenberg, and Rubinsztein-Dunlop [1], present both sides of the issue but, due to the structure of a review article, the two results appear in different sections of the article with the global result of a factor of n difference in the linear momentum being mentioned in Sec. III while the Minkowski momentum is described as (almost) conserved in Sec. VI-A in connection with the electromagnetic conservation law.
Next, we review the details of the argument using the familiar Maxwell-Minkowski formulation of macroscopic electrodynamics. Continuum electrodynamics can be described as a formal theory whose axioms are the Maxwell-Minkowski equations,
∇ × H − ∂D ∂(ct) = J f c (2.3a) ∇ × E + ∂B ∂(ct) = 0 (2.3b) ∇ · D = ρ f (2.3c) ∇ · B = 0 , (2.3d)
and constitutive relations,
D = εE (2.4a) B = µH (2.4b) n = √ εµ , (2.4c)
for the macroscopic fields E, B, D, and H in a simple linear magneto-dielectric medium. Later, the use of the wave equation will cause us to treat the relations between the vector potential A and the macroscopic fields (see Eqs. (2.17)) as axioms, as well. The free charge density ρ f and the free current density J f are macroscopic parameters. Also, ε is a continuum abstraction of the electric permittivity, µ is a continuum abstraction of the magnetic permeability, and n is the macroscopic refractive index. The physical system, as we have defined it, allows us to treat the material parameters ε, µ, and n in lowest order as depending on the center frequency of the quasimonochromatic field but are otherwise single-valued real constants [35][36][37].
Describing the theoretical viewpoint of physics, Rindler [38] states "a physical theory is an abstract mathematical model (much like Euclidian geometry) whose applications to the real world consist of correspondences between a subset of it and a subset of the real world". Continuum electrodynamics is constructed as a formal theory in this abstract mathematical framework by performing operations of algebra and calculus on the axioms. If any theorem of Eqs. Experimentalists [1,20] have a different viewpoint and are concerned about including the full set of physical effects that might affect measurements because their experimental conditions are not usually as pristine as a theoretical model. Real-world effects, like damping, material motion, higher orders of dispersion, electrostriction, non-linearity, etc, can be important in a general setting, but these effects are obviously not going to fix the essential contradiction between the (local) electromagnetic conservation law and global conservation in the physical theory of the specified model system. Adding these higher-order effects to a provably flawed model is a meritless appeal to complexity in the face of a contradiction between theorems of the Maxwell-Minkowski equations. Those higher-order effects can be incorporated later to align the theory with experiments over a broad range of conditions once the contradiction is resolved.
Derivations of the electromagnetic momentum density continuity equation (momentum conservation law) typically begin with the Lorentz force law [10,[35][36][37]. The free charge momentum density p mech imparted by the field to a distribution of free charges in the continuum limit can be calculated by postulating the Lorentz force density [10,[35][36][37]
dp mech dt = f L = ρ f E + J f c × B (2.5)
as a physical law [39,40]. The sources are eliminated in favor of the fields using the Gauss law, Eq. (2.3c), to eliminate ρ f and using the Maxwell-Ampère law, Eq. (2.3a), to eliminate J f . Then the momentum density p mech imparted to the free-charge density can be calculated by integrating [10,[35][36][37]
ρ f E+ J f c ×B = (∇·D)E+ ∇ × H − 1 c ∂D ∂t ×B . (2.6)
Substituting the calculus identity
∂ ∂t (D × B) = ∂D ∂t × B + D × ∂B ∂t , (2.7)
Faraday's law, Eq. (2.3b), Thompson's law, Eq. (2.3d), and Gauss's law into Eq. (2.6) yields the momentum continuity equation [35][36][37]
ρ f E + J f c × B = (∇ · D)E+ (∇·B)H−D×(∇×E)−B×(∇×H)− 1 c ∂ ∂t (D×B) . (2.8)
The textbook derivation is simple and the steps have obvious physical meaning. The textbook derivation is not as rigorous as we would like because we are unnecessarily postulating the Lorentz force density law, Eq. (2.5) [10, 35-37, 39, 40]. We propose an alternative derivation of the energy and momentum continuity equations as formal theorems of the Maxwell-Minkowski equations. We take the scalar product of Eq. (2.3b) with H and the scalar product of Eq. (2.3a) with E and subtract the results to produce a continuity equation [41,42] 1
c E · ∂D ∂t + H · ∂B ∂t + ∇ · (E × H) = − J f c · E (2.9)
that is a valid theorem (Poynting's theorem) of the formal theory of continuum electrodynamics. Adding the vector product of B with Eq. (2.3a), the vector product of D with Eq. (2.3b), the product of Eq. (2.3d) with −H, and the product of Eq. (2.3c) with −E produces the momentum continuity equation
1 c ∂ ∂t (D × B) + D × (∇ × E) + B × (∇ × H) − (∇ · D)E − (∇ · B)H = −ρ f E − 1 c J f × B (2.10)
that is also a formal theorem of Maxwellian continuum electrodynamics [42]. The free charge density and the free charge current density are parameters that are determined by the specification of the system configuration. Then we can specify a system that consists of a neutral magneto-dielectric medium situated in the vacuum and illuminated by a finite quasimonochromatic field. The Maxwell-Minkowski equations, Eqs. (2.3), become homogeneous Maxwell-Minkowski equations,
∇ × H − ∂D ∂(ct) = 0 (2.11a) ∇ × E + ∂B ∂(ct) = 0 (2.11b) ∇ · D = 0 (2.11c) ∇ · B = 0 , (2.11d)
for a neutral simple linear medium in the absence of the free charge density ρ f and the free current density J f . Reproducing the derivation of Eqs. (2.9) and (2.10) using the homogeneous Maxwell equations Eqs. (2.11), we obtain the homogeneous electromagnetic continuity equations, i ) The energy continuity equation (Poynting's theorem) and the momentum continuity equations are identities of the Maxwell-Minkowski equations. The usual derivation [10,[35][36][37] as equations of motion of free charge density and free charge current density, Eq. (2.5) to Eq. (2.8), is not appropriate when applied to a neutral medium in which these densities do not exist. Therefore, the usual derivation as equations of motion of the free charge density and free charge current density is not appropriate, in general.
E · ∂D ∂(ct) + H · ∂B ∂(ct) + ∇ · (E × H) = 0 (2.12a) ∂ ∂(ct) (D × B) + D × (∇ × E) + B × (∇ × H) − (∇ · D)E − (∇ · B)H = 0 ,(2.
ii ) The Lorentz force law is not a postulate of Maxwellian continuum electrodynamics [39,40]. Instead, the Lorentz force density law, Eq. (2.5), is a relation that is derived as part of a theorem, Eq. (2.10), of the macroscopic Maxwell-Minkowski equations, Eqs. (2.3), using the requirement that the change in mechanical momentum is equal and opposite to the change in electromagnetic momentum in a conservative system.
iii ) The divergence of the Poynting vector appears in the energy continuity equations, Eq. (2.9) and (2.12a), so that Poynting's vector is considered arbitrary to the extent that the curl of any vector field can be added to it [35,37]. Except the energy continuity equation is derived as an identity of the Maxwell-Minkowski equations that do not admit an arbitrary vector field in that manner.
iv ) The charge continuity equation (charge conservation law)
∂ρ f ∂t + ∇ · J f = 0 (2.13)
can be derived by substituting Eq. (2.3c) into the divergence of Eq. (2.3a). A continuity equation (conservation law), see Sec. III, describes the unimpeded, inviscid, incoherent, incompressible flow of non-interacting particles in the continuum limit through otherwise empty space (vacuum). The presence of a density of interacting charged material particles flowing unimpeded through a continuous polarizable/magnetizable material medium is not consistent with the conditions for a spacetime continuity equation that is derived for noninteracting particles in the continuum limit traveling unimpeded in the vacuum, Sec. III. We let ρ f = 0 and J f = 0 in order to treat the fundamental case of propagation of the field through a neutral linear medium.
v ) The theoretical procedure can be applied to derive analogous energy and momentum equations for the microscopic fields as identities of the microscopic Maxwell equations, instead of as equations of motion for the free charge density and free charge current density in the vacuum. The comments about the continuity equations apply in similar form to the field in the vacuum.
As a matter of linear algebra, Eqs. (2.12) can be written row-wise as a differential equation [42]. We write Eq. (2.12b) in component form as [35]
∂(D × B) i ∂(ct) + j ∂ ∂x j W c ij = − εE 2 2 ∇ε ε − µH 2 2
∇µ µ using the constitutive relations, Eqs. (2.4). Then [35],
∂ β T αβ M = f α M (2.14a) T αβ M = 1 2 (D · E + B · H) (E × H) 1 (E × H) 2 (E × H) 3 (D × B) 1 W 11 W 12 W 13 (D × B) 2 W 21 W 22 W 23 (D × B) 3 W 31 W 32 W 33 (2.14b) W ij = −D i E j − B i H j + 1 2 (D · E + B · H)δ ij (2.14c) f α M = 0, − εE 2 2 ∇ε ε − µH 2 2 ∇µ µ (2.14d) ∂ β = ∂ ∂(ct) , ∂ ∂x , ∂ ∂y , ∂ ∂z (2.14e)
is a formal theorem of the homogeneous electromagnetic continuity equations, Eqs. In the limit that the gradient Minkowski four-force density f α M is negligible, Eq. (2.14a) becomes
∂ β T αβ M = 0 , (2.15)
which is known as the electromagnetic conservation law. An equivalent statement is that the Minkowski momentum is 'almost' conserved based on the identity, Eq. (2.14), in the case the Minkowski four-force can be treated as negligible [1][2][3][4][5]. Conservation of the Minkowski energy and Minkowski momentum is an obviously correct implementation of the electromagnetic conservation law, Eq. (2.15), and there is a large body of work that is based on conservation of the Minkowski momentum [1][2][3][4][5].
In contradiction, there is a large body of scientific work that proves that the Minkowski momentum is neither conserved nor almost conserved [1,[6][7][8]. The wave equation
∇ × (∇ × A) + n 2 c 2 ∂ 2 A ∂t 2 = ∇µ µ × (∇ × A) (2.16)
is also a theorem of the Maxwell-Minkowski equations, Eq. (2.11), with the constitutive relations, Eq. (2.4), and the Coulomb-gauge definition of the macroscopic fields
E = − ∂A ∂(ct) (2.17a) B = ∇ × A (2.17b)
in terms of the vector potential A. The Coulomb gauge is suitable for a sourceless medium, ρ f = 0 and J f = 0, allowing the scalar potential Φ to be suppressed. The derivation of the wave equation theorem, Eq. (2.16), consists of substituting Eqs. (2.4) and (2.17) into the homogeneous Maxwell-Ampère law, Eq. (2.11a). Repeated analyses of the wave equation and wave propagation, for over a century, have disclosed that the Minkowski electromagnetic momentum in an antireflection-coated transparent linear dielectric is greater that the incident momentum by a non-negligible multiplicative factor of n [1, [6][7][8]. Acknowledgment of this easily verified theoretical fact is present in the scientific record, where the Minkowski pull-force is the hypothetical source of this momentum difference and there is no need to repeat the wave propagation analyses here.
In order to be complete, but concise, we provide a short demonstration using global conservation of energy to prove that Minkowski linear momentum is not conserved in a linear dielectric [1,[6][7][8]. For a monochromatic field of frequency ω f with refractive index n(ω f ) in which the vector potential amplitude of the incident field is A 0 , the Minkowski energy density [35][36][37] of the field in the medium is
u M = 1 2 (D · E + B · B) = ω 2 f n 2 2c 2 |A| 2 = ω 2 f n 2c 2 |A 0 | 2 (2.18)
in the plane-wave limit. Due to the reduced velocity of light in the dielectric, a quasimonochromatic field in the plane-wave limit has an extent along the propagation direction (the longitudinal width) that differs from the longitudinal extent of the incident field w by a factor of 1/n [42]. The Minkowski energy of a quasimonochromatic field of cross-sectional area σ
U M = 1 2 Σ (D · E + B · B)σdz = ω 2 f wσ 2c 2 |A 0 | 2 (2.19)
is constant in time as the field propagates from the vacuum (longitudinal field width w) and into the dielectric (width w/n) through a gradient-index antireflection coating in the plane-wave limit. For the same quasimonochromatic field, the Minkowski momentum is
G M = Σ D × B c σdz = ω 2 f nwσ 2c 2 |A 2 0 |k (2.20)
based on the Minkowski momentum density
g M = D × B c = ω 2 f n 2 2c 2 |A 2 0 |k . (2.21)
Comparing the formula for the Minkowski momentum, Eq. (2.20), with the formula for the conserved energy, Eq. (2.19), on the basis of the vector potential magnitude shows that the momentum of the electromagnetic field in the medium is not globally conserved by a factor of n for a finite field, even though this contradicts the (local) electromagnetic conservation law, Eq. (2.15). The fact that a theorem of Maxwellian continuum electrodynamics is proven false by another theorem of Maxwellian continuum electrodynamics proves that one or more of the axioms of the formal theory, the Maxwell-Minkowski equations, the constitutive relations, and the vector potential relations, are false. Incomplete is also false, nuanced false, but false nevertheless. The extant resolution of the Abraham-Minkowski controversy consists of adding a phenomenological material-motion energy-momentum tensor to a Maxwellian electromagnetic energy-momentum tensor (or adding a phenomenological material-motion momentum to a Maxwellian electromagnetic momentum) [1,2,6,[9][10][11][12][13][14][15][16][17]. The resolution is a tautology: the whole is the sum of the parts. However, the Maxwellian electromagnetic subsystem is still incomplete because the Maxwell-Minkowski equations are not coupled to the material subsystem equations of motion. Likewise, the material equations of motion remain incomplete. Instead of completing the subsystem equations of motion for both subsystems, the electrodynamic energy-momentum tensor is superficially coupled to the energy-momentum tensor for the material through the transient force term, f α M , of an arbitrarily long field.
The medium is typically modeled as dust [1], an unimpeded, inviscid, incoherent, incompressible flow of noninteracting particles of mass-bearing matter in the continuum limit through empty space. The total energy and total momentum are known quantities because the energy and momentum of the incident field are known. Then conservation of total energy and conservation of total momentum are used to derive the adjustable material parameters, the particle density and velocity [1]. We will see below that a microscopic model of the medium is not required because the total energy and total momentum are known by global conservation because the incident energy and the incident momentum are specified.
The material-motion momentum that supplements the Minkowski electromagnetic momentum is identified by Barnett and Loudon [15] as the material canonical momentum G medium canonical such that
G tot = G M + G medium canonical (2.22)
is the total momentum G tot . In the context of continuum electrodynamics, whatever microstructure of the material and field that exists in nature is treated in the continuum limit so that only the continuous linear response is left. Then, the particular microscopic model of the linear medium cannot matter and the material canonical momentum is given as G medium canonical = G tot − G M , where the total momentum G tot , the Minkowski momentum G M , and the material canonical momentum G medium canonical are all macroscopic quantities and are continuous at all length scales ( N n → dv) in the continuum limit. Using global conservation of total momentum in a closed system produces formulas for a total (field plus material) momentum [6]
G tot = Σ nE × B c σdz (2.23)
and a material canonical momentum
G medium canonical = Σ (n − n 2 )E × B c σdz (2.24)
based on the momentum of the incident field. The total (field plus material) momentum G tot was constructed by Gordon [6] to be constant in time for the field in a dielectric. However, Gordon uses the concept of pseudomomentum to reintroduce the extra factor of n in the total momentum. closed system and the Gordon total momentum successfully addresses the factor of n error in global conservation of linear momentum. The consensus resolution of the Abraham-Minkowski controversy is circular, accomplishing global conservation of linear momentum by fiat. A circular theory proves itself in the context in which it was derived. The total linear momentum, Eq. (2.23), that comes out of the system of subsystems treatment is provably correct because it was derived by global conservation principles [6]. For a linear dielectric medium, the penultimate result of the system of subsystems approach is the total (field plus material) energy-momentum tensor [1,21]
T αβ tot = 1 2 (n 2 E · E + B · B) (nE × B) 1 (nE × B) 2 (nE × B) 3 (nE × B) 1 W 11 tot W 12 tot W 13 tot (nE × B) 2 W 21 tot W 22 tot W 23 tot (nE × B) 3 W 31 tot W 32 tot W 33 tot (2.25a) W ij tot = −n 2 E i E j −B i B j + 1 2 (n 2 E·E+B·B)δ ij . (2.25b)
The total energy U tot = Σ T 00 tot σdz and the total momentum G tot = Σ (T 01 tot , T 02 tot , T 03 tot )σdz are demonstrably constant in time for our model system. However, substituting the total energy-momentum tensor, Eq. (2.25a), into the local electromagnetic conservation law (see Eq. (3.1) and compare Eq. (2.15))
∂ β T αβ tot = 0 , (2.26) one obtains ∂ ∂(ct) 1 2 (εE · E + B · B) + ∇ · (nE × B) = 0 (2.27)
for the α = 0 component. Equation (2.27) violates Poynting's theorem and the equation is self-inconsistent because the non-zero terms depend on different powers of n. Then the consensus resolution of the Abraham-Minkowski controversy in terms of the total (field plus material) energy-momentum tensor (or the total (field plus material) momentum) is demonstrably false, even though important portions have been proven true. The material subsystem conjecture has been disproved by showing that the total (field plus material) energymomentum tensor that heals the violation of the global conservation law introduces violations of the spacetime (local) conservation law (including Poynting's theorem). Although cast in terms of the Minkowski energymomentum tensor, the disproof works equally well with the Abraham energy-momentum tensor because the total (field plus material) energy-momentum tensor T αβ tot , Eq. (2.25a), is the same in both cases [1].
Because the Maxwell-Minkowski model is assumed to be incomplete, one can propose other physically motivated subsystems in an attempt to resolve the conservation issue. Dispersion has been suggested and phenomenologically added to the theoretical model [10]. The way our system is defined includes dispersion to lowest order so the inclusion of additional dispersion is an exercise in complexity for a second-order consequence. Because the total energy and the total momentum, including dispersion, are globally conserved, the total energymomentum tensor remains given by Eq. (2.25a), violating self-consistency, the Poynting theorem, and the local electromagnetic conservation law.
We can identify other inconsistent physics in the formal theory of continuum electrodynamics. In Ref. [43], the set of macroscopic field equations, The century-old momentum contradiction at the center of continuum electrodynamics stands very much unresolved. Moreover, issues with Maxwellian continuum electrodynamics now extend beyond angular momentum conservation and global linear momentum conservation to include consistency with the local energy conservation law, Poynting's theorem, and special relativity in a linear medium. The formal equivalence of incommensurate macroscopic field equations, Eqs. (2.11) and Eqs. (2.28), proves that the axioms of continuum electrodynamics, the Maxwell-Minkowski and constitutive equations, are manifestly false.
Einstein taught that fundamental physical principles are rooted in the vacuum. The vacuum was later formalized as an isotropic, homogeneous, flat, four-dimensional, Minkowski spacetime S M (ct, x, y, z). The microscopic Maxwellian model of a linear medium consists of tiny bits of matter embedded in the vacuum with the permittivity ε = 1 + χ e and the permeability µ = 1 + χ m defined in terms of the unit vacuum electric susceptibility, the unit vacuum magnetic susceptibility, the material electric susceptibility χ e , and the material magnetic susceptibility χ m . As long as the individual particles of the medium are localized and the interactions of each particle with the microscopic field are perturbative, the flat, fourdimensional, empty Minkowski spacetime is "regarded as the proper setting within which to formulate those laws of physics that do not refer specifically to gravitational phenomena" [44].
Optically transparent material are mostly empty space in which light travels at an instantaneous speed of c [45]. The tiny polarizable and magnetizable bits of matter that are embedded in the vacuum scatter and delay the light. Even if one intends to build a microscopic model of physical optics in Minkowski spacetime, there are far to many particles and far too many interactions to keep track of in 'real' materials. Consequently, in continuum electrodynamics, the phenomenological model of the medium is an abstraction that is continuous at all length scales from the very outset and the effective speed of light is c/n. The interstitial vacuum has no role in the continuum limit and a continuous medium with a macroscopic refractive index n cannot be re-discretized or un-averaged.
In this article, we use field theory to derive equations of motion for electromagnetic fields in continuous linear materials starting from identifiable and characterizable principles. We show that an isotropic, homogeneous, flat, four-dimensional, continuous non-Minkowski 'material' spacetime is the proper setting for continuum electrodynamics, conservation laws, special relativity, invariance, and other optical principles that take place in an isotropic, homogeneous, magneto-dielectric linear medium in which the effective speed of light is c/n. Each different isotropic, homogeneous, transparent, linear medium will be associated with a different continuous 'material' spacetime connected to other spacetimes by boundary conditions.
III. SPACETIME CONSERVATION LAWS
Special relativity, Laue's theorem [30,31], and Noether's theorem [32] constitute a powerful framework within which to analyze energy and momentum conservation of a continuous flow of light. In fact, so much of the physics is performed by the formalism that our problem with conservation of momentum in a simple linear medium is embedded in the re-application of the relativistic formalism of physics in a vacuum to a continuous medium.
The tensor energy-momentum formalism is wellknown when applied to continuum (fluid) dynamics. Before treating conservation laws in a linear medium, we review what is typically known about conservation laws in the vacuum of an otherwise empty Minkowski spacetime.
a) For a thermodynamically closed system, the local spacetime conservation law of the total system
∂ β T αβ tot = 0 (3.1)
is derived by applying the divergence theorem to a Tay-lor series expansion of the density field of the energy and momentum properties of an unimpeded, inviscid, incoherent, incompressible, flow of non-interacting particles (nonpolar fluid molecules, dust particles, or photons) in the continuum limit (fluid or light field (light fluid)) in an otherwise empty volume (vacuum) [23]. The local spacetime conservation law, Eq. (3.1), is a theorem of the field theory and is characteristic of a conserved flow in Minkowski spacetime S M (ct, x, y, z). The fourdivergence of the energy-momentum tensor must vanish as a condition for conservation of an unimpeded, inviscid, incoherent, incompressible flow of non-interacting particles in the continuum limit through empty space [23,24]. b) Under typical conditions, the energy density and the momentum density integrated over the total volume Σ of the thermodynamically closed system
U = Σ T 00 tot dv (3.2) G = 1 c Σ T 01 tot dv, 1 c Σ T 02 tot dv, 1 c Σ T 03 tot dv (3.3)
must be constant in time (global conservation). The system can be as large as is required to completely contain the matter and energy, but the boundaries of the closed system will still be definite (arbitrarily large). The conservation conditions, Eqs. (3.2) and (3.3), require no matter or energy crossing the boundary of the system as an initial condition (−∞ < t 0 ≤ t). (Zero-field boundary conditions for all time (−∞ < t < ∞) correspond to an empty or static system [31]). Examples of nonconservative systems for which the global conservation laws, Eqs. (3.2) and (3.3), fail include systems in which a source or sink is present, unbounded systems, subsystems of a complete system, and inconsistently defined systems. c) For typical conditions in which the energymomentum tensor of the initial flow is diagonally symmetric, or is transformed into a symmetric tensor, the energy-momentum tensor of a closed system must remain symmetric T αβ tot = T βα tot (3.4) in order to conserve angular momentum. This condition explicitly couples the rows of the energy-momentum tensor. Obviously, if the incident field contains angular momentum then the energy-momentum tensor of a conservative system will not be symmetric. It is possible to write, pro forma, a matrix-based differential equation from continuity equations of different systems or subsystems. Such a compound system is inconsistent and that is discovered by the non-symmetric matrix that results from the lack of coupling between the continuity equations. Pathological exceptions to symmetry may include non-symmetric initial and boundary conditions, unclosed systems (subsystems), inhomogeneous systems that include microstructure, non-isotropic systems, coordinate system changes, and inconsistently defined systems. Pathological conditions are not likely in the middle of free space, but the issue is presaged for the case of propagation of light from the vacuum into a simple linear medium where the non-symmetric Minkowski energy-momentum tensor has come to be viewed as acceptable.
d ) The trace of the energy-momentum tensor is the density of the fluid ρ = g αα T αα tot (3.5) with metric tensor g αβ for a non-pathological closed system.
e) The local conservation law is sometimes written as [1,9] ∂ α T αβ tot = 0 .
∂ β T αβ vac = 0 (3.7a) T αβ vac = 1 2 (E · E + B · B) (E × B) 1 (E × B) 2 (E × B) 3 (E × B) 1 W 11 vac W 12 vac W 13 vac (E × B) 2 W 21 vac W 22 vac W 23 vac (E × B) 3 W 31 vac W 32 vac W 33 vac (3.7b) W ij vac = −E i E j − B i B j + 1 2 (E · E + B · B)δ ij . (3.7c)
is a theorem of the energy and momentum continuity equations that are typically derived in electricity and magnetism/electrodynamics textbooks using the microscopic Maxwell equations for light fields in the vacuum [36]. Then Eq. (3.7a) is considered to be the electromagnetic conservation law based on the fact that Eq. (3.7) is a theorem of the fundamental (vacuum) Maxwell equations of electrodynamics, the similar appearance of Eqs. (3.7a) and (3.1), and a physical necessity argument that a closed system consisting of a finite quasimonochromatic field propagating in the vacuum is conserved. However, the principles of conservation are nowhere used in the derivation of Eq. (3.7) from the microscopic Maxwell equations and several important conditions are not incorporated into the derivation of Eq. (3.7). Therefore it is not strictly correct to identify Eq. (3.7) as 'the' electromagnetic conservation law unless the closed system satisfies all conservation laws, Eqs. (3.1)-(3.5), zerofield boundary conditions with the entire field contained within the boundaries of the system at a finite time in the past, and the predicate of unimpeded, inviscid, incoherent, incompressible flow of non-interacting photons in the continuum limit through empty space.
The spacetime conservation laws, Eqs. (3.1)-(3.5), are satisfied by a quasimonochromatic field propagating in the vacuum of free space in the plane-wave limit. This is easily demonstrated by substituting the elements of the vacuum-based energy-momentum tensor, Eq. (3.7b), into the conservation laws, Eqs. (3.1)-(3.5), with g αβ = diag(1, −1, −1, −1). Condition Eq. (3.5) shows that the trace of the energy-momentum tensor is zero corresponding to massless photons. Then Eq. (3.7) can indeed be considered to be the spacetime conservation law for the electrodynamics of fields in the vacuum.
Next, we switch from the propagation of electromagnetic fields in the vacuum to propagation in a linear medium. Consider the application of the spacetime conservation laws, Eqs. (3.1)-(3.5), to the propagation of a continuous light field in a continuous linear medium. Substituting elements of the Minkowski energy-momentum tensor, Eq. (2.14b), into the spacetime conservation laws, we find that the global momentum, Eq. (3.3), is not constant in time and that the symmetry law, Eq. (3.4), is violated, as expected based on the discussion in Sec. I. The recognized fix is to use global conservation to supplement the macroscopic Minkowski energy-momentum tensor with a phenomenological material motion energy-momentum tensor to create a total, field plus matter, energy-momentum tensor. Substituting elements of the total, field plus matter, energymomentum tensor, Eq. (2.25a), into the conservation laws, we find that the α = 0 element of the local spacetime conservation law, Eq. (3.1), reproduces Eq. (2.27) that is self-inconsistent and violates Poynting's theorem. The local conservation law and the global conservation law are inconsistent in this case because a continuous linear medium does not meet the condition of an otherwise empty volume for the application of the conservation laws.
IV. LAGRANGIAN DENSITY
Substituting the elements of the macroscopic Minkowski energy-momentum tensor, which is derived as a theorem from the Maxwell-Minkowski equations, into the spacetime conservation laws Eqs. (3.1)-(3.5), proves that the macroscopic system violates conservation of angular momentum and violates conservation of linear momentum. Rather than start anew, the accepted approach has been to treat the system as incomplete and propose supplemental energy-momentum tensors. The complete energy-momentum tensor is known by using global conservation of energy and momentum to derive the necessary elements of the total energy-momentum tensor [1,21,42]. Substituting these elements into the local conservation law produces a false statement, Eq. (2.27). Then the Maxwell-Minkowski equations are manifestly false and the equations of motion of the total (field plus material) system are also false. Having proven the existing macroscopic theory to be false, we are starting with a clean slate for the construction of an entirely new formalism of continuum electrodynamics.
Theoretical physics in a simple linear medium that is continuous at all length scales is a problem that is multiply connected with a large variety of places to start. But if we enforce consistency at the boundaries between electrodynamics, relativity, invariance, spacetime, electromagnetic boundary conditions, etc, then we should arrive at the same set of results no matter where we start.
Lagrangian field theory is a generalization of particle dynamics to a continuous field [24,25]. The classical Lagrangian is
L = 1 2 Σ (T − V )dv , (4.1)
where T is the kinetic energy density, V is the potential energy density, and integration is performed over a closed system Σ. For the electromagnetic field in a source-free simple linear medium, the classical Lagrangian, Eq. (4.1), can be written as
L = 1 2 Σ εE 2 − B 2 /µ dv (4.2)
in the common Maxwell-Minkowski formulation of Maxwellian continuum electrodynamics [24,25,35,37]. The corresponding Lagrangian density is
L = 1 2 εE 2 − B 2 /µ = 1 2 ε ∂A ∂(ct) 2 − (∇ × A) 2 µ .
(4.3) Now, we can use [43,46] n e = √ ε (4.4) to denote the electric component of the refractive index and the magnetic component of the refractive index can be denoted by
n m = √ µ . (4.5)
The electric refractive index n e , like the electric permittivity ε, is clearly associated with the kinetic energy density T of the Lagrangian. The magnetic refractive index n m and the magnetic permeability µ are clearly associated with the potential energy density component V of the Lagrangian. Using simple algebra, the classical Lagrangian, Eq. (4.2), can be written as
L = 1 2 Σ n e c ∂A ∂t 2 − ∇ × A n m 2 dv . (4.6)
The Lagrangian density,
L = 1 2 n e c ∂A ∂t 2 − ∇ × A n m 2 , (4.7)
is the integrand of the Lagrangian, Eq. (4.6). We consider an arbitrarily large region of space to be filled with an isotropic, homogeneous, transparent, continuous, linear magneto-dielectric medium that can be characterized by a macroscopic electric refractive index n e and a macroscopic magnetic refractive index n m . Treating dispersion in lowest order, the electric and magnetic refractive indices will depend on the center frequency of the quasimonochromatic field (or the frequency of a monochromatic field) that illuminates the medium.
We limit our attention to an arbitrarily large simple linear medium and we write a new time-like variablē
x 0 = ct n e (4.8)
and new spatial variables
x = n m x (4.9a) y = n m y (4.9b) z = n m z (4.9c)
based on the way the electric and magnetic indices of refraction appear in the Lagrangian density. Although we can retain the spatial and temporal dependencies of the components of the refractive index, in this work we have adopted the limit of an isotropic homogeneous medium in which these dependence's can be neglected in order to proceed with the fundamental physical issues. As always, we can treat a piecewise homogeneous medium by using the homogeneous theory plus boundary conditions. Further, we have specified conditions that allow dispersion to be treated to lowest order and velocity-dependent anisotropy to be neglected. We construct a 'material' Laplacian operator
∇ = ∂ ∂x , ∂ ∂ȳ , ∂ ∂z (4.10)
to be used in the abstract mathematical model of an arbitrarily large, isotropic, homogeneous simple linear medium. Substituting Eqs. (4.8)-(4.10) into Eqs. (4.6) and (4.7), we obtain a Lagrangian,
L = 1 2 Σ ∂A ∂x 0 2 − ∇ × A 2 dv , (4.11)
and a Lagrangian density, 12) in which the kinetic and potential terms are explicitly quadratic corresponding to a conservative system with real eigenvalues. In Lagrangian field theory, the Lagrangian density is not unique. In order to determine whether a given Lagrangian density is viable, we must derive the Lagrange equations of motion and determine whether the results agree with physical reality. Specifically, it should not be asserted that the hypothesis, Eq. (4.12), of our Lagrangian field theory is a priori wrong based on assumptions about the consequences before the theory is actually developed. Hundreds of years ago, it was wrong to assert that the hypothesis that parallel lines meet is manifestly false. Fortunately, non-Euclidian geometry managed to outlive its critics although several generations of its proponents expired before it was generally accepted. More recently, Einstein's theory of special relativity is a purely inductive theory that survived critics that advocated for 'obvious' or 'well-established' absolute simultaneity [47].
L = 1 2 ∂A ∂x 0 2 − ∇ × A 2 ,(4.
We take Eq. (4.12) as our hypothesis and apply field theory to systematically derive equations of motion for macroscopic fields in an arbitrarily large simple linear magneto-dielectric medium. We develop a cohesive physical theory of field theory-based electrodynamics, spacetime, relativity, tensor theory, and conservation laws for a region of space in which the speed of light is c/n, rather than c. The new theory is demonstrated to be in agreement with the physical world as is required.
In this work, the index convention for Greek letters is that they belong to {0, 1, 2, 3} and lower case Roman indices from the middle of the alphabet are in {1, 2, 3}. Cartesian coordinates (x 1 , x 2 , x 3 ) correspond to (x, y, z) as usual. The Einstein summation convention in which repeated indices on the same side of the equal sign are summed over is employed.
V. LAGRANGIAN EQUATIONS OF MOTION
The Lagrange equations for electromagnetic fields in the vacuum are [24,25] d dt
∂L ∂(∂A j /∂t) + i ∂ ∂x i ∂L ∂(∂A j /∂x i ) = ∂L ∂A j . (5.1)
We multiply and divide the first term of Eq. (5.1) by c/n e and the second term by n m . Using the re-parameterized temporal and spatial coordinates, Eqs. (4.8) and (4.9), the preceding equation corresponds to
d dx 0 ∂L ∂(∂A j /∂x 0 ) + i ∂ ∂x i ∂L ∂(∂A j /∂x i ) = ∂L ∂A j(∂L ∂(∂A j /∂x 0 ) = ∂A j ∂x 0 (5.3a) ∂L ∂A j = 0 (5.3b) i ∂ ∂x i ∂L (∂ i A j /∂x i ) = [∇ × (∇ × A)] j . (5.3c)
Substituting the individual pre-evaluated terms, Eqs. (5.3), back into Eq. (5.2), the equations of motion for the electromagnetic field in a simple linear magnetodielectric medium are the three orthogonal components of the vector wave equation
∇ × (∇ × A) + ∂ ∂x 0 ∂A ∂x 0 = 0 . (5.4)
The second-order equation, Eq. (5.4), can be written as a set of first-order differential equations. Writing portions of the wave equation as
Π = ∂A ∂x 0 (5.5a) β β =∇ × A (5.5b)
introduces macroscopic fields Π and β β. The macroscopic field variable Π, Eq. (5.5a), is the canonical momentum field [24] whose components were derived as Eq. (5.3a). We substitute the canonical momentum field Π, Eq. (5.5a), and the magnetic field β β, Eq. (5.5b), into the wave equation, Eq. (5.4), to obtain
∇ × β β + ∂Π ∂x 0 = 0 ,(5.6)
which is similar in form to the Maxwell-Ampère law in the Maxwell-Minkowski representation of continuum electrodynamics, but in a non-Minkowski 'material' spacetime. Applying the 'material' divergence operator (∇·) to Eq. (5.5b), we obtain
∇ · β β = 0 . (5.7)
Applying the 'material' curl operator (∇×) to Eq. (5.5a) produces a version of Faraday's Law,
∇ × Π − ∂β β ∂x 0 = 0 . (5.8) Finally,∇ · Π = 0 (5.9)
is a modified version of Gauss's law that is obtained by integrating the material divergence of Eq. (5.6) with respect to the new timelike coordinatex 0 . This completes the set of first-order equations of motion, Eqs. (5.6)-(5.9), for macroscopic fields in an arbitrarily large, isotropic, homogeneous, simple linear magneto-dielectric medium.
Grouping the field equations, Eqs. (5.6)-(5.9), for clarity and convenience, we havē
∇ × β β + ∂Π ∂x 0 = 0 (5.10a) ∇ · β β = 0 (5.10b) ∇ × Π − ∂β β ∂x 0 = 0 (5.10c) ∇ · Π = 0 (5.10d)
as the equations of motion for macroscopic electromagnetic fields in an arbitrarily large isotropic, homogeneous, simple linear magneto-dielectric medium. Equations (5.10) appear to violate Einstein's relativity, except Einstein's relativity was derived for events in the vacuum of Minkowski spacetime. On the other hand, Eqs. (5.10) are fully consistent with Rosen's dielectric special relativity [28] for an observer in the non-Minkowski continuous 'material' spacetime that is associated with a linear medium [26,28], see Sec. VI. The version of dielectric special relativity that was derived by Laue [27] using the relativistic velocity sum rule applies to an observer in a Laboratory Frame of Reference that is in the vacuum (or tenuous terrestrial atmosphere) that surrounds the dielectric [26].
Equations (5.10) will also apply to a piecewisehomogeneous material with Fresnel boundary conditions [46]. The vacuum Maxwell equations correspond to a special case of Eqs. (5.10) in Minkowski spacetime.
The continuum limit is a theoretical abstraction in which the linear medium is isotropic, homogeneous, and continuous at all length scales from the very outset. This is represented in field theory by defining both the Lagrange equations, Eq. (5.2), and the Lagrangian density, Eq. (4.12), for the non-Minkowski continuous 'material spacetime' S c (x 0 ,x 1 ,x 2 ,x 3 ), see Sec. VII. Because each linear material is associated with its own material spacetime, there is no need for independent material parameters like the permittivity and permeability (except as boundary conditions to identify and relate different spacetimes). Then, the Maxwell-Minkowski equations cannot be derived as an identity of Eqs. (5.10) despite having derived the similar appearing Eqs. (2.28) as an identity of the Maxwell-Minkowski equations in Refs. [21,43].
Fundamental physical processes are derived and defined for the vacuum. The microscopic Maxwell equations are fundamental laws of electrodynamics in the vacuum. Maxwellian continuum electrodynamics, which is obtained by adding a phenomenological material response as a perturbation of the vacuum Maxwell equations, has been known to be inconsistent with conservation laws for over a century and was proven to be manifestly false in Sec. I. Therefore, no formula, theorem, or other result of Maxwellian continuum electrodynamics can be used to disprove the new field equations, Eqs. (5.10), that are derived in an isotropic, homogeneous, flat, four-dimensional, non-Minkowski, continuous spacetime S c (x 0 ,x 1 ,x 2 ,x 3 ). Other fundamental physical processes like relativity and conservation are likewise rooted in the vacuum and phenomenologically transported into the continuous linear medium with a view to consistency with Maxwellian continuum electrodynamics. Consequently, these processes cannot be used to disprove the new field equations, Eqs. (5.10), either. The process of recasting these fundamental principles for a medium that is continuous at all length scales from the outset has begun and we can report success in demonstrating that Eqs. (5.10) are consistent with the Fresnel relations [46] and special relativity in a dielectric [26].
VI. SPECIAL RELATIVITY IN A MAGNETO-DIELECTRIC MEDIUM
In adapting Einstein's special theory of relativity to a dielectric medium, Laue [27] applied the relativistic velocity sum rule to a dielectric material moving uniformly in the vacuum-based local Laboratory Frame of Reference. Some four decades later, Rosen [28] considered a continuous dielectric medium that is sufficiently large that the vacuum is inaccessible from the interior (in the time it takes for an experiment to be performed) and derived a second form of dielectric special relativity by a phenomenological replacement of the speed of light by c/n. The phenomenological Rosen theory and its consequences are mostly ignored in the scientific literature and there is little or no discussion about the incompatibility of the two contradictory theories of relativity in a dielectric [26].
In Ref. [26], the current author used two inertial reference frames in uniform motion to prove that the two forms of dielectric special relativity are incommensurate and that both are correct, but are correct in different physical contexts. Placing the common origin of the inertial reference frames on the interface between a semiinfinite medium and the vacuum and restricting the direction of relative motion to the interface [26], it was found that the Laue [27] theory, with 'vacuum' Lorentz factor
γ v = 1 1 − v 2 /c 2 ,(6.1)
relates electrodynamics inside the material to a vacuumbased local Laboratory Frame of Reference outside of the material. In contrast, the Rosen [28] theory, with 'material' Lorentz factor,
γ d = 1 1 − εv 2 /c 2 , (6.2)
applies if both inertial reference frames are within an arbitrarily large isotropic, homogeneous, simple linear dielectric medium. In this section, we extend the theory of dielectric special relativity [26] to include simple linear magneto-dielectric media. We consider two inertial frames of reference,S(x,ȳ,z) andS (x ,ȳ ,z ), in a standard configuration. The origins of the reference frames are located inside an arbitrarily large isotropic homogeneous simple linear magnetodielectric medium. The origins of the two systems coincide at time t 0 = 0 and all clocks are synchronized. At time t c = t c = 0, a light pulse is emitted from the common origin along the positiveȳ andȳ −axes. In theS frame of reference, Fig. 1, the pulse is reflected by a mirror in the medium atȳ = D c and returns to the origin at time ∆t c = 2D c /c c , where c c is the speed of light in the medium. The trajectory of the light pulse in theS frame of reference is shown in Fig. 2. The translation of theS frame is transverse to theȳ-axis so the distance from the mirror at m c to thex -axis is D c , the same as the distance from the mirror at m c to thex-axis. Viewed from theS frame, the light pulse is emitted from the point o at time t c = 0, is reflected from the mirror at point m c , and is detected at the point d c at time t c = ∆t c . During that time, the point of emission/detection has moved a distance v c ∆t c .
By the Pythagorean theorem, we have
(c c ∆t c ) 2 = (c c ∆t c ) 2 + (v c ∆t c ) 2 ,(6.3)
where we have used reflection symmetry about the midpoint. We write the previous equation as [26] ∆t c = ∆t c
c c 2 /c 2 c − v 2 c /c 2 c (6.4)
and define the 'material' Lorentz factor γ c by ∆t c = γ c ∆t c (6.5) such that
γ c = 1 c c 2 /c 2 c − v 2 c /c 2 c . (6.6)
In the Laue model, the speed of light in the dielectric depends on the velocity of the block in the Laboratory Frame of Reference. Here, the isotropy of an arbitrarily large homogeneous continuous dielectric medium at rest in the local frame of reference leads us to postulate that light travels at a uniform speed c c in the simple linear magneto-dielectric medium, basically the same reasoning that led to the Einstein postulate. Substituting
c c = c c (6.7)
into Eq. (6.6), one obtains
γ c = 1 1 − v 2 c /c 2 c . (6.8)
Using the definitions of the renormalized coordinates, we have v c = n m v and c c = c/n e . Then,
γ c = 1 1 − n 2 e n 2 m v 2 /c 2 = 1 1 − n 2 v 2 /c 2 = 1 1 − v 2 /c 2 (6.9
) for a simple linear magneto-dielectric mediumc = c/n. There is a different material Lorentz factor for each isotropic homogeneous simple linear magneto-dielectric medium. The vacuum Lorentz factor, Eq. (6.1), is a special case of Eq. (6.9) for n e = n m = 1. The Rosen dielectric Lorentz factor, Eq. (6.2), is a special case of Eq. (6.9) for n m = 1.
VII. SPACETIME SETTING
If a light pulse is emitted from the origin at a time t = 0 into the empty vacuum of free space then spherical wavefronts are defined by
x 2 + y 2 + z 2 = (x 0 ) 2 (7.1)
in a flat, four-dimensional, vacuum Minkowski spacetime S M (x 0 = ct, x, y, z). Equation (7.1) underlies classical electrodynamics and its relationship to Einstein's special relativity.
Consider a quasimonochromatic light pulse that is emitted from the origin (x = 0,ȳ = 0,z = 0) at 'time' x 0 = 0 into an isotropic, homogeneous, simple linear magneto-dielectric medium, instead of the vacuum. In this medium, spherical wavefronts are defined bȳ x 2 +ȳ 2 +z 2 = (x 0 ) 2 (7.2) in an isotropic, homogeneous, flat, four-dimensional, non-Minkowski continuous material spacetime S c (x 0 ,x,ȳ,z). There will be a different material spacetime that is associated with each set of refractive indices, n e and n m . The four-dimensional 'material' light conē
x 2 +ȳ 2 +z 2 − (x 0 ) 2 = 0 (7.3)
is embedded in the flat four-dimensional non-Minkowski 'material' spacetime S c (x 0 ,x,ȳ,z) that is associated with a linear, isotropic, homogeneous, simple linear magneto-dielectric medium. The basis functions, exp(−i(ω/c)(x 0 −k ·r)) + c.c., wherer = (x,ȳ,z), define the null surface,x 0 =k ·r. Here,ω can be associated with n e ω. material light cone will always be α = π/4 in that spacetime. The unit slope of the null in thex 0 −x plane of the non-Minkowski material spacetime is related to the coordinate speed of light in an isotropic, homogeneous, simple linear medium by
dx dt = dx dx 0 dx 0 dt dx dx = 1· c n e · 1 n m = c n e n m = c n =c . (7.4)
This equation shows that the effective speed of light in a linear magneto-dielectric medium is attributable to two different effects: i) the renormalization of the timelike coordinate by n −1 e and ii) the renormalization of the spatial coordinates by n m .
The usual characterization of events inside the vacuum light cone as timelike and events outside the vacuum light cone as spacelike also applies to the 'material' spacetime with events inside the renormalized 'material' light cone being timeline and events outside are spacelike.Čerenkov radiation, spontaneous emission, and mass-bearing particle dynamics in a simple linear medium will have to be treated carefully, if at all, because an atom or a charged particle must displace some of the linear medium that is effectively continuous at all length scales.
VIII. LORENTZ TRANSFORMATION PROPERTIES
Maxwellian continuum electrodynamics is based on averaging the interaction of microscopic fields with tiny bits of polarizable and magnetizable matter embedded in the vacuum. In the vacuum, Lorentz transformations of the coordinates follow from the invariance of in the continuum approximation and to apply Lorentz invariance to 'real' magneto-dielectric materials that are comprised almost entirely of empty space. Except, a macroscopic linear medium is defined as being continuous at all length scales from the very outset because the performing of such a macroscopic average from microscopic particles, fields, and interactions requires assumptions, approximations, and limiting behavior that become dubious in their complexity. Consequently, the assumption of Lorentz invariance for a macroscopic linear medium appears to be well-founded, based on the microscopic model, but it is manifestly not well-founded for macroscopic fields in continuous linear media. By now, we should know that Lorentz invariance is not a symmetry of light in a macroscopic linear medium [29] and we cannot re-discretize or un-average the physics in order to impose Lorentz invariance. Instead, the invariance of
s 2 = (x 0 ) 2 − x 2 − y 2 − z 2 .(s) 2 = (x 0 ) 2 − (x 1 ) 2 − (x 2 ) 2 − (x 3 ) 2 (8.2)
for monochromatic light in an arbitrarily large, isotropic, homogeneous simple linear medium imposes the conditions for linear medium-specific Lorentz-like transformations of the coordinates (x 0 ,x 1 ,x 2 ,x 3 ) of a flat four-dimensional non-Minkowski material spacetime S c (x 0 ,x,ȳ,z). The special Lorentz-like transformations (with v parallel to thex-axis) take the form
x 0 = γ c x 0 − (n e n m v/c)x 1 = γ c x 0 − (nv/c)x 1 (8.3a) x 1 = γ c x 1 − (n e n m v/c)x 0 = γ c x 1 − (nv/c)x 0 (8.3b) x 2 =x 2 (8.3c) x 3 =x 3 (8.3d)
in a simple linear medium. The index-dependent Lorentz-like transformation confirms the observations of Ravndal [29] that the invariance properties of a linear medium differ from Lorentz invariance of the vacuum.
IX. TENSOR FORMULATION
With the advent of special relativity, the 3-vector formulation of electrodynamics became archaic. The tensor formalism allows the development of electrodynamics in the context of general properties of physical laws on Minkowski spacetime in a form that is manifestly invariant under Lorentz transformations. "Increasing the sophistication of the notation simplifies the appearance of the governing equations, revealing hidden symmetries and deeper meaning in the equations of electromagnetism" [48].
The fundamental laws of physics are formulated in the vacuum and Minkowski spacetime is empty. That is not considered to be a problem for continuous media because 'real' matter is mostly empty space. Except, Feynman's [49] pedagogy makes it clear that there is a place in physics for macroscopic descriptions of fields and matter. Physical theory that is formulated for enumerated localized microscopic particles and microscopic fields interacting in a vacuum and defined on an empty Minkowski spacetime is always correct, but it is manifestly unjustified for macroscopic fields in an effective medium that is isotropic, homogeneous, and continuous at all length scales from the very outset.
In this section, we will develop an expressly macroscopic tensor formulation of continuum electrodynamics based on the macroscopic field equations, Eqs. (5.10).
It is straightforward to use algebra and calculus in order to construct the energy-momentum tensor as a theorem of the macroscopic field equations. The derivation of the energy continuity equation follows the same procedure that was used to derive Poynting's theorem, Eq. (2.9), as an identity of the Maxwell-Minkowski equations. We combine Eqs. (5.10) to derive a theorem of macroscopic continuum electrodynamics in the form the scalar energy continuity equation ∂u c ∂x 0 +∇ · s c = 0 , (9.1) in the continuous material spacetime where
u c = 1 2 (Π 2 + β β 2 ) (9.2)
is the continuous energy density and
s c = cβ β × Π (9.3)
is the continuous 'material' energy-flux vector. Similarly we combine Eqs. (5.10) to form the vector momentum continuity theorem
∂g i c ∂x 0 + j ∂ ∂x j W c ij = 0 (9.4) in component form, where g c = β β × Π c (9.5)
denotes the 'material' momentum density and
W c ij = −Π i Π j − β i β j + 1 2 (Π 2 + β β 2 )δ ij (9.6)
are the elements of a rank 3 matrix. Readers that are not familiar with the procedure used to derive Eq. (9.4) can consult Ref. [10], Sec. 6.8 of Ref. [35], or similar reference. The scalar energy continuity equation, Eq. (9.1), and the scalar components of the vector momentum continuity equation, Eqs. (9.4), can be written, row-wise, as a single differential equation [42] ∂ β T αβ c = 0 (9.7)
as a matter of linear algebra, wherē
∂ β = ∂ ∂x 0 ,∇ (9.8)
is the 'material' four-divergence operator. The 'tensor' differential equation, Eq. (9.7), is a theorem of the macroscopic field equations, Eqs. (5.10), and the diagonally symmetric matrix T αβ c is
T αβ c = u s 1 c /c s 2 c /c s 3 c /c cg 1 c W 11 c W 12 c W 13 c cg 2 c W 21 c W 22 c W 23 c cg 3 c W 31 c W 32 c W 33 c ,(9.9)
by construction. Obviously, the intent is to identify W ij c as the continuum electrodynamic stress tensor, to identify T αβ c as the continuum electrodynamic energymomentum tensor, and to identify the differential equation, Eq. (9.7), with the local electromagnetic conservation law.
We can re-formulate most of the other features of tensor electrodynamics, c.f., Sec. 11 of Ref. [35]. We construct the field strength tensor
F αβ c = 0 Π x Π y Π z −Π x 0 −β β z β β y −Π y β β z 0 −β β x −Π z −β β y β β x 0 (9.10)
and the dual field-strength tensor
F αβ c = 0 −β β x −β β y −β β z β β x 0 −Π z Π y β β y Π z 0 −Π x β β z −Π y Π x 0 (9.11) such that∂ α F αβ c = 0 (9.12a) ∂ α F αβ c = 0 (9.12b)
constitute an identity of the macroscopic field equations, Eqs. (5.10). The energy-momentum tensor, Eq. (9.9), (9.13) and the Lagrangian density, Eq. (4.12), In 1953, Balazs [33] proposed a thought experiment to resolve the Abraham-Minkowski controversy. The thought experiment was based on the law of conservation of momentum and a theorem that the center of massenergy moves at a uniform velocity [50]. The application of this theorem indicates that microscopic constituents of the material that carry mass also travel with the field.
T αβ c = F αµ F β µ + 1 4 g αβ F µσ F µσL c = 1 2 Π 2 − β β 2 = − 1 4 F αβ F αβ
The relativistic total energy E = p · pc 2 + m 2 c 4 1/2 (10.1) becomes the Einstein formula E = mc 2 for massive particles in the limit v/c → 0. For massless particles, like photons, Eq. (10.1) becomes
p = E cê k = ω 0 cê k ,(10.2)
whereê k is a unit vector in the direction of motion. Equation (10.2) defines the instantaneous momentum of a photon traveling at speed c. Consider a photon that enters a material that is composed of electric and magnetic dipoles embedded in the vacuum. The photon travels at speed c between scattering events [49], but due to scattering the effective speed of the photon in the incident direction is c/n. Due to the reduced effective speed of individual photons, an unimpeded, inviscid, incoherent, incompressible flow of noninteracting photons in the continuum limit travels at an average speed of c/n and the longitudinal extent of the field in the direction of the flow is reduced by a factor of n. Then the photon density, the energy density, and the momentum density are increased by a factor n. Integrating the quantities over the reduced longitudinal width of the field, the energy and momentum are constant as the field leaves vacuum and enters a dielectric medium without needing to assume any material motion. Due to the decreased velocity and increased density, the energy velocity of the ensemble is c = n · c/n. Then the energy velocity of a field into and through a linear medium moves at a constant speed c because of the enhanced photon density. The mass-polariton (MP) model [51] of propagation of the electromagnetic field in which the propagating field combines with mass-bearing particles of a continuous dielectric is not supported. Because the density of photons is larger due to the reduced average velocity, the higher density of photons corresponds to an increase in energy density and momentum density in the macroscopic field. Then, the macroscopic field in a linear medium that corresponds to the energy of a single photon occupies a smaller volume than the volume occupied in the vacuum. Likewise, a smaller volume of the field is associated with the momentum of a single photon. An additional issue with the photon description of light propagation in a continuous dielectric is illustrated by the commingling of macroscopic fields and the macroscopic refractive index with microscopic photon momentum and momentum states in a description of photon recoil momentum in a medium [52].
As an electromagnetic field propagates from vacuum into a simple linear medium, the 'effective' velocities of photons in the field are reduced creating an enhancement of the classical energy density u c = (Π 2 + β β 2 )/2 and the classical momentum density g c = β β × Π/c, compared to the vacuum. For finite pulses in a dielectric, the enhanced energy density is offset by a narrowing of the pulse so that the electromagnetic energy
U total = Σ 1 2 Π 2 + β β 2 c dv ,(10.3)
is time independent for quasimonochromatic fields in the plane-wave limit. The electromagnetic energy is the total energy by virtue of being constant in time. Likewise, the electromagnetic momentum,
G total = Σ β β × Π c dv ,(10.4)
is time independent and is the total momentum. Invoking the Einstein mass-energy equivalence, it is argued that some microscopic constituents of the dielectric must be accelerated and then decelerated by the field; otherwise the theorem that the center of mass-energy moves at a constant velocity is violated [14]. For a distribution of particles of mass m i and velocity v i , the total momentum
P total = i m i v i (10.5)
is the sum of the momentums of all the particles i in the distribution. If the mass of each particle m i is constant, the statement that the velocity of the center of mass
v CM = i m i v i i m i (10.6)
is constant is a statement of conservation of total momentum.
Because of the enhanced momentum density of the field in a dielectric, the differential of electromagnetic momentum
δp = β β × Π c δv (10.7)
that is contained in an element of volume δv (a 'particle'), is a factor of n greater than in the vacuum. Then the mass-energy of each 'particle' m i is not constant as would be required for the center-of-mass theorem. For a finite pulse, the narrower pulse width offsets the enhanced momentum density allowing the macroscopic electromagnetic momentum, like the macroscopic electromagnetic energy, to be constant in time as the field enters, and exits, the simple linear medium through the gradientindex antireflection coating. Consequently, there is no need to hypothesize mass-polariton quasiparticles [51] or any other material constituents of the continuous linear medium to be in motion in order to preserve the conservation of linear momentum. Even though the velocity of light slows to c/n, the hypothetical Minkowski pull-force is also disproved.
B. The Jones-Richards experiment
One of the enduring questions of the Abraham-Minkowski controversy is why the Minkowski momentum is so often measured experimentally while the Abraham form of momentum is so favored in theoretical work. We now have the tools to answer that question. The Minkowski momentum is not measured directly, but inferred from a measured index dependence of the optical force on a mirror placed in a dielectric fluid [1,15,34]. The force on the mirror is (10.8) which depends on the total momentum density, Eq. (10.4). If we were to assume F = 2dG M /dt, which is the relation between momentum and force in an otherwise empty Minkowski spacetime, then we would write Then one might use Eq. (10.9) of interpret the results of an experiment in such a way that the momentum density of the field in the dielectric fluid is the Minkowski momentum density.
F = d dx 0 (2cG total ) = d dx 0 V 2β β × Π δ(z)dv ,
The measured force on the mirror in the Jones-Richards experiment [34] is consistent with both Eqs. (10.8) and Eqs. (10.9), depending on what theory you use to interpret the results. Clearly an experiment that measures force, instead of directly measuring the change in momentum in the dielectric, will not conclusively distinguish the momentum density. Specifically, the Jones-Richards experiment does not prove that the Minkowski momentum density is the momentum density in the dielectric, as has been argued, nor does it prove that the continuum momentum density, Eq. (9.5), is the momentum density in the dielectric.
XI. SUMMARY
It has been said that physics is an experimental science and that physical theory must be constructed on the solid basis of observations and measurements. That is certainly true for serendipitous discoveries like x-rays and radioactivity; But Maxwell [53] used inductive reasoning to modify the Ampère law and construct the laws of electrodynamics two decades before Hertz [54] demonstrated the existence of electromagnetic waves. Later, Einstein's theory of relativity was criticized for violating the 'well-established' principle of absolute simultaneity [47]. The law of conservation of mass became the law of conservation of mass-energy long before any measure-ments of relativistic mass effects. Mathematics is the language of physics and there are many other examples (nonlinear optics, high-energy physics, negative refraction, etc.) in which theoretical physics leads experiments by a substantial period of time.
Axiomatic formal theory is a cornerstone of abstract mathematics. The contradiction of valid theorems of Maxwellian continuum electrodynamics proves, unambiguously, that Maxwellian continuum electrodynamics is false. Having proven Maxwellian continuum electrodynamics to be manifestly false, as it has been proven false by the Abraham-Minkowski momentum contradiction for over a century, we established a reformulation of theoretical continuum electrodynamics by deriving equations of motion for the macroscopic fields from a generalized Lagrangian field theory. For every simple linear medium there is a different set of equations of motion based in a different continuous 'material' spacetime with coordinates that are renormalized by the linear permittivity and linear permeability. The Abraham-Minkowski controversy is trivially resolved because the tensor total energy-momentum continuity theorem, the total energymomentum tensor, the total momentum, and the total energy are fully electromagnetic, unique, and conserved for a closed (complete) model system consisting of a simple linear dielectric block draped with a gradientindex antireflection coating that is illuminated by quasimonochromatic light.
(2.3), (2.4), and (2.17) is proven false, then one or more of the axioms are proven false and all other theorems that are derived from the axioms are unproven.
12b) that are formal theorems of the homogeneous Maxwell-Minkowski equations for a neutral magneto-dielectric linear medium. The derivations of the electromagnetic continuity equations, Eqs. (2.9) and (2.10), and the homogeneous electromagnetic continuity equations, Eqs. (2.12), are straightforward theorems of the Maxwell-Minkowski equations and constitutive relations. The derivations and results present some significant features:
(2.12), as well as a formal theorem of the homogeneous Maxwell-Minkowski equations, Eqs. (2.11). Specifically, the Minkowski energy-momentum tensor (matrix) T αβ M , Eq. (2.14b), the Minkowski energy density u = T 00 M , and the Minkowski momentum density g M = (T 10 M , T 20 M , T 30 M ) are formally derived from the axioms of continuum electrodynamics, the Maxwell-Minkowski and constitutive equations, as part of the theorem for the continuity equation, Eq. (2.14a).
∇ n m × (n m H) + ∂(−n e E) ∂(ct/n e ) = ∇n m n m n m × (n m H) (2.28a) derived as an identity of the homogeneous Maxwell-Minkowski equations, Eqs. (2.11), with constitutive relations, Eqs. (2.4), for a simple linear medium with macroscopic fields −n e E and n m H. The derivation [43] is simple, reproducible, and correct. Equations (2.28a)-(2.28d) are isomorphic to the vacuum Maxwell field equations with a timelike coordinate of ct/n e , instead of ct, and spatial coordinates (n m x, n m y, n m z) in the limit that the gradients of the permittivity and permeability may be neglected. Then, Eqs. (2.28) are inconsistent with Laue's implementation of Einstein's relativity in a continuous linear medium [27]. Clearly, there is an existential inconsistency associated with Eqs. (2.28) and (2.11) because a simple application of algebra and calculus changes the new field equations back to the Maxwell-Minkowski equations and the two expressions of the identity correspond to two different relativities with different timelike coordinates, x 0 = ct andx 0 = ct/n e . The Maxwell-Minkowski equations, Eqs. (2.3)-(2.4) and (2.28), are proven false by contradiction.
is typically true for a conserved system, derived by substituting Eq. (3.4) into Eq. (3.1). However, Eq. (3.6) cannot be considered a conservation law in the sense of Eq. (3.1) because it implicitly includes an additional condition, namely diagonal symmetry of the energy-momentum tensor.The conservation law, Eq. (3.1), is derived[23] using spacetime coordinates (ct, x, y, z) and it is manifestly not dependent on the Maxwell field equations. To be sure, the energy and momentum of an inviscid, incoherent, incompressible flow of non-interacting photons propagating unimpeded in the vacuum are conserved and therefore must be consistent with the spacetime conservation laws, Eqs. (3.1)-(3.5). Now,
FIG. 1 .
1Fig. 3is a depiction of the intersection of the 'material' light cone with thex 0 −x plane in the flat material spacetime showing the nullx 0 =x. There will be a different material spacetime for each pair of material constants, n e and n m , but the half-opening angle of the Path of light in the unprimed coordinate frame.
( 8 . 1 )FIG. 2 .
812Then it makes sense to apply Lorentz invariance to the microscopic theory before the interactions are averaged Path of light in the primed coordinate frame.FIG. 3. Null cone for light depicted in thex 0 −x plane of a flat, non-Minkowski material spacetime that corresponds to a linear magneto-dielectric medium.
demonstrated by substitution of the field strength tensor, Eq. (9.10). Combining the definitions Eq. (9.5) and Eq. (9.3) we obtain s = c 2 g . a continuous flow of massless particles. Substituting s = c 2 g into the energy-momentum tensor, Eq. (9.9), we have the symmetry property
Returning to Maxwellian continuum electrodynamics for a moment, the well-known Minkowski energymomentum tensor can be constructed by linear algebra from macroscopic energy and momentum continuity equations that are theorems of the Maxwell-Minkowski equations, but the Minkowski energy-momentum tensor is not symmetric. The Faraday law, Eq. (2.3b), in a linear medium has the same timelike coordinate as the Faraday law in the vacuum Minkowski spacetime, S v (x 0 = ct, z, y, z). In contrast, the Maxwell-Ampère law, Eq. (2.3a), has a renormalized temporal coordinate that we would associate with the non-Minkowski spacetime, S ? (ct/n 2 , z, y, z). These equations cannot be combined self-consistently to form valid energy and momentum continuity equations because they are based on different coordinate systems and belong in different spacetimes. The Minkowski energy-momentum tensor is not symmetric because the macroscopic Maxwell-Minkowski equations are inconsistently defined in a pathological spacetime. In contrast, all of the new macroscopic field equations are in the continuous 'material' spacetime, S c (x 0 ,x,ȳ,z) that is consistent with the electric and magnetic properties of the linear medium.X. EXPERIMENTAL CONFIRMATIONA. The Balazs thought experimentT αβ
c = T βα
c .
(9.17)
Using the symmetry property, Eq. (9.17), of the energy-
momentum tensor, we obtain
∂ α T αβ
c = 0
(9.18)
from Eq. (9.7).
∇ × E −
∂B
∂(ct)
= 0
(9.19)
∇ × B +
∂E
∂(ct/n 2 )
= 0
(9.20)
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| AbstrakSalah satu pekerjaan yang ada di dalam mengelola dokumen adalah bagaimana menemukan intisari dari dokumen. Topic modeling merupakan teknik yang dikembangkan untuk menghasilkan representasi dokumen berupa kata-kata kunci dari dokumen. Kata-kata kunci tersebut yang akan digunakan dalam proses pengindeksan serta pencarian dokumen untuk ditemukan kembali sesuai kebutuhan pengguna.Pada penelitian ini, akan dibahas secara spesifik mengenai Probabilistic Latent Semantic Analysis (PLSA). Pembahasan akan meliputi mekanisme bagaimana PLSA yang juga melibatkan algoritma Expectation Maximization (EM) sebagai pelatihan diterapkan pada sekumpulan data korpus, serta bagaimana melakukan uji coba dan memperoleh akurasi hasil penggunaan PLSA. | null | [
"https://arxiv.org/pdf/1512.00576v1.pdf"
]
| 13,178,524 | 1512.00576 | ff9c317f5d5bcbe3368996c189f487e29c202f98 |
Probabilistic Latent Semantic Analysis (PLSA) untuk Klasifikasi Dokumen Teks Berbahasa Indonesia DERWIN SUHARTONO Technical Report Fakultas Ilmu Komputer
Studi Program
Universitas
2014Indonesia Desember
Ilmu Doktor
Universitas
2014Indonesia Desember
Komputer
Universitas
2014Indonesia Desember
Probabilistic Latent Semantic Analysis (PLSA) untuk Klasifikasi Dokumen Teks Berbahasa Indonesia DERWIN SUHARTONO Technical Report Fakultas Ilmu Komputer
Technical Report Program Studi Doktor Ilmu Komputer Fakultas Ilmu Komputer Universitas Indonesia, Desember 2014Keyword: topic modellingProbabilistic Latent Semantic AnalysisExpectation Maximization
AbstrakSalah satu pekerjaan yang ada di dalam mengelola dokumen adalah bagaimana menemukan intisari dari dokumen. Topic modeling merupakan teknik yang dikembangkan untuk menghasilkan representasi dokumen berupa kata-kata kunci dari dokumen. Kata-kata kunci tersebut yang akan digunakan dalam proses pengindeksan serta pencarian dokumen untuk ditemukan kembali sesuai kebutuhan pengguna.Pada penelitian ini, akan dibahas secara spesifik mengenai Probabilistic Latent Semantic Analysis (PLSA). Pembahasan akan meliputi mekanisme bagaimana PLSA yang juga melibatkan algoritma Expectation Maximization (EM) sebagai pelatihan diterapkan pada sekumpulan data korpus, serta bagaimana melakukan uji coba dan memperoleh akurasi hasil penggunaan PLSA.
Pendahuluan
Teks merupakan salah satu media yang mampu mengkomunikasikan berbagai macam hal. Apabila dibandingkan dengan gambar, video, audio, animasi, dan lain sebagainya, maka teks merupakan bagian utama yang ada di dalam dokumen. Surat, majalah, dan koran merupakan contoh media yang menggunakan teks dan sudah banyak dikenal dengan baik oleh berbagai kalangan. Jika dilakukan observasi dan diamati lebih lanjut, dokumen teks yang sebelumnya dipublikasikan pada berbagai media cetak mulai beralih menjadi dokumen teks yang dipublikasikan dalam bentuk elektronik. Tentunya dengan pergantian bentuk dokumen tersebut, jumlah dokumen elektronik mengalami peningkatan jumlah yang cukup pesat. Kondisi ini mendorong pada munculnya keperluan akan pengelolaan dokumen elektronik yang baik.
Salah satu pekerjaan yang ada di dalam mengelola dokumen adalah bagaimana menemukan intisari dari dokumen. Hal ini terkait erat dengan melakukan perangkuman atau membuat representasi yang ringkas dari dokumen. Panjang dokumen sangat bervariasi, dari bentuk dokumen yang singkat/pendek seperti memo atau dokumen yang panjang seperti berita Technical Report Program Studi Doktor Ilmu Komputer Fakultas Ilmu Komputer Universitas Indonesia, Desember 2014 elektronik, buku, skripsi, tulisan ilmiah dan sejenisnya. Pencarian dokumen umumnya dilakukan dengan menggunakan kueri berupa kata, frase, kalimat atau judul dokumen yang dibutuhkan.
Apabila pencarian dilakukan dengan pencocokan kata per kata dari kueri ke dokumen, tentu akan memakan waktu komputasi yang lama. Hal ini mendorong pada keperluan dalam pembentukan representasi dari dokumen yang menghilangkan kata-kata yang tidak mewakili arti dari dokumen serta mempertahankan kata-kata yang memiliki nilai tinggi sehingga dianggap bisa mewakili arti dari dokumen aslinya.
Topic modeling merupakan teknik yang dikembangkan untuk menghasilkan representasi dokumen berupa kata-kata kunci dari dokumen. Kata-kata kunci tersebut yang akan digunakan dalam proses pengindeksan serta pencarian dokumen untuk ditemukan kembali sesuai kebutuhan pengguna. Latent Semantic Analysis (LSA) muncul sebagai teknik pertama yang bisa menghasilkan representasi dokumen berupa kumpulan kata-kata. LSA merupakan metode yang paling banyak dikenal dengan melekatkan ciri Bag-of-Words (Landauer, Foltz, dan Laham, 1998)
Landasan Teori
Probabilistic Latent Semantic Analysis
Sebelum kemunculan Probabilistic Latent Semantic Analysis (PLSA) pada tahun 1999, 1 (satu) tahun sebelumnya, Landauer, Foltz dan Laham (1998) menciptakan satu teknik yang dinamakan Latent Semantic Analysis (LSA). LSA adalah sebuah teknik statistik terotomasi untuk membandingkan kesamaan semantik dari beberapa kata atau beberapa dokumen. LSA bukanlah sebuah program yang tradisional dari natural language processing ataupun artificial intelligence.
Teknik ini digunakan dalam menganalisis dokumen untuk menemukan arti atau konsep dari dokumen tersebut. LSA lahir karena didasarkan pada pemikiran bahwa syntaxdan style saja tidak mencukupi untuk menilai sebuah esai. Yang menjadi kesulitan mendasar adalah ketika kita hendak membandingkan kata-kata untuk menemukan dokumen yang relevan. Sebenarnya yang dibandingkan adalah arti atau konsep di balik kata-kata tersebut. Metode LSA melakukan mapping dari kata ataupun dokumen menjadi sebuah concept space dan perbandingan dilakukan pada space ini.Concept space tersebut atau yang lebih sering disebut sebagai latent semantic space merupakan hasil mapping dari matriks dimensi tinggi menjadi dimensi yang lebih kecil.
Meskipun dalam dimensi yang lebih kecil, matriks tersebut merupakan matriks yang merepresentasikan isi dari keseluruhan dokumen. Ciri khas dari LSA adalah teknik yang dinamakan Singular Value Decomposition (SVD). SVD digunakan untuk melakukan dekomposisi matriks setelah diberikan pembobotan untuk kemudian diukur kesamaannya dengan data yang akan diujicobakan.
Inti pekerjaan dari topic modeling adalah menghasilkan representasi dokumen yang baik sehingga dapat digunakan untuk berbagai macam task. Hofmann(1999) mengemukakan bahwa titik permulaan dari ide PLSA adalah pada sebuah model statistik yang disebut sebagai aspect model. Sebuah dokumen teks terdiri dari kumpulan kata-kata. Apabila ditarik satu hal diantara dokumen dan kata tersebut, bisa diperoleh kata kunci (keyword) yang menjadi jembatan diantara dokumen dan kata.Kata kunci ini yang disebut sebagai aspect model. Aspect modeldidefinisikan sebagai sebuah variabel yang tidak terlihat (latent variable) dari sebuah dokumen. Matriks yang dibentuk hasil dari PLSA terdiri dari P(d|z), P(w|z) dan P(z). Matriks P(d|z) adalah matriks probabilitas yang menggambarkan sebaran nilai topik pada sebuah dokumen, sedangkan matriks P(w|z) menggambarkan nilai probabilitas topik pada setiap kumpulan katakata, lalu P(z) merupakan nilai probabilitas dari topik itu sendiri.
P(d|z) =
Gambar 2. Matriks P(d|z)
Dari skema matriks di atas, bisa dilihat hubungan antara baris dan kolom yaitu bahwa pada setiap dokumen akan memiliki probabilitas topik masing-masing. Sebagai contohnya,d 1 (dokumen pertama) memiliki probabilitas dari z 1 (topik pertama), z 2 , hingga z n , begitu juga dengan d 2 , d 3 , d 4 , dan seterusnya.
Data yang digunakan untuk diinputkan ke PLSA adalah dokumen yang menggunakan representasi berupa term-document matrix. Misalnya, dimensi awal dari term-document matrix adalah n buah dokumen, dan m buah kata. Apabila kita memiliki 100 dokumen dan 1000 kata, maka komponen dari term-document matrix akan berjumlah 100.000. Dari ukuran 100x1000, bisa dibuat distribusi probabilitas P(d|w). Sebagai contoh, dari distribusi data tersebut, apabila hendak diketahui berapa jumlah kata "partai" pada dokumen pertama, maka dapat diperoleh dari sel perpotongan antara dokumen pertama (pada kolom pertama matriks) dengan kata "partai" (pada
Logistic Regression
Penggunaan logistic regression untuk memprediksi class probability adalah sebuah pemilihan pemodelan, sama seperti pemilihan pemodelan untuk memprediksi variabel kuantitatif
Document Preprocessing
Document preprocessing merupakan proses pengolahan dokumen ke dalam bentuk yang lebih padat, dimana bentuk tersebut mewakili makna dari dokumen secara utuh. Keseluruhan data yang digunakan akan melewati tahap document preprocessing terlebih dahulu.
Tokenisasi
Tokenisasi merupakan proses pemotongan dari bentuk kalimat menjadi kumpulan kata-kata yang disebut sebagai token. Token akan digunakan sebagai representasi data pada tiap baris yang ada di matriks probabilitas.
Pembuangan stop word
Sebelum token yang sudah dihasilkan dari proses tokenisasi digunakan pada matriks probabilitas, akan diproses terlebih dahulu token mana saja yang merupakan stop word untuk kemudian dihilangkan. Stop word merupakan kata yang tidak mewakili makna dari suatu konteks. Beberapa contoh dari stop word pada bahasa Indonesia adalah "yang", "dari", "kemudian" dan lain sebagainya. Hasil dari pembuangan stop word ini yang kemudian akan menjadi representasi baris pada matriks probabilitas, baik pada data training ataupun data testing.
Proses Training Data
Terdapat kurang lebih 1000 dokumen yang akan digunakan untuk melakukan training pada model PLSA. Data yang disediakan untuk training tersebut akan divariasikan menjadi 400, 700, dan 1000 dokumen. Hal ini dilakukan untuk mengamati perbedaan yang terjadi apabila jumlah dokumen yang digunakan untuk training berbeda. 1000 dokumen tersebut terdiri dari masing-masing 250 dokumen dari 4 kategori, yaitu ekonomi, internasional, politik, dan olahraga.
Berita tersebut diambil dari kumpulan artikel Tempo, Kompas, dan Republika.
Supaya lebih bisa menarik kesimpulan dari berbagai macam parameter dan konfigurasi, maka dibuat skenario untuk pelatihan data sebagai berikut:
1. Variasi dengan menggunakan jumlah topik yang akan dihasilkan (3, 4, dan 5 topik) 2. Variasi dengan jumlah iterasi yang berbeda (1, 3, 5, 7, 9, 10 dan 20 iterasi) 3. Variasi jumlah data training (400, 700, dan 1000 dokumen)
Perulangan konfigurasi yang sama sebanyak 2 kali
Training dengan menggunakan algoritma EM dilakukan pada matriks P(d|z) yang dilatih sesuai dengan jumlah topik, jumlah iterasi, jumlah data training dan perulangan konfigurasi yang sama sesuai korpus yang sudah tersedia. Matriks P(d|w) tidak dimasukkan ke dalamnya karena yang sedang diuji adalah mengenai seberapa efektif reduksi dimensi yang dilakukan oleh PLSA (seperti dijelaskan pada bagian 2), sehingga hanya P(z|d) saja yang terlibat.
Setelah dilakukan pemodelan menggunakan algoritma EM, maka diperoleh matriks yang baru. Matriks tersebut merupakan model yang sudah terbentuk dari pelatihan dari sejumlah data.
Pada saat training, setiap dokumen yang diproses akan diberikan identitas khusus supaya ketika proses klasifikasi dokumen berlangsung, semuanya bisa berjalan dengan baik. Contoh identitas khusus adalah apabila diambil data dari kategori ekonomi, maka pada saat training data selesai dijalankan, dokumen tersebut tetap dikenali sebagai kategori ekonomi. Caranya adalah dengan memberikan penanda pada dokumen tersebut.
Proses Testing Data
Data yang disediakan untuk keperluan uji coba berjumlah 100 dokumen, masing-masing berjumlah 25 dokumen. Baik kategori ekonomi, internasional, politik ataupun olahraga jumlahnya adalah sama. Berita tersebut juga diambil dari kumpulan artikel Tempo, Kompas, dan Republika.
Mekanisme yang dibuat mirip dengan data training. Dari dokumen testing yang disediakan, akan diekstrak topik-topiknya. Matriks yang terbentuk akan terdiri dari banyak matriks probabilitas. Namun, karena pada data testing tidak menggunakan algoritma PLSA, maka Technical Report Program Studi Doktor Ilmu Komputer Fakultas Ilmu Komputer Universitas Indonesia, Desember 2014 ekstraksi topiknya adalah dilakukan dengan mencocokkan kata-kata yang ada di dalam dokumen testing dengan kata-kata yang ada pada model hasil dari training data. Pasti akan ditemukan katakata yang ada pada training data dan ada juga yang tidak terdapat pada training data. Apabila kata-kata tersebut memiliki kesamaan dengan kata yang ada pada training data, maka kata tersebut akan dinilai sebagai topik dari dokumen testing. Nilai probabilitas yang diperoleh juga datang dari proses lookup kepada matriks probabilitas dari masing-masing kata yang disebut sebagai topik dari dokumen testing. Sama halnya seperti data training, setiap dokumen yang diproses akan diberikan identitas dari mana kategori asli dari data tersebut. Informasi ini yang nanti digunakan sebagai pembanding apakah algoritma PLSA berhasil untuk menghasilkan representasi dokumen dengan nilai akurasi yang baik. Hasil dari pengolahan dokumen testing itu akan menghasilkan sebuah matriks P(z|d). Matriks yang dihasilkan dari hasil lookup kurang lebih bisa dicontohkan pada gambar 3.
Gambar 3. Matriks dari Testing Documents
Pada gambar 3, diasumsikan bahwa kata ke-17, ke-48, ke-211, dan ke-286 merupakan kata-kata yang muncul pada training documents, sehingga matriks tersebut dibentuk dari nilai probabilitas masing-masing kata. Dari keseluruhan nilai probabilitas z 1 hingga z k pada semua kata yang muncul di training documents, selanjutnya akan dihitung nilai rata-ratanya dan menghasilkan satu vektor probabilitas yang mewakili dokumen testing.
Pengujian
Pengujian terhadap algoritma PLSA ini dikerjakan dengan melibatkan classifier.
Implementasi
Implementasi dari metode PLSA diterapkan menggunakan Pythonprogramming language. Sedangkan untuk melakukan klasifikasi dokumen, software yang digunakan adalah
Hasil dan Pembahasan
Setelah dilakukan eksperimen secara lebih komprehensif dari keseluruhan variasi yang ditetapkan. Eksperimen dengan konfigurasi yang sama dilakukan sebanyak 2 kali. Hasil akurasi yang diperoleh dari kedua macam uji coba tersebut bisa dilihat pada tabel 1 dan 2.
Tabel 1. Hasil Uji Coba pada Percobaan Pertama
Jumlah Topik
Dari masing-masing jumlah topik pada berbagai konfigurasi yang sudah dibuat maka akan dihitung rata-rata akurasinya. Data hasil perhitungan rata-rata dari masing-masing jumlah topik
Jenis Classifier
Dari dua jenis classifier yang digunakan pada berbagai konfigurasi yang sudah dibuat maka akan dihitung rata-rata akurasinya. Data hasil perhitungan rata-rata dari masing-masing classifier adalah:
-SVM Classifier: 45% -Logistic Regression Classifier: 55% Bisa dilihat data secara statistik bahwa classifier Logistic Regression memberikan kinerja yang lebih baik dari classifier Support Vector Machine (SVM).
Gambar 5. Perbandingan Akurasi dari Jenis Classifier
Jumlah Dokumen Training
Dari variasi jumlah dokumen training pada berbagai konfigurasi yang sudah dibuat maka akan dihitung rata-rata akurasinya. Data hasil perhitungan rata-rata dari berbagai variasi jumlah dokumen
Jumlah Iterasi
Dari variasi jumlah iterasi pada berbagai konfigurasi yang sudah dibuat maka akan dihitung ratarata akurasinya. Data hasil perhitungan rata-rata dari berbagai variasi jumlah iterasi adalah seperti terlihat pada tabel 3.
Tabel 3. Data dan Rata-rata dari Variasi Jumlah Iterasi
Jumlah Iterasi Rata-rata 1 iterasi 35% 3 iterasi 43% 5 iterasi 47% 7 iterasi 52% 9 iterasi 53% 10 iterasi 57% 20 iterasi 65%
Dari pengamatan pada hasil iterasi yang berbeda dari data yang sama, diperoleh hasil bahwa nilai akurasi mengindikasikan ciri-ciri meningkat.
Kesimpulan
Dari beberapa variasi yang dilakukan pada data hasil training dan testing, dapat disimpulkan beberapa hal sebagai berikut:
1. Jumlah topik yang didefinisikan sebagai output dari algoritma PLSA sangat dipengaruhi oleh topik yang didefinisikan. Apabila beberapa topik yang didefinisikan merupakan topik yang cukup beririsan maka nilai akurasi dari variasi jumlah topik tidak akan jauh berbeda.
.
Semua variabel yang digunakan untuk memodelkan aspect model tersebut melibatkan asumsi conditional independence. Dokumen dan kata dalam kondisi conditional independencedihubungkan dengan topik seperti tergambarkan pada gambar 1. Joint probability antara dokumen (D) dengan kata (W) itu tergambarkan pada persamaan berikut: Technical Report Program Studi Doktor Ilmu Komputer Fakultas Ilmu Komputer Universitas Indonesia, Desember 2014 Model tersebutbisa digambarkan juga dengan persamaan: Semua variabel yang ada di atas merupakan matriks probabilitas. Matriks yang digunakan pada PLSA tidak sama seperti term-documentmatrix yang ada pada LSA. Matriks yang digunakan pada PLSA sudah merupakan likelihood dari faktor topik terkait dengan dokumen (document given topic) dan juga terkait dengan kata (word given topic), sedangkan kebalikannya pada LSA, fungsi matriksnya hanya berfungsi sebagai likelihood term-frequency. Hubungan antar Dokumen, Topik, dan Kata Kehadiran topik (Z) di antara kata (W) dan dokumen (D) sebenarnya menyatakan mengapa ketika berada pada dokumen (D), maka kata-kata yang muncul adalah word (W). Hal ini menunjukkan adanya sebuah ketergantungan (dependence), namun karena ketergantungannya tidak secara langsung, sehingga hal ini dinamakan sebagai sebuah conditional independence. Pembentukan model PLSA untuk menghasilkan kumpulan topik (Z) beserta dengan nilai probabilitasnya diawali dengan pembuatan nilai probabilitas secara acak (random) atau bisa juga dengan dilakukan inisialisasi matriks secara greedy. Pembuatan nilai secara acak ini disebut sebagai proses stokastik karena proses ini mengandung ketidakpastian. Apabila dilakukan inisialisasi secara greedy, hal ini dimaksudkan untuk membentuk gradient descent untuk mencapai probabilitas maksimum (maximum probability). Setelah diperoleh matriks awal probabilitas, matriks tersebut akan diproses ke dalam training dengan jumlah iterasi tertentu untuk memperoleh probabilitas yang terbaik dengan menggunakan algoritma Expectation Maximization (EM). Algoritma ini terdiri dari 2 tahap yaitu: 1) Expectation (E), dimana probabilitas posterior dihitung untuk latent variable. Nilai probabilitas yang diperoleh dari langkah Expectation adalah: 2) Maximization (M), dimana parameter-parameter yang ada akan diperbarui nilainya. Nilai yang diperoleh dari langkah Maximization adalah: Technical Report Program Studi Doktor Ilmu Komputer Fakultas Ilmu Komputer Universitas Indonesia, Desember 2014 Algoritma EMakan mengusahakan untuk memperoleh nilai error yang semakin kecil.Apabila nilai error masih tinggi, nilai bobot akan dioptimasikan supaya nilai error yang ada semakin kecil. Sehingga akan sangat memungkinkan bahwa apabila dilakukan semakin banyak training, maka topik serta probabilitas yang dibentuk akan semakin baik.
2.1. Support Vector Machine (SVM) Support Vector Machine (SVM) adalah sebuah metode klasifikasi dan regresi yang mengkombinasikan algoritma komputasional dengan hasil teoretikal; kedua karakteristik ini memberikannya reputasi yang bagus dan menaikkan pamornya dalam penggunaannya di berbagai area. (Cortes and Vapnik, 1995). SVM merupakan sebuah teknik yang baru yang cocok untuk binary classification task, yang terkait dan memuat elemen-elemen statistik terapan non parameterik, jaringan saraf tiruan, dan machine learning. (Auria and Moro, 2008).
Classifier yang digunakan adalah Support Vector Machine (SVM) dan Logistic Regression. Data hasil training yang merupakan model PLSA akan dijadikan patokan di dalam penilaian akurasi algoritma. Dari semua testing documents yang sudah diproses menjadi representasi matriks akan diukur dengan menggunakan classifier. Seperti sebelumnya sudah disebutkan bahwa setiap matriks pada testing documents memiliki identitas khusus yang berisi informasi dari mana kategorinya. Di sisi lain, pada saat classifier melakukan proses pada testing documents dengan berpatokan pada model data training, akan dihasilkan output yang menginformasikan dokumen tersebut masuk ke dalam kategori yang mana. Perbandingan antara kedua data ini yang nantinya Technical Report Program Studi Doktor Ilmu Komputer Fakultas Ilmu Komputer Universitas Indonesia, Desember 2014 diukur satu persatu sehingga akan diperoleh nilai akurasi dari algoritma PLSA dalam melakukan klasifikasi dokumen teks.
Weka 3. 6 .
6Di dalam Weka ini terdapat banyak classifier yang bisa digunakan untuk pengolahan data. Dari banyaknya opsi yang diberikan oleh Weka, maka Support Vector Machine (libSVM) dan Logistic Regression (Logistic) yang digunakan sebagai classifier di penelitian ini. Sesuai dengan keterangan sebelumnya, korpus yang digunakan merupakan koleksi dari artikel berita berbahasa Indonesia yang diperoleh dari Kompas, Tempo, dan Republika. Korpus dibagi menjadi 4 kategori yaitu ekonomi, internasional, politik, dan olahraga. Berbagai variasi konfigurasi untuk testing dilakukan supaya bisa ditarik kesimpulan dari berbagai perspektif. Bagaimana perbandingan hasil akurasi dari classifier SVM dan Logistic, kemudian bagaimana hasilnya jika training dilakukan hanya dengan 5 iterasi dan 20 iterasi, dan lain sebagainya. Technical Report Program Studi Doktor Ilmu Komputer Fakultas Ilmu Komputer Universitas Indonesia, Desember 2014
jumlah topik diperoleh bahwa akurasi tertinggi ada pada jumlah topik 4 dan 5.
Gambar
Gambar 4. Perbandingan Akurasi dari Variasi Jumlah Topik
sebagai representasi dokumen. Probabilistic Latent Semantic Analysis (PLSA) yang dikembangkan oleh Hoffman (1999) merupakan LSA yang menggunakan nilai probabilistik sebagai penentu bobot topik dari setiap dokumen yang ada.Sebagai varian baru dari LSA, teknik GLSA diajukan oleh Islam dan Hoque (2012) yang mengubah keberadaan term di dalam termdocument matrix menjadi n-gram. Sedangkan, metode bernama Multidimensional Latent Semantic Analysis (MDLSA) yang diusulkan oleh Zhang, Ho, Wu, dan Ye (2013) merupakan metode yang meninjau kepada hubungan term dan distribusi spasial. Teknik yang melibatkan aspek sintaksis secara langsung adalah SELSA (Syntactically Enhanced Latent Semantic Analysis) yang diajukan oleh Kanejiya, Kumar dan Prasad (2003). modeling masih berupa model sehingga masih luas area cakupan penerapannya dalam berbagai jenis aplikasi. Pada laporan ini, akan dibahas secara spesifik mengenai salah satu teknik topic modeling yaitu Probabilistic Latent Semantic Analysis (PLSA). Pembahasan akan meliputi mekanisme bagaimana PLSA yang juga melibatkan algoritma Expectation Maximization (EM) sebagai pelatihan diterapkan pada sekumpulan data korpus, serta bagaimana melakukan uji coba dan memperoleh akurasi hasil penggunaan PLSA. Data korpus yang digunakan adalah data teks bahasa Indonesia yang diperoleh dari beberapa sumber yakni Kompas, Tempo, dan Republika. Hasil akurasi diukur dengan menggunakan tool untuk klasifikasi dengan membandingkan matriks hasil training data dengan matriks dari sekumpulan data yang hendak diujicobakan. Kemudian, pada bagian akhir dari laporan ini dikemukakan kesimpulan yang diperoleh serta saran untuk penelitian serta eksperimen selanjutnya. Technical Report Program Studi Doktor Ilmu Komputer Fakultas Ilmu Komputer Universitas Indonesia, Desember 2014Topic
Technical Report Program Studi Doktor Ilmu Komputer Fakultas Ilmu Komputer Universitas Indonesia, Desember 2014 baris tertentu pada matriks). Hal ini membuat pengolahan data terlihat praktis dan hasilnya pun pasti baik dan relevan, namun permasalahan yang terjadi pada dimensi 100 x 1000 tersebut adalah dimensi matriks yang terlalu besar. Dimensi yang terlalu besar akan memperlambat proses komputasi dan banyak proses yang tidak perlu dilakukan akan tetap dilakukan. Matriks yang terlalu sparse akan mengakibatkan pemborosan memori di dalam prosesnya. PLSA menjadi salah satu solusi reduksi dimensi. Hal ini terjadi karena PLSA memiliki jembatan berupa topik Z yang membagi probabilitas menjadi 3 matriks tetapi dengan dimensi yang lebih kecil. Pada proses reduksi dimensi ini, Singular Value Decomposition (SVD) akan berperan secara aktif.Berikut ini diberikan contoh untuk memudahkan penggambaran reduksi dimensi pada matriks probabilitas PLSA. Misalnya terdapat 100 dokumen dan 1000 kata di dalam dokumen tersebut, kemudian didefinisikan ada 10 topik (Z=10) yang dihasilkan, maka data yang ada menjadi 100 dokumen, 1000 kata, dan 10 topik. Pada term-document matrix sebelum direduksi akan dibutuhkan 100000 entri. Namun, apabila dengan menggunakan PLSA (melibatkan adanya topik) dengan melakukan reduksi dimensi, maka akan diperoleh matriks dengan dimensi 100x10 dan 1000x10. Apabila jumlah entri dari kedua matriks tersebut dijumlahkan, maka akan menjadi 11000 entri. Perubahan jumlah entri dari 100000 entri menjadi 11000 entri ini yang dimaksudkan dengan reduksi dimensi pada PLSA. Reduksi dimensi tersebut membuat algoritma akan berjalan lebih efisien.
Technical Report Program Studi Doktor Ilmu Komputer Fakultas Ilmu Komputer Universitas Indonesia, Desember 2014 dengan menggunakan linear regression. Logistic regression adalah salah satu dari tool yang paling umum digunakan pada statistik terapan dan analisis data diskrit(Shalizi, 2012). PLSA dapat diterapkan ke dalam berbagai aplikasi seperti penilai esai otomatis, peringkas dokumen, dan lain sebagainya. Pada penelitian ini, yang dikerjakan oleh algoritma PLSA adalah hanya terbatas pada pekerjaan untuk melakukan klasifikasi dokumen teks. Dari sejumlah data training yang sudah disediakan, maka algoritma EM akan menjalankan proses training. Proses training akan dilakukan dengansejumlah angka iterasi tertentu. Output dari algoritma EM merupakan model dari hasil training yang dilakukan oleh PLSA. Perlakuan yang sama akan ditujukan pada data testing. Dari data testing yang sudah disediakan, akan diambil representasi dokumen yang nantinya akan dibandingkan dengan model PLSA. Dengan menggunakan classifier, maka akan dihitung keakuratan metode PLSA dalam melakukan klasifikasi dokumen teks. Data yang digunakan merupakan korpus yang merupakan koleksi dari artikel berita berbahasa Indonesia dari Kompas, Tempo, dan Republika. Korpus dibagi menjadi 4 kategori yang khusus yaitu ekonomi, internasional, politik, dan olahraga.3. Metodologi
Technical Report Program Studi Doktor Ilmu Komputer Fakultas Ilmu Komputer Universitas Indonesia, Desember 2014
SVM Log SVM Log SVM Log SVM Log SVM Log SVM Log SVM Log SVM Log SVM LogTabel 2. Hasil Uji Coba pada Percobaan KeduaBerdasarkan data yang disediakan pada tabel 1 dan 2, dapat diukur beberapa hal sebagai perbandingan yang komprehensif mengenai variasi jumlah topik, variasi classifier, variasi jumlah dokumen training, dan variasi jumlah iterasi.Eksperimen I
400 dokumen
700
1000
3 topik
4 topik
5 topik
3 topik
4 topik
5 topik
3 topik
4 topik
5 topik
1 iterasi
36% 40% 47% 41% 35% 36% 26% 43% 20% 29% 33% 33% 40% 34% 38% 39% 42% 34%
3 iterasi
43% 42% 41% 59% 35% 48% 45% 37% 45% 48% 38% 55% 37% 40% 29% 35% 28% 42%
5 iterasi
45% 57% 39% 53% 58% 66% 29% 41% 49% 53% 59% 60% 31% 47% 51% 43% 40% 48%
7 iterasi
36% 51% 49% 68% 45% 57% 33% 69% 55% 66% 68% 77% 39% 56% 43% 62% 33% 61%
9 iterasi
31% 43% 37% 63% 39% 63% 37% 60% 45% 68% 60% 76% 54% 64% 35% 57% 42% 62%
10 iterasi
44% 61% 69% 64% 41% 63% 57% 74% 58% 68% 42% 62% 44% 59% 61% 68% 55% 71%
20 iterasi
52% 60% 63% 62% 59% 68% 55% 62% 69% 71% 66% 76% 61% 68% 65% 69% 67% 68%
Technical Report Program Studi Doktor Ilmu Komputer
Fakultas Ilmu Komputer Universitas Indonesia, Desember 2014
Eksperimen II
400 dokumen
700
1000
3 topik
4 topik
5 topik
3 topik
4 topik
5 topik
3 topik
4 topik
5 topik
SVM Log SVM Log SVM Log SVM Log SVM Log SVM Log SVM Log SVM Log SVM Log
1 iterasi
43% 37% 27% 24% 24% 35% 31% 39% 25% 33% 42% 45% 30% 32% 31% 33% 35% 44%
3 iterasi
40% 50% 37% 38% 34% 41% 36% 46% 50% 55% 48% 61% 39% 48% 42% 47% 37% 41%
5 iterasi
46% 53% 46% 63% 39% 66% 26% 39% 42% 53% 38% 52% 33% 45% 42% 56% 30% 43%
7 iterasi
66% 72% 50% 56% 49% 57% 26% 44% 29% 54% 71% 71% 25% 47% 43% 60% 26% 49%
9 iterasi
60% 67% 55% 51% 51% 77% 34% 57% 31% 59% 63% 75% 41% 57% 50% 61% 33% 48%
10 iterasi 33% 54% 67% 67% 35% 51% 58% 63% 59% 68% 73% 70% 38% 57% 43% 60% 50% 61%
20 iterasi 40% 59% 75% 81% 55% 72% 54% 75% 65% 74% 55% 68% 44% 43% 69% 84% 77% 82%
Technical Report Program Studi Doktor Ilmu Komputer
Fakultas Ilmu Komputer Universitas Indonesia, Desember 2014
4. Perbandingan Akurasi dari Variasi Jumlah TopikPerbandingan Akurasi dari Variasi Jumlah TopikTechnical Report Program Studi Doktor Ilmu Komputer Fakultas Ilmu Komputer Universitas Indonesia, Desember 201444%
45%
46%
47%
48%
49%
50%
51%
52%
53%
training adalah :
adalah-400 dokumen training: 51% -700 dokumen training: 52% -1000 dokumen training: 48% Bisa dilihat bahwa penambahan jumlah dokumen untuk data trainingmengindikasikan tidak adanya pengaruh secara signifikan pada hasil akurasi. Dari eksperimen yang dilakukan, hasil yang Perbandingan Akurasi dari Jenis Classifier Technical Report Program Studi Doktor Ilmu Komputer Fakultas Ilmu Komputer Universitas Indonesia, Desember 2014 ditunjukkan pada training dengan menggunakan data yang lebih banyak cenderung sama dan justru lebih kecil dari data yang lebih sedikit. Melihat kecenderungan yang tidak umum pada hasil data ini yakni dengan bertambahnya training data seharusnya akurasi akan lebih meningkat. Hipotesis yang muncul karena hal ini adalah bahwa jumlah data pada setiap sumber artikel tidak seimbang, sehingga model yang terbentuk tidak proporsional atau seimbang untuk mewakili berbagai gaya bahasa yang ada pada ketiga media tersebut yaitu Kompas, Tempo, dan Republika. Akan tetapi hal tersebut masih harus diuji kembali supaya terbukti validitasnya. Gambar 6. Perbandingan Akurasi dari Jumlah Dokumen Training Technical Report Program Studi Doktor Ilmu Komputer Fakultas Ilmu Komputer Universitas Indonesia, Desember 20140%
10%
20%
30%
40%
50%
60%
SVM
Logistic Regression
46%
47%
48%
49%
50%
51%
52%
53%
400 dokumen
700 dokumen
1000 dokumen
Perbandingan Akurasi dari
Jumlah Dokumen Training
2 .
2Classifier Logistic memiliki performa yang lebih baik dari classifier SVM. Hal ini sesuai dengan data yang digunakan untuk uji coba. Karena data yang digunakan merupakan angka diskrit, klasifikasi dengan Logistic Regression memberikan hasil yang lebih baik daripada SVM. Hal ini sesuai dengan teorinya bahwa SVM lebih bagus untuk klasifikasi data untuk bilangan real. 3. Jumlah dokumen trainingseharusnya berpengaruh signifikan untuk uji coba penggunaan algoritma PLSA, akan tetapi cara bagaimana data training dan testing disiapkan butuh menjadi perhatian yang khusus. 4. Semakin banyak iterasi yang dilakukan maka nilai akurasi akan terus meningkat. Hal ini akan terus berlanjut sampai ditemukan kondisi konvergen dimana nilai akurasi tidak akan meningkat lagi
SupportVector Machines (SVM) as a Technique for Solvency Analysis. L Auria, R A Moro, Discussion Papers. Auria, L. & Moro, R.A., SupportVector Machines (SVM) as a Technique for Solvency Analysis. Discussion Papers, Berlin, 2008
Support-vector networks. C Cortes, V Vapnik, Machine Learning. 203Cortes, C. & Vapnik, V., Support-vector networks, Machine Learning 20 (3), 273-297. 1995
Probabilistic Latent Semantic Analysis. T Hofmann, Proceedings of the Fifteenth conference on Uncertainty in Artificial Intelligence. the Fifteenth conference on Uncertainty in Artificial IntelligenceHofmann T.. Probabilistic Latent Semantic Analysis. Proceedings of the Fifteenth conference on Uncertainty in Artificial Intelligence, 1999.
Automated Essay Grading Software Stirs Debate. Information Week: Connecting the Business Technology Community. Michael Fitzgerald, Michael Fitzgerald. Automated Essay Grading Software Stirs Debate. Information Week: Connecting the Business Technology Community.
36-402 Undergraduate Advanced Data Analysis. C Shalizi, Shalizi C.. 36-402 Undergraduate Advanced Data Analysis, Spring 2012
An Introduction to Latent Semantic Analysis. T K Landauer, P W Foltz, D Laham, Discourse Processes. 25Landauer, T.K., Foltz, P.W., Laham, D. An Introduction to Latent Semantic Analysis. Discourse Processes, 25, 259-284. 1998.
| []
|
[
"CLOUDS ON THE HOT JUPITER HD189733B: CONSTRAINTS FROM THE REFLECTION SPECTRUM",
"CLOUDS ON THE HOT JUPITER HD189733B: CONSTRAINTS FROM THE REFLECTION SPECTRUM"
]
| [
"J K Barstow ",
"S Aigrain ",
"P G J Irwin ",
"T Hackler ",
"L N Fletcher ",
"J M Lee ",
"N P Gibson ",
"\nDepartment of Physics\nInstitute for Theoretical Physics\nUniversity of Oxford\nOxfordUK\n",
"\nEuropean Southern Observatory\nUniversity of Zürich\nCH-8057, 85748Zürich, Garching bei MünchenSwitzerland, Germany\n"
]
| [
"Department of Physics\nInstitute for Theoretical Physics\nUniversity of Oxford\nOxfordUK",
"European Southern Observatory\nUniversity of Zürich\nCH-8057, 85748Zürich, Garching bei MünchenSwitzerland, Germany"
]
| []
| The hot Jupiter HD 189733b is probably the best studied of the known extrasolar planets, with published transit and eclipse spectra covering the near UV to mid-IR range. Recent work on the transmission spectrum has shown clear evidence for the presence of clouds in its atmosphere, which significantly increases the model atmosphere parameter space that must be explored in order to fully characterise this planet. In this work, we apply the NEMESIS atmospheric retrieval code to the recently published HST /STIS reflection spectrum, and also to the dayside thermal emission spectrum in the light of new Spitzer /IRAC measurements, as well as our own re-analysis of the HST /NICMOS data. We first use the STIS data to place some constraints on the nature of the cloud on HD 189733b, and explore solution degeneracy between different cloud properties and the abundance of Na in the atmosphere; as already noted in previous work, absorption due to Na plays a significant role in determining the shape of the reflection spectrum. We then perform a new retrieval of the temperature profile and abundances of H 2 O, CO 2 , CO and CH 4 from the dayside thermal emission spectrum. Finally, we investigate the effect of including cloud in the model on this retrieval process. We find that the current quality of data does not warrant the extra complexity introduced by including cloud in the model; however, future data are likely to be of sufficient resolution and signal-to-noise that a more complete model, including scattering particles, will be required. | 10.1088/0004-637x/786/2/154 | [
"https://arxiv.org/pdf/1403.6664v1.pdf"
]
| 30,699,047 | 1403.6664 | 7c2177f2d88b8ce549ab30f2443ccbadf2c7f2c3 |
CLOUDS ON THE HOT JUPITER HD189733B: CONSTRAINTS FROM THE REFLECTION SPECTRUM
26 Mar 2014 , Draft version March 27, 2014 March 27, 2014
J K Barstow
S Aigrain
P G J Irwin
T Hackler
L N Fletcher
J M Lee
N P Gibson
Department of Physics
Institute for Theoretical Physics
University of Oxford
OxfordUK
European Southern Observatory
University of Zürich
CH-8057, 85748Zürich, Garching bei MünchenSwitzerland, Germany
CLOUDS ON THE HOT JUPITER HD189733B: CONSTRAINTS FROM THE REFLECTION SPECTRUM
26 Mar 2014 , Draft version March 27, 2014 March 27, 2014Preprint typeset using L A T E X style emulateapj v. 08/22/09 Draft versionSubject headings: Methods: data analysis -planets and satellites: atmospheres -radiative transfer
The hot Jupiter HD 189733b is probably the best studied of the known extrasolar planets, with published transit and eclipse spectra covering the near UV to mid-IR range. Recent work on the transmission spectrum has shown clear evidence for the presence of clouds in its atmosphere, which significantly increases the model atmosphere parameter space that must be explored in order to fully characterise this planet. In this work, we apply the NEMESIS atmospheric retrieval code to the recently published HST /STIS reflection spectrum, and also to the dayside thermal emission spectrum in the light of new Spitzer /IRAC measurements, as well as our own re-analysis of the HST /NICMOS data. We first use the STIS data to place some constraints on the nature of the cloud on HD 189733b, and explore solution degeneracy between different cloud properties and the abundance of Na in the atmosphere; as already noted in previous work, absorption due to Na plays a significant role in determining the shape of the reflection spectrum. We then perform a new retrieval of the temperature profile and abundances of H 2 O, CO 2 , CO and CH 4 from the dayside thermal emission spectrum. Finally, we investigate the effect of including cloud in the model on this retrieval process. We find that the current quality of data does not warrant the extra complexity introduced by including cloud in the model; however, future data are likely to be of sufficient resolution and signal-to-noise that a more complete model, including scattering particles, will be required.
INTRODUCTION
Since its discovery in 2005 (Bouchy et al. 2005), the hot Jupiter HD 189733b has been repeatedly observed as it transits and is eclipsed by its parent star, leading to excellent coverage in both its transmission and eclipse spectra from the visible to mid-infrared. This has resulted in HD 189733b being probably the best characterised of all the known exoplanets; it is known from the observed slope of the reflectance spectrum that this unresolved planet would appear a deep shade of blue (Evans et al. 2013), a fact that testifies to the power of the transit spectroscopy technique. The transmission spectrum presented by shows clear evidence for haze or cloud in the atmosphere of this planet, making it the first transiting planet outside the solar system that is known to be cloudy. and Line et al. (2012) analyse the dayside spectrum from secondary eclipse observations, and retrieve the temperature-pressure profile and abundances of H 2 O, CO 2 , CO and CH 4 . Knutson et al. (2012) use Spitzer /IRAC phase curves to investigate the longitudinal temperature variability, and de Wit et al. (2012) combine the phase curves with an analysis of the ingress/egress shape in eclipse to place further constraint on spatial variability.
The recent albedo spectrum from HST /STIS obtained Electronic address: [email protected] by Evans et al. (2013) provides an opportunity to investigate the cloud structure on the dayside. Unlike the transmission spectrum investigated by and which probes the limb of the exoplanet amosphere, the dayside reflection spectrum has near-nadir geometry, and so it is sensitive to deeper regions of the atmosphere. If the cloud is similar at the terminators and on the dayside, we can use this to further constrain its properties. Alternatively, we may see an entirely different cloud layer on the hotter dayside from that observed at the terminator. The albedo spectrum is useful for placing constraint on the cloud, as scattering particles have a significant effect on the optical reflectivity of an atmosphere. We expect there to be fewer gaseous absorbing species in the visible part of the spectrum than in the infrared; distinct absorption features of Na and K are seen in the transmission spectrum, but otherwise cloud appears to be the dominant opacity source in this region . We also expect little, if any, thermal contribution from the planet itself at these wavelengths due to its temperature.
In this work, we use the Non-linear optimal Estimator for MultivariateE spectral analySIS (NEMESIS) software (Irwin et al. 2008) to calculate synthetic spectra using a simple cloudy model atmosphere for HD 189733b, including multiple scattering. We vary cloud parameters and Na volume mixing ratio (VMR) and compare the resultant spectra to the STIS measurement; calculating the Gas Source H 2 O HITEMP2010 (Rothman et al. 2010) CO 2 CDSD-1000 (Tashkun et al. 2003) CO HITRAN1995 (Rothman et al. 1995) CH 4 STDS (Wenger and Champion 1998) Na VALD (Heiter et al. 2008) K VALD (Heiter et al. 2008) Table 1 Sources of gas absorption line data.
χ 2 goodness-of-fit parameter allows us to determine the region of model parameter space that provides the best match. We then use a subset of our best-fitting models to examine the effect of clouds on our ability to accurately retrieve temperature and molecular abundances from the infrared dayside emission spectrum, following the work of .
NEMESIS
We use the NEMESIS spectral retrieval tool to produce forward models (predicted spectra for a range of model atmospheres) for comparison with the HST /STIS spectra, and also to retrieve the atmospheric state from the thermal emission spectrum as in . NEMESIS was developed by Irwin et al. (2008) for atmospheric retrieval of solar system planets, and has since been extended to enable the same analysis for observations of transiting and directly imaged extrasolar planets . NEMESIS is an Optimal Estimation retrieval model (Rodgers 2000) and uses a correlated-k radiative transfer model (Lacis and Oinas 1991;Goody and Yung 1989). NEMESIS is not a radiative equilibrium model; instead, it simply uses the atmospheric model provided to compute the incoming and outgoing radiative flux. In an irradiated case, it will compute the incoming and scattered/reflected flux from the star, but it will not take into account the heating effect of the incoming stellar flux on the atmosphere. For the multiple scattering runs, NEMESIS uses beams over a user-specified number of zenith angles; azimuthal dependence is accounted for using Fourier decomposition. For more details of the scattering calculation in this work, see Section 1.1.1.
We use the same line and collision-induced absorption data as , and Barstow et al. (2013a). A list of sources for absorption line data is given in Table 1. H 2 -He collision-induced absorption data are taken from the models of ; ; Borysow and Frommhold (1990); Borysow et al. (1997) and Borysow (2002). The reference stellar spectrum is taken from the model set made available by Kurucz 1 , and the stellar radius is taken from Baines et al. (2008). We use the same planetary mass and radius as , and a H 2 :He ratio of 9:1. As found by , the precise value of this does not have a large effect on secondary eclipse retrievals. In order to reproduce an accurate reflection spectrum in the presence of optically thick cloud, it is necessary to include multiple scattering as many scattering events are likely. NEMESIS uses the matrix operator algorithm of Plass et al. (1973) for multiple scattering calculations, where the zenith angle integration is achieved using a 5point Gaussian-Lobatto quadrature scheme and azimuth angle integration is achieved through Fourier decomposition, with the necessary number of Fourier components determined by the stellar and emission zenith angles. The analytical disc-averaged integration scheme used in for eclipse spectra is not used here as it is not applicable to scattering situations; instead, we represent the disc average for the synthetic STIS spectra by running multiple scattering calculations with the stellar zenith angles set to each of the 5 Gaussian-Lobatto quadrature angles and the azimuth angle set for backscattering, since during secondary transit the observer is located in the same direction as the star and the stellar zenith angle is equal to the emission angle. The discaverage is then determined using a weighted average of these 5 different calculations, assuming that the atmospheric conditions are the same at all points on the disc. Retrievals including multiple scattering are computationally expensive. Therefore, we anticipate that the majority of the thermal emission retrieval calculations in the future will still be performed with an extinction-only approximation using the disc integration described by . However, we also tested the effect of including multiple scattering, to test the sensitivity of the retrieval to differences in our modelling approach. The 5-angle approach used to calculate the STIS synthetic spectra is still too time-consuming for the emission spectrum retrieval, so in this case we approximate further by calculating the spectrum for a stellar zenith angle of 45 • only, which represents the average angle of weighting function used to compute the disc-averaged spectrum sinceR = π 2 0 2R(θ)sin(θ)cos(θ) dθ = π 2 0 R(θ)sin(2θ) dθ. The scattering phase function of the particles is calculated using Mie theory and approximated by the Henyey-Greenstein parameterisation (Henyey and Greenstein 1941), as in previous work on planetary clouds (e.g. Irwin et al. 2009, Barstow et al. 2012; expected deviations from the true phase function are small compared with the errors on the observed spectrum, so we consider this approximation to be valid. We use the double-peaked version of the phase function:
P (θ) = 1 4π f 1−g 2 1 (1+g 2 1 −2g1µ) 3/2 + (1 − f ) 1−g 2 2
(1+g 2 2 −2g2µ) 3/2 , which represents the phase function as the sum of forward and backward scattering peaks. Here, µ is the cosine of the scattering angle, g 1 and g 2 are scattering asymmetry parameters for the forward and backward peaks respectively, and f is the fractional contribution of the forward peak to the total phase function. For smaller particles approaching the Rayleigh scattering limit, the parameter f is close to 0.5 and there are approximately equal contributions from the forward and backward scattering peaks; as the particle size increases relative to the wavelength of light the value of f increases and the scattering becomes more asymmetric. However, in no case is the scattering completely isotropic.
We use the enstatite refractive index values of Scott and Duley (1996) and the MnS values from Huffman and Wild (1967) in our calculations of the scattering parameters. It is worth noting that, as we model realistic particles, they absorb as well as scatter incident radiation. The fraction of light that is absorbed rather than scattered is dependent on the particle size and also on the composition of the particles. In general, more of these realistic particles would be required than idealised, perfectly scattering particles to produce an equivalent planetary albedo in an atmospheric model, since for realistic particles some of the incident radiation is absorbed by the particles rather than scattered back to space. The single scattering albedo is also wavelength dependent, and this can therefore affect the shape of modelled planetary albedo and thermal emission spectra.
DATA
The HST /STIS data used in this work are presented in Evans et al. (2013). We use the tabulated eclipse depths binned in 6 channels, spanning the wavelength range 290-570 nm, to place constraint on the cloud properties for HD 189733b.
We also investigate the impact of including clouds on retrievals of temperature structure and atmospheric composition from the available thermal emission spectra. We use the same data as , with the exception that we include the new values for the Spitzer /IRAC 3.6 and 4.5 µm points presented in Knutson et al. (2012). Data used as in are the Spitzer /IRS spectrum from Grillmair et al. (2008), the Spitzer /IRS broadband point originally measured by Deming et al. (2006) and the broadband Spitzer MIPS and Spitzer /IRAC 5.8/8.0 µm points from .
We also include a reanalysis of the HST /NICMOS observations of Swain et al. (2009). Gibson et al. (2011) re-analysed a number of exoplanet transmission spectra obtained with NICMOS, including that of XO-1b, originally published by Tinetti et al. (2010). This re-analysis showed that the shape of the NICMOS spectra is highly sensitive to the assumptions made about the instrumental systematics, and that the errors may have been underestimated in the original publications. This was recently confirmed by new observations of XO-1b in transit with HST /WFC3 (Deming 2013), which are less affected by systematics, and showed significant discrepancies with the original NICMOS dataset. We expect that the NIC-MOS emission spectrum of HD189733b may suffer from the same problems as the aforementioned transmission spectra, but it is the only dataset published to date in the crucial near-infrared wavelength range. We therefore re-analysed it using a method proposed by Gibson et al. (2012), which makes minimal assumptions about the nature of the systematics. This re-analysis, which we describe briefly in Section 2.1, leads to larger, but somewhat more robust error bars, and thus more conservative results for the retrieval analysis.
We omit the ground-based K-and L-band data of Waldmann et al. (2012), as these data contain an extremely strong emission feature at 3.3 µm which is attributed to non-Local Thermodynamic Equilibrium (non-LTE) processes; modelling this feature is therefore beyond the limits of our code. The attribution of this feature as resulting from the atmosphere of HD 189733b has also been disputed by Mandell et al. (2011) as these authors did not detect it; they suggest it may be a result of telluric absorption. This conflict is so far unresolved and as a result we do not include these data in our analysis.
Reanalysis of NICMOS emission spectrum
We re-analysed the NICMOS emission spectrum presented in Swain et al. (2009) using the method presented by Gibson et al. (2012). We briefly outline the process here, but refer the interested reader to Gibson et al. (2012) for further information on the data reduction and the details of the systematics modelling method. We model the secondary eclipse and systematics simultaneously in each of 18 wavelength channels, using Gaussian Processes (GP) to model the dependence of the systematics on the instrumental parameters which are known to affect them: the orbital phase of HST φ and its square φ 2 ; the x and y-position of the spectral trace on the detector, its width w and its angle θ relative to the x-axis. The ephemeris, planet-to-star radius ratio, inclination and system scale (a/R ⋆ ) were fixed to the values obtained by Winn et al. (2006), These are also consistent with the more recent determination of these parameters by Southworth (2008). The eccentricity was assumed to be zero, as there is no evidence in the literature for a non-zero eccentricity. We used three Markov-Chain Monte Carlo (MCMC) chains of 40,000 steps each to marginalise over all the parameters except the eclipse depth in each channel. Figure 1 shows the spectrum obtained after re-analysis, with the original from Swain et al. (2009) for comparison. The two versions of the spectrum are largely similar in shape, but our error bars are about a factor of 3 larger, which is consistent with the findings of Gibson et al. (2012). The wavelength range probed by NIC-MOS is particularly sensitive to the gaseous abundances on the dayside of HD 189733b , so the increased error bars will impact the constraints we can extract from the thermal emission spectrum. The spectrum and 1σ error bars are provided in numeric form in Table 2.
We note that the choice of which instrumental parameters to include in the systematics model for NICMOS affects both the shape of the resultant spectrum and the errors, whether using traditional linear de-correlation methods (Gibson et al. 2011) or our GP approach. However, in a GP model, the length scale η associated with each parameter gives an indication of the relevance of that parameter (see Gibson et al. 2012 for details). In the present case, the best-fit length scales for all the instrument parameters we tried including in the model were found to be of the same order of magnitude, which is why we included them all in the final model. Nonetheless, this highlights the fact that the NICMOS results should be viewed with caution, as the spectrum is extremely sensitive to the details of the systematics removal.
The reanalysed spectrum and 1σ errors are provided in Table 2.
CLOUD MODELS
We use the HST /STIS data to investigate the range of cloud models that provide a reasonable match with HD Figure 1. The NICMOS spectrum as analysed by Swain et al. (2009;stars, dotted line) and in this work (crosses, solid line). We find that the spectrum has a similar shape but the feature amplitude is decreased and the error bars are increased.
Wavelength Flux ratio (10 −4 ) Error 189733b's reflection spectrum. We use the best-fit atmospheric temperature profile retrieved by Lee (2012); the other parameters with a significant effect in the STIS wavelength range are all related to cloud, except for the abundance of sodium. Sodium has been detected in transmission spectra of HD 189733b (Redfield et al. 2008;Huitson et al. 2012), but it is not possible to obtain an absolute constraint on the sodium abundance from these measurements. Atmospheric models by Sudarsky et al. (2000) and Heng and Demory (2013) suggest that Na should have a measurable effect on the reflection spectrum. The abundance is left as a free parameter in the model.
Clouds are extremely complex phenomena in radiative transfer. The resultant spectrum for a cloudy atmosphere depends on the scattering properties of the cloud, which in turn are affected by the composition and size
Variable
Values Base pressure (mbar) 1000, 100, 10, 1, uniform Particle size (µm) 0.01, 0.03, 0.1, 0.3, 1, 3, 10 Optical depth at 0.25 µm 0.1, 0.2, 0.4, 0.6, 0.8, 1, 10 Na VMR (ppmv) 0.5, 5, 50, 500 Table 3 The values used for each parameter in the cloud model, and for Na volume mixing ratio. A synthetic spectrum is generated for every combination of these parameters, making a total of 980 models. All cloud models span a decade in pressure, except for the uniform case -so if the base pressure is 1000 mbar, the top pressure is 100 mbar. Figure 2. The 6-channel STIS spectrum from Evans et al. (2013), with four cloud-free models overlaid. Models with 50 ppmv Na produce the best fit to the observed spectrum without the requirement for cloud; models with 0.5 ppmv Na give a poor fit. 5 ppmv is approximately solar abundance.
of the cloud particles. The altitude and optical depth of the cloud affect how far light from the star will penetrate the atmosphere, which affects the fraction of the light that is absorbed before it can be reflected back. In order to investigate the potential cloud structure in the simplest possible way, the cloud has been modelled for a fixed set of particle sizes, optical depths and cloud altitudes; the free parameters in our model for the STIS spectrum are listed in Table 3, with the range of values taken by each. The scattering efficiency and phase function for the cloud are calculated using Mie theory with a double-peaked Henyey-Greenstein approximation to the phase function; the scattering properties are therefore a consequence of the size and composition of the cloud particles. All other parameters, including the atmospheric temperature and H 2 O, CO 2 , CO and CH 4 VMRs, are fixed during this analysis at the best-fit values of Lee (2012); none of these gases are expected to have strong absorption features between 0.3 and 0.6, so their only effect on the spectrum is a small effect on the mean molecular weight. We also include Rayleigh scattering from H 2 and He, and the abundances of these gases are also fixed.
For most of the tests described below, we assume that the cloud particles are made of enstatite, MgSiO 3 (Lecavelier Des Etangs et al. 2008; refractive indices used are taken from Scott and Duley 1996). This is likely to Figure 3. Contour plots of χ 2 goodness-of-fit parameter for (bottom left): cloud bottom pressure, optical depth, particle size and Na volume mixing ratio; (top right): cloud optical depth, particle size and Na volume mixing ratio for a uniformly mixed cloud. The Na VMR is quoted as a multiplication of the solar value (∼5 ppmv). Also plotted are the percentages of models with χ 2 of <10 (top line) and <5 (bottom line) for each parameter. Each contour plot represents a 2-D cut through the 4-D parameter space, with two parameters varied and the other two held fixed in each plot. The dashed lines indicate the values used for each parameter when held fixed.
be condensed at the stratospheric temperature on HD 189733b (Fortney et al. 2010), but it is not the only chemical species that may be relevant; Morley et al. (2012) suggest that constituents such as MnS may also form clouds on hot Jupiters, and we consider the impact of changing the composition of the cloud particles in Section 3.2. Conversely, we do not expect to see absorption due to gaseous metal oxides such as TiO in the atmosphere as Ti is likely to be in condensate form at the expected photospheric temperature of HD 189733b (Fortney et al. 2010). This may mean that any condensates present also contain Ti oxides, as suggested by Helling and Woitke (2006).
To test the effect of particle size, we use a range of monodisperse (single-size) cloud populations from 10 nm to 10 µm in radius.For the STIS wavelength range, the Rayleigh scattering limit is approached for a particle size of 10 nm, so the scattering behaviour as a function of wavelength will be similar for any particles smaller than 10 nm; the only difference would be that a larger number of smaller particles would be required to achieve the same optical depth. The nadir optical depths (referenced at 0.25 µm) are varied between 0.1 and 10, and the cloud is located in a layer 1 decade thick in pressure coordinates (i.e. between 1000 and 100 mbar, or 100 and 10 mbar, etc.). We locate the cloud base at 1000, 100, 10 and 1 mbar, and also test the effect of distributing the cloud particles uniformly throughout the atmosphere. The cloud particle number density scale height is assumed to be the same as the pressure scale height in all cases.
Synthetic reflection spectra are generated for all combinations of these three cloud parameters and the Na abundance parameter, using NEMESIS. We list the values used for each parameter in Table 3.
We also generate a series of cloud-free models to investigate the impact of changing the sodium abundance by itself. It is clear that for a range of sodium abundances the STIS spectrum can be reproduced without the need for clouds ( Figure 2; the effect of sodium on the spectrum is as predicted by Sudarsky et al. 2000) but if the atmosphere in fact contains more/less sodium additional scatterers/absorbers would be required (Heng and Demory 2013).
For each of the cloudy models, the χ 2 goodness-of-fit parameter is computed to indicate the quality of the fit to the 6 data points in the STIS spectrum. We use this as a means of comparing models across our parameter space; our 4 variables are Na VMR, cloud particle radius, cloud altitude and cloud optical depth. For a good fit, therefore, the χ 2 should approach 2 (N data − N variables ). Contour plots of the χ 2 are shown for each parameter combination in Figure 3, as well as the number of models for each value of each single parameter with a χ 2 of less than 10 and less than 5. The contour plots show two-dimensional cuts through the four-dimensional parameter space, with two parameters varying in each plot and the other two held fixed. The values for each fixed parameter are indicated by the dashed lines in the histogram plots. This indicates that the best-fit region of parameter space occurs mostly for either small particles (< 0.3 µm in size) or 10 µm particles, 10× solar Na (50 ppmv) and optical depths of 1 or less. The preference for small particles (0.01-0.1 µm) is consistent with the findings of Lecavelier Des Etangs et al. (2008).
A subsolar sodium abundance (0.1×, or 0.5 ppmv) is insufficient to reproduce the spectrum for the cases we test here. Examples of models with different cloud properties and Na VMRs are shown in Figure 4; models with 5-500 ppmv Na could all produce an adequate fit to the spectrum depending on the assumed cloud properties, but a model with 0.5 ppmv Na does not produce a sufficiently large absorption feature at wavelengths >0.5 µm. In general, the cloud altitude is the model parameter with the smallest effect, except for the case where the sodium abundance is 100× solar ( Figure 5). The degeneracy between sodium abundance and the cloud properties becomes complex when the sodium abundance is high, because the cloud must make the atmosphere optically thick at precisely the right altitude in order for the model to fit. For 50 ppmv sodium a cloud-free model can fit the spectrum with a χ 2 of less than 5 (Figure 2), but the increase in sodium abundance means that an additional reflective component is required in the model atmosphere for 500 ppmv.
The data do not allow us to distinguish between the finite deck-type models, where the cloud is confined to a certain pressure range, and models where the cloud particles are uniformly distributed throughout the atmosphere as for many cases we can obtain a good fit using either (Figure 3). Based on the results of Parmentier et al. (2013), a uniform cloud of particles up to 0.1 µm in size is plausible; we may expect a cloud formed of larger particles to be located lower in the atmosphere due to sedimentation and rain out. For particle sizes of up to 0.1 µm, the uniform cloud model contains fewer assumptions (as we do not define a cloud base or cloud top pressure) and Occam's razor suggests that this should be preferred for any given values of Na VMR, particle radius and optical depth, where the quality of fit is equal. Heng and Demory (2013) also compare a set of cloud models to the HST /STIS spectrum, and find similar degeneracy between Na abundance and cloud. They find that the abundance of Na relative to the abundance of cloud particles decreases by a factor 10 5 if the cloud particle size decreases by a factor 10 from 100 nm to 10 nm. For sub-µm particles, we find a similar result. The best fit Na VMR, and therefore the number of Na atoms, is roughly constant as a function of particle size if optical depth and cloud altitude are fixed (Figure 3), and we require a factor 10 5 increase in the number of cloud particles as the size decreases from 100 nm to 10 nm to maintain a fixed optical depth. As in the work of Heng and Demory (2013), we find that the shape of the reflection spectrum is largely dictated by the particle size and the Na VMR.
K, TiO and VO
So far, we have only examined the effect of Na on the visible spectrum, and have ignored other species. We include K in the model atmosphere (0.1 ppmv), and Figure 6 shows the effect of varying the abundance over 4 orders of magnitude for a cloud-free model. It can be seen that K has a much smaller effect on the shape of the spectrum than Na (Figure 2), and the magnitude of the change in flux ratio is smaller than the error bars on the STIS spectrum, justifying our decision not to vary K in our main analysis.
HD 189733b is not expected to have a hot enough photosphere for absorption due to gaseous TiO and VO to be present in spectra (Fortney et al. 2010), and no features due to these gases have been observed in the transmission spectrum; however, it is still worthwhile testing the effect of these gases on the spectrum should they unexpectedly be present. We show a series of cloud-free synthetic spectra including absorption due to TiO and VO (using absorption data from Freedman 2011) in Figure 7. The effect is much more significant than the effect of varying the K abundance, and including some TiO and VO as well as Na (10 ppb) improves the spectral fit from the case without TiO and VO. This should not be taken as evidence that TiO and VO are present on HD 189733b, as the problem is degenerate and we still expect this planet to be too cold for these species to be present in the gas phase; in addition, TiO and VO have not yet been detected in the atmospheres of hotter planets (e.g. , so the presence of these molecules in planetary atmospheres is questionable.
Cloud composition
We have so far assumed that any cloud on HD 189733b is made of enstatite. Morley et al. (2012) propose a range of other minerals that are likely to form clouds on brown Figure 5. χ 2 contour plots for 100× solar Na and a cloud from 1000-100 mbar (top) and 100-10 mbar (bottom). The bestfitting models for the first case occur for an optical depth of 1 and a particle size between 0.3 and 3 µm; for the second case, the bestfit model occurs for an optical depth of 10 and a particle size of 0.03 µm. dwarfs, and MnS is a possible candidate for brown dwarfs with similar temperatures to HD 189733b. We repeat the analysis for a sodium VMR of 5 ppmv, with clouds composed of MnS. The results are presented in Figure 8.
It is clear that the particle size/optical depth parameter space still encompasses the most important spectral variability for the cloud, as discussed in the Appendix of . However, choosing MnS instead of enstatite results in a greater range of cloud models that produce a fit to the data with χ 2 <10, but no models that provide a very good fit to the data (χ 2 <5). This is due to the fact that MnS has a different refractive index to enstatite, which affects the scattering efficiency and phase function of the particles, and serves to illustrate the complexity of the cloud parameter space. Figure 7. Synthetic, cloud-free STIS spectra including TiO and VO. The presence of TiO and VO is unlikely in HD 189733b as it is expected to be too cold for TiO and VO condensates to evaporate, but including these species in the model produces a good fit to the spectrum.
Comparison with terminator cloud models
The results presented here illustrate the highly degenerate nature of any cloudy atmosphere solutions for HD 189733b. If we are to assume that the dayside atmosphere is similar to the terminator atmosphere, then we can place some further constraint on the cloud by comparing with results from the transmission spectrum. The analysis of shows that the best-fit models for the terminator atmosphere, for a uniformlydistributed cloud, have particles ∼0.1 µm in size, with an optical depth of 0.01 at 1.0 µm. This corresponds to our set of solutions for uniformly-distributed 0.1 µm particles with an optical depth of 0.5 at 0.25 µm, as the extinction cross-section of 0.1 µm enstatite particles is a factor ∼50 smaller at 1.0 µm than at 0.25 µm. A χ 2 of <6 is achieved for these particles if the Na VMR is 5 ppmv, and χ 2 <3 for a VMR of 50 ppmv. If we can expect the cloud structure to be the same at the terminator and on the dayside, we could therefore place some limits on the Na VMR.
Whether or not we expect the same cloud to form is a question for general circulation models of HD 189733b to solve. So far, few have dealt with cloudy atmospheres, with some exceptions. Parmentier et al. (2013) present a 3D circulation model for the somewhat hotter planet HD 209458b, including passive tracer particles of various sizes to examine the relative effects of circulation and sedimentation on aerosols and condensates; they find that, whereas larger particles sediment out and show significant longitudinal variability in abundance, the number of 0.1 µm-sized tracer particles remains relatively uniform with longitude. This indicates that a cloud composed of 0.1 µm particles could easily be approximately uniform throughout the atmosphere of a hot Jupiter, and therefore we may be observing the same cloud deck on the dayside as observe at the terminator, although it is likely that different altitudes are probed.
THERMAL EMISSION RETRIEVAL
We now take a subset of the cloud models used in Section 3 and examine their effect on retrievals of temper- Figure 8. Contour plots of χ 2 goodness-of-fit parameter for cloud bottom pressure, optical depth and particle size for MnS clouds. Also plotted are the percentages of models with χ 2 of <10 for each parameter. The dashed lines indicate the values used for each parameter when held fixed. Figure 9. The data for the thermal emission retrieval. NICMOS points are as reanalysed in this work; 3.6 µm and 4.5 µm IRAC points are from Knutson et al. (2012); the IRS spectrum is from Grillmair et al. (2008); the IRS broadband point was originally measured by Deming et al. (2006); 5.8 and 8.0 µmSpitzer /IRAC and MIPS points are as measured by , along with a reanalysis of the broadband IRS point. ature and molecular abundances from the infrared secondary eclipse spectrum (Figure 9). The retrieval is performed as in using hemispheric integration for non-scattering runs and a 45 • emission angle approximation for scattering runs. We also use our reanalysed NICMOS spectrum and the warm Spitzer 3.6, 4.5 and 8 µm points presented in Knutson et al. (2012). We include the cloud model in the model atmosphere, but do not make any attempt to retrieve it; instead, we examine its effect on the other parameters in the retrieval.
Firstly, we repeat the cloud-free analysis of with the addition of the reanalysed NICMOS data points; state that the NICMOS spectrum provides information about the temperature profile in the deep atmosphere, and also the abundances of trace gases, and as these data have changed we expect to retrieve a different temperature profile and abundances. We perform the retrieval of temperature as a function of pressure, over 50 atmospheric levels with a correlation length of 1.5 in ln(pressure), and altitude-independent abundances of H 2 O, CO 2 , CO and CH 4 using a range of different temperature and gas abundance priors; since the problem is underconstrained, it is necessary to test the influence of the prior on the retrieved property. We present the results of this analysis for temperature in Figure 10 and Table 4. The best-fit retrieved temperature profile (Figure 10 top left) and gas abundances are averaged over all runs. The temperature sensitivity weighting function averaged over all wavelengths is shown qualitatively in Figure 10 by a shaded bar, with the peak (lightest colour) occuring between 1 and 100 mbar; this corresponds to the region of the atmosphere probed by the majority of the data, and therefore the temperature is least dependent on the temperature prior in this region; this is shown in Figure 10 top right, in which the retrieved profiles for different priors converge between 1 and 100 mbar.
The temperature structure is most sensitive to the Knutson et al. (2012) Spitzer /IRAC 3.6 and 4.5 µm points as these data have by far the smallest fractional error, meaning that they drive the temperature retrieval. This accounts for the differences in the shape of the retrieved T-p profile compared with that of . The 3.6 µm weighting function peaks at around 1 mbar and constrains the stratospheric temperature, whereas the 4.5 µm weighting function peaks at the tropopause, at around 100 mbar.
In Figure 10 it can be seen that the greatest degeneracy between temperature and gas abundance occurs for H 2 O and CO 2 , with the precise shape of the retrieved T- Figure 10. Retrieved temperature profiles for a range of different priors. The mean and standard deviation are shown in the top left; the shaded bar shows the average sensitivity to temperature at each altitude, across all wavelengths, with black being low sensitivity and white high sensitivity. The maximum sensitivity occurs between 1 and 100 mbar. The varying temperature prior is the best-fit case from Lee (2012), shifted at 50 K intervals from -250K (lightest grey line) to +250K (darkest grey line), with a prior error of ±50 K. We also vary the prior altitude-independent gas VMR for H 2 O, CO 2 , CO and CH 4 . The darkest-lightest grey colours correspond to priors for each gas of 1 ppmv, 10 ppmv, 100 ppmv and 1000 ppmv. It can be seen that the greatest degeneracy occurs for CO 2 . The temperature prior used where the gas priors are varying is the thick black line, with a prior error of ±50 K. p profile between 1 and 100 mbar varying as a function of the H 2 O and CO 2 abundance priors. This is because the peak of the temperature weighting function and therefore the pressure probed varies with atmospheric opacity, and these two gases are the most spectrally active. The increase in the error bars for the NICMOS spectrum means that the gas abundance retrievals are more sensitive to the prior than found by , so for a small H 2 O or CO 2 prior abundance the retrieved abundance is smaller, the atmospheric opacity is less and the pressure probed is higher. The reverse is true for high prior abun-dances of H 2 O or CO 2 . The effects of changing the CO and CH 4 abundances are smaller, but similar.
The average temperature profile retrieved is largely similar to that of Line et al. (2013), except that we see a slight decrease in temperature above the 1 mbar level compared with Line et al. (2013). This is probably due to the fact that we retrieve temperature for each atmospheric level individually, with an assumed correlation length (1.5 in log-pressure coordinates), whereas Line et al. (2013) parameterise the temperature profile. Therefore, the upper atmosphere temperature retrieved 10 mbar 10 0.03 µm Table 5 The parameters for the cloud models included in thermal emission retrievals, to test the dependence on cloud properties.
by Line et al. (2013) cannot vary freely with respect to the temperature between 1 and 100 mbar, where the sensitivity is greatest, whilst we retrieve a continuous profile.
The mean, standard deviation and range of retrieved gas VMRs are given in Table 4. The standard deviations are highest for CO 2 and CO, implying a high dependence on the prior. The retrieved values for H 2 O and CO 2 are similar to that of , but the value for CO is smaller, although due to the dependence on the prior this result must be viewed with caution. However, we retrieve a much higher CH 4 abundance than , roughly a factor of 10 higher than the Spitzerderived upper limit of Madhusudhan and Seager (2009). Our higher value for CH 4 is similar to the result of Line et al. (2013), and so it is likely to be a result of the revised 3.6 and 4.5 µm points which are also used by these authors.
We now seek to investigate the effect of cloud on the retrieved temperature profile and gas abundances. We use the average temperature prior and a priori gas VMRs of 10 −4 for all retrieved species. The parameters for the set of models we use are shown in Table 5. These correspond to some of the best-fitting solutions from the previous analysis that have very different infrared optical depths, so are likely to affect the thermal emission retrieval differently.
We show the retrieved temperature profiles and gas VMRs for each of the cloud models, as well as the gas abundances for the cloud-free case ( Figure 11, Table 6). We perform each of the retrievals twice, first using the fast, extinction-only disc integration technique as in , and secondly using multiple scattering as in Section 3, except we approximate the disc with a single 45 • emission/incidence angle as the 5-angle calculation is extremely time-consuming for a full retrieval.
It is clear from Figure 11 and Table 6 that the thermal emission retrieval is largely insensitive to the inclusion of clouds corresponding to our best-fitting models. This is due to the fact that the majority of best-fitting models with optical depths greater than 1 at 0.25 µm have small particle sizes, and therefore the extinction drops rapidly as a function of wavelength. These particles have little effect at wavelengths longer than 1 µm because the particle size is small with respect to the wavelength of the light, and so the optical depth in the infrared is negligible even though it is significant at shorter wavelengths. This means that even higher optical depths of 0.1 µm-sized particles would be unlikely to change the temperature retrieval.
The exception is the retrieved H 2 O and CO 2 abundances and temperature profile for cloud model 4, using an extinction-only assumption; including a high (10-1 mbar) cloud of 0.03 µm-sized particles results in a larger retrieved CO 2 abundance, which in turn affects the shape of the temperature profile in this pressure range (as in Figure 10). Where solutions are degenerate, as in this case, it is possible for small changes in one model variable (here, the cloud optical depth) to result in different best-fit solutions being obtained. In Figure 12, we show the synthetic spectrum for the model 4 retrieved case, and the spectrum for a forward model with retrieved temperature profile and gas abundances as for a cloud-free model but including a model 4 cloud. The spectra are very similar, with χ 2 of 138 and 142 respectively, demonstrating that the difference in retrieved CO 2 abundance for this case should not be treated as significant. Additionally, the multiple scattering retrieval for model 4 yields a lower abundance of CO 2 and a temperature profile more similar to those retrieved using other cloud models. For the case of HD 189733b, therefore, it seems that an accurate retrieval of temperature and atmospheric composition from the thermal emission spectrum may not be critically dependent on an understanding of the cloud properties.
DISCUSSION
We have used a very simple set of cloud models to investigate the acceptable parameter space for the HD 189733b STIS spectrum. The quality of the data at this stage do not warrant more detailed modelling, but it is instructive to consider the most important steps that could be taken to improve this model in anticipation of future results.
Monodisperse particles
We have assumed that all the particles in the cloud are of a single size. This is of course not likely to be the case in practice; cloud models based on microphysical processes indicate that particle sizes will change as a function of altitude, and at a given altitude particle size distributions may be relatively broad, for example as discussed by Helling et al. (2008). Coalescence processes, condensational growth and sedimentation will result in the presence of larger grains towards the bottom of the cloud deck.
Broadening the size distribution will make the cloud optical depth more uniform as a function of wavelength; extinction due to cloud particles is most efficient when the particle size is comparable to the wavelength of light. This may result in the cloud having a larger effect on the thermal emission spectrum than we find in this work; although in most cases we find that particle sizes of larger than 0.1 µm do not provide a good fit to the data at short wavelengths, this does not preclude larger particles being Table 6 The retrieved gas VMRs for each of the four cloud models, including the cloud-free case (model 5), for extinction-only (left) and multiple scattering (right). The cloud-free case was modelled using an extinction-only approximation.
present as they would scatter the short wavelength light less strongly than smaller particles. When the data are available that would allow us to test a more complex model, we consider that investigating different particle size distributions would be the most important step to take, as particle size is the property that has the largest effect on the spectrum.
Particle composition
The composition of clouds on cool brown dwarfs and hot exoplanets is treated somewhat differently in different models. Morley et al. (2012) consider a range of possible cloud compositions for specific temperatures, under the assumption that if clouds of more than one species are present they will form at different temperatures, and therefore altitudes. Helling et al. (2008) have a very different model, in which multiple species condense onto the same TiO 2 'seed particle' (Helling and Woitke 2006), thus forming grains of mixed composition. This significantly complicates the calculation of spectral properties for these clouds; however, Helling et al. (2008) find that, for a low-gravity atmosphere with T ef f ∼1300 K, the enstatite is the constituent with the largest effect on the spectrum, meaning that enstatite probably represents the best approximation to the multi-component grains. Figure 12. We show synthetic spectral fits for a forward model containing the cloud from model 4 but the retrieved parameters from model 5 (grey) and the retrieved parameters and cloud from model 4 (black). The difference between the two spectra is small, with χ 2 of 142 and 138 respectively. The numbers of spectral points (71) and degrees of freedom (54) are the same for both cases. The circles show the high-resolution synthetic spectra convolved at the Spitzer IRAC, broadband IRS and MIPS bandpasses.
We tested the effect of including MnS, as proposed by Morley et al. (2012) for cool brown dwarfs, and find that the different spectral properties of the material do affect the spectrum, but to a lesser extent than the size of the particles. Therefore, particle composition is an interesting route for further investigation, but is of secondary importance to developing more detailed representations of particle size.
Effect of temperature profile
We have modelled the STIS reflection spectrum using the best-fit temperature profile from Lee (2012), but have then retrieved a different profile using the thermal emission spectrum. We tested the effect on the best-fit cloud properties of including the altered temperature profile in our analysis of the cloud. We found that the changes to the spectral shape were negligible compared with the error bars on the STIS spectrum (Figure 13), and the χ 2 values for different cloud models do not change greatly. Therefore small inaccuracies in the temperature profile we use would not significantly affect our conclusions; the same cloud models provide good/poor fits to the measured spectrum regardless of the temperature profile used.
The reason for the limited effect of temperature is likely to be that the absorption due to Na at the altitudes to which we are sensitive is dominated by pressurebroadened line wings. Temperature variation is more likely to affect the line core absorption at higher altitudes, but at the resolution of the STIS spectrum such variation would not be seen. The STIS measurements also lie in a wavelength region where the signal is dominated by reflected starlight, rather than light emitted from the planet.
CONCLUSION
We find that a large range of enstatite cloud models can fit the measured HST /STIS spectrum. Cloud-free model atmospheres are also acceptable solutions, although we consider this to be less plausible due to the clear evi- Figure 13. We show two STIS spectrum models, both containing the model 1 cloud properties. One is the model generated using the original temperature profile, the other uses the retrieved temperature profile for model 1. The synthetic spectra are indistinguishable from each other.
dence from the transmission spectrum that HD 189733b is cloudy ) and the likelihood that small particles are evenly distributed in hot Jupiter atmospheres (Parmentier et al. 2013). Small enstatite particles (<0.1 µm) and 50 ppmv of Na provide the best fit to the STIS data of the examples we test; we find an overlap with the models of for a uniformly-distributed 0.1 µm cloud with an optical depth of 0.5 at 0.25 µm. However, the problem is extremely degenerate and we cannot exclude solutions with larger cloud particles.
The retrieval of temperature and atmospheric composition from the thermal emission spectrum is relatively insensitive to the inclusion of cloud in the model atmosphere, for our best-fitting models. This suggests that for the case of HD 189733b accurate retrieval of temperature and gaseous abundances from the thermal emission spectrum is possible, even without detailed knowledge of the cloud properties. Solution degeneracy prevents firm conclusions from being drawn about the nature of the cloud on HD 189733b; the current quality and coverage of spectroscopic data for most exoplanets is therefore insufficient to simultaneously constrain temperature structure, gaseous abundances and multiple cloud properties. Given the additional complexity (and therefore number of degenerate solutions) introduced to the retrieval problem when clouds are included, this implies that the best approach with the currently available secondary eclipse data is to use cloud-free model atmospheres for temperature retrieval. As data quality improves, alternative, more detailed cloud models must be explored and their effect on the emission spectrum reassessed.
Figure 4 .
4Spectra for a selection of cloud models with different properties and Na VMRs. The 0.1 µm models are uniformly distributed in altitude, and the 10 µm models are between 1 and 0.1 mbar. The 0.1 µm model with 10×Na and the 10 µm model with 1×Na fit the spectrum well (χ 2 <5), whereas the other two models do not.
Figure 6 .
6Synthetic, cloud-free STIS spectra containing different abundances of K. Varying the amount of K in the model does not significantly affect the spectrum.
Figure 11 .
11Retrieved temperature profiles for each cloud model. Thick solid lines show the a priori temperature, dotted lines the extinction-only retrieval and dashed lines the multiple scattering retrieval. Thick lines show the retrieved value and thin lines show the error envelope.
1.1.1. Multiple scattering calculations 1 http://kurucz.harvard.edu/stars/HD189733/
The reanalysed NICMOS spectrum.1.4638
1.53
1.81
1.5214
-5.11
2.31
1.5790
-0.762
2.34
1.6366
-1.01
2.99
1.6941
2.98
1.96
1.7518
3.73
2.44
1.8094
4.16
1.33
1.8670
3.93
1.84
1.9246
2.65
1.52
1.9822
0.834
1.86
2.0398
2.93
2.68
2.0974
1.83
1.20
2.1550
1.76
2.38
2.2126
4.37
0.941
2.2702
3.44
0.937
2.3278
4.79
3.06
2.3854
3.61
2.17
2.4430
-0.588
5.20
Table 2
Table 4
4The mean retrieved values for each gas VMR, and the minimum and maximum retrieved values over all priors. H 2 O and CH 4 can be reasonably well constrained, but CO 2 and CO have σ larger than 100% of the mean value. The best-fit values for Lee et al.(2012) are given for comparison.
Model
Base p
0.25 µm O. D. Particle radius
1
Uniform
10
0.1 µm
2
1000 mbar
1
0.01 µm
3
1000 mbar
1
3 µm
4
Model H 2 O VMR (10 −4 ) CO 2 VMR (10 −4 ) CO VMR (10 −4 ) CH 4 VMR (10 −4 ) 1 9.7±4.9/11±6 8.9±4.6/8.1±4.2 1.4±1.4/1.4±1.4 0.25±0.11/0.26±0.11 2 9.6±4.9/11±6 9.0±4.7/8.2±4.3 1.4±1.4/1.4±1.4 0.25± 0.11/0.26±0.11 3 8.4±4.4/11±5 8.1±4.2/7.8±4.1 1.4±1.4/1.4±1.4 0.24±0.10/0.25±0.11
4
3.8±2.0/6.8±3.7
31±17/13±7
1.3±1.3/1.3±1.3 0.31± 0.09/0.31± 0.11
5
9.7±4.9
8.9±4.6
1.4±1.4
0.25±0.11
ACKNOWLEDGEMENTSJKB acknowledges the support of the John Fell Oxford University Press (OUP) Research Fund for this research and LNF is supported by a Royal Society Research Fellowship. We thank the anonymous reviewer for his/her report, and Tom Evans, Frédéric Pont, David Sing and Kevin Heng for helpful discussions about the visible albedo observations.
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| []
|
[
"THE ORIGIN OF HEAVY ELEMENT CONTENT TREND IN GIANT PLANETS VIA CORE ACCRETION",
"THE ORIGIN OF HEAVY ELEMENT CONTENT TREND IN GIANT PLANETS VIA CORE ACCRETION"
]
| [
"Yasuhiro Hasegawa \nJet Propulsion Laboratory\nCalifornia Institute of Technology\n91109PasadenaCAUSA\n",
"Geoffrey Bryden \nJet Propulsion Laboratory\nCalifornia Institute of Technology\n91109PasadenaCAUSA\n",
"Masahiro Ikoma \nDepartment of Earth and Planetary Science\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan\n",
"Gautam Vasisht \nJet Propulsion Laboratory\nCalifornia Institute of Technology\n91109PasadenaCAUSA\n",
"Mark Swain \nJet Propulsion Laboratory\nCalifornia Institute of Technology\n91109PasadenaCAUSA\n"
]
| [
"Jet Propulsion Laboratory\nCalifornia Institute of Technology\n91109PasadenaCAUSA",
"Jet Propulsion Laboratory\nCalifornia Institute of Technology\n91109PasadenaCAUSA",
"Department of Earth and Planetary Science\nThe University of Tokyo\n7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan",
"Jet Propulsion Laboratory\nCalifornia Institute of Technology\n91109PasadenaCAUSA",
"Jet Propulsion Laboratory\nCalifornia Institute of Technology\n91109PasadenaCAUSA"
]
| []
| We explore the origin of the trend of heavy elements in observed massive exoplanets. Coupling of better measurements of mass (M p ) and radius of exoplanets with planet structure models enables estimating the total heavy element mass (M Z ) in these planets. The corresponding relation is characterized by a power-law profile, M Z ∝ M 3/5 p . We develop a simplified, but physically motivated analysis to investigate how the power-law profile can be produced under the current picture of planet formation. Making use of the existing semi-analytical formulae of accretion rates of pebbles and planetesimals, our analysis shows that the relation can be reproduced well if it traces the final stage of planet formation. In the stage, planets accrete solids from gapped planetesimal disks and gas accretion is limited by disk evolution. We also find that dust accretion accompanying with gas accretion does not contribute to M Z for planets with M p < 10 3 M ⊕ . Our findings are broadly consistent with that of previous studies, yet we explicitly demonstrate how planetesimal dynamics is crucial for better understanding the relation. While our approach is simple, we can also reproduce the trend of a correlation between planet metallicity and M p that is obtained by detailed population synthesis calculations, when the same assumption is adopted. Our analysis suggests that pebble accretion would not play a direct role at the final stage of planet formation, whereas radial drift of pebbles might be important indirectly for metal enrichment of planets. Detailed numerical simulations and more observational data are required for confirming our analysis. | 10.3847/1538-4357/aad912 | [
"https://arxiv.org/pdf/1807.05305v1.pdf"
]
| 119,088,936 | 1807.05305 | 1f3765165de40b2c0482427a911297dfc0dc7b2f |
THE ORIGIN OF HEAVY ELEMENT CONTENT TREND IN GIANT PLANETS VIA CORE ACCRETION
July 17, 2018 Draft version July 17, 2018
Yasuhiro Hasegawa
Jet Propulsion Laboratory
California Institute of Technology
91109PasadenaCAUSA
Geoffrey Bryden
Jet Propulsion Laboratory
California Institute of Technology
91109PasadenaCAUSA
Masahiro Ikoma
Department of Earth and Planetary Science
The University of Tokyo
7-3-1 Hongo, Bunkyo-ku113-0033TokyoJapan
Gautam Vasisht
Jet Propulsion Laboratory
California Institute of Technology
91109PasadenaCAUSA
Mark Swain
Jet Propulsion Laboratory
California Institute of Technology
91109PasadenaCAUSA
THE ORIGIN OF HEAVY ELEMENT CONTENT TREND IN GIANT PLANETS VIA CORE ACCRETION
July 17, 2018 Draft version July 17, 2018Draft version Preprint typeset using L A T E X style emulateapj v. 12/16/11methods: analytical -planets and satellites: composition -planets and satellites: formation -planets and satellites: gaseous planets -protoplanetary disks
We explore the origin of the trend of heavy elements in observed massive exoplanets. Coupling of better measurements of mass (M p ) and radius of exoplanets with planet structure models enables estimating the total heavy element mass (M Z ) in these planets. The corresponding relation is characterized by a power-law profile, M Z ∝ M 3/5 p . We develop a simplified, but physically motivated analysis to investigate how the power-law profile can be produced under the current picture of planet formation. Making use of the existing semi-analytical formulae of accretion rates of pebbles and planetesimals, our analysis shows that the relation can be reproduced well if it traces the final stage of planet formation. In the stage, planets accrete solids from gapped planetesimal disks and gas accretion is limited by disk evolution. We also find that dust accretion accompanying with gas accretion does not contribute to M Z for planets with M p < 10 3 M ⊕ . Our findings are broadly consistent with that of previous studies, yet we explicitly demonstrate how planetesimal dynamics is crucial for better understanding the relation. While our approach is simple, we can also reproduce the trend of a correlation between planet metallicity and M p that is obtained by detailed population synthesis calculations, when the same assumption is adopted. Our analysis suggests that pebble accretion would not play a direct role at the final stage of planet formation, whereas radial drift of pebbles might be important indirectly for metal enrichment of planets. Detailed numerical simulations and more observational data are required for confirming our analysis.
INTRODUCTION
The detection of a large amount (> 3000) of confirmed exoplanets has rapidly filled out a greater area in the mass-semimajor axis diagram (e.g., Mayor et al. 2011;Borucki et al. 2011;Mayor et al. 2014;Twicken et al. 2016). These observations unveil a huge diversity of exoplanetary systems that gives a number of challenges to the current theory of planet formation. These include the presence of hot Jupiters that were first discovered by the radial velocity technique (Mayor & Queloz 1995), the rich population of close-in super-Earths that is confirmed by both doppler and transit methods (e.g., Mayor et al. 2011;Howard et al. 2010), the existence of distant giant planetary systems that is revealed by direct imaging (e.g., Marois et al. 2010), and the prediction of a significant population of free-floating planets made by microlensing observations (e.g., Sumi et al. 2011;Mróz et al. 2017).
A number of improvements have been made so far for better understanding observed exoplanetary systems and eventually developing a complete picture of planet formation. One of the biggest leaps achieved was planetary migration (e.g., Goldreich & Tremaine 1980). This process arises from gravitational, tidal resonant interaction between planets and their gas disks, and was initially invoked for explaining the presence of hot Jupiters (Lin et al. 1996). However, subsequent studies showed that migration is inevitable for planets in a wide mass range (M p > 1M ⊕ ), and that the migration rate is generally [email protected] much faster than the growth rate of planets (e.g., Ward 1997;Nelson et al. 2000;Masset 2001;Tanaka et al. 2002;Paardekooper et al. 2010;Hasegawa & Pudritz 2011a). As a result, whereas some mechanisms for slowing down or even stopping migration have been proposed (e.g., Masset et al. 2006;Hasegawa & Pudritz 2011b;Kretke & Lin 2012;Dittkrist et al. 2014), the fundamental role of planetary migration is still unclear (see Kley & Nelson 2012, for a review). The general consensus in the community is that planet-forming materials move through protoplanetary disks, and hence planet formation is a global process involved with the entire region of the disks, rather than a local process.
Characterization of exoplanets is crucial for making further progress. For instance, influence of the host stellar metallicity on the occurrence rate of planets has been explored to specify the formation mechanism of observed exoplanets (e.g., Santos et al. 2004;Fischer & Valenti 2005;. Mass measurements by the radial velocity coupled with radius measurements by transit allow one to estimate the bulk density of exoplanets (e.g., Gettel et al. 2016;Jontof-Hutter et al. 2016). More recently, observations of exoplanets' atmospheres have become feasible, and one can now detect some molecules in the atmospheres (e.g., Tinetti et al. 2007;Swain et al. 2008;Madhusudhan et al. 2011;Kreidberg et al. 2014;Wakeford et al. 2018). Accompanying such observations, theoretical studies have been undertaken for making a link with the observations and obtaining insights into the formation and migration histories of planets (e.g., ; Mor-arXiv:1807.05305v1 [astro-ph.EP] 13 Jul 2018 dasini et al. 2012;Madhusudhan et al. 2014;Hasegawa & Pudritz 2014;Mordasini et al. 2016;Madhusudhan et al. 2017). For example, Guillot et al. (2006) directly computed the total heavy element mass in planets, using mass and radius measurements of observed hot Jupiters (also see Miller & Fortney 2011).
In this paper, we develop a consistent view of how accretion of gas and solids takes place onto growing planets in protoplanetary disks. We focus on the total heavy element mass (M Z ) in observed exoplanets that is calculated by Thorngren et al. (2016, hereafter T16). In their study, the radius evolution of warm Jupiters is computed, utilizing their thermal evolution models of planets (see Section 4.1 for the detail). By comparing their computed planet radii with observed ones, they specify the value of M Z for warm Jupiters and derive correlations between M Z and M p and between M p and the planet metallicity (Z p = M Z /M p , see Figure 1). Hereafter these two correlations are referred to as the M Z − M p and the Z p − M p relations. In this work, we examine both planetesimal and pebble accretion within a single framework to account for the results of T16. More specifically, we make use of the existing semi-analytical formulae for the accretion rates of planetesimals and pebbles, and compute the power-law indices for the M Z − M p and Z p − M p relations. As clearly demonstrated below, we find that the subsequent planetesimal accretion after core formation is the most plausible case for better reproducing the relations. This is consistent with the results of previous studies (e.g., Pollack et al. 1996;Mordasini et al. 2014Mordasini et al. , 2016). Yet, our follow-up work pins down the importance of planetesimal dynamics on the M Z − M p relation.
The plan of this paper is as follows. In Section 2, we describe the core accretion scenario and summarize some key quantities and equations. In Section 3, We develop a framework to investigate how both gas and solid accretion onto growing planets determine the power-law indices of the M Z − M p and Z p − M p relations in the core accretion picture. We treat core formation, planetesimal accretion, pebble accretion, and the effect of gas accretion separately, and examine their contributions to these two relations. In Section 4, we introduce the results of T16 and reanalyze them. We also compare the results of our theoretical analysis with those of T16. In Section 5, we summarize the limitation of our analysis. We also discuss other physical processes that are not included in our analysis, and compare our findings with those of previous studies. We propose a classification of observed exoplanets. We finally list up potential roles of the current and future observations. A brief summary and conclusions of this work are presented in Section 6.
PLANET FORMATION VIA CORE ACCRETION
We here consider the basic picture of core accretion.
The key quantities of this work are summarized in Table 1.
Core formation & gas accretion
The core accretion scenario is the widely accepted picture of how planets form in protoplanetary disks (e.g., Mordasini et al. 2009;Benz et al. 2014).
In this scenario, planetary cores form first and then gas accretion onto the cores proceeds with simultaneous accretion of non-negligible amounts of solids (e.g., Pollack et al. 1996). Currently, two scenarios of core formation are actively investigated: one is runaway and oligarchic growth and the other is pebble accretion. For the former, planetesimals are the dominant form of solids to build planetary cores, and their size is generally considered as a few hundred km (e.g., Wetherill & Stewart 1989;Kokubo & Ida 1998). In this scenario, core formation is terminated when cores accrete all the planetesimals in their feeding zone and achieve the so-called isolation mass that is a function only of the solid surface density. For the latter, pebble-sized (∼ cm-m) particles that are weakly coupled with the disk gas provide the main contribution to core formation through the radial drift of such particles (e.g., Ormel & Klahr 2010;Lambrechts & Johansen 2012). In this case, mass growth of planetary cores shuts off when the cores become massive enough to open up a gap in their gas disks (e.g., Lambrechts et al. 2014;Bitsch et al. 2018). 1 In other words, the cores are not exposed to the pebble flux anymore due to blocking out of pebbles by a gas gap formed around the cores. Both the scenarios therefore lead to the final core mass that is a function only of disk parameters.
One of the key quantities in core accretion is the critical core mass that regulates the onset of efficient gas accretion onto planetary cores (e.g., Mizuno 1980;Ikoma et al. 2000). The critical core mass is defined such that gaseous envelopes around the cores cannot maintain a hydrostatic equilibrium and runaway gas accretion takes place. Under the assumption that the grain opacity of the envelopes is comparable to the ISM value, the canonical value of ∼ 10M ⊕ has been widely adopted in the literature (e.g., Pollack et al. 1996;Ikoma et al. 2000;Mordasini et al. 2009). Recent studies, however, show that when dust grain growth in planetary envelopes is properly taken into account, the value of the critical core mass tends to decrease considerably. This arises from a lower value of the grain opacity in planetary envelopes, which leads to rapid cooling of the envelopes and their resulting, efficient contraction (e.g., Movshovitz & Podolak 2008;Hori & Ikoma 2010;Movshovitz et al. 2010;Ormel 2014). It is interesting that a lower value ( 5 − 10M ⊕ ) of the critical core mass is in favor of theoretically reproducing the trends of observed exoplanet population (e.g., Mordasini et al. 2014;Hasegawa & Pudritz 2014). This can be readily seen by considering gas accretion onto planetary cores (see below). Another interesting feature of the critical core mass is that it may be used as one of the tracers to differentiate the origin of super-Earths from that of gas giants. Given that one clear difference between these two types of planets is the envelope mass and that the formation mechanism(s) of super-Earths is still unclear (e.g., Hansen & Murray 2013;Chiang & Laughlin 2013;Hasegawa 2016), it is of fundamental importance to identify the value of the critical core mass using the observational data of exoplanets. their Table 1), and the black solid line is their best fit (see their Figure 7). For the right panel, the planet metallicity (Zp = M Z /Mp) that is normalized by the stellar metallicity (Zs) is shown as a function of Mp. As in the left panel, the black points and the black solid line are adopted from T16. Gas accretion onto planets begins once planetary cores become massive enough. In principle, the gas accretion process can be modeled as the Kelvin-Helmholtz timescale (τ g,KH ). This timescale is written as (e.g., Ikoma et al. 2000;Hasegawa & Pudritz 2012)
τ g,KH = 10 c f grain M p 10M ⊕ −d yr,(1)
where f grain 1 is the acceleration factor due to the reduction of grain opacity in planetary envelopes, resulting from grain growth there. In this paper, we adopt that c = 7 and d = 4, following Tajima & Nakagawa (1997, see their equation (26)). As clearly seen in equation (1), τ g,KH becomes much shorter than the typical disk lifetime of a few 10 6 yrs (e.g., Williams & Cieza 2011) when the initial core mass exceeds ∼ 10M ⊕ . This is one of the reasons why smaller core masses are preferred for reproducing the observed population of exoplanets.
One would notice that τ g,KH keeps decreasing as M p increases (see equation (1)). This can eventually lead to an unrealistically high value of the gas accretion rate (dM XY /dt) for massive planets ( 100M ⊕ ). Accordingly, an upper limit is generally imposed for limiting dM XY /dt. In this paper, we adopt the results of Tani-gawa & Watanabe (2002). In their work, 2D hydrodynamical simulations are performed, and gas accretion flow onto planets from protoplanetary disks and the fine structure of circumplanetary disks are resolved with high spatial resolution simulations. They find that the upper limit of the gas accretion rate is given as (Tanigawa & Watanabe 2002, see their equation (20)) τ g,hydro 1.1 × 10 3 σ −1 p,acc a p 5 au
1.5 M p 10M ⊕ −d yr,
(2) where σ p,acc is the normalized surface density of gas that participates in gas accretion, a p is the semimajor axis of planets, and d = 1/3. Note that when a gap is opened up in gas disks due to disk-planet interaction (e.g., Nelson et al. 2000;Crida et al. 2006;Kley & Nelson 2012), σ p,acc becomes a function of M p (Tanigawa & Ikoma 2007).
In summary, the mass growth rate of planets via gas accretion can be written as
dM XY dt dM p dt = M p τ g,acc ,(3)
where
τ g,acc = max [τ g,KH , τ g,hydro ] .(4)
The above equations are valid mainly at the final stages of planet formation in which core formation nearly ends and solid accretion onto planets is insignificant, compared with gas accretion.
2.2. Additional solid accretion It has been suggested for a long time that additional solid accretion is essential for fully understanding the total heavy element mass of gas giant planets (e.g., Pollack et al. 1986;Podolak et al. 1988). For instance, the enhanced metallicity in the atmosphere of Jupiter and Saturn claims the need of additional solid accretion during the process of forming (e.g., Pollack et al. 1996;Saumon & Guillot 2004). As another example, Mordasini et al. (2014) show that planetesimal accretion after core formation completes is important for reproducing the Z p − M p relation of observed exoplanets (also see Mordasini et al. 2016).
In order to examine at what stage, how solid accretion occurs for growing planets in protoplanetary disks, we explore the mass contribution (M Z,solid ) arising from solid accretion by decomposing it into three components:
M Z,solid = M core + M pl + M pe ,(5)
where M core is the initial, seed core mass of a protoplanet at which the subsequent gas accretion begins, M pl is the total heavy element mass that is obtained via planetesimal accretion, and M pe is the total heavy element mass that is gained during accretion of small bodies such as pebbles (see Table 1). Accretion of both planetesimals and pebbles onto (proto)planets would be possible during the gas accretion stage (e.g., Pollack et al. 1996;Rafikov 2004;Alibert et al. 2005;Tanigawa et al. 2014). As described in equation (5), we treat them separately in this paper.
2.3. Mass budget in planets Finally, the mass budget of a planet can be written as
M p = M XY + M Z ,(6)M Z = M Z,solid + M Z,gas ,(7)
where M XY is the total envelope mass of the planet, M Z is the total heavy element mass of the planet, M Z,solid is the total heavy element mass that is accumulated in the planet through accretion of solids such as pebbles and planetesimals (see equation (5)), and M Z,gas is the total heavy element mass that is accreted through gas accretion (see Table 1). Note that the disk gas accreted onto planets contains small (∼ µm -mm) dust particles. Such solids are well coupled with the disk gas and hence follow the gas motion. Accordingly, these solids are also accumulated in planets as the gas is accreted onto the planets. We take into account this contribution by including the term of M Z,gas .
THEORETICAL ANALYSIS
We develop a simplified, but physically motivated analysis to understand how accretion of gas and solids takes place onto growing protoplanets in protoplanetary disks. We make use of the equations in the above section.
Basic formulation
We first formulate the basic equation exploring the M Z − M p relation.
As discussed in Section 2.2, additional solid accretion would be plausible during the gas accretion stage. It is nonetheless important to point out that the actual efficiency is currently under active investigation (e.g, Zhou & Lin 2007;Johansen & Lambrechts 2017) and is most likely determined by disk parameters. In order to shed light on the underlying physics, we focus only on the power index of the M Z −M p relation in this paper. While this simplification provides some limitations for our analysis (see Section 5.1), we then need to care only about the M p dependence on each valuable.
To proceed, we adopt the approach originally developed by Shiraishi & Ida (2008). In this approach, the derivative of M Z is examined, which is given as
dM Z dM p = dM Z dt dt dM p ≈ dM Z dt τ g,acc M p ∝ M Γ p ,(8)
where it is assumed that mass growth (dM p /dt) of planets is dominated by gas accretion (dM XY , also see equation (3)). This assumption would be valid at the final stages of planet formation. Then, we simplify the gas accretion timescale (τ g,acc , see equation (4)) as
τ g,acc = max [τ g,KH , τ g,hydro ] ∝ M D p ,(9)
where D = −d = −4 when τ g,KH > τ g,hydro , and D = −d = −1/3 when τ g,KH < τ g,hydro . Note that we pay attention only to the M p dependence in this analysis. Also, we neglect the effect of gas gaps that can be opened up by disk-planet interaction. We discuss this effect in Section 5.2 and Appendix A.
In the following, we utilize equation (8) and investigate how the power index of the M Z − M p relation changes as a function of forms (planetesimals vs pebbles) of solids that are accreted onto planets.
Contribution from planetesimal accretion
In this section, we consider the contribution arising from M pl to M Z,solid , that is, how planetesimal accretion proceeds in a post-stage of (initial) core formation. Equivalently, (see equations (5), (7), and (8))
M Z ≈ M Z,solid ≈ M pl ,(10)dM Z dM p ≈ dM pl dt τ g,acc M p ∝ M Γ pl p .(11)
The remarkable recognition that continuous accretion of planetesimals is important for planet formation is made by the milestone work of Pollack et al. (1996). In this study, it is assumed that such accretion originates from the expansion of planets' feeding zone as the planets grow in mass and their Hill radius increases. Adopting the most efficient accretion rate of planetesimals, they can reproduce the trend of the enhanced atmospheric metallicity of Jovian planets in the solar system such as Jupiter and Saturn. Such efficient accretion of planetesimals leads to emergence of the so-called "phase 2", where the planetesimal accretion rate is so high (∼ 10 −6 M ⊕ yr −1 ) that the onset of runaway gas accretion is postponed for ∼ a few Myr. Despite of the success achieved by their model, a number of follow-up studies pose a question about their assumption that the most efficient planetesimal accretion would be realized and continue for a long (∼ Myr) time (e.g., Fortier et al. 2007;Zhou & Lin 2007;Shiraishi & Ida 2008;Hasegawa & Pudritz 2014). This is because, following mass growth of planets, planetesimals in their feeding zone will be used up, and some of them will be even scattered out of the zone due to the gravitational interaction with the planets. Coupled with the eccentricity dumping by the disk gas, this scattering process can end up with the creation of a gap in planetesimal disks around planets. In fact, the common conclusion of these studies is that when both the dynamics of planetesimals in gas disks and the effect of planetary growth are considered realistically, efficient planetesimal accretion cannot be established.
To appropriately take into account the dynamics of planetesimals around a growing planet in a gas disk and to reliably derive the power-law index (Γ pl ) of dM Z /dM p (see equation (11)), we here make use of the results of Shiraishi & Ida (2008). In their study, a number of N −body simulations are carried out to investigate how planetesimal accretion takes place for planets that undergo gas accretion, and to derive a semi-analytical accretion rate of planetesimals (dM pl /dt). Based on their results, dM pl /dt is determined by the interplay among excitation of planetesimals' eccentricity by a growing planet, dumping of their eccentricity by the disk gas, and the expansion of the Hill radius of the planet. When the dumping efficiency of planetesimals' eccentricity by the disk gas is less than the expansion rate of the Hill radius of a growing planet, the growth rate of the planet is so fast that the planet can keep accreting planetesimals in its expanding feeding zone. In other words, a gap is not generated in the planetesimal disk. For this case, the planetesimal accretion rate is given as (see equations (22) and (24)
in Shiraishi & Ida (2008)) dM pl dt nogap ∝ R 2 p M −α/3 p τ −α g,acc ∝ M (2−α)/3 p τ −α g,acc ,(12)
where α 4/5. Note that dM pl /dt is a function of τ g,acc . This originates from that planetary growth is regulated mainly by gas accretion. On the other hand, when the eccentricity dumping of scattered planetesimals by the disk gas is more significant than the Hill radius expansion, then planetary growth is slow enough that planetesimals can leave from the feeding zone of a planet before they will be accreted. Equivalently, a gap can open up in planetesimal disks. Under this situation, the accretion rate of planetesimals is written as (see equations (23) and (25) in Shiraishi & Ida (2008)
) dM pl dt gap ∝ R 2 p M −α /6 p τ −α g,acc ∝ M (4−α )/6 p τ −α g,acc (13)
where α 7/5. Again, dM pl /dt is related to τ g,acc . Thus, the planetesimal accretion rate is a function of both M p and τ g,acc , and the functional forms of dM pl /dt are different, depending on the creation of a gap in planetesimal disks.
We are now in a position to derive the power-law index of dM Z /dM p , which is given as (with equation (9))
Γ pl = − 1 + α 3 + D(1 − α) = D − 3 5 (14)
without planetesimal gaps, and
Γ pl = − 2 + α 6 + D(1 − α ) = − 12D + 17 30(15)
with planetesimal gaps. Given that there are two modes in gas accretion (see equation (4)), one of which is regulated by the Kelvin-Helmholtz timescale, the other of which is limited by disk evolution, the corresponding power-law indices are summarized in Table 2. By integrating dM Z /dM p , we find the resulting power-law in-
dices of M Z (∝ M Γ pl p ) for planetesimal accretion (see Ta- ble 3): Γ pl =
−2/5 with no gap and τ g,acc = τ g,KH 1/3 with no gap and τ g,acc = τ g,hydro 2 with a gap and τ g,acc = τ g,KH 3/5 with a gap and τ g,acc = τ g,hydro .
(16) Based on the above analysis, the power-law index of Z p (∝ M β pl p ) is the same as Γ pl and is given as (also see Table 2)
β pl ∝
−7/5 with no gap and τ g,acc = τ g,KH −2/3 with no gap and τ g,acc = τ g,hydro 1 with a gap and τ g,acc = τ g,KH −2/5 with a gap and τ g,acc = τ g,hydro .
(17)
It is interesting that our analysis predicts that β pl = 1 for the case with planetesimal gaps and τ g,acc = τ g,KH , which is inconsistent with the current trend of observed exoplanets. We consider that this inconsistency suggests that such a case never occurs in planet formation. In fact, it can be expected readily that if planets are massive enough to open up a gap in planetesimal disks, the corresponding τ g,KH should be smaller than τ g,hydro (see equation (9)). Our case study therefore would be useful for specifying the mass growth path of planets without any detailed calculations.
Thus, we find that the M Z − M p relation has different slopes, depending on the planetesimal distribution around planets and their gas accretion rates.
Contribution from pebble accretion
We here examine the case of pebble accretion. Equivalently, we consider the following case (see equations (5), (7), and (8):
M Z ≈ M Z,solid ≈ M pe . (18) dM Z dM p ≈ dM pe dt τ g,acc M p ∝ M Γ pe p .(19)
Substantial progress is currently being made for pebble accretion since the first realization of its importance on planet formation (see Johansen & Lambrechts 2017, as a most recent review). For the completeness of this paper, we will utilize the most recent results of pebble accretion and develop a formulation, which is similar to that of planetesimal accretion (see Section 3.2). It is nonetheless fair to mention that pebble accretion is not explored at the final stages of gas giant formation very much, compared with that of planetesimal accretion. In fact, even in the most recent studies, the primary target is the role of pebble accretion on core formation (e.g., Bitsch et al. 2015;Madhusudhan et al. 2017). Furthermore, these studies essentially treat accretion of gas and pebbles onto planets separately. In other words, the adopted pebble accretion rate (dM pe /dt) is independent of the gas accretion rate. The following analysis, therefore, should be viewed as a reference one, rather than the final results.
Once the similar level of complexity is included in numerical simulations of pebble accretion, one can undertake a more comprehensive calculation to examine the importance of pebble accretion on the M Z − M p and the Z p − M p relations more realistically.
Keeping this caveat in mind, we discuss the accretion rate of pebbles onto growing planets. In practice, dM pe /dt is written as (see equation (34)
in Johansen & Lambrechts 2017) dM pe dt ∝ M 2/3 p ,(20)
where the so-called Hill regime is considered. This is because our analysis assumes that (initial) core formation is almost completed and the core mass should be relatively large ( 1−5M ⊕ ). For this case, the growth mode is regulated by the relative velocity of Keplerian shear, rather than the azimuthal drift (e.g., Ormel Table 2 summarizes the results for both the cases of gas accretion (τ g,acc = τ g,KH and τ g,acc = τ g,hydro ). When integrating the above equation, we obtain the power-law index of M Z (∝ M Γpe p ) for pebble accretion, which is given as (see Table 3)
Γ pe = −10/3 with τ g,acc = τ g,KH 1/3 with τ g,acc = τ g,hydro .(22)
Also, the power-law index of Z p (∝ M βpe p ) is written as
β pe = −13/3 with τ g,acc = τ g,KH −2/3 with τ g,acc = τ g,hydro .(23)
As in the case with planetesimal accretion, the M Z − M p relation has different slopes for different gas accretion recipes.
3.4. Contribution from planetary cores In this section, we focus on the contribution of M Z,soild arising from core formation (see equation (5)):
M Z ≈ M Z,solid ≈ M core .(24)
As discussed in Section 2.1, both the oligarchic growth and pebble accretion scenarios lead to the core mass that is independent of M p . Then the power-law indices of
M Z (∝ M Γcore p ) and Z p (∝ M βcore p ) are readily computed as Γ core = constant,(25)β core = −1.(26)
It is interesting that these profiles are inconsistent with the trend of observed exoplanets (see Figure 1, also see Section 4).
3.5. Contribution arising from gas accretion Finally, we examine the contribution (M Z,gas ) originating from gas accretion (see equation (7)).
For this case, we can directly compute the total amount of M Z,gas . Assuming that the dust abundance in the gas accreted onto planets is comparable to Z s , the value of M Z,gas can be given as (using equation (6))
M Z,gas ≡ Z s M XY = Z s (M p − M Z ).(27)
Given that M p M Z for gas giant planets, the contribution of M Z,gas is only about 1 % (∼ Z s ) of the total planet mass. We thus can conclude that dust accretion accompanying with gas accretion is not significant to M Z for planets with the mass of M p 10 3 M ⊕ . As shown below (see Section 4.4), this conclusion is justified for observed massive exoplanets.
REANALYSIS OF THE RESULTS OF T16
We here turn our attention to the results obtained by T16. We reanalyze their computed values of the total heavy element mass in observed exoplanets and investigate how they are useful for developing a better understanding of planet formation.
The results of T16
We first introduce the results of T16 (see Figure 1, also see Miller & Fortney 2011).
In the study, observed exoplanets that have better measurements of mass and radius are chosen from the Extrasolar Planets Encyclopedia (exoplanets.eu Schneider et al. 2011) and the NASA Exoplanet Archive (Akeson et al. 2013). Especially, 47 exoplanets are selected from larger samples based on the criterion of a relatively low value of stellar insolation (F * < 2 × 10 8 erg s −1 cm −2 ). This criterion is adopted in order to filter out potentially inflated hot Jupiters, the origin of which is still unknown.
Through the careful examination of the data from both the original sources and the websites, they obtain the values of the planet mass (M p ) and radius (R p ), and the host star age and metallicity (Z s ). They make use of these values to combine their planet structure model and to compute the thermal evolution of planets. Such computations allow one to trace the radius evolution of planets.
More specifically, they adopt 1D planet structure models that are composed of an inert core (a 50/50 rock-ice mixture), homogenous convective envelope (a H/He-roclice mixture), and a radiative atmosphere as the upper boundary condition. For the atmosphere model, the solar metallicity grids are interpolated from Fortney et al. (2007). Their calculations employ a number of assumptions and simplifications. A more detailed model should include a self-consistent treatment of atmospheres, the composition of heavy elements, the treatment of thermal properties of cores (see section 3 of T16). They however find that uncertainties from observations (mass, radius, and host star age) are still dominant over those from model uncertainties (see Section 5.1). By comparing the computed radius of planets with the observational data, they identify the values of M Z in the planets that can distribute in both their cores and envelopes.
Here we simply summarize their derived M Z − M p and Z p − M P relations (also see their Figures (7) and (11)):
M Z ∝ M Γ T 16 p ,(28)Z p Z s = M Z M p 1 Z s ∝ M β T 16 p ,(29)
where Γ T 16 = 0.61 ± 0.08 and β T 16 = −0.45 ± 0.09. In this paper, we adopt that Γ T 16 ≈ 3/5 and β T 16 ≈ −2/5, respectively. For clear presentation, we do not show error bars in figures in this and following sections. It is interesting that β T 16 ≈ Γ T 16 − 1. This suggests that both M Z and M p are almost independent of or only very weakly dependent on Z s for observed exoplanets.
In fact, exoplanet observations confirm that while the occurrence rate of exoplanets is correlated with stellar metallicity (e.g., Fischer & Valenti 2005;Hasegawa & Pudritz 2014), the maximum mass of planets is not related to Z s . Note that T16 found that the Z p − M p relation becomes clearer when the planet metallicity is normalized by the host stellar metallicity (see their figures 10 and 11). Accordingly, we adopt the same convention.
In the following, we reanalyze the results of T16 in order to derive some constraints on planet formation and to examine how the M Z −M p relation can be reproduced.
The envelope mass and the critical core mass
We begin with computing the envelope mass (M XY , see equation (6)) and considering the critical core mass. Figure 2 depicts the computed value of M XY (= M p − M Z ) and the mass fraction (M XY /M p ) as a function of M p on the left and right panels, respectively. Our simple calculations show that the envelope mass becomes comparable to the total mass of planets when they are more massive than ∼ 100M ⊕ (see the green dots on the left panel). This suggests that efficient gas accretion occurred for all of the observed exoplanets that have masses larger than ∼ 100M ⊕ , which is also confirmed by the mass fraction of M XY (see the right panel). Importantly, we find that some of planets in the mass range of 20M ⊕ M p 100M ⊕ did not experience efficient gas accretion. Given that previous studies demonstrate that the critical core mass is about 5 − 10M ⊕ (see Section 2.1), our computations indicate that some mechanisms would be needed to postpone the onset of efficient gas accretion for some exoplanets until their masses reach ∼ 20 − 100M ⊕ . Note that the upper value of M p ( 100M ⊕ ) comes from only two points (see Figure 2). This critical value may change when more and improved results of planet structure models would become available.
The effect of solid accretion
We here examine the effect of solid accretion on the M Z − M p and Z p − M p relations. Given that solid accretion can divide into the core formation stage (M core ) and the post-core formation stage (M pl and M pe , see equation (5)), we subtract M core from M Z and explore the resulting behavior of the heavy element mass (= M Z − M core ). Note that as discussed in Section 3.5, the contribution of M Z,gas is negligible (also see Section 4.4). Figure 3 shows the results of our analysis. Since it is unknown what is the initial core mass for these planets, 2 we adopt a parameterized approach. In this approach, three plausible values (1M ⊕ , 5M ⊕ , and 10M ⊕ ) of the core mass are subtracted. We find that as the subtracted core mass increases (from the top to the bottom panel of Figure 3), the slope of the heavy element mass becomes steeper, especially at the less massive (M p 10 3 M ⊕ ) region (see the red dots). This is simply because when planets are not so massive, the total heavy element mass is also relatively small. If a certain value of the core mass is removed from M Z , then the reduction in M Z becomes more enhanced for lower mass planets than massive ones. Thus, our analysis indicates that the slope tends to be steeper (> 3/5) for planets with the mass of 10 3 M ⊕ and to be shallower ( 3/5) for more massive planets when the core mass is subtracted from the total heavy element mass (M Z ).
We now turn our attention to the Z p −M p relation. For this case, we utilize the results of our analysis to develop an interpretation that is different from the above one. More specifically, we assume that the metallicity computed from M Z − M core represents the envelope metallicity. This assumption would be valid if planetary cores do not dissolve into their envelopes and if solids accreted in the post-core formation stage fully dissolve into the envelopes due to thermal ablation. Figure 4 shows the results. We have adopted the same parameterized approach as above. From top to bottom, the assumed core mass that is removed from M Z is altered from 1M ⊕ , 5M ⊕ , and 10M ⊕ , respectively. Our analysis shows that subtraction of the core mass from M Z tends to wash out the Z p − M p relation, especially for planets that have masses of < 20−100M ⊕ (see the red dots). This occurs simply because the value of planetary metallicity (Z p ) is more affected for lower-mass planets, as discussed above. If the above assumption would be reasonable for observed exoplanets and envelope metallicity is determined only by the subsequent solid accretion, then our results can be interpreted that a correlation between envelope metallicity and planet mass should have a shallower slope than that of the Z p − M p relation. Also, there should be a transition in envelope metallicity as the planet mass increases. This transition would be related to the core mass. In Section 5.3, we will discuss more about how these interpretations are related to the current observations of exoplanets' atmospheres.
4.4. The effect of gas accretion We here consider the effect of gas accretion (M Z,gas ) on the total heavy element mass (M Z ) and the planet metallicity (Z p ).
As already shown in Section 3.5, the contribution of M Z,gas is readily computed for given values of M Z , M p , and Z s (see equation (27)). Figure 5 shows the resulting values (see the blue dots). Our analysis confirms that dust accretion accompanying with gas accretion is not crucial for understanding the total heavy element mass of observed planets (see the left panel). We also find that the contribution of dust accretion is an order of unity for massive ( 100M ⊕ ) planets (see the right panel). This can be viewed as a verification of the assumption that the dust abundance in the accreted gas is about Z s .
Comparison with our theoretical analysis
We now compare our theoretical results (see Section 3) with those of T16 (see Figure 1). To proceed, we summarize our results in Table 3.
We find that if the core mass of observed exoplanets is relatively small ( 1M ⊕ ), the best fit is achieved for the case where planetesimal accretion is slowed down due to gap formation and gas accretion is also limited by disk evolution (see Figure 3 and Table 3). This implies that the M Z − M p relation would be determined predominantly by the final stage of planet formation. Even if the core mass of these planets would be relatively large ( 5 − 10M ⊕ ), the trend for observed exoplanets can be reproduced well only in the case of gap formation in planetesimal disks (see Section 4.3): as the value of M p increases, the slope for the correlation between the heavy element mass and M p becomes shallower with increasing the planet mass. Mathematically, the power-law index changes from 2 to 3/5 for this case (see Table 3).
As a conclusion, our analysis suggests that the trend found by T16 would be understood well if it traces the final stage of planet formation: Planets are already mas-sive enough to generate a gap in their surrounding planetesimal disks and the gas accretion rate onto the planets is considerably reduced and mainly regulated by disk evolution.
DISCUSSION
We first list up the limitations of our analysis. We then discuss other physical processes that are not considered in the above analyses, and examine their effects on our conclusions. Also, we summarize previous studies which are directly related to this work, and compare them with our finding. We provide a comprehensive picture of planet formation that is derived from our analysis, and finally discuss some implications for the current and future observations of exoplanets and their atmospheres.
Limitation of our analysis
We here discuss the limitations of our analysis. The first limitation is that the trend discovered by T16 is based on only 47 exoplanets (see equations (28) and (29)). It is well known that while hot Jupiters are statistically rare, actually observed planets are not rare since they are readily observed by both radial velocity and transit methods (e.g., Winn & Fabrycky 2015;Dawson & Johnson 2018). This limitation indeed originates from an incomplete understanding of inflation mechanisms of hot Jupiters (T16). Once the dominant mechanism is identified, a similar analysis will be carried out to such hot Jupiters. Furthermore, the current and future observations attempt to improve measurements of both the mass and radius of detected exoplanets. Such better data will make it possible to apply a similar analysis not only to hot/warm Jupiters but also to smaller sized planets. Thus, it is currently not obvious that the M Z − M p and the Z p − M p relations derived by T16 are universal for various types of planets, which remains to be explored in the future work.
The second limitation is that our analysis heavily relies on the computed value of M Z . As discussed in Section 4.1, both better observational data and modeling are needed to constrain the value of M Z tightly. T16 −2/5 (τg,acc = τ g,KH ) 2 (τg,acc = τ g,KH ) −10/3 (τg,acc = τ g,KH ) 0.61 ± 0.08 1/3 (τg,acc = τ g,hydro )
3/5 (τg,acc = τ g,hydro ) 1/3 (τg,acc = τ g,hydro ) 3/5 β −7/5 (τg,acc = τ g,KH ) 1 (τg,acc = τ g,KH ) −13/3 (τg,acc = τ g,KH ) −0.45 ± 0.09 −0.68 −1.1 −2/3 (τg,acc = τ g,hydro )
−2/5 (τg,acc = τ g,hydro ) −2/3 (τg,acc = τ g,hydro ) −2/5 −2/3 a see equation (30). b see equation (31).
pointed out that the present observational data are still not good enough (see their section 4.1). As a result, the error bars of M Z are currently determined mainly by uncertainties in mass and radius measurements of observed exoplanets. Even if the observational data become better, uncertainties in model parameters cannot be fully removed.
The third limitation is involved with our approach. In this approach, we focus only on the power-law indices of the M Z − M p and the Z p − M p relations, in order to elucidate the underlying physics. This simplification needs to be examined carefully by detailed numerical simulations. In particular, recent studies show that the gas accretion process behaves differently with different assumptions and numerical setups (e.g., Machida et al. 2010;D'Angelo & Bodenheimer 2013;Venturini et al. 2016;Lambrechts & Lega 2017). We however emphasize that our adopted formula fits well the results of numerical simulations that are performed by different groups such as Tanigawa Tanigawa & Tanaka (2016), the formula works well for planets with the mass range of 10M ⊕ M p 30M ⊕ with a specific disk model that has the gas surface density of 140 g cm −2 , the aspect ration of 0.05, and the turbulent parameter α of 4 × 10 −3 (Shakura & Sunyaev 1973) at the planet position of r = 5.2 au. This implies that once gas accretion is regulated by disk evolution (see equation (2)), the formula would become reasonable until a (clear) gap is curved in gas disks. Note that Lissauer et al. (2009) investigate gas accretion onto planetary cores, taking into account disk-planet interaction. While they derive a different form of the gas accretion recipe (see their equation (2)), they adopt simulations of D' Angelo et al. (2003). Thus, our formula should be broadly consistent with theirs. A severer limitation of our approach is that we cannot compute the absolute value of M Z directly. The value would be determined by the combination of model and disk parameters. We will leave such a detailed study for the future work.
Other physical processes
In this section, we consider the effect of other physical processes that are not included in our analyses. These include orbital evolution due to planetary migration, gas gap formation by the migration, and the effect of nearby forming planets.
First, we point out that our analyses do not take into account the orbital evolution of planets by planetary migration (e.g., Kley & Nelson 2012). It is expected that planetary migration allows protoplanets to replenish planetesimals in their feeding zones. This is because the protoplanets can sweep up a new region of their planetesimal disks. In fact, Alibert et al. (2005) show that migrating protoplanets can have more chance to accrete a larger number of planetesimals in the disks, which speeds up core formation. It is however important to emphasize that more detailed simulations with a direct N −body integrator suggest that the planetesimal accretion rate and gap formation in planetesimal disks depend on the migration speed, which is a function of planet mass (Tanaka & Ida 1999). A more self-consistent simulation is needed to investigate how gaps form around growing, migrating planets in their planetesimal disks, and how semianalytical formulae can be affected due to planetary migration (see equations (12) and (13)).
Second, we discuss gap formation in gas disks that is the inevitable outcome of migration, especially for massive planets (e.g., Nelson et al. 2000;Crida et al. 2006;Hasegawa & Ida 2013;Dürmann & Kley 2015). As described in Sections 3.1, the M Z − M p relation is determined not only by solid accretion, but also gas accretion onto planets (see equation (8)). In the above analyses, the effect of gas gaps has not been considered explicitly. This is because our analyses heavily rely on the results of Shiraishi & Ida (2008), and their results are obtained under the assumption of no gap formation in gas disks for simplicity. One might consider that the presence of gas gaps would affect our conclusion very much since the gas surface density can now become a function of planet mass (see σ p,acc in equation (2)). In order to address this point, we develop a similar analysis in Appendix A.
Here we briefly summarize the results. We find that the trend found by T16 can be reproduced only when gaps are present in gas disks but no gap in planetesimal disks (see Table 5). We argue that this situation is very unlikely to be achieved. This is because gap formation takes place more readily in planetesimal disks than gas disks due to the lack of the pressure term. Furthermore, even if planets accrete gas and solids from gapped gas disks, the total amounts of accreted gas and solids would not be significant, compared with those accreted from gas disks without any gap (e.g, Tanigawa & Ikoma 2007;Tanigawa & Tanaka 2016). Accordingly, it would be reasonable to consider that the trend of M Z is determined predominantly before gap formation takes place in gas disks and such a trend does not change very much after gas gap formation. Thus, our conclusion would be maintained even if gap formation in gas disks is properly taken into account, while a more self-consistent simulation is needed to fully justify this consideration.
Third, we have so far assumed implicitly that planet formation proceeds in an isolated region, that is, we consider formation of single planets. We must admit that From the top to the bottom, the assumed core mass (1M ⊕ , 5M ⊕ , and 10M ⊕ ) is subtracted from M Z , respectively. This parameterized approach shows that the power-law index for the M Z − Mp relation tends to be smaller with increasing Mp. This trend is well reproduced when observed exoplanets formed under the condition that gap formation is achieved in planetesimal disks around the planets and gas accretion onto the planets is controlled by disk evolution (τg,acc = τ g,hydro , see Table 3). Our analysis therefore implies that the relationship discovered by T16 provides the useful information for the final stage of planet formation. We again adopt the parameterized approach for the core mass, in order to examine how subtraction of possible values (1M ⊕ , 5M ⊕ , and 10M ⊕ ) of the core mass affects the Zp − Mp relation from the top to the bottom panel, respectively (as done in Figure 3). Under the assumption that the envelope metallicity of planets is purely determined by solid accretion in the post-core formation stage, our results can be viewed that a correlation between envelope metallicity and planet mass should be characterized by a shallower slope. Also, some transition in envelope metallicity should be present at the mass range of 10M ⊕ Mp 100M ⊕ , which may be related to the core mass. . This indicates that heavy elements that are accreted following gas accretion are not crucial for the value of M Z until the planet mass exceeds 10 3 M ⊕ . Also, we confirm that the disk gas accreted onto planets contains the dust abundance that is similar to the stellar metallicity (see the blue line).
this is a highly idealized situation. In reality, multiple planets form in single disks at the same time, and the gravitational interaction arising from nearby growing planets would affect the dynamics of planetesimals there. This can change the spatial distribution of planetesimals and hence the condition of gap formation in planetesimal disks. It is interesting to investigate how the M Z − M p relation can be altered when formation of multiple planets is considered appropriately.
Thus, while some improvements would be required in our analyses for developing a more complete picture of planet formation, our present results are still useful for understanding a number of the currently known observational trends.
Comparison with previous studies
In this section, we touch on recent studies that are relevant to this work and compare their findings with ours.
One of the most advanced models that compute the total heavy element mass in planets are Mordasini et al. (2014Mordasini et al. ( , 2016. In this model, the standard core accretion picture is adopted to trace mass growth of planets. By coupling with planetary migration, they also make use of an enhanced planetesimal accretion rate, following the approach of Alibert et al. (2005). While they do not treat dust physics in planetary envelopes self-consistently, they mimic this effect by artificially reducing the grain opacity there . Covering a large parameter space and performing population synthesis calculations, they find that the Z p − M p relation is given as (see Table 7 in Mordasini et al. 2016)
Z p Z s M 14 = 7.2 M p M J −0.68 ,(30)
where M J is the Jupiter mass. Note that this relationship is derived from the total heavy element mass (M Z , see Mordasini et al. 2014). It is interesting that this slope is steeper than the results of T16 (see Figure 6, also see table 3). As discussed in Section 3.2, the slope is regu- . Comparison with previous studies. As in Figure 5, the black dots and the black solid line represent the results of T16 and its best fit, respectively. For comparison purpose, the results of Mordasini et al. (2014) and of Kreidberg et al. (2014) are denoted by the red dotted and the green dashed lines, respectively. Note that the former computes the total heavy element mass while the latter is for atmospheric metallicities. It is interesting that the slope of M14 is well reproduced by our simple analysis when the same assumption is adopted, that is, solid accretion proceeds from planetesimal disks without any gap and gas accretion is limited by disk evolution (see Table 3). The slope of KB14 is the most steepest. This may suggest that dust grain growth and settling is more efficient for more massive planets. In other words, a difference in slopes between the total heavy element mass (the black solid line) and the atmospheric metallicity (the green dashed line) may be used as a tracer of metallicity evolution of exoplanets' atmospheres.
lated by both planetesimal dynamics and gas accretion onto planets. In their model, the disk-limited gas accretion (τ g,acc = τ g,hydro ) is taken into account, but the effect of gap formation in planetesimal disks is not. As a result, their simulations lead to a steeper slope. In fact, our analysis predicts the value of their slope, which is about −2/3 (see the case of no planetesimal gap in Table 3). Thus, Mordasini et al. (2014Mordasini et al. ( , 2016) undertook a pioneering work and indicate the importance of planetes-imal accretion for understanding the Z p − M p relation. And our follow-up work reproduces the results of T16 better and derive a clearer view of how the Z p − M p relation can be used for obtaining better understanding of planet formation. While we focus mainly on the total heavy element mass (M Z ) in this paper, it would be interesting to consider atmospheric metallicity as done in Section 4.3 (see Figure 4). To proceed, we here discuss a correlation between envelope/atmospheric metallicity and planet mass. As an example, we adopt the result of Kreidberg et al. (2014), which is given as (see Table 3, also see Table 7 of Mordasini et al. 2016)
Z atm p Z s KB14 = 2.75 M p M J −1.1 .(31)
In their work, the metallicity of a hot Jupiter's atmosphere is estimated based on the precise determination of the water abundance in the atmosphere. Combining the data points of four giant planets in the solar system, they obtain the above trend (see the green dashed line in Figure 6). It is obvious that their slope is much steeper than that of T16. Since such a steep slope cannot be explained by removing the initial core mass (see Figure 4), we propose that the results of Kreidberg et al. (2014) are very likely to trace the metallicity evolution in exoplanet atmospheres: dust grain growth and settling take place in planetary atmospheres, namely, in the top, thin layer of planetary envelopes, and their effects are more pronounced for massive planets. If this would be the case, comparison between the total heavy element mass (M Z ) and atmospheric metallicity can be used as an indicator of how atmospheric metallicity of planets evolves with time. Given that most of heavy elements should be present in planetary envelopes for massive planets, not in the core (see Figure 3), they would be kept in the inner region of these envelopes. It is interesting that numerical simulations already show that these processes operate efficiently in planetary envelopes even during the process of forming (e.g., Movshovitz & Podolak 2008;Movshovitz et al. 2010). Note that the primordial envelope of planets should be more tenuous than the present one due to larger sizes, which principally leads to inefficient dust growth and settling there. Finally, we discuss pebble accretion. As already pointed out in Section 3.3, recent studies focus mainly on core formation (e.g., Bitsch et al. 2015;Johansen & Lambrechts 2017), and application of their results to the final stage of planet formation may not be reasonable. In fact, we find that the resulting power-law profile of the M Z − M p relation is not consistent with the result of T16 (see Table 3.) It is nonetheless important to point out that there is significant potential that pebble accretion may play a role in understanding the M Z − M p and Z p − M p relations. For example, it can be anticipated that a large amount of pebbles would accumulate at the outer edge of the gas gaps after core formation is nearly terminated due to gas gap formation. If this would be the case, such accumulation of pebbles would lead to planetesimal formation there. Then, it would be possible to trigger the subsequent planetesimal accretion onto planets, which can eventually achieve enrichment of heavy elements in the planets. In fact, high abundance of heavy elements in observed exoplanets requires a large amount of supplies that can potentially be delivered to the feeding zone of planets via radial drift of pebbles. Thus, while new numerical simulations of pebble accretion are desired, pebble accretion might not play a direct role at the final stage of planet formation.
A comprehensive picture
In this section, we combine the analyses and discussions done in the above sections. Keeping the limitations of our analysis in mind (Section 5.1), we develop a comprehensive picture of how observed exoplanets likely formed and of how our understanding of planet formation can be improved (see Figure 7 and Table 4).
We begin with the M Z − M p relation (see the left panel of Figure 7). We first divide the mass range explored by T16 into two regimes, based on the gas accretion process (see the blue dots and the green and grey regions, also see Section 4.2). Our analysis suggests that the behavior of M Z for planets with the mass of 100M ⊕ M p 3 × 10 4 M ⊕ can be understood well if the observed planets keep the formation histories at their final stages. It is interesting to point out that this region can be extended to the mass range of brown dwarfs, beyond which gas accretion plays a more important role in determining the value of M Z (see the shaded region with the yellow color). This implies that while their formation efficiency may not be high, some brown dwarfs may form via the same mechanisms of forming planets. As M p increases, the contribution (M Z,gas ) coming from gas accretion becomes more significant (see the blue line). When M p reaches 3 × 10 4 M ⊕ ( 10 2 M J ), M Z ≈ M Z,gas (see the yellow region). In this paper, we tentatively call objects residing in the yellow region as "stars". Note that the theoretical distinction between a star and a planet should come from formation mechanisms (e.g., Chabrier et al. 2014;Hatzes & Rauer 2015).
We now discuss the lower mass region. We first mention that the interpretation that is developed for the grey shaded region is still applicable to planets that have masses of 15M ⊕ M p 100M ⊕ (see the green region). In fact, the best fit is obtained for exoplanets with the mass range of 20M ⊕ M p 3 × 10 3 M ⊕ in the original analysis (T16). It is nonetheless important to emphasize that some planets in the green region did not undergo efficient gas accretion (see the blue dots). Additional explanations would be needed to fully understand these planets, which remains to be explored in the future work.
Another interesting point on the left panel of Figure 7 is that the line of M Z ∝ M 3/5 p and the straight line of M Z = M p intersect at M p 4M ⊕ (see the red solid line). This indicates that planets with the mass of > 4M ⊕ can essentially obtain gaseous envelopes from their natal protoplanetary disks. This in turn implies that the critical core mass for the onset of gas accretion is about 4M ⊕ . It should be noticed that this value of the core mass is roughly consistent with previous studies which show that exoplanets with the radius of larger than 1.6 earth radii (the corresponding mass of 5 − 6M ⊕ ) are unlikely to be purely rocky (e.g., Marcy et al. 2014;Rogers 2015;Hasegawa 2016). We can therefore suggest that planets less massive than 4M ⊕ tend to be made mostly of rocky (or solid) materials. Thus, the M Z − M p Table 4 Classification of observed exoplanets and the key physical processes of forming these planets Name
Mass range Color in Figure 7 Key process a Rocky planets
Mp 4M ⊕ Blue Significant solid accretion with (or (super)Earth-type) an almost negligible amount of gas Gas-poor sub-giants 4M ⊕ Mp 100M ⊕ Green Planetesimal accretion with a gap (or Neptune-type) & slowed-down gas accretion Gas-rich giants 100M Table 4). On both panels, the computed values and the best fit of T16 are denoted by the black dots and the black solid line, respectively. In addition, four planets in the solar system (Jupiter, Saturn, Uranus and Neptune) are shown by the red dots for comparison purpose (Saumon & Guillot 2004;Helled et al. 2011;Wahl et al. 2017). Note that the error bars of Uranus and Neptune are so small that they are almost invisible in these plots. On the left panel, the computed M Z,gas and the straight line of M Z = Mp are plotted by the blue dots and the green dashed line, respectively. In addition, the upper limit of M Z,gas (= 0.03Mp) is shown by the blue solid line. The mass range investigated by T16 is divided into two regions (gray and green), following the gas accretion process (see the blue dots and Section 4.2). Based on the behavior of M Z,gas and the intersection between M Z ∝ M 3/5 p and M Z,gas = 0.03Mp, the region of gas-rich giant planets is identified (see the grey region). The region of gas-poor sub-giant planets is determined by the value of M Z,gas and the intersection between M Z ∝ M 3/5 p and M Z = Mp (see the green region). Since the intersection between M Z ∝ M 3/5 p and M Z = Mp defines the boundary beyond which planets can contain gaseous atmospheres, we can suggest that the critical core mass for initiating gas accretion is about 4M ⊕ (see the red horizontal line). In other words, rocky super-Earths will distribute in the blue region. On the right panel, the computed M Z,gas /(MpZs) and the result of Kreidberg et al. (2014) are plotted by the blue dots and the green dashed line, respectively. Also, the straight line of Zp/Zs = 1 is denoted by the blue solid line for the reference. Exoplanets in the grey region can be used for studying the metallicity evolution in these planets' atmospheres, while planets in the green region may suggest a possibility of dissolving planetary cores into their envelopes. diagram is useful for developing a better understanding of planet formation.
Finally, we turn our attention to the Z p − M p relation (see the right panel of Figure 7). Our analysis suggests that the evolution of atmospheric metallicity in (exo)planets can be explored in the grey region, by comparing the total planet metallicity (Z p , the black solid line) with the atmospheric metallicity (the green dashed line). In the entire grey region, gas accretion provides only a minor contribution to Z p (see the blue dots). As a result, in order to fully understand the composition of (exo)planet atmospheres and to reliably make a link with planet formation, a number of physical processes should be taken into account self-consistently. These are not only gas and solid accretion onto growing planets, but also the subsequent processes such as planetesimal ablation in planetary envelopes (e.g, Podolak et al. 1988), and dust growth and settling there (e.g., Movshovitz et al. 2010;Mordasini et al. 2014;Ormel 2014). For the low mass region, there is not a clear difference between Z p and the atmospheric metallicity, which might imply that most masses of planetary cores would potentially dissolve into their envelopes. Further analysis and/or modeling are certainly required for carefully examining exoplanets in the green region.
In summary, we propose that observed exoplanet populations can be classified into three categories, depending on their masses (see Table 4). When M p 4M ⊕ , planets are made predominantly of rocky (or solid) materials, and they can be regarded as (super-)Earths. When 4M ⊕ M p 100M ⊕ , they contain some amount of gas, so that they can be called as gas-poor, sub-giant planets like Neptune in the solar system. Note that some mechanisms and/or fine-tuning of the formation timing are necessary for fully reproducing these planets. This additional requirement leads to a prediction that the population of planets in this category may not be so common (e.g., Mordasini et al. 2009). Such a prediction would be effective only for massive ( 10M ⊕ ) planets, since a couple of formation mechanisms are proposed for mini-Neptune mass planets (e.g., Hansen & Murray 2013;Chiang & Laughlin 2013). Finally, planets that have the mass of 0.4M J M p 10 2 M J are viewed as gas-rich giant planets. It is important to realize that most of their masses originate from their gaseous disks, while most of heavy elements in these planets are determined by solid accretion such as planetesimals in the last formation stage. 5.5. Potential roles of the current and future observations We finally discuss potential roles of the current and future observations of exoplanets and their atmospheres that can be deduced from this work.
We begin with listing up these roles. Observations of exoplanets' atmospheres will allow one to explore evolution of atmospheric metallicity as shown in the right panel of Figure 7 (see the grey region). Especially, comparison between hot and warms Jupiters would be invaluable for investigating how dust growth and settling take place in exoplanetary atmospheres. This is because the atmospheres of hot Jupiters are considered as fully radiative (e.g., Fortney et al. 2007), and atmospheric metallicity may have a steeper slope due to efficient dust growth and settling (see the green dashed line). On the contrary, warm Jupiters would tend to have convective atmospheres because they are far away from their host stars. If this would be the case, a larger amount of heavy elements may be able to stay in planetary atmospheres, and their trend in atmospheric metallicity may differ from that of hot Jupiters. In addition, observations taken towards brown dwarfs would be interesting for examining their formation origins. Finally, the dots in the green region are not large enough to fully understand why some planets in this region did not undergo runaway gas accretion. More observations are obviously needed to investigate how understanding of planetesimal accretion can be extended to this green region (e.g., Mordasini et al. 2016;Espinoza et al. 2017) and/or how pebble accretion comes into play to develop better understanding of planet formation (e.g., Madhusudhan et al. 2017).
How can we examine the effect of planetary migration through observations of exoplanets and their atmosphere? In order to address this issue, we consider the bulk density of observed exoplanets. Figure 8 shows the data points obtained from the NASA Exoplanet Archive (Akeson et al. 2013) with our classification of planets (see Table 4). One interesting feature of this figure is that the data points in the green region have scatter. As demonstrated clearly in Figure 3, planets in this region are most sensitive to subtraction of the assumed core mass. This in turn suggests that the total heavy element mass is regulated predominantly by the core mass itself. Given that the contribution of planets' atmospheres is not so significant to their total mass (see Figure 2), this scatter may be interpreted as a potential signature of planetary migration: when core formation takes place in various regions of protoplanetary disks, their bulk densities posses diversity. If the subsequent planetary migration delivers these cores in the current positions, the observed diversity can be used as a fossil record of where they form in the disks. Another interpretation of Figure 8 is that Table 4). The bulk densities of observed exoplanets are computed directly by adopting the values of planet mass and radius taken from the NASA Exoplanet Archive (Akeson et al. 2013, see the black dots). It is interesting that the data points in the green region show scatter, which might be related to planetary migration. Planets in the grey region line up with the straight line with a large band. This straight line involves the equation of state for metallic hydrogen.
while the contribution of H/He-dominated atmospheres is not substantial to the total mass, their contribution to the planet radius is crucial, leading to diversity in the bulk density of planets (e.g., Wolfgang & Lopez 2015).
SUMMARY & CONCLUSIONS
We have investigated how accretion of gas and solids onto growing planets determines the trend of the total heavy element mass (M Z ) in observed exoplanets. This work is motivated by T16 which shows that observed exoplanets have the correlations of M Z ∝ M 3/5 p and Z p /Z s ∝ M −2/5 p (see Figure 1 and Table 3). We have made use of the existing semi-analytical formulae that are derived from more detailed studies, and explored how accretion of planetesimals and pebbles proceeds onto planets with simultaneous gas accretion (see Table 3). We have demonstrated that the M Z − M p relation discovered by T16 is understood well if the relation traces the final stage of planet formation. At the stage, planets accrete solids from their gapped planetesimal disks and gas accretion is limited by disk evolution. We have also found that core formation and pebble accretion cannot reproduce the power-law index derived by T16. It is interesting that this work suggests that pebble accretion might not play a direct role at the final formation stage. Moreover, our analysis has showed that the contribution arising from gas accretion is negligible to the total heavy element mass in planets (see Figure 5).
We have then reanalyzed the results of T16 to consider how they can be used for deriving some insights about planet formation. We have found that the envelope mass becomes comparable to the total planet mass at M p > 100M ⊕ (see Figure 2). It is interesting that some planets in the mass range of 20M ⊕ M p 100M ⊕ have less massive envelopes, compared with the total mass. This indicates that they did not undergo runaway gas accretion. Some mechanisms and/or fine tuning of for-mation timing are needed for postponing the onset of runaway gas accretion for these planets. Furthermore, we have applied the results of our analysis to the atmospheric metallicity of exoplanets. Our analysis has suggested that the evolution of metallicity in exoplanets' atmospheres can be examined by comparing the total heavy element mass in planets and the heavy element mass in their atmospheres (see Figure 6).
We have compared our results with those of previous studies (see Table 3). We have found that despite of the simplicity of our analysis, we can reproduce the powerlaw index of the Z p − M p relation that is obtained by Mordasini et al. (2014), when the same assumption is employed. Note that the power-law index of Mordasini et al. (2014) is different from that of T16. We can therefore conclude that our simplified, but physically motivated framework provides a clearer view of under what conditions the correlations of M Z ∝ M 3/5 p and Z p ∝ M −2/5 p are generated in the course of planet formation.
We have listed up the limitations of our analysis that should be examined by detailed numerical simulations and the future observations. We have discussed other physical processes that are not included in our analysis, such as planetary migration and the effect of multiple planet formation. Combining all the analyses done in this paper, we have proposed a classification of observed exoplanets. We have finally summarized potential roles of the current and future observations of exoplanets and their atmospheres. It is important to detect exoplanets' atmospheres more for exploring the evolution of metallicity there.
Thus, we conclude that investigation of the the M Z − M p relation is very important for understanding the final stage of planet formation. And further detailed modeling, numerical simulations, and dealing with a larger number of observational data are required for confirming our results and drawing a more complete picture of planet formation. by the disk gas. Once such planetesimals enter the gas rich region that is beyond gas gaps, however, their eccentricity can rapidly decrease, and hence gap formation in planetesimal disks can be accelerated. Thus, when gaps are present around planets in gas disks, it would be reasonable to consider that planetesimal disks also have gaps. Under such a condition, the M Z − M p relation that is derived from observed exoplanets cannot be reproduced.
Figure 1 .
1Reproduction of the figures made by T16. The left panel shows the total heavy element mass (M Z ) as a function of planet mass (Mp). The black dots are the computed values adopted from T16 (see
Figure 2 .
2The computed envelope mass of observed exoplanets as a function of planet mass. Our reanalysis shows that most of the observed exoplanets experienced efficient gas accretion (see the green dots). This trend is clearly seen on both plots of the envelope mass (M XY on the left panel) and of the mass fraction (M XY /Mp on the right panel). On both panels, the green dashed line denotes the straight line of M XY = Mp for the reference. It is interesting that some planets that have the mass of ∼ 20 − 100M ⊕ have low values of M XY /Mp, indicating that they did not undergo runaway gas accretion. The value of ∼ 20 − 100M ⊕ is larger than the canonical value of the critical core mass that is about 10M ⊕ in the literature. Our simple calculations therefore suggest that efficient gas accretion tends to be postponed for some exoplanets.
& Watanabe (2002); D'Angelo et al. (2003); Machida et al. (2010). As clearly shown in figure 1 of
Figure 3 .
3Heavy element mass as a function of Mp for observed exoplanets. As inFigure 1(left), the computed values and the best fit derived by T16 are denoted by the black dots and the black solid line, respectively on each panel. We also plot the straight line of M Z = Mp for reference (see the green dashed line).
Figure 4 .
4Metallicity as a function of Mp. As inFigure 1(right), the computed values of Zp and the best fit of T16 are plotted as the black dots and the black solid line, respectively on each panel.
Figure 5 .
5Gas accretion and its contribution to M Z . As in Figure 1, the black dots and the black solid line represent the estimated values of T16 and its best fit, respectively. For comparison purpose, the straight line of M Z = Mp is denoted by the green dashed line on the left panel, and the straight line of Zp = Zs is by the blue solid line on the right panel. Our analysis shows that the computed value of M Z,gas (= ZsM XY ) is much smaller than that of M Z (see the blue dots)
Figure 6
6Figure 6. Comparison with previous studies. As in Figure 5, the black dots and the black solid line represent the results of T16 and its best fit, respectively. For comparison purpose, the results of Mordasini et al. (2014) and of Kreidberg et al. (2014) are denoted by the red dotted and the green dashed lines, respectively. Note that the former computes the total heavy element mass while the latter is for atmospheric metallicities. It is interesting that the slope of M14 is well reproduced by our simple analysis when the same assumption is adopted, that is, solid accretion proceeds from planetesimal disks without any gap and gas accretion is limited by disk evolution (see Table 3). The slope of KB14 is the most steepest. This may suggest that dust grain growth and settling is more efficient for more massive planets. In other words, a difference in slopes between the total heavy element mass (the black solid line) and the atmospheric metallicity (the green dashed line) may be used as a tracer of metallicity evolution of exoplanets' atmospheres.
⊕Figure 7 .
7Mp 3 × 10 4 M ⊕ Grey Planetesimal accretion with a gap (or Jovian-type) (0.4M J Mp 10 2 M J ) & slowed-down gas accretion Stars 10 2 M J Mp YellowCollapse of self-gravitating gas a Detailed numerical simulations and further modeling for the observations of exoplanets are obviously needed to confirm our prediction. Characterization of observed exoplanets based on our analyses (also see
Figure 8 .
8Planet density as a function of planet mass. As in Figure 7, our classification of planets is shown as the shaded regions (see
Table 1
1List of key quantitiesName
Symbol
Related process
Host stellar metallicity
Zs
Total planet mass
Mp
Radius of planets
Rp
Total envelope mass in planets
M XY
Gas accretion
Gas accretion timescales
τg,acc(∝ M D
p )
Kelvin-Helmholtz timescales
τ g,KH (∝ M −d
p )
Envelope contraction (d = 4)
Upper limit of τg,acc
τ g,hydro (∝ M −d
p
)
Disk evolution (d = 1/3)
Total heavy element mass in planets
M Z
Solid accretion
Planet metallicity
Zp(= M Z /Mp)
Heavy element mass via gas accretion
M Z,gas (= ZsM XY ) Accretion of dust via gas accretion
Heavy element mass due to solid accretion
M Z,solid
Accretion of pebbles and planetesimals
Core mass of planets
Mcore
Accretion of pebbles and planetesimals
Heavy element mass via planetesimal accretion M pl
Heavy element mass via pebble accretion
Mpe
Table 2
2Power-law indices of dM Z /dMp(∝ M Γ p ) for both cases of planetesimal and pebble accretionGas accretion mode
Planetesimal Accretion Planetesimal Accretion Pebble Accretion
No Gap
Gap
Kelvin-Helmholtz (D = −4)
−7/5
31/30 1
-13/3
Limited by disk evolution (D = −1/3)
−2/3
−13/30 −2/5
-2/3
& Klahr 2010; Lambrechts & Johansen 2012; Ida et al. 2016). Then, the power-law index of dM pe /dM p (∝ MΓ pe
p ) can be calcu-
lated as
Γ pe = D −
1
3
(21)
Table 3
3Summary of power-law indices of M Z (∝ M Γp ) and Zp(∝ M β
p )
Table 5
5Power-law indices for the M Z − Mp relation when the effect of gas gaps is taken into account 2r H > xm 2r H < xm 2r H < xm 2r H < xm (D = 1) First term (D = 2/3) Second term (D = 11/15) Third term (D = 4/5) No gap in planetesimal disks3/5
8/15
1/2
41/75
8/15
1/2
14/25
3/5
gap in planetesimal disks
1/30
1/6
7/50
1/7
17/150
3/25
1/8
More recently,Brouwers et al. (2017) have investigated direct core growth via pebble accretion. Through the calculations of envelope structures around planetary cores, they have found that the maximum mass of rocky cores that can form directly via pebble accretion is only up to 0.6M ⊕ . They have also shown that this value is relatively insensitive to the position of the cores. Such a small core mass arises from ablation of pebbles in planetary envelopes that prevents pebbles from reaching planetary cores.
T16 treated the core mass as a free parameter with the upper limit of 10M ⊕ , and their best fit values are not provided in their paper.
The authors thank an anonymous referee for useful comments on our manuscript. Y.H. thanks Jonathan Fortney for his encouragement of this work and Gennaro D'Angelo for stimulating discussions. This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. Y.H. is supported by JPL/Caltech.APPENDIXEFFECT OF GAS GAPS ON THE M Z − M P RELATIONHere we briefly discuss how the presence of gas gaps will affect the M Z − M p relation. To proceed, we make use of the results obtained byTanigawa & Ikoma (2007). In this study, mass growth of planets via gas accretion is investigated. In order to reliably take into account the effect of gas gaps that are opened up by planets, they consider both tidal interaction arising from the planets and viscous diffusion of gas disks, and compute the resulting value of σ p,acc that regulates gas accretion onto the planets (see equation(2)). They find that the profile of σ p,acc can divide into two regions. Provided that M s , r p , Ω p , h, and ν are the stellar mass, the position of a planet, the angular frequency at r = r p , the disk pressure scale height, and the disk viscosity, respectively, the the Hill radius of the planet (r H ), the characteristic lengths, l and x m , are given asandThen, the profile of σ p,acc is determined by the balance between tidal torque and viscous diffusion for the case of 2r H > x m . For the case of 2r H < x m , its profile becomes steep enough that the Rayleigh instability condition eventually regulates the behavior of σ p,acc . Finally, the gas accretion timescales can be given as (see equation(B3)Combining these timescales with the solid accretion rate (dM Z,solid /dt), we obtain tthe M Z − M p relation (see equations(14)and(15)).Table 5summarizes the results. One may wonder that the trend found by T16 can be reproduced if no gap is formed in planetesimal disks for both the cases of 2r H > x m and 2r H < x m . We argue that if gas gaps are already opened up by planets, then it can be anticipated that gap formation would take place in planetesimal disks as well. This is because under the presence of gas gaps, planetesimals can obtain a higher value of eccentricity there, which arises from both the high mass of planets and a reduced efficiency of eccentricity damping
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| []
|
[
"On design of hybrid diffractive optics for achromatic extended depth-of-field (EDoF) RGB imaging",
"On design of hybrid diffractive optics for achromatic extended depth-of-field (EDoF) RGB imaging"
]
| [
"Seyyed Reza \nComputing Sciences Unit\nFaculty of Information Technology and Communication Sciences\nTampere University\nFI-33720TampereFinland\n",
"Miri Rostami \nComputing Sciences Unit\nFaculty of Information Technology and Communication Sciences\nTampere University\nFI-33720TampereFinland\n",
"Samuel Pinilla \nComputing Sciences Unit\nFaculty of Information Technology and Communication Sciences\nTampere University\nFI-33720TampereFinland\n",
"Igor Shevkunov \nComputing Sciences Unit\nFaculty of Information Technology and Communication Sciences\nTampere University\nFI-33720TampereFinland\n",
"Vladimir Katkovnik \nComputing Sciences Unit\nFaculty of Information Technology and Communication Sciences\nTampere University\nFI-33720TampereFinland\n",
"Karen Egiazarian \nComputing Sciences Unit\nFaculty of Information Technology and Communication Sciences\nTampere University\nFI-33720TampereFinland\n"
]
| [
"Computing Sciences Unit\nFaculty of Information Technology and Communication Sciences\nTampere University\nFI-33720TampereFinland",
"Computing Sciences Unit\nFaculty of Information Technology and Communication Sciences\nTampere University\nFI-33720TampereFinland",
"Computing Sciences Unit\nFaculty of Information Technology and Communication Sciences\nTampere University\nFI-33720TampereFinland",
"Computing Sciences Unit\nFaculty of Information Technology and Communication Sciences\nTampere University\nFI-33720TampereFinland",
"Computing Sciences Unit\nFaculty of Information Technology and Communication Sciences\nTampere University\nFI-33720TampereFinland",
"Computing Sciences Unit\nFaculty of Information Technology and Communication Sciences\nTampere University\nFI-33720TampereFinland"
]
| []
| A hybrid imaging system is a simultaneous physical arrangement of a refractive lens and a multilevel phase mask (MPM) as a diffractive optical element (DOE). The favorable properties of the hybrid setup are improved extended-depth-of-field (EDoF) imaging and low chromatic aberrations. We built a fully differentiable image formation model in order to use neural network techniques to optimize imaging. At the first stage, the design framework relies on the model-based approach with numerical simulation and end-to-end joint optimization of both MPM and imaging algorithms. In the second stage, MPM is fixed as found at the first stage, and the image processing is optimized experimentally using the CNN learning-based approach with MPM implemented by a spatial light modulator. The paper is concentrated on a comparative analysis of imaging accuracy and quality for design with various basic optical parameters: aperture size, lens focal length, and distance between MPM and sensor. We point out that the varying aperture size, lens focal length, and distance between MPM and sensor are for the first time considered for end-to-end optimization of EDoF. We numerically and experimentally compare the designs for visible wavelength interval [400-700] nm and the following EDoF ranges: [0.5-100] m for simulations and [0.5-1.9] m for experimental tests. This study concerns an application of hybrid optics for compact cameras with aperture [5-9] mm and distance between MPM and sensor [3-10] mm. | null | [
"https://arxiv.org/pdf/2203.16985v1.pdf"
]
| 247,839,551 | 2203.16985 | c2b2d0a1d0d15507e17d2dbbb871ee4d82e2c827 |
On design of hybrid diffractive optics for achromatic extended depth-of-field (EDoF) RGB imaging
Seyyed Reza
Computing Sciences Unit
Faculty of Information Technology and Communication Sciences
Tampere University
FI-33720TampereFinland
Miri Rostami
Computing Sciences Unit
Faculty of Information Technology and Communication Sciences
Tampere University
FI-33720TampereFinland
Samuel Pinilla
Computing Sciences Unit
Faculty of Information Technology and Communication Sciences
Tampere University
FI-33720TampereFinland
Igor Shevkunov
Computing Sciences Unit
Faculty of Information Technology and Communication Sciences
Tampere University
FI-33720TampereFinland
Vladimir Katkovnik
Computing Sciences Unit
Faculty of Information Technology and Communication Sciences
Tampere University
FI-33720TampereFinland
Karen Egiazarian
Computing Sciences Unit
Faculty of Information Technology and Communication Sciences
Tampere University
FI-33720TampereFinland
On design of hybrid diffractive optics for achromatic extended depth-of-field (EDoF) RGB imaging
Diffractive imagingencoded phase maskhybrid diffractive opticsFourier opticsinverse imagingjoint design of diffractive optics and image processing
A hybrid imaging system is a simultaneous physical arrangement of a refractive lens and a multilevel phase mask (MPM) as a diffractive optical element (DOE). The favorable properties of the hybrid setup are improved extended-depth-of-field (EDoF) imaging and low chromatic aberrations. We built a fully differentiable image formation model in order to use neural network techniques to optimize imaging. At the first stage, the design framework relies on the model-based approach with numerical simulation and end-to-end joint optimization of both MPM and imaging algorithms. In the second stage, MPM is fixed as found at the first stage, and the image processing is optimized experimentally using the CNN learning-based approach with MPM implemented by a spatial light modulator. The paper is concentrated on a comparative analysis of imaging accuracy and quality for design with various basic optical parameters: aperture size, lens focal length, and distance between MPM and sensor. We point out that the varying aperture size, lens focal length, and distance between MPM and sensor are for the first time considered for end-to-end optimization of EDoF. We numerically and experimentally compare the designs for visible wavelength interval [400-700] nm and the following EDoF ranges: [0.5-100] m for simulations and [0.5-1.9] m for experimental tests. This study concerns an application of hybrid optics for compact cameras with aperture [5-9] mm and distance between MPM and sensor [3-10] mm.
INTRODUCTION
End-to-end optimization of diffractive optical element (DOE) profile (e.g., binary/multi-level phase elements [3,16,17]; meta-optical elements included [4][5][6]23,24]) has gained an increasing attention in emerging applications such as photography [8,22], augmented reality [15], spectral imaging [12], microscopy [1], among others that are leading the need for highly miniaturized optical systems [2,25], etc. The design methodology is performed by building numerical differentiable models for propagation of light fields through the physical setup in order to employ for modeling and optimization neural networks methods. In particular, the power-balanced diffractive hybrid optics (lens and MPM) is proposed and studied in [20], the methodology that is intended to be followed in this work, where a spatial light modulator (SLM) is used in experiments for implementation of MPM encoding of light fields.
In this work, the elements of interest to be jointly designed are MPM and image processing algorithms. The techniques and algorithms used for this design take advantage of those developed in [18,20]. As in [20], the targeted imaging problem is Extended Depth-of-Field (EDoF) with reduced chromatic aberrations. We exploit a fully differentiable image formation model for joint optimization of optical and imaging parameters for the designed computational camera using neural networks. In particular, for the number of levels and Fresnel order features, we introduce a smoothing function because both parameters are modeled as piecewise continuous operations. The paper is concentrated on pragmatical aspects of the design, especially, on the imaging quality and accuracy as functions of basic optical parameters: aperture size, lens focal length, thickness of MPM, distance between MPM and sensor, F -number. We numerically and experimentally compare the designed systems for visible wavelength interval (400 − 700) nm and depth-of-field range defined as (0.5-100) m for numerical and (0.5-2) m for experimental tests. The study concerns application of the hybrid optics for compact cameras with aperture (5 − 9) mm and lens focal length (3 − 10) mm. We point out that the variables aperture size, lens focal length, and distance between MPM and sensor are for the first time considered for end-to-end optimization of EDoF.
The contribution of this work can be summarized as follows.
• End-to-end optimization methodology for the joint design of MPM in the hybrid optics and imaging algorithms, showing high efficiency in terms of image accuracy and visual quality. • Optimal hybrid setup in terms of the optimal balance between aperture size and lens focal length concluded from multiple simulated experiments. • Algorithms for using SLM as MPM in the hybrid optics with learning-based CNN optimization of inverse imaging. • The advanced achromatic EDoF imaging of the designed system as compared with conventional compound multi-lens cameras such as in iPhone Xs Max.
END-TO-END OPTIMIZATION OF IMAGING WITH HYBRID OPTICS
The optical setup of the imaging system is depicted in Figure 1, object, aperture, and sensor are 2D flat, where d 1 is a distance between the object and the aperture, d 2 is a distance from the aperture to the sensor (d 2 d 1 ), f λ0 is a lens focal length. In what follows, we use coordinates (x, y), and (u, v) for aperture, and sensor planes, respectively. In this section, we mainly follow the image formation modeling and design optimization presented in [20]. These results are included for completeness of presentation and in order to give a clear picture of our approach, methodology, and algorithms.
Image Formation Model
PSF-based RGB imaging
Based on the Fresnel diffraction wavefront propagation, the response of an optical system to an input wavefront is modeled as a convolution of the system's PSF and a true object-image. Let us assume that there are both a lens and MPM in the aperture, then a generalized pupil function of the system for intensity imaging shown in Figure 1 is of the form (see Eqs. (5-23)-(5-28) in [11])
P λ (x, y) = P A (x, y)e jπ λ 1 d 1 + 1 d 2 − 1 f λ (x 2 +y 2 )+jϕλ 0 ,λ (x,y) .(1)
In (1), f λ is a lens focal length for the wavelength λ, P A (x, y) represents the aperture of the optics and ϕ λ0,λ (x, y) models the phase delay enabled by MPM for the wavelength λ provided that λ 0 is the wavelength design-parameter for MPM. In this formula, the phase jπ λ 1 d1 + 1 d2 x 2 + y 2 appears due to propagation of the coherent wavefront from the object to the aperture (distance d 1 ) and from the aperture to the sensor plane (distance d 2 ), and −jπ λf λ x 2 + y 2 is a quadratic phase delay due to the lens. For the lensless system
P λ (x, y) = P A (x, y)e jπ λ 1 d 1 + 1 d 2 (x 2 +y 2 )+jϕλ 0 ,λ (x,y) ,(2)
and for the lens system without MPM, ϕ λ0,λ (x, y) ≡ 0 in (1).
In the hybrid system, which is the topic of this paper, the generalized aperture takes the form
P λ (x, y) = P A (x, y)e jπ λ 1 d 1 + 1 d 2 − 1 f λ (x 2 +y 2 )+jϕλ 0 ,λ,α (x,y) ,(3)
where the optical power of the hybrid is shared between the lens with the optical power 1/f λ and the MPM due to the quadratic phase component included in the phase delay of MPM. The magnitude of the latter phase is controlled by a real-valued parameter α.
The PSF of the coherent monochromatic optical system for the wavelength λ is calculated by the formula [11]
P SF coh λ (u, v) = F P λ u d 2 λ , v d 2 λ ,(4)
where F P λ is the Fourier transform of P λ (x, y). Then, PSF for the corresponding incoherent imaging, which is a topic of this paper, is a squared absolute value of P SF coh λ (u, v). After normalization, this PSF function takes the form
P SF λ (u, v) = P SF coh λ (u, v) 2 ∞ −∞ P SF coh λ (u, v) 2 dudv .(5)
We calculate PSF for RGB color imaging assuming that the incoherent radiation is broadband and the intensity registered by an RGB sensor per c-band channel is an integration of the monochromatic intensity over the wavelength range Λ with the weights T c (λ) defined by the sensor color filter array (CFA) and spectral response of the sensor. Normalizing these sensitivities on λ, i.e. Λ T c (λ)dλ = 1, we obtain RGB channels PSFs
P SF c (u, v) = Λ P SF λ (u, v)T c (λ)dλ ∞ −∞ Λ P SF λ (u, v)T c (λ)dλdudv , c ∈ {r, g, b},(6)
where the monochromatic P SF λ is averaged over λ with the weights T c (λ).
Thus, for PSF-based RGB imaging, we take into consideration the spectral properties of the sensor and in this way obtain accurate modeling of image formation [13]. The OTF for (6) is calculated as the Fourier transform of P SF c (u, v) :
OT F c (f x , f y ) = ∞ −∞ P SF c (u, v)e −j2π(fxu+fyv) dudv,(7)
where (f x , f y ) are the Fourier frequency variables.
From PSFs to Imaging
Let us introduce P SF s for defocus scenarios with notation P SF c,δ (x, y), where δ is a defocus distance in d 1 , such that d 1 = d 0 1 + δ with d 0 1 equal to the focal distance between the aperture and the object. Introduce a set D of defocus values δ ∈ D defining area of the desirable EDoF. It is worth noting that the corresponding optical transfer functions are used with notation OT F c,δ (f x , f y ). The definition of OT F c,δ (f x , f y ) corresponds to (7), where P SF c is replaced by P SF c,δ . Thus, let I s c,δ (u, v) and I o c (u, v) be wavefront intensities at the sensor (registered focused/misfocused images) and the intensity of the object (true image), respectively. Then, I s c,δ (u, v) are obtained by convolving the true object-image I o c (u, v) with P SF c,δ (u, v) forming the set of misfocused (blurred) color images I s c,δ (x, y) = P SF c,δ (x, y) I o c (x, y),
where stays for convolution. In the Fourier domain we have
I s c,δ (f x , f y ) = OT F c,δ (f x , f y ) · I o c (f x , f y ).(9)
The indexes (o, s) stay for object and sensor, respectively.
EDoF Image Reconstruction
For image reconstruction from the blurred data {I s,k c,δ (f x , f y )}, we use a linear filter with the transfer function H c which is the same for any defocus δ ∈ D. We formulate the design of the inverse imaging transfer function H c as an optimization problem H c ∈ arg min
Hc 1 σ 2 δ,k,c ω δ ||I o,k c − H c · I s,k c,δ || 2 2 + 1 γ c ||H c || 2 2 J ,(10)
where k ∈ K stays for different images, I o,k c and I s,k c,δ are sets of the true and observed blurred images (Fourier transformed), c for color, σ 2 stands for the variance of the noise, and γ is a Tikhonov regularization parameter. The parameters ω δ > 0 are the residual weights in (10). We calculate these weights as the exponential function ω δ = exp(−µ · |δ|) with the parameter µ > 0. The norm || · || 2 2 is Euclidean defined in the Fourier domain for complex-valued variables.
Thus, we aimed to find H c such that the estimates H c · I s,k c,δ would be close to FT of the corresponding true images I o,k c . The second summand stays as a regularizer for H c . Due to (9), minimization on H c is straightforward leading toĤ
c (f x , f y ) = δ∈D ω δ OT F * c,δ (f x , f y ) δ∈D ω δ |OT F c,δ (f x , f y )| 2 + reg k |I o,k c (f x , f y )| 2 ,(11)
where the regularization parameter reg stays for the ratio σ 2 /γ. Therefore, the reconstructed images are calculated aŝ
I o,k c (x, y) = F −1 {Ĥ c · I s,k c,δ },(12)
where F −1 models the inverse Fourier transform. For the exponential weight ω δ = exp(−µ · |δ|), µ > 0 is a parameter that is optimized. The derived OTFs (11) are optimal to make the estimates (12) efficient for all δ ∈ D, in this way, we are targeted on EDoF imaging.
MPM Modeling and Design Parameters
In our design of MPM, we follow the methodology proposed in [13]. The following parameters characterize the free-shape piece-wise invariant MPM: h is a thickness of the varying part of the mask, N is a number of levels, which may be of different height.
Absolute Phase Model
The proposed absolute phase ϕ λ0,α for our MPM takes the form
ϕ λ0,α (x, y) = −πα λ 0 f λ0 (x 2 + y 2 ) + β(x 3 + y 3 ) + R r=1,r =4 ρ r P r (x, y).(13)
The factor with λ 0 in this equation is introduced for a proper scaling of the MPM's quadratic phase with the phase delay of the refractive lens. The parameter α in this factor controls the optical power sharing between the lens and MPM. The cubic phase of a magnitude β is a typical component for EDoF, the third group of the items is for parametric approximation of the free-shape MPM using the Zernike polynomials P r (x, y) with coefficients ρ r to be estimated. We exclude from this approximation the fourth Zernike polynomial defining the quadratic defocus term because it is considered as the first item in ϕ λ0,α (x, y).
Fresnel Order (thickness of MPM)
In radians, the mask thickness is defined as Q = 2πm Q , where m Q is called 'Fresnel order' of the mask which in general is not necessarily integer. The phase mask profile of the thickness Q is calculated aŝ
ϕ λ0,α (x, y) = mod(ϕ λ0,α (x, y) + Q/2, Q) − Q/2.(14)
The operation in (14) returnsφ λ0,α (x, y) taking the values in the interval [−Q/2, Q/2). The parameter m Q is known as 'Fresnel order' of the mask. For m Q = 1 this restriction to the interval [−π, π) corresponds to the standard phase wrapping operation.
Number of Levels
The mask is defined on 2D grid (X, Y ) with the computational sampling period (computational pixel) ∆ comp . We obtain a piece-wise invariant surface for MPM after the non-linear transformation of the absolute phase. The uniform grid discretization of the wrapped phase profileφ λ0,α (x, y) to the N levels is performed as
θ λ0,α (x, y) = φ λ0,α (x, y)/N · N ,(15)
where w stays for the integer part of w. The values of θ λ0,α (x, y) are restricted to the interval [−Q/2, Q/2). Q is an upper bound for thickness phase of θ λ0,α (x, y).
The introduced discretization and modulo functions are not differentiable, therefore we use a smoothing approximation to be able of optimizing the thickness and the number of levels of MPM by gradient descent algorithms. The details of this approximated function can be found in [20] .
The mask is designed for the wavelength λ 0 . Thus, the piece-wise phase profile of MPM for the wavelength λ is calculated as
ϕ M P M λ 0 ,λ,α (x, y) = λ 0 (n(λ) − 1) λ(n(λ o ) − 1) θ λ0,α (x, y),(16)
where θ λ0,α is the phase shift of the designed MPM and n(λ) is the refractive index of the MPM material,
x ∈ X, y ∈ Y . The MPM thickness h in length units is of the form h λ0 (x, y) = λ 0 (n(λ o ) − 1) θ λ0,α 2π .(17)
Optimization Framework
The framework which is presented in Figure 2 is developed to optimize the proposed optical system using the iterative NN algorithms with stochastic gradient ADAM optimizer. It can be downloaded from PyTorch with an optimized tensor library for Neural Network (NN) learning using GPUs * . Some details concerning this framework are given in what follows in this section. Figure 2. The optimal design framework of phase-encoded optics and image reconstruction algorithms for achromatic EDoF. The spectral PSFs are convolved with batches of RGB ground-truth images. The inverse imaging provides estimates of these images. Finally, a quality/accuracy loss L, such as mean squared error with respect to the ground-truth images (or PSNR criterion), is defined on reconstructed images.
Loss Function
Let Θ be a full set of the optimization parameters defined as
Θ = (α, β, ρ r , reg).(18)
Then, we use the following multi-objective formulation of our optimization goalŝ
Θ = arg max Θ (P SN R(Θ, δ), δ ∈ D).(19)
In this formulation, we maximize all P SN R(Θ, δ), δ ∈ D, simultaneously, i.e. to achieve the best accuracy for all focus and defocus situations. Here, P SN R(Θ, δ) is calculated as the mean value of P SN R k (Θ, δ) over the set of the test-images, k ∈ K:
P SN R(Θ, δ) = mean k∈K (P SN R k (Θ, δ)).(20)
There are various formalized scalarization techniques reducing the multi-objective (vector) criterion to a scalar one. Usually, it is achieved by aggregation of multiple criteria in a single one (e.g. [9]). In this paper, we follow pragmatical heuristics comparing P SN R(Θ, δ) as the 1D functions of δ in order to maximize P SN R(Θ, δ) for each δ ∈ D. Here,Θ are estimates of the optimization parameter. In this heuristic, we follow the aim of the multi-objective optimization (19). The key challenges in developing of the proposed optimization framework were to satisfy manufacturing constraints, finding stable optimization algorithms, and fitting models within memory limits.
Parameters for MPM design and simulation tests
The sensor's parameters used in simulation correspond to the physical sensor used in our experiments: pixel size 3.45 µm and resolution 512 × 512 pixels. The Fourier transform for PSFs calculations are produced on the grid 3000 × 3000 of the computational pixel size ∆ comp =2 µm, defining discretization of lens and MPM. We fixed the number of MPM levels to N = 31 and Fresnel order to m Q = 1, the latter restricts the MPM phase wrapping to the interval [−π, π). The optimization stage includes finding the optimal α, β, ρ r for the MPM design and reg for the image inverse reconstruction using the Adam stochastic gradient descent solver with the step-size 5 × 10 −3 .
We analyze and compare the hybrid optics of different lens diameter (aperture size of hybrid) taking values (5,6,7,9) mm and lens focal length taking values f = (3, 5, 7, 10) mm. The focus imaging distance for the hybrid is fixed to d 0 1 = 1 m. For each lens focal length f , d 2 is calculated according to the focusing equation 1
d 0 1 + 1 d2 = 1 f
. These values of d 2 are very close to f . It was concluded from our tests that R = 14 (Zernike coefficients excluding the fourth polynomial in (13)) is enough and larger values of R do not improve image quality significantly. The design wavelength is λ 0 = 510 nm. An additive white Gaussian noise is included in observations with variance equal to 1 × 10 −4 . We choose 31 wavelengths, with step 10 nm, covering the visual interval (400 − 700) nm to model RGB imaging. To enable EDoF imaging, we use Wiener filtering with d 1 = 0.5, 0.6, 0.7, 1.0, 1.9, 10, and 100.0 m. These d 1 define the defocus parameter δ in (11) as δ = d 1 − d 0 1 . The optimization stage employs 200 epochs, which takes approximately 6 hours on NVIDIA GeForce RTX 3090 GPU with memory of 24GB.
Data sets for optimization and tests
For optimization and training, we chose 1244 high-resolution RGB images from databases † . For testing of the designed systems, we used 200 high-resolution RGB images from the same databases which are not included in the training set. In what follows, all illustrative materials (tables, curves, and images) are given for these test images.
SIMULATION TESTS
In this section, we design the phase profiles for MPM in the hybrid optical setup with different aperture sizes (5, 6, 7, and 9) mm and lens focal lengths (3. 5, 7, and 10) mm. Our intention is to find combinations of these physical parameters for the best achromatic EDoF imaging. The corresponding numerical results obtained by simulation using the end-to-end joint optimization of optics and inverse imaging algorithms are presented in Table 1. The reported P SN Rs are averaged over 7 depth (defocus) distances d 1 from the interval (0.5 -100.0) m and over 200 RGB test-images. The imaging accuracy is evaluated and reported in two versions: P SN R RGB calculated for each of the color channels separately (column 4), and P SN R total calculated for all three color channels jointly (column 3). The best result (highest values of PSNR) is achieved by the setup with 6 mm aperture size and 5 mm lens focal length. These physical parameters result in F -number=0.83 and 70.5-degree field of view (FOV). The P SN R total value for this case is equal to 44.23 dB, but it degrades dramatically for larger and smaller focal lengths within the fixed diameter. If we compare the PSNR for the color channels separately, the values for 6mm diameter designed hybrid optics are highest (all above 41 dB) and more or less the same for all color channels.
Note also, that for each lens diameter there is an optimal lens focal length and this optimal value is close to the diameter size. The optimal focal lengths for the diameters (5, 6, 7, and 9) mm are (5, 5, 7, and 7) mm, respectively. We may conclude that the lens focal length plays a crucial role in hybrid optics and there is a trade-off between imaging quality and FOV. Smaller focal length (in Table 1, 3 mm) gives wider FOV at expense of less imaging accuracy. This conclusion is valid for all lens diameters in Table 1. Further information on the comparative performance of the imaging system with the optimized hybrid optics can be seen in Figure 3. Here we present PSNR curves as functions of d 1 (distance between the object and optics) averaged over 200 test images. The four curves are given for the four values of lens diameter with the corresponding optimal lens focal length as shown in Table 1.
The uniformly best performance is achieved by the 6 mm aperture hybrid optics with f 0 = 5mm. For this case, the PSNR value is about 37dB for the defocus point d 1 = 0.5m. The peak of this curve is at d 1 = 1.0m with PSNR=50dB. Remind, that this is a focus point of the system. For larger defocus distances, d 1 > 1, PSNR takes lower values which are nevertheless are close to 45 dB, which guarantees a high-quality imaging. The hybrid with the 5 mm aperture and f 0 = 5mm also demonstrates a very good performance with slightly lower PSNR values. For the two other cases: D = 7, f 0 = 7 mm and D = 9, f 0 = 7 mm, we can see a much worse performance with PSNR values lower from 5 to 10 dB as compared with the best ones.
The spectral performance of the best-optimized hybrid system (D = 6mm and f 0 = 5mm) characterized by PSNRs calculated for the RGB channels as functions of d 1 is presented in Figure 4. These curves with P SN Rs averaged over 200 test-images show the accuracy of imaging for each color channel and depth d 1 .
The P SN R total , black curve in Figure 4, shows the accuracy as function of d 1 calculated for the all spectral channels simultaneously as averaged over 200 test-images. The color channel curves mainly follow the behavior Figure 5. Visual performance of the designed hybrid systems is illustrated for different diameters (D = 5, 6, 7, 9) mm with the optimal lens focal length as defined in Table 1. The reconstructed images and their small fragments are shown for the distances d1 = (0.5, 1.0, 100.0) m. The color channels PSNR values are shown in these images. Thus, the comparison can be produced visually and numerically. The high-quality imaging for different colors and depths is achieved by the optical hybrid setups with the 6 and 5 mm diameter and 5mm focal length (columns 2 and 3). In contrast, the results for 7 and 9 mm diameters (columns 4 and 5) are suffering from strong chromatic aberration and the performance is degrading especially for off-focus distances d1 = 0.5 and d1 = 100.0 m. The optimized phase profiles of MPMs are shown in this first row of the image.
of P SN R total . All these spectral curves are well above the 35 dB line confirming high-accuracy imaging for all d 1 and all spectral channels. Figure 5 illustrates a visual performance of the designed hybrid systems of different diameters (D = 5, 6, 7, 9) mm with the optimal lens focal length as defined in Table 1 Figure (a) illustrates the architecture of the optical setup with SLM and the photo of the corresponding hardware. P is a polarizer, BS is a beamsplitter, SLM is a spatial light modulator. The lenses L1 and L2 form the 4f -telescopic system projecting wavefront from the SLM plane to the imaging lenses L3 and L4, CMOS is a registering camera. d1 is a distance between the scene and the plane of the hybrid optics (L3 and L4) and d2 is a distance between this hybrid optics and the sensor. shown in this first row of Figure 5.
Comparing these results, we may conclude, that the best results are achieved by the 5 mm and 6 mm diameter aperture sizes (columns 2 and 3) with an advantage of the latter one. For instance, for d 1 = 0.5m, the improvement in PSNR is about 2 to 4 dB for color channels in favor of the hybrid optics with 6 mm lens diameter. Moreover, details and colors are better preserved in this case. This best setup provides uniformly better imaging quality for various depths and colors. The zoomed fragments of the reconstructed images visually reveal clearly that the hybrid optics with 7 and 9 mm diameters (columns 4 and 5) are suffering from strong chromatic aberrations and quite blurry.
The advantage of the best hybrid optics with D = 6 mm and f 0 = 5 mm is well seen as compared with its counterparts, what is in direct agreement with the results shown in Table 1. Additionally in Figure 7, for this best hybrid system, we show the cross-sections of PSFs for the three RGB channels and for the distances d 1 used in Figure 5. These cross-section curves are well consolidated, which explain a source of a good performance of the imaging system for different distances d 1 and different color channels.
EXPERIMENTAL TESTS
Optical Setup and Equipment
In this work, to implement our hybrid optics and in order to avoid building several MPM to physically analyze the performance of our camera, we build an optical setup based on a programmable phase SLM to exploit its phase capabilities to investigate the performance of the designed hybrid setup. The optical setup is depicted in Figure 6(a), where 'Scene' denotes objects under investigation; the polarizer, 'P', keeps the light polarization needed for a proper wavefront modulation by SLM; the beamsplitter, 'BS', governs SLM illumination and further light passing; the lenses 'L 1 ' and 'L 2 ' form a 4f-telescopic system transferring the light wavefront modified by SLM to the lenses 'L 3 ' and 'L 4 ' plane; the lenses 'L 3 ' and 'L 4 ' forms an image of the 'scene' on the imaging detector, 'CMOS'. We use two lenses 'L 3 ' and 'L 4 ' tightly a fixed to each other in order to get the hybrid's lens, as in Figure 1, of a smaller focal length: f 0 = f 1 /2, where f 1 is the focal length of 'L 3 ' and 'L 4 '.
For physical modeling of MPM phase delay, we use SLM: the Holoeye phase-only GAEA-2-vis SLM panel, resolution 4160 × 2464, pixel size 3.74 µm. 'L 1 ' and 'L 2 ' are achromatic doublet lenses with diameter 12.7 mm and focal length 50 mm; Two BK7 glass lenses 'L 3 ' and 'L 4 'are of diameter 6 mm and focal length 10.0 mm which results in f 0 = 5.0 mm; 'CMOS' Blackfly S board Level camera with the color pixel matrix Sony IMX264, 3.45 µm pixel size and 2448 × 2048 pixels. This SLM allows us to experimentally study the optical hybrid imaging with an arbitrary phase-delay distribution for the designed MPM. The MPM phase was created as an Figure 7. For the best hybrid system (D = 6 mm, f0 = 5 mm), we show the cross-sections of the spectral PSFs for the three RGB channels and for the distances d1 used in Figure 5. These cross-section curves are well consolidated, what which explains a good performance of the imaging system for different distances d1 and different color channels.
8-bit *.bmp file and imaged on SLM. We calibrated the SLM phase-delay response to the maximum value of 2.0π for wavelength equal to 510 nm. This 2.0π corresponds to the value 255 of *.bmp file for the phase-delay image of MPM.
Optimization of image reconstruction provided a fixed MPM: learning-based approach
This optimization is used in our physical experimental works provided that the optimized phase-delay profile of MPM with D = 6 mm, obtained in the model-based approach, is implemented by SLM. The outputs of the sensor are blurred images registered for a sequence of the train dataset images displayed on three monitors at three different depths, d 1 = (0.5, 1.0, 1.8 m). Convolutional Neural Network (CNN) is used to fit these blurred images to the known true target images. In this way, CNN designs the inverse imaging algorithm defined by the CNN parameters. For optimization, we exploit the stochastic gradient ADAM optimizer. The training process is running for 1244 high-quality images on three monitors which give in a total of 3732 registered images. The network has been trained for 320 epochs which takes two weeks on NVIDIA GeForce RTX 3090 GPU. Figure 8 illustrates an architecture of CNN used in our experiments (DRUNet CNN [26]). We remark that this network has the ability to handle various noise levels for an RGB image, per channel, via a single model. The backbone of DRUNet is U-Net which consists of four scales. Each scale has an identity skip connection between 2 × 2 strided convolution (SConv) downscaling and 2 × 2 transposed convolution (TConv) upscaling operations. The number of channels in each layer from the first scale to the fourth scale are 64, 128, 256, and 512, respectively. Four successive residual blocks are adopted in the downscaling and upscaling of each scale. Each residual block only contains one ReLU activation function. The proposed DRUNet is bias-free, which means no bias is used in all the Conv, SConv and TConv layers [26].
An appropriate loss function is required to optimize the inverse imaging to provide the desired output. Thus, we use a weighted combination of PSNR between estimated and ground truth images ( L P SN R ), perceptual loss, and adversarial loss which are given below.
Perceptual loss: To measure the semantic difference between the estimated output and the ground truth, we use a pre-trained VGG-16 [21] model for our perceptual loss [14]. We extract feature maps between the second convolution (after activation) and second max pool layers ϕ 22 , and between the third convolution (after Figure 8. Inverse imaging UNet-based neural network architecture. The generator model is a U-net architecture that has seven scales with six consecutive downsampling and upsampling operations [26]. We adopt a weighted combination of PSNR between estimated and ground truth images, LP SN R, and perceptual losses L Adv and LP ercep, with weights σ1, σ2, and σ3. Figure 9. Performance of CNN for design of inverse imaging algorithm. The quality achieved by CNN starts from 20.5 dB and reaches 23.8 dB of PSNR for the training image set. The reconstructed images over the testing dataset are presented for three epochs 0, 100, and 320 for visualization of the training process. It could be seen that the trained network performs well for this task and the output image for epoch 320 is sharp enough. activation) and the fourth max pool layers ϕ 43 . Then, the loss L P ercep is the averaged PSNR between the outputs of these two activation functions for both estimated and ground truth images.
Adversarial loss: Adversarial loss [10] was added to further bring the distribution of the reconstructed output close to those of the real images. Given the swish activation function [19] as our discriminator D, this loss is given as L Adv = − log(D(I est )) where I est models the estimated image.
Our total loss for the proposed CNN inverse imaging while training is a weighted combination of these three losses and is given as, L CN N = σ 1 L P SN R + σ 2 L P ercep + σ 3 L Adv , where, σ 1 , σ 2 and σ 3 are empirical weights assigned to each loss. In this work, these constant are fixed as σ 1 = 1.0, σ 2 = 0.6, and σ 3 = 0.1. Lastly, the parameters of this networks to be optimized.
In Figure 9 we report an evaluation of PSNR versus a number of epochs. From these results, we can see that the quality achieved by CNN for the designed hybrid system is quite high for the training data set. Illustrating reconstructed images chosen among the testing dataset are presented for epochs 0, 100, and 320. It could be seen that the trained network performs well and the output image for epoch 320 is sharp enough. The practical value of this approach to image processing design follows from using physical modeling of image formation including in particular wavefront propagation and mosaicing/demosaicing operations.
Experimental Results
In this section, we present the results of two types of experiments. In the first one, the test-images are displayed on the three monitors as in Figure 6(b), the observations are blurred and the images are reconstructed by the trained CNN. These results are shown in Figure 10. In this scenario, we presented and evaluated the quality of reconstructions visually as well as numerically by PSNR values for each of the RGB color channels.
In the second type of experiments, we image a scene composed of different objects arbitrarily located within the range (0.4-1.9) m from the hybrid optics. This optical setup is used to evaluate the performance of the designed system in a real-world scenario for the EDoF imaging task. The performance of the designed system is compared with the compound multi-lens commercial iPhone Xs Max camera. These results can be seen in Figure 11.
The results in Figure 10 are presented in 7 columns for three depth distances: d 1 = 0.5 m (columns 2 and 3), d 1 = 1.0 m (columns 4 and 5), and d 1 = 1.8 m (columns 6 and 7). The Groundtruth column shows the true images. Two images from the test dataset are presented in this figure for comparison (rows 1 and 3) with one zoomed region (rows 2 and 4). We can see the zoomed fragments of the blurred noisy images on the sensor used for CNN image reconstruction as well as the corresponding reconstructed images. The zoomed sections for blurry and reconstructed images visually reveal that the images are sharp and clear enough and the quality of imaging is high and more or less the same for different depths. Besides, the colors are well preserved properly along with distances. If we compare the results numerically by PSNR, we could conclude that the PSNR values for different colors and depths are more or less the same at about 23 dB. It confirms that the designed hybrid imaging indeed demonstrates achromatic EDoF imaging.
The imaging results for the scene scenario are presented in Figure 11. The scene consists of 5 objects located at different distances from 0.4 m to 1.9 m, approximately: d 1 = 0.4 m (Train Wagon), 0.65 m (locomotive), 1.15 m (ThorLab snack box), 1.2 m (Dwarf Christmas Santa Claus Doll), and 1.9 m (Panda toy). It is worth mentioning, that for iPhone (compound optics), we adjusted the focusing distance to d 1 = 1.0 m as it is for the Figure 11. Comparison of the designed hybrid diffractive imaging versus the compound lens camera of iPhone Xs Max. For the designed hybrid, two image reconstruction approaches are employed (columns 2 and 3) to recover the blurred image on the sensor: Model-based and learning-based. The obtained images are presented in row 3 with their enlarged fragments in rows 4, 5, 6, and 7 corresponding to four off-focus distances d1 = 0.4, 0.65, 1.2, 1.9 m, respectively. By comparing the results over the recovering approaches in the designed hybrid (columns 2 and 3), the advantage of using a deep UNet-style CNN is clear. For the iPhone camera, the imaging quality is not good for the close and far distances. The visual advantage in sharpness and color preservation is clearly in favor of the designed hybrid imaging.
hybrid system. The hybrid diffractive imaging is compared with imaging by the mobile phone camera iPhone Xs Max (column 1).
For the designed hybrid, two image reconstruction techniques are demonstrated: model-based (column 2) and learning-based (column 3). In the model-based algorithm, the scene is reconstructed using the calculated color channel PSFs and the inverse imaging according to Eq. (12). After this step, a denoising process equipped with a sharpening procedure [7] is performed over the estimated scene to improve the quality of imaging. This final denoised image is returned as the estimated scene from experimental data. Contrary to it, the learning-based inverse imaging uses the trained UNet. For a detailed comparison, the four zoomed fragments of the images are shown in rows 4, 5, 6, and 7 which correspond to the scene's objects of different out-of-focus distances. Comparing columns 2 and 3, we may note an obvious advantage of the learning-based inverse imaging. The model-based approach is not able to recover all details, the output image is still blurry, and chromatic aberrations are strong. There are a number of reasons for this advantage. First of all, it concerns a mismatch between reality and the analytical modeling of image formation by PSFs. Second, the mosaicing/demosaicing are not included in our modeling. The leaning-based approach allows successfully compensate these drawbacks of the analytical modeling.
Comparison of the learning-based hybrid imaging (column 3) versus the iPhone camera imaging (column 1) results in an exciting conclusion about a quite clear advantage of the hybrid diffractive imaging. This advantage is obvious for sharpness of images for all distances. Thus, hybrid imaging demonstrates high-quality all-in-focus imaging. Concerning color aberrations: red and green perhaps not be properly presented by the hybrid but the white color is definitely perfect. Thus overall, the hybrid diffractive imaging can be tread at least as quite competitive and even advanced with respect to the commercial iPhone with multi-lens optics. Here we need to note that the white balance and γ correction procedure have been produced for the images reconstructed by the hybrid system in order to have a fair comparison with the iPhone camera.
CONCLUSION
For the first time, aperture size, lens focal length, and distance between MPM and sensor are considered as optimization variables for diffractive achromatic EDoF imaging. It is shown in this paper that the design and end-to-end optimization proposed for computational imaging with optics composed from a single refractive lens and a diffractive phase-encoded MPM is quite successful. In particular, comparison versus imaging by the multilens iPhone camera is definitely in favor of the designed imaging system. One of the novelties proposed and exploited in this paper is a the physical modeling of MPM by SLM which allows an on-line design of free-shape phase encoding for diffractive optics excluding a mismatch of theoretical image formation modeling and physical reality. As further work, we consider three lines of development. First, an improved design of SLM phase delay based on the Hardware-In-the-Loop approach. Second, a design of 'thick' MPMs (Fresnel order much larger than 1) is not possible using SLM but allows to get more efficient phase encoding and imaging. Third, an implementation of MPM as a physical phase mask.
Figure 1 .
1A light wave with a given wavelength and a curvature for a point source at a distance d1 propagates to the aperture plane containing MPM (refractive index n) to be designed. The MPM modulates the phase of the incident wavefront. The resulting wavefront propagates through the lens to the aperture-sensor, distance d2, via the Fresnel propagation model. The intensities of the sensor-incident wavefront define PSFs of the diffractive hybrid optical system.
Figure 3 .
3PSNR curves of the optimized hybrid setups with 4 different aperture size D= (5, 6, 7, and 9) mm as a function of distance from the scene to the optics (d1). The optimized hybrid setups with 5 and 6 mm diameters perform in the best way with more or less uniform PSNR values which are well above the good imaging quality line, PSNR = 35 dB, for all depths. The advantage of hybrid optics with D = 6 mm versus D = 5 mm is obvious of about 1 to 2 dB of PSNR values for each distance. The imaging with D = (7 and 9) mm shows good results in the vicinity of the system focal point (d1 = 1m), but the performance is dropped for far and even quite close distances.
Figure 4 .
4The spectral performance of the best-optimized hybrid system (D = 6mm and f0 = 5mm) is characterized by PSNRs calculated for the RGB channels as functions of d1. All curves are above the good imaging quality line 35 dB. The curves for color components mainly follow the behavior of the total PSNR curve (black).
. The reconstructed images and their small fragments are shown for the distances d 1 = (0.5, 1.0, 100.0) m. The color channels PSNR values are shown in these images. Thus, the comparison can be produced visually and numerically. The optimized phase profiles of MPMs are
Figure 6 .
6Experimental optical setup.
Figure (b) shows the photo of this hardware with three monitors displaying the scenes (images) of three fixed distances d1 = (.5, 1.0.1.8) m.
Figure 6 (
6a) illustrates the architecture of the developed optical setup with SLM and the photo of the corresponding hardware. Figure 6(b) shows the photo of this hardware with three monitors displaying the scenes (images) with three fixed distances d 1 = (0.5, 1.0, 1.8) m. The imaging monitors have a resolution of 1920 × 1080 and 570ppi. The distance d 1 = 1.0 m is the focal point of the optical system.
Figure 10 .
10Results for three monitor setup over the testing dataset. The reconstructed images with zoomed region at three different distances (imaging monitor-SLM): d1 = 0.5, 1.0, 1.8 m, for the optimized hybrid system. The PSNR values are reported for each depth and each color channel separately. The high-quality imaging with PSNR values of about 23 dB for different imaging depths and colors is achieved by the designed hybrid.
Table 1 .
1Comparative performance of the hybrid optics: different lens diameter (aperture size) and lens focal length.Diameter (mm) Focal length (mm) P SN R total (dB)
PSNR per channel
F -number
FOV
(degree)
R
G
B
5
3
31.48
28.43
34.61
31.64
0.6
99.3
5
41.58
43.20
44.65
39.82
1
70.5
7
38.75
39.44
42.71
35.98
1.4
53.6
10
36.29
36.98
40.11
31.84
2
38.9
6
3
25.64
23.12
27.21
22.89
0.5
99.3
5
44.23
44.92 46.81 41.74
0.83
70.5
7
36.61
39.41
40.87
30.29
1.17
53.6
10
33.66
32.22
34.07
29.46
1.66
38.9
7
3
25.8
25.29
29.77
21.58
0.43
99.3
5
33.28
29.09
37.09
29.47
0.71
70.5
7
36.14
34.61
39.26
29.93
1
53.6
10
31.65
28.41
33.20
29.86
1.43
38.9
9
3
24.21
24.74
27.46
19.59
0.33
99.3
5
26.43
22.45
28.40
26.05
0.54
70.5
7
34.79
29.08
39.10
30.19
0.76
53.6
10
30.97
31.00
36.25
26.09
1.08
38.9
* The Pytorch library https://pytorch.org/
† https://data.vision.ee.ethz.ch/cvl/DIV2K/, and http://cv.snu.ac.kr/research/EDSR/Flickr2K.tar.
ACKNOWLEDGMENTSThis work is supported by the CIWIL project funded by Jane and Aatos Erkko Foundation, Finland.
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[
"New photometric investigation of the double ringed galaxy ESO474-G26. Unveiling the formation scenario",
"New photometric investigation of the double ringed galaxy ESO474-G26. Unveiling the formation scenario"
]
| [
"M Spavone \nDipartimento di Fisica e Astronomia\nUniversitá di Padova\nVicolo dell'Osservatorio 2I-35122PadovaItaly\n",
"E 3⋆ \nINAF-Astronomical Observatory of Naples\nvia Moiariello 16I-80131NapoliItaly\n",
"Iodice \nINAF-Astronomical Observatory of Naples\nvia Moiariello 16I-80131NapoliItaly\n",
"D Bettoni \nINAF-Astronomical Observatory of Padova\nVicolo dell'Osservatorio 5I-35122PadovaItaly\n",
"G Galletta \nDipartimento di Fisica e Astronomia\nUniversitá di Padova\nVicolo dell'Osservatorio 2I-35122PadovaItaly\n",
"P Mazzei \nINAF-Astronomical Observatory of Padova\nVicolo dell'Osservatorio 5I-35122PadovaItaly\n",
"V Reshetnikov \nSt.Petersburg State University\nUniversitetskii pr. 28Petrodvoretz198504 Russia\n\nIsaac Newton Institute of Chile\nSt Petersburg Branch\n"
]
| [
"Dipartimento di Fisica e Astronomia\nUniversitá di Padova\nVicolo dell'Osservatorio 2I-35122PadovaItaly",
"INAF-Astronomical Observatory of Naples\nvia Moiariello 16I-80131NapoliItaly",
"INAF-Astronomical Observatory of Naples\nvia Moiariello 16I-80131NapoliItaly",
"INAF-Astronomical Observatory of Padova\nVicolo dell'Osservatorio 5I-35122PadovaItaly",
"Dipartimento di Fisica e Astronomia\nUniversitá di Padova\nVicolo dell'Osservatorio 2I-35122PadovaItaly",
"INAF-Astronomical Observatory of Padova\nVicolo dell'Osservatorio 5I-35122PadovaItaly",
"St.Petersburg State University\nUniversitetskii pr. 28Petrodvoretz198504 Russia",
"Isaac Newton Institute of Chile\nSt Petersburg Branch"
]
| [
"Mon. Not. R. Astron. Soc"
]
| We present a detailed photometric study of the peculiar double ringed galaxy ESO474-G26. Near-Infrared (NIR) and optical data have been used, with the main goal to constrain the formation history of ESO474-G26. NIR photometry is fundamental in this kind of study, because gives better constraints on the Spectral Energy Distribution (SED) and well traces the older stellar population of the galaxy. This galaxy presents a very complex structure, with two almost orthogonal rings, one in the equatorial and another in the polar plane, around an elliptical-like object. Due to the peculiar morphology of ESO474-G26, we used both NIR images (J and K bands) to derive accurate analysis of the stellar light distribution, and optical images (in the B, V and R bands) to derive color profiles and color maps to study the structure of the rings. The observational characteristic of ESO474-G26 are typical of galaxies which have experienced some kind of interactions during their evolution. We investigated two alternatives: a merging process and an accretion event. | 10.1111/j.1365-2966.2012.21815.x | [
"https://arxiv.org/pdf/1207.6559v1.pdf"
]
| 118,464,591 | 1207.6559 | d7a427f7359221c4750e0070a9e5860adbc1d098 |
New photometric investigation of the double ringed galaxy ESO474-G26. Unveiling the formation scenario
27 Jul 2012. 2012
M Spavone
Dipartimento di Fisica e Astronomia
Universitá di Padova
Vicolo dell'Osservatorio 2I-35122PadovaItaly
E 3⋆
INAF-Astronomical Observatory of Naples
via Moiariello 16I-80131NapoliItaly
Iodice
INAF-Astronomical Observatory of Naples
via Moiariello 16I-80131NapoliItaly
D Bettoni
INAF-Astronomical Observatory of Padova
Vicolo dell'Osservatorio 5I-35122PadovaItaly
G Galletta
Dipartimento di Fisica e Astronomia
Universitá di Padova
Vicolo dell'Osservatorio 2I-35122PadovaItaly
P Mazzei
INAF-Astronomical Observatory of Padova
Vicolo dell'Osservatorio 5I-35122PadovaItaly
V Reshetnikov
St.Petersburg State University
Universitetskii pr. 28Petrodvoretz198504 Russia
Isaac Newton Institute of Chile
St Petersburg Branch
New photometric investigation of the double ringed galaxy ESO474-G26. Unveiling the formation scenario
Mon. Not. R. Astron. Soc
00027 Jul 2012. 2012Accepted 2012 July 27. Received 2012 July 23; in original form 2012 June 12Printed 2 (MN L A T E X style file v2.2)Galaxies: photometry -Galaxies: evolution -Galaxies: formation - Galaxies: individual: ESO474-G26 -Galaxies: peculiar -Methods: data analysis
We present a detailed photometric study of the peculiar double ringed galaxy ESO474-G26. Near-Infrared (NIR) and optical data have been used, with the main goal to constrain the formation history of ESO474-G26. NIR photometry is fundamental in this kind of study, because gives better constraints on the Spectral Energy Distribution (SED) and well traces the older stellar population of the galaxy. This galaxy presents a very complex structure, with two almost orthogonal rings, one in the equatorial and another in the polar plane, around an elliptical-like object. Due to the peculiar morphology of ESO474-G26, we used both NIR images (J and K bands) to derive accurate analysis of the stellar light distribution, and optical images (in the B, V and R bands) to derive color profiles and color maps to study the structure of the rings. The observational characteristic of ESO474-G26 are typical of galaxies which have experienced some kind of interactions during their evolution. We investigated two alternatives: a merging process and an accretion event.
INTRODUCTION
The main aim of the extragalactic astrophysics is to understand how galaxies formed and evolved: the advent of the new all-sky surveys, covering a wide wavelength range, and the high resolution data from the large ground-based and space telescopes have strongly confirmed that gravitational interactions and mergers affect the morphology and dynamics of galaxies from the Local Group to high-redshift universe (Conselice et al. 2003;HDF and HUDF;SDSS). From this kind of studies, there is a growing evidence that mergers play a major role in the formation of early-type galaxies (Ellipticals and S0s), both in the field and in clusters. The traditional debate about the formation of spheroids (Es) and disk (S0s) galaxies, is today re-addressed to understand the origin of slow and fast rotators (see Emsellem et al. 2011 andKhochfar et al. 2011) in which ETGs are today divided. Emsellem et al. (2011) show that about 66% of elliptical galaxies are fast rotators, i.e. they might have a disk component. Tal et al. (2009) found that 73% of nearby ellipticals show morphological feature of interactions. ⋆ E-mail: [email protected] (MS) These observational results support the Cold Dark Matter scenario for galaxy formation (Cole et al. 2000): it is based on the hierarchical mass assembly, where the observed galaxies and their dark halo were formed through repeated mergings of small systems. In this framework, the study of peculiar and interacting galaxies, both at low and at high redshift, has a special role to investigate on the main processes at work during gravitational interactions between galaxies and between galaxies and their environment. In particular, in latest ten years, a big effort has given to study the morphology and kinematics of Polar Ring Galaxies (PRGs) and related objects: in these systems, the existence of two orthogonal components of the angular momentum is a consequence of a "second event" happened in their formation history, thus, PRGs can be considered as the ideal laboratory to study both the gravitational interactions among galaxies and the dark halo shape.
In the PRG catalogue made by Whitmore et al. (1990), the included objects are all classified as polar ring galaxies, where the morphology of central host resemble that of an early-type galaxy (Elliptical or S0) and the polar structure is a ring made up of gas, stars and dust that orbits in a nearly perpendicular plane with respect to central com-ponent (Schweizer et al. 1983;Bertola et al. 1985). By taking advantage of high resolution spectroscopy and photometry, only subsequent studies on the prototype of PRGs, NGC4650A, have revealed for the first time that the polar structure in this object has the morphology and kinematics of a disk, rather than a ring (see Arnaboldi et al. 1997;Gallagher et al. 2002;Swaters & Rubin 2003). From that moment on, by comparing observations and theoretical predictions, studies on polar ring/disk galaxies have tried to address how different kind of interactions (i.e. between galaxies and with environment) let to different galaxy morphologies and kinematics. Currently, in order to account both for the featureless morphology of the central spheroidal galaxy and for the more complex structure of the polar ring/disk, the main formation processes proposed are: i) a major dissipative merger; ii) tidal accretion of material (gas and/or stars) by outside; iii) cold accretion of pristine gas along a filament. In the merging scenario, the PRG results from a "polar" merger of two disk galaxies with unequal mass: the morphology and kinematics of the merger remnants depends on the merging initial orbital parameters and the initial mass ratio of the two galaxies (Bekki 1998a;Bekki 1998b;Bournaud et al. 2005). In the accretion scenario, the polar ring/disk may form by a) the disruption of a dwarf companion galaxy orbiting around an early-type system, or by b) the tidal accretion of gas stripping from a disk galaxy outskirts, captured by an early-type galaxy on a parabolic encounter (Reshetnikov & Sotnikova 1997;Bournaud & Combes 2003;Hancock et al. 2009). The cold accretion scenario has been proposed very recently for the formation of a wide disk-like polar rings: a long-lived polar structure may form through cold gas accretion along a filament, extended for ∼ 1 Mpc, into the virialized dark matter halo (Macciò et al. 2006;Brook et al. 2008). In this formation scenario, there is no limits to the mass of the accreted material, thus a very massive polar disk may develop either around a stellar disk or a spheroid. From the "observational" side, as suggested by very recent studies on PRGs (Iodice et al. 2006;Spavone et al. 2010Spavone et al. , 2011, the critical physical parameters that allow to discriminate among the three formation scenarios are 1) the total baryonic mass (stars plus gas) observed in the polar structure with respect to that in the central spheroid; 2) the kinematics along both the equatorial and meridian planes; 3) the metallicity and SFR in the polar structure. By studying the chemical abundances in the polar structure of three polar disk galaxies, NGC4650A, UGC7576 and UGC9796, Spavone et al. (2010Spavone et al. ( , 2011 have traced the formation history of these objects by accounting for all the three parameters mentioned above. In particular, the cold accretion scenario was successfully tested for the first time.
In the present paper, we address the formation history of the multiple ring galaxy ESO474-G26, by comparing the observed structure with the predictions from different formation scenarios. Together with the previous one (cited above), this work is part of an ongoing research project which aim to study the morphology, kinematics and SFR of a statistically significant sample of polar ring/disk galaxies and related objects, selected from both the Whitmore's PRG catalogue and from the new PRG catalogue compiled by Moiseev et al. (2011) based on SDSS data.
Properties of the PRG ESO474-G26
ESO474-G26 ( Figure 1) has been classified as a possible candidate for polar ring galaxy (PRC C-3) by Whitmore et al. (1990). This object has a heliocentric radial velocity of V = 15802 km s −1 , which implies a distance of about 211 Mpc, based on H0 = 75 km s −1 M pc −1 , and with this distance 1 arcsec ≃ 1 kpc.
This galaxy has two perpendicular and almost irregular rings, one in the equatorial and one in the polar plane, surrounding a central nearly spherical galaxy. Reshetnikov et al. (2005) found very blue optical colors for the rings, typical of late type spirals. Moreover, since both rings rotate around the central galaxy (Whitmore et al. 1990;Reshetnikov et al. 2005), they conclude that ESO474-G26 can be classified as a kinematically confirmed PRG.
Optical spectra also show that the ionized gas rotates with the north and west side receding. Galletta et al. (1997) by analyzing the relatively strong CO signal in ESO474-G26 found that the outer ring (north-south) has the northern side receding, indicating that the molecular gas rotates in the same direction as the ionized one.
The field around ESO474-G26 does not show any object within 10 arcmin or 600 kpc, since the galaxy visible on the NW side is a background object . Reshetnikov et al. (2005) analyzing the luminosity profiles of ESO474-G26 in the optical bands were able to distinguish three components: i ) the main central body with almost round isophotes (b/a ∼ 0.94), ii ) a narrow ring, with a diameter of ≃ 37 ′′ = 37 kpc, in the equatorial plane, and iii ) a second (larger) polar ring, with a diameter of ≃ 58 ′′ = 58 kpc, in the polar plane. Both rings are very irregular.
They also found that the galaxy colors, the ratio of the H2 mass to blue luminosity and the HI content, correspond to those of an Sb-Sbc spiral, even if the Spectral Energy Distribution (SED) for a prototype advanced merger remnant well fit the observed SED for ESO474-G26. The central galaxy appears to be redder than the galaxy as a whole, but bluer than typical ellipticals, while both rings are bluer than the central body and have colors typical of PRG rings (Reshetnikov et al. 1994(Reshetnikov et al. , 1995.
The main properties of ESO474-G26 are listed in Table 1.
Radio emission and star formation rate
In Figure 2 is shown the POSS image of the galaxy with the isocontours of the 1.4 GHz continuum emission from the NRAO VLA Sky Survey (NVSS, Condon et al. 1998) superimposed. The radio emission is elongated in the Northern direction and Condon et al. (1998) reported three sources (the two peaks and the flux in between). If we consider only the emission closest to ESO474-G26 the flux is F1.4 = 4.65×10 −28 W m −2 Hz −1 corresponding, with our adopted distance (see Table 1), to a luminosity L1.4= 2.43×10 23 WHz −1 . The 1.4 GHz luminosity is insensitive to dust obscuration and for this reason is a good tracer of the star formation rate (SFR1.4). We adopt the calibration of Hopkins et al. (2003) and we found a SFR=130M⊙y −1 (see Table 2). 0.20 Reshetnikov et al. (2005) a NASA/IPAC Extragalactic Database Reshetnikov et al. (2005), converting the far-infrared luminosity to a star formation rate, also found a high rate of star formation (SF RF IR = 43M⊙/yr).
OBSERVATION AND DATA REDUCTION
Near-Infrared data -ESO474-G26 belongs to a selected sample of peculiar galaxies observed in the Near-Infrared (NIR) J and K bands in December 2002, with the SofI infrared camera at the ESO-NTT telescope. The field of view was 4.92 × 4.92 arcmin 2 with a pixel scale of 0.292 arcsec/pixel. Images were acquired in the offsetting mode: a cycle was defined by several images on the target, interspersed with sky frames and with an integration time of 60 seconds; each object frame was taken with a small offset from the galaxy center and the sky frames were taken before and after each galaxy frame. More cycles were obtained in the K band than in the J band, in order to have a better estimate of the background level. A total exposure time of 360 sec was obtained on the target in the J band and of 1080 sec in the K band. The average seeing during the observing time is about FWHM ≃ 1.1 arcsec. The data reduction was carried out using the CCDRED package in the IRAF 1 (Image Reduction and Analysis Facility) environment. The main strategy adopted for each data-set included dark subtraction 2 , flatfielding correction, sky subtraction and rejection of bad pixels. Finally, all frames were registered and co-added to form the final science frames.
Several standard stars, from Persson et al. (1998), observed at the beginning, middle and end of each observing night, were used to transform instrumental magnitudes into the standard J and K band systems. The obtained photometric zero points are ZP (J) = 23.04 ± 0.02 mag/arcsec 2 for the J band and ZP (K) = 22.35 ± 0.02 mag/arcsec 2 for the K band.
The calibrated J and K band images of ESO474-G26 are shown in Figure 3. The ring-like structure of the galaxy is still visible in the J band image, while it disappears in the K band one.
Optical data -Photometric observations in the Johnson B, V and Cousins R bands were obtained in August 2002 on the 1.6 m telescope of the Observatorio do Pico dos Dias (operated by the MCT/Laboratorio Nacional de Astrofisica, Brazil), equipped with direct imaging camera and a CCD detector with a pixel scale of 0.18 arcsec/pixel. The average seeing during the observing time is about FWHM ≃ 1.3 arcsec. Reduction of the CCD frames was performed as described in Reshetnikov et al. (2005). The photometric calibration was made by using standard stars from the Landolt (1983) and Graham (1982) lists, obtaining the following photometric zero points: ZP (B) = 23.29 ± 0.06 mag/arcsec 2 , ZP (V ) = 23.23 ± 0.06 mag/arcsec 2 and ZP (R) = 23.26 ± 0.06 mag/arcsec 2 , for the B, V and R bands respectively.
HOST GALAXY AND RINGS MORPHOLOGY
The NIR images of ESO474-G26 ( Figure 3) show that most of the NIR light comes from the host galaxy and its morphology resembles that of an almost round elliptical object. The equatorial ring, with a diameter of about 40 kpc, is within the optical radius of the central galaxy (∼ 40 kpc), while the polar one has a diameter of about 60 kpc and so it is more extended in radius than the host galaxy. Both rings are more clearly visible in the optical B band image, while they gradually disappear in the J and K bands. However, both NIR and optical images show that the host galaxy is the dominant luminous component, while the rings appear knotty and dusty.
To examine the inner structure of the central host galaxy, and to identify the high frequency residuals with respect to the homogeneous light distribution, we create a residual image produced by taking the ratio of the original reduced image with a smoothed one, where each original pixel value is replaced with the median value in a rectangular window. This has the effect of remove the large-scale structure in the image and emphasize the galaxy substructure. We use the IRAF task FMEDIAN to smooth the original reduced image, by using a two-dimensional window. The window size (7 × 7) is chosen to best emphasize the inner structure of the central host. The final un-sharp masked im-2 Bias frame is included in the Dark frame.
age is shown in Figure 4 and it represents the high frequency residual image of ESO474-G26.
The most important result obtained by this analysis is the absence of any disk-like structures associated with the host galaxy major axis. The absence of a disk in the host galaxy suggested the use of a Sersic law for the 2D fit of the light distribution in this component (see Sec. 5.1).
PHOTOMETRY: LIGHT AND COLOR DISTRIBUTION
The overall morphology of ESO474-G26 is very tricky, due to the presence of two, almost perpendicular, ring-like structures. NIR photometry is necessary to reduce as much as possible the dust absorption that affect the starlight distribution and to accurately analyse it as well as to easily identify the inner structure of ESO474-G26. In addition, optical images are used to derive optical versus NIR color profiles and color maps to study the peculiar structure of this galaxy.
Isophotal analysis
We used the IRAF-ELLIPSE task on the NIR images to perform the isophotal analysis for ESO474-G26 and the results are shown in Figure 5. The average surface brightness extends up to about 25 and 15 arcsec from the galaxy center for the J and K band respectively; in the K band the half-light radius is Re = 9.3 arcsec, while in the J band is Re = 10.5. For a semi-major axis r, in the range 2 r 15 arcsec, the ellipticity and the Position Angle (P.A.) are almost constant and equal to 0.05 and ∼ 40 • , that indicates that in this regions the isophotes are almost round and coaxial. For 0 r 2 arcsec the ellipticity shows the presence of a flatter structure in the center, with a P.A. of ∼ 80 • and a twisting of the isophotes of about 50 degrees. For r 15 the profiles for the J band result perturbed by the presence of the rings. The shape parameters ( Figure 6) are all consistent with zero, thus the isophotes do not significantly deviate from purely elliptical shape. Figure 7 shows that the radial surface brightness between 1 ′′ and 10 ′′ is well reproduced by a de Vaucouleur profile, while in the outer regions we observe a bump, in both J and K profiles, which reflect the presence of the ringlike structures and clearly stands out also in the light profiles (see Figure 15).
In Table 3 we give the total integrated magnitudes within two circular apertures, derived for the NIR J and K bands. The apertures were chosen in order to make easier the comparison with the magnitudes of 2MASS data.
Color distribution and integrated magnitudes
We have derived the mean J-K color profile (Figure 8), and B-K (Figure 9, left panel) color profiles along both photometric axes of ESO474-G26, and the 2-dimensional B-K color map (Figure 9 right panel). On average, the central regions of the galaxy have redder colors, with a maximum value of J-K ∼ 1.14±0.04 and B-K ∼ 4.50±0.08. As already showed by the un-sharp masked image, also in the 2D B-K color map we find no trace of a disk-like structure.
We also derived the integrated magnitudes and J-K and B-K colors in 5 rectangles, as shown in Figure 10: one including the central region of ESO474-G26 and 4 including different regions of the rings. The rectangles are determined from the B band image, using the IRAF task POLYMARK, and used for all bands after the images were registered and scaled. The integrated magnitudes inside each rectangle are evaluated using the IRAF task POLYPHOT. The derived magnitudes and colors are reported in Table 4.
USING COLORS TO DATE THE STELLAR POPULATION
We analyze the integrated colors (optical vs NIR) derived for the rings and spheroid in ESO474-G26 in order to date the average stellar populations of these main components.
Taking into account that the integrated colors are the result of both old and young stellar populations, by studying them one can only obtain an indication on how much one is more prominent than the other in the two galaxy components, i.e. Host Galaxy (HG) and rings. As a consequence, the age estimate is the average value relative to all stellar populations present, which is strongly biased by the last burst of star formation. In the case of ESO474-G26, the central spheroidal component dominates the light in the NIR bands, particularly in the Ks band, while the rings emission becomes weaker from J to Ks bands: this strongly suggests that most of the light relative to an old and evolved stellar population comes from the central HG and in the rings a stellar population as old as that in the HG is absent.
The integrated colors derived for ESO474-G26 are compared with those of PRGs in the sample of Iodice et al. (2002a,b), in order to check whether there are differences in colors and average stellar population age estimates between the main components of this galaxy and other polar rings/disks, which could give some hints on the formation mechanism. As explained in the previous studies on PRGs (Iodice et al. 2002a,b), the B-K versus J-K diagram is used to break the age-metallicity degeneracy; the J-K color is a good estimate of the metallicity and it is quite insensitive to the presence of a young stellar population.
The stellar population synthesis model by
Component
Region Bruzual & Charlot (2003) were used to reproduce the integrated colors in the selected regions, in order to derive an estimate of the average (i.e. old plus the new bursts) stellar population ages in the central component and in the ring-like structures. We selected a set of models that were able to reproduce the average integrated colors observed for the main components of ESO474-G26.
m B (mag) m J (mag) m K (mag) M B M J M K B-K J-K±0
The key input parameters for GISSEL (Galaxies Isochrone Synthesis Spectral Evolution Library, Bruzual & Charlot 2003) are the Initial Mass Function (IMF), the Star Formation Rate (SFR) and the metallicity. For the central galaxy we adopted a star formation history with an exponentially decreasing rate, that produces a reasonable fit of the photometric properties of early-type galaxies in the local Universe. It has the following analytical expression: SF R(t) = 1/τ exp(−t/τ ), where the τ parameter quantifies the "time scale" when the star formation was most efficient. Adopting τ = 1 Gyr and τ = 7 Gyr, the correspondent evolutionary tracks were derived for different metallicities (Z=0.1, Z=0.02, Z=0.05, Z=0.008 and Z=0.0004), which were assumed constant with age. For the ring-like structures of ESO474-G26 instead, since they have bluer colors than the host galaxy, which suggests even a younger age for this component, we used models with constant SFR computed for the same metallicities as above, because these models reproduce the integrated colors of local spiral galaxies, in which star formation is still active. In every model it has been assumed that stars Figure 11 shows that the central galaxy is bluer in B-K color than the average values for PRGs in the sample of Iodice et al. (2002a,b), so a younger average age is to be expected, while figure 12 shows that the rings are bluer than the central galaxy. The colors of the NW component are on comparable with other narrow PRGs and not with NGC4650A, which is a wide polar disk, while the SE side appear redder in J-K color (see Table 4), but this could be the effect of a contamination due to the presence of a very bright star in this direction; for this reason the colors corresponding to the South and East side of the rings are not reliable, and so they are not reported in Figure 12. To account for the B-K and J-K colors the best model is that obtained for Z = 0.1 for the central galaxy and Z = 0.05 for the North and West side of rings, from which we derived an average age, of less than 1 Gyr for the inner region and of less than 0.03 Gyrs for the outer ones. Such values for ages and metallicity turn to be comparable with those derived for the narrow PRGs in the sample of Iodice et al. (2002a,b). This metallicity values are inside the range observed for early type galaxies (Z4 Z Z1, Bothun & Gregg 1990), but they are higher with respect to the average value, which is around Z2. Furthermore, the J-K color for ESO474-G26 (and also for other PRGs) are also consistent with the J-K colors derived by Rossa et al. (2007) for the interacting galaxies at an intermediate merger stage (0.5 J − K 1.5): such range in Figures 11 and 12 correspond to metallicities between Z3 and Z2.
We will show in section 6 that, for both HG and rings, the average ages derived by the integrated colors are comparable with the epoch of formation for these galaxy components estimated by simulations.
2-Dimensional model of the host galaxy light distribution
Two-dimensional model. We performed a 2-dimensional model of the light distribution of the host galaxy in the Ks and in the B bands. To this aim, we used the GALFIT task Peng et al. (2002) and the resulting structural parameters are listed in Table5. The Ks image is used to better constrain the structure of the central spheroidal component, since it dominates the light in this band and the emission from the rings and the dust absorption are very weak (see Table 5). The best model for this component is obtained by fitting the galaxy light through a single Sersic law (Sersic 1968). The result is shown in Figure 13: except for the center, where the residuals show the effect of the seeing, there are no evident features, only a diffuse low-luminosity emis- sion is detectable in the North and SE directions, which is the weak residual light coming from the rings.
In order to analyze the complex ring-like structure in ESO474-G26, which is very luminous in the B band, we have also derived the 2D model in this band. By taking into account the constraints for the central galaxy obtained by the 2D model in the Ks band and by accurately mask all the ring structures (which need to be excluded by the fit), the best 2D model in the B band is shown in Figure 14. The residuals appear very different from those in the Ks band: the whole structures of both polar and equatorial rings stands out very clear. Furthermore, it is evident another ring-like structure in the very central regions: it approaches to the galaxy center in the SW quadrant, it extends in the SE tracing a parabolic shape which seems to be connected with the Northern arm of the polar ring.
The comparison between the observed and fitted light profiles along the galaxy major and minor axis (P.A.=0 and P.A.=90 degrees, respectively) is shown in Figure 15 and in Figure 16, for Ks and B band respectively. In all the profiles and at both position angles, the "additional" light coming from the rings is evident at about 10 arcsec from the galaxy center in the Ks band, and it is much more luminous and extended in the B band, from 10 to 20 arcsec. Except for these regions, inside 10 arcsec, residuals are better than 0.2 mag. In the outer regions (r 15 arcsec in the Ks band and r 35 arcsec in the B band) light is dominated by background fluctuations.
COMPARING THE DATA WITH MODELS
To address the question on the formation scenario for ESO 474-G26 we attempt to best fit its overall SEDs and global properties, analyzing a large set of SPH simulations.
Our SPH simulations of galaxy formation and evolution start from the same initial conditions described in Mazzei & Curir (2003) (and references therein) i.e., collapsing triaxial systems composed of dark matter (DM) and gas with density distribution ρ ∝r −1 , in different proportions and different total masses. We then allowed a large set of galaxy encounters involving systems with a range of mass ratios from 1:1 to 1:10. In order to exploit a vast range of orbital parameters, we carried out different simulations for each pair of interacting systems, varying the orbital initial conditions in order to have, for the ideal Keplerian orbit of two mass points, the first peri-center separation, p, ranging from the initial length of the major axis of the dark matter triaxial halo of the primary system to 1/10 of the same (major) axis. For each of these separations, we changed the eccentricity in order to have hyperbolic orbits of different energies. For the most part we studied direct encounters, where the spins of the systems are equal (MC03), generally parallel to each other, and perpendicular to the orbital plane. However, we also analyzed some cases with misaligned spins in order to enhance the effects of the system initial rotation on the results. Moreover, for a given set of encounters with the same orbital parameters we also examined the role of increasing initial gas fractions. All simulations include self-gravity of gas, stars and DM, radiative cooling based on standard cooling function including metals, hydrodynamical pressure, shock heating, artificial viscosity, star formation (SF) and feedback from evolved stars and type II SNe, and chemical enrichment. The minimum temperature reached is 10 3 K to save time computing.
Here, as in the following, cold gas is gas with temperature lower than 10 4 K, given that its cooling timescale is shorter than the snapshot time-range, as we will point out later on. The gravitational softening is taken to be 0.5 and 1 Kpc respectively for the gas and DM particles. The Initial Mass Function (IMF) is of Salpeter's type with upper mass limit of 100 M⊙ and lower mass limit of 0.01 M⊙ (Salpeter 1955; see MC03 and references therein for a discussion). All our simulations provide the synthetic SED at each evolutionary step. The SED accounts for chemical evolution, stellar emission, internal extinction and re-emission by dust in a self-consistent way, as described in Spavone et al. (2009) and references therein, and extends over almost four orders of magnitude in wavelength, i.e., from 0.05 to 1000 µm. So, each simulation self-consistently accounts for morphological, dynamical and chemo-photometric evolution.
The whole SED of ESO 474-G26 and its global properties are well matched by a major merger, i.e. with a 1:1 mass ratio. The total initial mass of the systems is 4×10 12 M⊙ with gas fraction 0.10, so that the total initial mass of the gas is 4×10 11 M⊙. The mass particle resolution is 1.33×10 7 M⊙ for gas and 1.2×10 8 M⊙ for DM particles. This requires 60000 initial particles. The first pericenter separation, 101.4 Kpc, corresponds to 1/10 of the major axis of their halo; the orbit eccentricity is 1.3 and the anomaly corresponds to 200 degrees. The initial haloes have perpendicular spins (λ = 0.058) and triaxiality ratio 0.84 (Mazzei & Curir 2003). Stars are born in the inner regions of their halos after about 2.5 Gyr from the beginning. Galaxies grow changing their shapes step by step as their trajectories are approaching and their halos mixing.
Evolution of ESO 474-G26
We match the global properties of ESO 474-G26 during a very active phase, due to the strong interaction between the systems, i.e. 11.2 Gyrs from the beginning of the simulation. The older stellar populations are 8.5 Gyr old, however the large majority of star clusters are younger than 6.5 Gyr. By averaging population ages inside ∼r ef f (B) and ∼4r ef f (B), our fit corresponds respectively to a galaxy age of 2 Gyr and 4 Gyr. The B band absolute magnitude of the model, -22.31 mag, agrees well with the cosmologically corrected value from HyperLeda and the value in Table 1, as so as the K band absolute of the model, -25.32 (see Table 1). The mean velocity distribution along the galaxy major axis of cold and warm gas, i.e. gas with temperature lower than 10 4 K whose cooling timescale is shorter than the snapshot time-range (38Myr), compares with the same distribution derived from Hα measurements at the major axis position angles (P.A) of Whitmore et al. (1990), as shown in Figure 17; measurements in Whitmore et al. (1990) are corrected to account for a line of sight inclination angle of 52.2 degree (HyperLeda). Vertical dashed line corresponds to the inner region (R 1 Kpc) where model sampling could affect the results. We point out that the slightly difference, of almost 10 km/s, can be accounted for the error in the recession velocity of the galaxy (12 Km/s, by HyperLeda). At the selected snapshot the cold and warm gas, which will form new stars, is distributed in a ring (see Figure 18) surrounding the central spheroid. Figure 19 shows the best fit of the predicted SED by the simulation for ESO474-G26 to the available data. Dust components, warm and cold with PAH as discussed in Mazzei et al. (1992) with the same average properties as derived by and for a complete sample of nearby early-type galaxies are included in the far-IR SED. The SFR inside 6 r ef f (B) (∼ 60 Kpc) is 41 M⊙/yr, in agreement with radio estimates (see Sec. 1.2). The total mass inside r ef f (B) is 1.30×10 11 M⊙ with 14.4% of dark matter, inside 2r ef f (B) raises to 3.50×10 11 M⊙ with 25% of DM and to 4.650×10 11 M⊙ with 44% of DM inside 4 ef f (B). The predicted Mtot/LB ratio goes from 3 M⊙/L⊙ at r ef f (B) to 6.5 M⊙/L⊙ at 4r ef f (B).
By inspecting the simulation at the snapshot corresponding to 0.5 Gyr after the merging (Figure 20), the remnant becomes smooth and the morphology resembles that of spheroidal early-type system.
DISCUSSION AND CONCLUSIONS
We have presented a detailed photometric study of the double ringed galaxy ESO474-G26, based on new NIR observations. The main results of the present work are: i) the central spheroidal component dominates the light in the NIR bands, particularly in the Ks band, while the rings emission becomes weaker from J to Ks bands; ii) by using the stellar population synthesis model (see sec. 5), the last burst of star formation is dated to be less than 1 Gyr for the central galaxy, and between 0.1 and 0.03 Gyr for the ring-like structures, comparable with those of other PRGs with nar- Figure 18. YZ map, 30 Kpc × 30 Kpc, of cold and warm gas (see text) at the snapshot corresponding to the final merging stage (∼ 9 Gyr). The mass of cold and warm gas is normalized to its total mass within the map; its density contrast is 200 with 60 equispaced levels and spatial resolution 0.4 Kpc.
row rings (see Figure 11 and 12); iii) the 2D model of the light distribution suggests that one of the two rings, most probably the polar one, extends till the galaxy center (see Figure 14); iv) the SED turns to be well matched by a major merger (see Figure 19).
The main goal of the analysis presented in this work is to address the most reliable formation scenario for ESO474-G26, by reconciling the observed properties for this peculiar object with those predicted by different formation mechanisms for such kind of systems. As widely discussed into sec. 1, the up-to-date scenarios proposed for PRGs formation are (1) the disruption of a dwarf companion galaxy orbiting around an early-type system, or the tidal accretion of gas stripping from a disk galaxy outskirts, (2) a dissipative merging of two disk galaxies, (3) accretion of cold gas from cosmic web filaments. In the case of ESO474-G26, we expect that the most likely formation scenario has to account for i) the morphology, i.e. the presence of two orthogonal ringlike structures around an almost spherical early-type object, ii) the observed colors and ages for both components, i.e. a good fit for the observed SED; iii) the observed kinematics; iv) the gas content and distribution. To this aim, we used the infrared photometry which has the main advantages to be less affected by dust absorption so it well traces the older stellar population.
In order to reproduce the structure of ESO474-G26, Reshetnikov et al. (2005) have performed N-body simulations of a low-velocity head-on collision between two galaxies with orthogonal spiral disks. In their simulations, the main galaxy was a giant spiral galaxy with a low gas fraction, Figure 19. Continuous line (red) show the prediction of our model (see text). (Blue) filled circles correspond to data from NED and from Table 1. Arrows show upper limits; error bars account for band width and 3σ uncertainties. Dust components (warm and cold with PAH as discussed in Mazzei et al. (1992) with the same average properties as derived by for a complete sample of nearby early-type galaxies are included (i.e. I 0 =46I local , Iw=110I local , Rwc=0.27 and r d =30rc, as in . Figure 20. B band intrinsic flux map, 60 Kpc × 60 Kpc, of XY, YZ, and XZ projections at the snapshot corresponding to 0.5 Gyr after the merging. The remnant becomes smooth and the morphology resembles that of spheroidal early-type system. Each panel is normalized to its total flux; the density contrast is 200 with 60 equispaced levels, the spatial resolution 1.2 Kpc. The color scale is the same as in Fig. 18. while the second galaxy was less massive but with a higher gas content. They found that, 350 Myr after the first crossing, the merger remnant shows an equatorial ring made up of stars coming from the intruder galaxy, and an expanding collisional polar ring. They found that the polar ring is a transient feature and that 1 Gyr after the full merging, the remnant becomes an elliptical-like object.
On the basis of these previous results and by taking into account the new NIR photometry, we have developed new SPH simulations of galaxy formation and evolution (Sect.6), in order to test the major merger scenario for ESO474-G26.
Differently from Reshetnikov et al. (2005), in our simulations no galaxies exist at the beginning, but only collapsing triaxial halos composed of dark matter and gas, in equal proportions in both the systems, with equal total masses (1:1) and perpendicular spins (Sect.6 for details). In the paper of Reshetnikov et al. (2005) galaxies with mass ratio 2.5:1 are separated by 70 Kpc and their relative velocity was 70 km/s whereas, in our case, the initial relative velocity of their centre of mass is ≃104 Km/s and its position corresponds to 233 Kpc in the XY, orbital plane. The total inital mass in Reshetnikov et al. (2005) is 8.8×10 11 M⊙, while in our simulation it is 4.5 times more. We confirm that a major merger of two haloes with a 1:1 mass ratio is able to well reproduce the observed structure and the properties of ESO474-G26: in particular, i) an excellent fit of the SED (see Figure 19), which is well-constrained by the NIR fluxes; by inspecting the simulation at the snapshot corresponding to 0.5 Gyr after the merging, ii) the remnant becomes smooth, the ring-like structure vanishes and the morphology resembles that of spheroidal early-type system (see Figure 20); iii) the predicted gas kinematics is consistent with the observed one along the galaxy major axis (see Figure 17); iii) the cold gas, which will form new stars, is distributed in a ring (see Figure 18) surrounding the central spheroid. As already pointed out by Reshetnikov et al. (2005), also this new simulation suggests that the structure of ESO474-G26 could be a transient phase and not a stable dynamical configuration.
In order to complete our analysis on the evolution history of ESO474-G26, we now examine if the other possible formation scenarios, could also equally well match its observed properties. The tidal accretion scenario, in which gas is stripped from a gas-rich donor in a particular orbital configuration (Bournaud & Combes 2003), is able to produce wide rings and/or disks both around a disk or an elliptical galaxy: in this framework, in the field around the new forming ring galaxy the gas-rich donor galaxy is still present. In the case of ESO474-G26, inside a radius of about five times its diameter, as suggested by Brocca et al. (1997), there are no close companions as possible donor galaxy candidates. Moreover, in the tidal accretion scenario, the total amount of accreted gas by the early-type object is about 10% of the gas in the disk donor galaxy, i.e. up to 10 9 M⊙: the high baryonic mass (star plus gas) in the rings of ESO474-G26, which is about 10 10 M⊙, turns to be not consistent with this limit and further confirms that the tidal accretion need to be ruled out. The gradual disruption of a dwarf satellite galaxy cannot reconcile with a multiple ring structure.
Finally, the cold accretion scenario predict the formation of wide disk-like structures around an host galaxy. In this scenario, a long-lived polar structure may form through cold gas accretion along a filament extended for ∼ 1 Mpc into the virialized dark matter halo (Macciò et al. 2006). In this formation scenario, there are no limits to the mass of the accreted material, thus, a very massive polar disk may develop either around a stellar disk or a spheroid. Brook et al. (2008), by using high-resolution cosmological simulations of galaxy formation, have confirmed and strengthened the formation scenario proposed by Macciò et al. (2006). However, Brook et al. (2008) in their study have referred to objects characterized by polar structures with disk galaxy charac-teristics. In other galaxies, such as ESO474-G26, the polar structure is better described as a ring, with gas and stars in a narrow annulus. For this reason, the study of Brook et al. (2008) is not conclusive on this issue, and the classic merging and accretion models remain viable explanations for the formation of narrow polar ring structures. Spavone et al. (2011), by studying the formation mechanism of the PRG UGC7576, were able to test and confirm the cold accretion for this object, whose polar structure is a ring rather than a disk. However, differently from ESO474-G26, UGC7576 have a wide ring-like structure. Moreover, the narrow rings observed in ESO474-G26 are very faint structures, and they are not able to survive to the repeated accretion and mergers predicted by Brook et al. (2008).
We can thus conclude that the major merger is the most plausible scenario for the formation of the complex double ringed structure of ESO474-G26.
Figure 1 .
1Color composite image of ESO474-G26 assembled from images in the B (blue channel), V (green channel) and R (red channel) bands. The north is up, while the east is on the left of the image.
Reshetnikov et al. (2005) M(H 2 )(M ⊙ ) 2.3 × 10 10 Galletta et al. (1997) M (HI)/L B (M ⊙ /L ⊙ ) 0.15 Reshetnikov et al. (2005) M (H 2 )/L B (M ⊙ /L ⊙ )
Figure 2 .
221 cm contours superimposed to the B band image of ESO474-G26. The north is up, while the east is on the left of the image.
Figure 3 .
3Left panel-J-band image of ESO474-G26. Right panel-K-band image of ESO474-G26.
Figure 4 .
4High frequency residual B band image for ESO474-G26. Lighter colors correspond to brighter features. The image size is 90 ′′ × 115 ′′ , and the north is up while the east is on the left.
Figure 5 .
5Position Angle (P.A.), Ellipticity (ǫ) and mean surface brightness profile in the J and K bands. The error bar for the surface brightness profile (± 0.02) is within the dimensions of data points.form according to theSalpeter (1955) IMF, in the range from 0.1 to 125 M⊙.
Figure 6 .
6Shape parameters in the J and K bands.
Figure 7 .
7de Vaucouleur radial surface brightness profile. The error bar (± 0.02) is within the dimensions of data points.
Figure 8 .
8Mean J-K color profile. The error bar (± 0.04) is within the dimensions of data points.
Figure 9 .
9Left panel -B-K color profiles along the minor (top panel) and major (bottom panel) axis. The error bar (± 0.08) is within the dimensions of data points. Right panel -B-K color map. The North is up, while the east is on the left of the image. Lighter colors correspond to redder galaxy regions.
Figure 10 .
10B band image with superimposed the five rectangles limiting the different areas where the integrated magnitudes have been computed. The North is up, while the east is on the left of the image.
Figure 11 .
11B-K vs J-K diagram of the evolutionary tracks for the stellar synthesis models optimized for the central component of ESO474-G26. We used models with a characteristic time scale τ = 1 Gyr (heavy dotted lines) and models with τ = 7 Gyrs (heavy dashed lines), computed for different metallicities as shown on the figure. Light dotted and and light dashed lines indicates loci of constant age for the different models; different ages are reported on the plot. The red star corresponds to the central object of ESO474-G26, while the other points correspond to the sample of PRGs in Iodice et al. (2002a,b).
Figure 12 .
12B-K vs J-K diagram of the evolutionary tracks for the stellar synthesis models optimized for the central component of ESO474-G26. For the polar structures of ESO474-G26 we used models with constant SFR computed for different metallicities (heavy lines). Light lines are loci of constant age; different ages are quoted on the plot. Red stars correspond to different regions of the polar rings in ESO474-G26, while the other points correspond to the sample of PRGs in Iodice et al. (2002a,b).
Figure 13 .
132D fit of ESO474-G26. Left panel -K band image of ESO474-G26. Middle panel -K band model of the galaxy. Right panel -Residual of the subtraction of the model to the K band image.
Figure 14 .
142D fit of ESO474-G26. Left panel -B band image of ESO474-G26. Middle panel -B band model of the galaxy. Right panel -Residual of the subtraction of the model to the B band image.
Figure 15 .
15Top left panel-2-D fit of ESO474-G26 light distribution in the K band. The observed light profile along the minor axis (EW), is compared with those derived by the fit (dashed line). Bottom left panel -Residuals between the observed and the fitted light profiles. Top right panel-The same as above but for major axis (NS). Bottom right panel -Residuals between the observed and the fitted light profiles.
Figure 16 .
16Top left panel-2-D fit of ESO474-G26 light distribution in the B band. The observed light profile along the minor axis (EW), is compared with those derived by the fit (dashed line). Bottom left panel -Residuals between the observed and the fitted light profiles. Top right panel-The same as above but for major axis (NS). Bottom right panel -Residuals between the observed and the fitted light profiles.
Figure 17 .
17Blu dashed line shows the mean velocity distribution predicted for the cold and warm gas in our model (see text) along the galaxy major axis. Magenta solid line corresponds to the averaged velocity distribution as derived from Hα measurements along the galaxy major axis, corrected for the inclination. Vertical dashed line corresponds to the inner region (R 1 Kpc) where model sampling could affect the results.
Table 1 .
1Global properties of ESO474-G26.Parameter
Value
Ref.
Morphological type
Sc peculiar
NED a
R.A. (J2000)
00h47m07.5s
NED
Decl. (J2000)
-24d22m14s
NED
Helio. radial velocity
15802 km/s
NED
Redshift
0.052710
NED
Distance
211 Mpc
Axial ratio
0.65
Reshetnikov et al. (2005)
Absolute magnitude M B
-22.04
Reshetnikov et al. (2005)
Absolute magnitude M J
-24.67
This paper
Absolute magnitude M K
-25.52
This paper
M (HI)(M ⊙ )
1.7 × 10 10
Table 2 .
2Radio fluxes, luminosities and SFRs of the three sources detected by NVSS.NVSS
RA (J2000)
Dec (J2000)
S1.4 (mJy)
err(mJy) L 1.4 (W/Hz)
SFR (M ⊙ /yr)
004709-242140
00 47 09.06
-24 21 40.4
46.5
2.4
2.4291e+23
1.3e+2
004710-241949
00 47 10.51
-24 19 49.3
9.1
1.3
4.8475e+22
26.8
004712-241823
00 47 12.67
-24 18 23.5
40.9
2.0
2.1787e+23
1.2e+2
Table 3 .
3Magnitudes for ESO474-G26 in circular apertures. The errors on 2MASS magnitudes are 0.01, for both J and K, for the 14 arcsec aperture, and 0.02 and 0.04, respectively for J and K, for the 24 arcsec aperture.Aperture radius
m J
m K
m J (2M ASS) m K (2M ASS)
(arcsec)
±0.02
±0.02
-a
-
14.7
12.16
11.10
12.40
11.30
24.0
11.95
-
12.23
11.16
a
Table 4 .
4Integrated and absolute magnitudes and colors of different regions of ESO474-G26.
Table 5 .
5Galfit parameters.Parameter
Value
Parameter
Value
K band
B band
Component
Sersic
Component
Sersic
Integrated magnitude
11.30
Integrated magnitude
17.78
Effective radius
7 ± 0.01 arcsec
Effective radius
15.52 ± 0.05 arcsec
Sersic index
1.52 ± 0.01
Sersic index
3.13 ± 0.01
Axis ratio (b/a)
0.97 ± 0.01
Axis ratio (b/a)
0.99 ± 0.01
Position Angle
81.3 ± 1.7
Position Angle
101.0 ± 5.6
IRAF is distributed by the National Optical Astronomy Observatories, which is operated by the Associated Universities for Research in Astronomy, Inc. under cooperative agreement with the National Science Foundation.
ACKNOWLEDGEMENTSThe authors thank the referee, Frederic Bournaud, for the detailed and constructive report, which allowed them to improve the paper. E.I. wish to thank E. Pompei for the support given during the data acquisition. This work is based on observations made with ESO Telescopes at the Paranal Observatories under programme ID < 70.B -0253(A) > and < 74.B -0626(A) >. We acknowledge financial contribution from the agreement ASI-INAF I/009/10/0 and M.S. acknowledges financial support from the "Fondi di Ateneo 2011" (ex 60 %) of Padua University. V.R. acknowledges partial financial support from the RFBR grant 11 − 02 − 00471−a.This paper has been typeset from a T E X/ L A T E X file prepared by the author.
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| []
|
[
"Moduli of Representations of Quivers",
"Moduli of Representations of Quivers"
]
| [
"Markus Reineke "
]
| []
| []
| An introduction to moduli spaces of representations of quivers is given, and results on their global geometric properties are surveyed. In particular, the geometric approach to the problem of classification of quiver representations is motivated, and the construction of moduli spaces is reviewed. Topological, arithmetic and algebraic methods for the study of moduli spaces are discussed. | 10.4171/062 | [
"https://arxiv.org/pdf/0802.2147v1.pdf"
]
| 14,931,087 | 0802.2147 | 07a10eb63ed49f5e31d1056e8236ad4c946d5d2f |
Moduli of Representations of Quivers
15 Feb 2008
Markus Reineke
Moduli of Representations of Quivers
15 Feb 2008Mathematics Subject Classification (2000) Primary 16G20secondary 14D2014L24 Keywords Representations of quiversinvariant theorymoduli spacesstabilityBetti numberslocalizationrational pointsHall algebrasHilbert schemes
An introduction to moduli spaces of representations of quivers is given, and results on their global geometric properties are surveyed. In particular, the geometric approach to the problem of classification of quiver representations is motivated, and the construction of moduli spaces is reviewed. Topological, arithmetic and algebraic methods for the study of moduli spaces are discussed.
Introduction
One of the fundamental problems in the representation theory of algebras is the classification up to isomorphism of the finite-dimensional representations of a given algebra. But "most" algebras are wild, in the sense that the problem of classification of their representations is as difficult as the classification of representations of free algebras, or of any wild quiver. This is sometimes referred to as a "hopeless", or "impossible" problem (see however e.g. [24,47] for more optimistic opinions). The main obstacle in this case is the dependence of the isomorphism classes of representations on arbitrarily many continuous parameters, to which many of the classical tools of the representation theory of algebras do not apply.
Nevertheless, one can approach the classification problem geometrically, by "materializing" the continuous parametrization phenomena in spaces whose points correspond naturally to isomorphism classes of representations -these are the moduli spaces we consider in the present paper. We will focus on global qualitative geometric properties of such spaces, to hopefully derive a qualitative understanding of the classification problem.
The first aim of this paper is to motivate the geometric approach to the classification problem, the definition of the moduli spaces, and in particular the notion of stability of representations. To this end, we first collect some very basic observations and examples in section 2. We recall the basic concepts of Geometric Invariant Theory as in [48], and use them to construct the moduli spaces of stable representations, following [37], in section 3. In section 4, we discuss the purely algebraic aspects of the notion of stability, following [21,29,60]. In this latter section, complete proofs are given.
Our second aim is to discuss the kind of questions one can ask about the moduli spaces. We first collect several of their geometric properties, mainly following work of H. Derksen, A. King, L. Le Bruyn, A, Schofield, M. Van den Bergh, J. Weyman and others, in section 5. The notion (and utility) of cell decompositions is discussed in section 6, as a motivation for the topological and arithmetic considerations in all following sections.
Based on this, our third main theme is the discussion of the methods and results of the author's (and others) work [19,46,52,53,54,55,56,57,58]. We focus on the (essentially easy) methods which are used in these papers, namely torus localization in section 7, counting points over finite fields in section 8, Hall algebras in section 9, and framings of moduli in section 10. As applications of these techniques, the main results of the above papers are stated, together with indications of the proofs. Several conjectures, open questions and potential directions for further research are discussed.
The classification problem for representations of quivers
2.1. Quivers and their representations. We start by fixing some standard notation for quivers and their representations (for all basic notions of the representation theory of quivers and of finite dimensional algebras, we refer to [4,5]). Let Q be a finite quiver, possibly with oriented cycles, which is given by a finite set of vertices I and a finite set of arrows Q 1 . The arrows will be denoted by (α : i → j) ∈ Q 1 . We denote by ZI the free abelian group generated by I. An element d ∈ ZI will be written as d = i∈I d i i. On ZI, we have the Euler form of Q, a non-symmetric bilinear form, defined by We fix an algebraically closed field of characteristic 0. Let mod kQ be the abelian category of finite-dimensional representations of Q over k (or, equivalently, finite dimensional representations of the path algebra kQ). Its objects are thus given by tuples M = ((M i ) i∈I , (M α : M i → M j ) α:i→j ) of finite dimensional k-vector spaces and k-linear maps between them. The dimension vector dimM ∈ NI is defined as dimM = i∈I dim k M i i. We denote by Hom(M, N ) (resp. Ext 1 (M, N )) the k-vector space of homomorphisms (resp. extension classes) between two representations M, N ∈ mod kQ (all higher Ext's vanish since the category mod kQ is hereditary).
2.2. The classification problem. The basic problem in the representation theory of Q is the following: This problem is solved in the case of Dynkin quivers and extended Dynkin quivers (i.e. the unoriented graph underlying Q is a disjoint union of Dynkin graphs of type A n , D n , E 6 , E 7 or E 8 or their extended versions A n , D n , E 6 , E 7 or E 8 ) in [23] and [18,50], respectively (see also [4,65]). By the Krull-Schmidt theorem, the problem is reduced to the classification of indecomposable representations. These are classified by a discrete parameter (their dimension vector, which is a positive root for the associated root system), together with at most one continuous parameter. The precise statement will be given in section 2.3 in the form of Kac's theorem.
For all other quivers, called wild, the classification problem is unsolved [66, Chapters XVII and XIX] (see however [36] for the study of discrete aspects of the problem, i.e. the Auslander-Reiten theory of wild quivers ). The main focus of the present paper lies on approaching the classification problem with geometric methods, and on trying to understand its nature qualitatively.
Even the very statement of Problem 2.1 leaves much room for interpretation: what could be the nature of a solution of the classification problem? By what kind of object, and by how explicit an object, should the representations be classified?
For example, is it a solution to parametrize a certain class of representations by the points of some algebraic variety which (at least in principle) can be described by explicit defining (in-)equalities in some projective space? This is what moduli spaces of representations can achive (at least for the class of stable representations defined in section 4), see Corollary 3.6.
And what is a normal form for a representation? It could mean a recipe for producing a representative of an isomorphism class of a representation M (i.e. describing the matrices M α representing the arrows α of Q) from a certain set of discrete and continuous invariants. But should these invariants determine M uniquely? And might there be different set of invariants producing isomorphic representations? This is unavoidable even in trivial examples, see section 2.4 below. Or does an algorithm for constructing representations explicitely (like the reflection functors of [6] in the case of Dynkin quivers) qualify as a normal form?
In section 10.3, we will see that it is indeed possible -in a rather direct combinatorial way -to obtain a classification together with explicit normal forms for a problem closely related to Problem 2.1, namely the classification of representations of quivers together with a fixed presentation as a quotient of a given finitely generated projective representation.
2.3. Kac's Theorem. We will now recall Kac's theorem, which shows that for wild quivers, the classification problem for (indecomposable) representations depends on arbitrarily many continuous parameters. This gives an explanation for the essentially different nature of the classification problem for wild quivers. For more details on the theorem and its proof, see [33,34,40].
Let Q be a quiver without oriented cycles. We denote by The last part of the theorem roughly means that, for an imaginary root d, the isomorphism classes of indecomposables of dimension vector d form a family depending on 1 − d, d continuous parameters. We will not need the precise definition of the notion of numbers of parameters in the following (this notion associates to an action of an algebraic group G on a constructible subset X of an algebraic variety a number p, which equals the dimension of a quotient variety X/G in case it exists, which is not true in general. In our situations, quotient varietes -the moduli spaces -will exist, thus there is no need for this more general concept).
Generic classification of representations -an example.
The following trivial example shows that it is often rather easy to classify representations "generically", and that this has consequences for the type of questions that should be asked on the nature of Problem 2.1.
Consider the three-arrow Kronecker quiver K 3 , and consider the dimension vector d = (n, n) for n ≥ 1. In view of Kac's theorem, we expect the classification of indecomposable representations of dimension vector d to depend on 1 − d, d = n 2 + 1 continuous parameters. Such a representation is given by a tripel (A, B, C) of n×nmatrices. Assume that A = I n is the identity matrix, that B = diag(λ 1 , . . . , λ n ) is diagonal with pairwise different diagonal entries, and that C is of the form
C = a 1,1 a 1,2 . . . a 1,n−1 a 1,n 1 a 2,2 .
. . a 2,n−1 a 2,n a 3,1 1 . . . a 3,n−1 a 3,n . . . a n,1 a n,2 . . . 1 a n,n (thus C k+1,k = 1 for all k = 1, . . . , n − 1), and that all a i,j are non-zero. Call this representation M (λ * , a * * ). All the M (λ * , a * * ) are indecomposable with trivial endomorphism ring. Given a tuple (λ * , a * * ) as above, there are only finitely many (in fact, at most n!) such tuples (λ ′ * , a ′ * * ) such that M (λ ′ * , a ′ * * ) ≃ M (λ * , a * * ).
Proof. Let M be an arbitrary representation of K 3 of dimension vector d, given by matrices A, B, C. Generically, we can assume the matrix A to be invertible (since this is characterized by the open condition of non-vanishing of its determinant), and we can assume the second matrix to be diagonalizable with pairwise different eigenvalues (since this is characterized by the open condition that its characteristic polynomial has no multiple zeroes). Up to the action of the base change group GL n (k) × GL n (k), we can thus assume that A = I n , the identity matrix, and that B = diag(λ 1 , . . . , λ n ). The subgroup of GL n (k) × GL n (k) fixing the matrices A and B is the diagonally embedded subgroup T n (K) of diagonal matrices, acting on C via conjugation. Generically, all entries of C = (a i,j ) i,j are non-zero, and T n (K) acts on C by
diag(t 1 , . . . , t n ) * C = ( t i t j C i,j ) i,j .
Thus, we can assume without loss of generality that C i+1,i = 1 for all i. In other words, almost all representations M are isomorphic to a representation M (λ * , a * * ) as defined above. Now assume that two such representations M (λ * , a * * ) and M (λ ′ * , a ′ * * ), given by triples of matrices (I n , diag(λ 1 , . . . , λ n ), C), resp. (I n , diag(λ ′ 1 , . . . , λ ′ n ), C ′ ), are isomorphic. This means that there exists a matrix g ∈ GL n (k) such that g diag(λ 1 , . . . , λ n ) = diag(λ ′ 1 , . . . , λ ′ n ) g and gC = C ′ g. The first condition implies that g is a monomial matrix, i.e. there exists a permutation σ ∈ S n such that g i,j = 0 for j = σ(i), and that λ ′ i = λ σ(i) . Setting
g i,σ(i) = p i , the condition gCg −1 = C ′ reads p i p j a σ(i),σ(j) = a ′ ij
for all i, j = 1, . . . , n. In particular, this holds for the pairs (i + 1, i), thus the p i are given inductively as
p k = ( l<k a σ(l+1),σ(l) ) −1 p 1 .
This shows that at most n! (depending on the permutation σ) such matrices C ′ are conjugate to C under g.
Finally, consider the case that λ ′ * = λ * and a ′ * * = a * * componentwise. The above computation shows that σ is the identity, and that all p i are determined by p 1 . Thus, the endomorphism ring of M ( λ * , a * * ) reduces to the scalars. This elementary example is instructional for several reasons. First of all, we see that it is often very easy to classify almost all indecomposables of a given dimension vector -in fact, the above example can be easily generalized to other dimension vectors for a generalized Kronecker quiver, or to other quivers. But such a generic classification does not capture the "boundary phenomena" which constitute the essence of the classification problem. In a very coarse analogy, one could argue that for the one-loop quiver L 1 , a generic classification reduces to the classification of diagonalizable matrices by their eigenvalues, completely missing the Jordan canonical form (i.e. all higher-dimensional indecomposable representations). Second, in the example we can see that there is no easy way to refine the given matrices to a normal form, in the sense that different systems of parameters give non-isomorphic representations. This phenomenom also can be seen in an even more simple example: Example 2.4. In parametrizing semisimple representations of the one-loop quiver up to isomorphism, i.e. diagonalizable matrices up to conjugation, the most natural normal form is diag(λ 1 , . . . , λ n ) for λ 1 , . . . , λ n ∈ k, but this is only unique up to permutation. On the other hand, we have a classification up to isomorphism by the characteristic polynomial, but this does not yield a normal form.
This phenomenom of "normal forms up to a finite number of identifications" is in fact closely related to the rationality problem for moduli spaces of quivers, to be discussed in section 6.3.
Definition of moduli spaces and basic geometric properties
3.1. Geometric approach. Our basic approach to the continuous parametrization phenomena in the classification of quiver representations is to "materialize" the inherent continuous parameters, by defining geometric objects parametrizing isomorphism classes of certain types of representations (or, as we will consider in section 10, representations together with some appropriate additional structure). See [10,25,39,53] for overviews over the use of geometric techniques in the representation theory of algebras.
One basic decision we have to make a priori is in which geometric category we want to work. For example, it is possible almost tautologically to define stacks [41] parametrizing isomorphism classes of arbitrary representations. But the geometric intuition seems to be lost almost completely for these objects.
Therefore, we will try to define such parameter spaces in the context of algebraic varieties. Thus, we will use in the following the language of varieties, the Zariski topology on them, etc..
The basic idea behind this geometric approach is very simple: Fix a dimension vector d, and fix k-vector spaces M i of dimension d i for all i ∈ I. Consider the affine k-space
R d = R d (Q) = α:i→j Hom k (M i , M j ).
Its points M = (M α ) α obviously parametrize representations of Q on the vector spaces M i . The reductive linear algebraic group
G d = i∈I GL(M i )
acts on R d via the base change action
(g i ) i · (M α ) α = (g j M α g −1 i ) α:i→j .
By definition, the G d -orbits G d * M in R d correspond bijectively to the isomorphism classes [M ] of k-representations of Q of dimension vector d. Our problem of defining parameter spaces can therefore be rephrased as follows:
Problem 3.1. Find a subset U ⊂ R d , an algebraic variety X and a morphism π : U → X, such that the fibres of π are precisely the orbits of G d in U .
More precisely, we ask the subset U to be "as large as possible" (and in particular to be a Zariski open subset of R d ), to capture as much of the "boundary phenomena" as possible, as motivated by section 2.4.
3.2.
Indecomposables and geometry -an example. We will now exhibit an example that, in general (even for quivers without oriented cycles and indivisible dimension vectors), a set U as in Problem 3.1 cannot be the set of indecomposable representations, or even the set of Schur representations.
Consider the 5-arrow Kronecker quiver K 5 and the dimension vector d = (2, 3). Define a family of representations M (λ, µ) for (λ, µ) = (0, 0) by the five matrices
1 0 0 0 0 0 , 0 0 0 1 0 0 , 0 0 0 0 0 1 , 0 0 λ 0 0 0 , 0 µ 0 0 0 0 .
The following lemma is proved by a direct calculation.
Lemma 3.2. All M (λ, µ) for (λ, µ) = (0, 0) are Schur representations. We have M (λ, µ) ≃ M (λ ′ , µ ′ )
if and only if λ ′ = tλ and µ ′ = 1 t µ for some t ∈ k * . Now suppose there exists a subset U of R (2,3) (Q) which contains all M (λ, µ) for (λ, µ) = (0, 0), together with a variety X and a morphism π : U → X as above. We have lim Applying the continuous map π, we get a contradiction: we have
M (t, 1) ≃ M (1, t) for all t = 0, but M (0, 1) ≃ M (1, 0).
. Thus, in trying to define a set U as in Problem 3.1, we have to make a choice between M (0, 1) and M (1, 0). It will turn out that these two representations can be distinguished by the structure of their submodule lattices. For example, M (0, 1) admits a subrepresentation of dimension vector (1, 1), whereas M (1, 0) does not.
The above example is just one of the typical problems in defining quotients: it shows that the potential "quotient variety U/G d " would be a non-separated scheme. In fact, even more severe problems are encountered, for example situations with "nice" actions of the group G d on an open subset U of R d , where nevertheless the desired X cannot be realized as a quasiprojective variety, see [10].
The above example, most importantly, tells us that the class of indecomposable (or Schurian) representations is not well-adapted to our geometric approach to the classification problem! 3.3. Review of Geometric Invariant Theory. Geometric Invariant Theory provides a general way of defining subsets U as in Problem 3.1. We follow mainly the presentation of [48], since the generality of the standard text [47] is not used in the following.
We will first consider the general problem of constructing quotient varieties which parametrize orbits of a group acting linearly on a vector space.
Suppose we are given a vector space V together with a linear representation of a reductive algebraic group G on V . A regular (that is, polynomial) function
f : V → k on V is called an invariant if f (gv) = f (v) for all g ∈ G and v ∈ V.
We denote by k[V ] G the subring of invariant functions of the ring k[V ] of all regular functions on V .
Since the group G is reductive, a theorem of Hilbert asserts that the invariant ring k[V ] G is finitely generated, thus it qualifies as the coordinate ring of a variety
V //G := Spec(k[V ] G ).
The embedding of k[V ] G in k[V ] dualizes to a morphism (associating to an element v ∈ V the ideal of invariants vanishing on v) denoted by π : V → V //G. This morphism is G-invariant by definition, and it fulfills the following universal property:
Given a G-invariant morphism h : V → X, there exists a unique map h : V //G → X such that h = h • π.
This property shows that π can be viewed as the optimal approximation (in the category of algebraic varieties) to a quotient of V by G.
Moreover, one can prove that any fibre of π (which is necessarily G-stable) contains exactly one closed G-orbit. The quotient variety X therefore parametrizes the closed orbits of V in G.
To obtain an open subset U as in Problem 3.1, we define V st as the set of all points v ∈ V such that the orbit Gv is closed, and such that the stabilizer of v in G is zero-dimensional (thus, finite). Then we have the following: gives a morphism whose fibres are exactly the G-orbits in V st .
We consider now a relative version of this construction: Choose a character of G, that is, a morphism of algebraic groups χ :
G → k * . A regular function f is called χ-semi-invariant if f (gv) = χ(g)f (v) for all g ∈ G and v ∈ V.
We denote by k[V ] G,χ the subspace of χ-semi-invariants, and by Let V χ−sst //G be the quasiprojective variety defined as Proj(k[V ] G χ ). Associating to v ∈ V χ−sst the ideal of functions f vanishing on v gives -by definition of semistability -a well-defined map π from V χ−sst to V χ−sst //G. Then the following properties, similar to Theorem 3.3, hold:
k[V ] G χ := n≥0 k[V ] G,
Theorem 3.4. The variety V χ−sst //G parametrizes the closed orbits of G in V χ−sst , and the restriction of π to V χ−st has as fibres precisely the G-orbits in
V χ−st .
Therefore, we see that the open subsets of χ-stable points in V with respect to a given character χ of G give us candidates for open subsets as in Problem 3.1. The case of Theorem 3.3 corresponds to the trivial choice χ = id of character -in this case, all points are semistable.
We summarize the basic geometric properties of quotients in the following diagram:
V χ−st ⊂ V χ−sst ⊂ V ↓ ↓ ↓ V χ−st /G ⊂ V χ−sst //G → V //G || || Proj(k[V ] G χ ) → Spec(k[V ] G )
Proposition 3.5. The following properties hold:
• V χ−st ⊂ V χ−sst ⊂ V is a
chain of open inclusions,
• the variety V //G is affine,
• the morphism V χ−sst //G → V //G is projective, • if the action of G on V χ−st is free, the morphism V χ−st → V χ−st /G is a G-principal bundle, • if V χ−st /G is non-empty, its dimension equals dim V − dim G.
3.4. Geometric Invariant Theory for quiver representations. We will now apply the above constructions to the action of the group G d on the vector space R d (Q); the details can be found in [37].
First of all, we note that the (diagonally embedded) scalar matrices in G d act trivially on R d (Q) (and therefore are contained in the stabilizer of any point). This means that we have to pass to the action of the factor group P G d = G d /k * first to admit orbits with zero-dimensional stabilizers.
An orbit G d * M = P G d * M is closed in R d (Q)
if and only if the corresponding representation M is semisimple by [3]. The quotient variety R d (Q)//P G d therefore parametrizes isomorphism classes of semisimple representations of Q of dimension vector d. This shows that the concept of quotients in its original form gives interesting moduli spaces only in the case of quivers with oriented cycles, since for a quiver without oriented cycles, the only simple representations are the one-dimensional ones S i of dimension vector i for i ∈ I. Therefore, the generality of the relative setup is necessary to obtain interesting moduli spaces.
The characters of the general linear group are just integer powers of the determinant map, thus the characters of the group P G d are of the form
(g i ) i → i∈I det(g i ) mi ,
for a tuple (m i ) i∈I such that i∈I m i d i = 0 to guarantee well-definedness on P G d .
Thus, let us choose a linear function Θ : ZI → Z and associate to it a character
χ Θ ((g i ) i ) := i∈I det(g i ) Θ(d)−dim d·Θi
of P G d (this adjustment of Θ by a suitable multiple of the function dim has the advantage that a fixed Θ can be used to formulate stability for arbitrary dimension vectors, and not only those with Θ(d) = 0). We will denote the corresponding sets of χ Θ -(semi-)stable points by
R sst d (Q) = R Θ−sst d (Q) = R d (Q) χΘ−sst and R st d (Q) = R Θ−st d (Q) = R d (Q) χΘ−st .
The corresponding quotient varieties will be denoted as follows:
M (Θ−)st d (Q) = R Θ−st d (Q)/G d and M (Θ−)sst d (Q) = R Θ−sst d (Q)//G d .
As an immediate application of the concepts of section 3.3, we get:
Corollary 3.6. The variety M Θ−st d (Q) parametrizes isomorphism classes of Θ- stable representations of Q of dimension vector d.
The closed orbits of G d in R Θ−sst d (Q) correspond to the so-called Θ-polystable representations. These are defined as direct sums of stable representations of the same slope. They can also be viewed as the semisimple objects in the abelian subcategory mod µ kQ N of semistable representations of fixed slope µ -see section 4 for details. Therefore, we get: 3.5. Basic geometric properties. We summarize the varieties appearing in the definition of quiver moduli in the following diagram:
R Θ−st d (Q) ⊂ R Θ−sst d (Q) ⊂ R d (Q) ↓ ↓ ↓ M Θ−st d (Q) ⊂ M Θ−sst d (Q) → M ssimp d (Q)
As an application of Proposition 3.5, we have the following geometric properties:
• R Θ−st d (Q) ⊂ R Θ−sst d (Q) ⊂ R d (Q) is a chain of open inclusions, • M Θ−st d (Q) is a smooth variety, • M ssimp d (Q) is an affine variety, • M Θ−sst d (Q) → M ssimp d (Q) is a projective morphism, • R Θ−st d (Q) → M Θ−st d (Q) is a P G d -principal fibration. • The dimension of the variety M Θ−st d (Q), if non-empty, equals 1 − d, d .
The relation between the moduli spaces M st d (Q) and M sst d (Q) can be made more precise using the Luna stratification; see [1] for more details:
Given a polystable representation
M = U m1 1 ⊕ . . . ⊕ U ms s
for pairwise non-isomorphic stables U k of dimension vectors d k (necessarily of slope µ(d)), we denote its polystable type by
(d * ) m * = ((d 1 ) m1 . . . (d s ) ms ).
The subset S d * of M sst d (Q) of all polystable representations of type (d * ) m * is locally closed, yielding a finite stratification of M sst d (Q), such that the stratum corresponding to type (d 1 ) is precisely M st d (Q). This can be used, for example, to determine the singularities in M sst d (Q), see [1]. Other applications will be mentioned in sections 8.2,10.1.
As noted before, in case the quiver Q has no oriented cycles, the only simple representations are the one-dimensional ones S i attached to each vertex i ∈ I. Thus, in this case, M ssimp d (Q) consists of a single point, corresponding to the semisimple representation i∈I S di i of dimension vector d. Consequently, the variety M Θ−sst d is projective in this case. Now assume that d is coprime for Θ, which by definition means that µ(e) = µ(d) for all 0 = e < d. For generic Θ, this is equivalent to d being coprime in the sense that gcd(d i , i ∈ I) = 1:
Suppose that d is Θ-coprime, and that k ∈ N is a common divisor of all entries of d. Then Θ( 1 k d) = Θ(d), and thus k = 1. Conversely, suppose that d is coprime. Then the conditions µ(e) = µ(d) for all 0 = e < d define finitely many proper hyperplanes, and a generic choice of Θ avoids all of them.
Then, by assumption, every semistable representation is already stable, thus the variety M Θ−sst
d = M Θ−st d is smooth.
Thus, in the case of a Θ-coprime dimension vector for a quiver without oriented cycles, we end up which smooth projective moduli spaces, which therefore classify for a consideration with classical techniques of algebraic geometry -see the following sections 6, 7, 8 for some applications of this principle.
We have thus found many solutions to Problem 3.1. The next step is to characterize the stable representations in algebraic terms:
Define the slope of a non-zero dimension vector as
µ(d) := Θ(d) dim d ,
and define the slope of a representation M = 0 as the slope of its dimension vector µ(M ) := µ(dimM ). The following characterization of χ Θ -(semi-)stable points in R d (Q) is given in [37]:
Theorem 3.8. A representation M ∈ R d (Q) is χ Θ -semistable if and only if µ(U ) ≤ µ(M ) for all non-zero subrepresentations U of V . The representation M is χ Θ -stable if and only if µ(U ) < µ(M ) for all non-zero proper subrepresentations U of V .
Thus, (semi-)stability can be characterized in terms of the submodule structure of a representation. Since this simple description of stable representations leads to some very interesting representation-theoretic aspects, we devote the following section to details of this concept.
Algebraic aspects of stability
As before, let Q be an arbitrary finite quiver, and let Θ : ZI → Z be a linear function, called a stability in the following. We also consider, as before, the functional dim on ZI defined by dim d = i∈I d i (this could be replaced by any other strictly positive linear function on dimension vectors). For a non-zero dimension vector d ∈ NI, we define its slope by
µ(d) := Θ(d) dim d ∈ Q.
We define the slope of a non-zero representation X of Q over some field k as the slope of its dimension vector, thus µ(X) := µ(dimX) ∈ Q. The set
NI µ = {d ∈ NI \ {0} : µ(d) = µ} ∪ {0}
forms a subsemigroup of NI.
We call the representation X semistable if µ(U ) ≤ µ(X) for all non-zero subrepresentations U of X, and we call X stable if µ(U ) < µ(X) for all non-zero proper subrepresentations U of X.
Lemma 4.1. Let 0 → X → Y → Z → 0 be a short exact sequence of non-zero k-representations of Q.
Then the following holds:
(1) µ(X) ≤ µ(Y ) if and only if µ(X) ≤ µ(Z) if and only if µ(Y ) ≤ µ(Z).
(2) The same holds with ≤ replaced by <.
(3) min(µ(X), µ(Z)) ≤ µ(Y ) ≤ max(µ(X), µ(Z)).
Proof. Let d and e be the dimension vectors of X and Z, respectively. Then the dimension vector of Y equals d + e, and thus the slope of Y equals
µ(Y ) = Θ(d) + Θ(e) dim d + dim e .
It is now trivial to verify that
Θ(d) dim d ≤ Θ(d) + Θ(e) dim d + dim e ⇐⇒ Θ(d) dim d ≤ Θ(e) dim e ⇐⇒ Θ(d) + Θ(e) dim d + dim e ≤ Θ(e) dim e ,
and the same statement with ≤ replaced by <. This proves the first two parts of the lemma. The third part follows immediately.
This lemma shows that semistability of a representation X can also be characterized by the condition µ(X) ≤ µ(W ) for any non-zero factor representation W of X.
Denote by mod µ kQ the full subcategory of mod kQ consisting of semistable representations of slope µ ∈ Q.
Lemma 4.2. The following properties of the subcategories mod µ kQ hold:
(1) Let 0 → X → Y → Z → 0 be a short exact sequence of non-zero k- representations of Q of the same slope µ.
Then Y is semistable if and only if X and Z are semistable.
(2) mod µ kQ is an abelian subcategory of mod kQ.
(3) If µ > ν, then Hom(mod µ kQ, mod ν kQ) = 0.
(4) The stable representations of slope µ are precisely the simple objects in the abelian category mod µ kQ. In particular, they are indecomposable, their endomorphism ring is trivial, and there are no non-zero morphisms between non-isomorphic stable representations of the same slope.
Proof. Suppose that X and Z are semistable, and let U be a subrepresentation of Y . This yields an induced exact sequence
0 → U ∩ X → U → (U + X)/X → 0
of subrepresentations of X, Y and Z, respectively. By semistability of X and Z,
we have µ(U ∩ X) ≤ µ(X) = µ and µ((U + X)/X) ≤ µ(Z) = µ.
Applying the third part of the previous lemma, we get
µ(U ) ≤ max(µ(U ∩ X), µ((U + X)/X))) ≤ µ = µ(Y ),
proving semistability of Y .
Conversely, suppose that Y is semistable. A subrepresentation U of X can then be viewed as a subrepresentation of Y , and thus µ(
U ) ≤ µ(Y ) = µ = µ(X), proving semistability of X. A subrepresentation U of Z induces an exact sequence 0 → X → V → U → 0 by pullback, and thus µ(V ) ≤ µ(Y ) = µ = µ(X).
Applying the first part of the previous lemma, we get µ(U ) ≤ µ(V ) ≤ µ = µ(Z), proving semistability of Z. This proves the first part of the lemma. It also proves that the subcategory mod µ kQ is closed under extensions.
Given a morphism f :
X → Y in mod µ kQ, we have µ = µ(X) ≤ µ(Im(f )) ≤ µ(Y )
= µ by semistabililty of X and Y , and thus µ(Im(f )) = µ. Thus, Ker(f ), Im(f ) and Coker(f ) all have the same slope µ, and they are all semistable by the first part. This proves that mod µ kQ is abelian. The same argument proves the third part: if f : X → Y is a non-zero morphism, then µ(X) ≤ µ(Im(f )) ≤ µ(Y ).
By the definition of stability, a representation is stable of slope µ if and only if it has no non-zero proper subrepresentation in mod µ kQ, proving that the stables of slope µ are the simples in mod µ kQ. The remaining statements of the fourth part follow from Schur's Lemma.
Definition 4.3.
A subrepresentation U of a representation X is called scss (short for strongly contradicting semistability) if its slope is maximal among the slopes of subrepresentations of X, that is, µ(U ) = max{µ(V ) | V ⊂ X}, and it is of maximal dimension with this property.
Such a subrepresentation clearly exists, since there are only finitely many dimensions and slopes of subrepresentations. By its defining property, it is clearly semistable.
Lemma 4.4. Every representation X admits a unique scss subrepresentation.
Proof. Suppose U and V are scss subrepresentations of X, necessarily of the same slope µ. The exact sequence 0
→ U ∩ V → U ⊕ V → U + V → 0 yields µ(U ∩ V ) ≤ µ = µ(U ⊕ V ), thus µ ≤ µ(U + V ) by Lemma 4.1.
By maximality of the slope µ among subrepresentations of X, we have µ(U + V ) = µ. By maximality of the dimension of U and V , we have dim U + V ≤ dim U, dim V , and thus U = V .
Remark 4.5. The uniqueness of the scss of a representation X has some interesting applications: for example, the scss has to be fixed under arbitrary automorphisms ϕ of X, since applying ϕ to a subrepresentations does not change its dimension vector, and thus also its slope and dimension.
This implies the compatibility of semistability with base extension: let k ⊂ K be a Galois extension, and let X be a semistable k-representation of Q. The scss V ofX = K ⊗ k X is fixed under all automorphism ofX, thus in particular under the Galois group of the extension k ⊂ K. Thus, it descends to a sub-
representation U of X (that is, V ≃ K ⊗ k U ). By semistability of X, we have µ(V ) = µ(U ) ≤ µ(X) = µ(X), thusX is semistable. Definition 4.6. A filtration 0 = X 0 ⊂ X 1 ⊂ . . . ⊂ X s = X of a representation X is called Harder-Narasimhan (abbreviated by HN) if the subquotients X i /X i−1 are semistable for i = 1, . . . , s, and µ(X 1 /X 0 ) > µ(X 2 /X 1 ) > . . . > µ(X s /X s−1 ).
Lemma 4.7. Every representation X possesses a unique Harder-Narasimhan filtration.
Proof. Existence is proved by induction on the dimension of X. Let X 1 be the scss of X. By induction, we have a HN filtration 0
= Y 0 ⊂ Y 1 ⊂ . . . ⊂ Y s−1 = X/X 1 .
Via the projection π : X → X/X 1 , we pull this back to a filtration of X defined by
X i = π −1 (Y i−1 ) for i = 1, . . . , s. Now X 1 /X 0 is semistable since X 1 is the scss of X, and X i+1 /X i ≃ Y i /Y i−1 is semistable by the choice of the Y i for i = 1, . . . , s−1.
We also infer µ(X 2 /X 1 ) > . . . > µ(X s /X s−1 ) from the corresponding property of the slopes of the subquotients in the HN filtration of X/X 1 . Since X 2 is a subrepresentation of X of strictly larger dimension than X 1 , we have µ(X 1 ) > µ(X 2 ) since X 1 is scss in X, and thus µ(X 1 /X 0 ) = µ(X 1 ) > µ(X 2 /X 1 ).
To prove uniqueness, we proceed again by induction on the dimension. Assume that 0 = X ′ 0 ⊂ . . . ⊂ X ′ s = X is a HN filtration of X. Let t be minimal such that X 1 is contained in X ′ t , thus the inclusion induces a non-zero map from X 1 to X ′ t /X ′ t−1 . Both representations being semistable and X 1 being scss, we have
µ(X ′ 1 ) ≤ µ(X 1 ) ≤ µ(X ′ t /X ′ t−1 ) ≤ µ(X ′ 1 ), thus µ(X 1 ) = µ(X ′ 1 ) and t = 1, which means X 1 ⊂ X ′ 1 .
Again since X 1 is scss, we conclude that X 1 = X ′ 1 . By induction, we know that the induced filtrations on the factor X/X 1 coincide, thus the filtrations of X coincide.
Interpreted properly, the HN filtration is even functorial: index the HN filtration by Q by defining
X (a) = X k if µ(X k /X k−1 ) ≥ a > µ(X k+1 /X k ) for all a ∈ Q. Lemma 4.8. Any morphism f : X → Y respects the HN filtration, in the sense that f (X (a) ) ⊂ Y (a) for all a ∈ Q.
Proof. We will prove by induction on k the following property:
If f (X k ) ⊂ Y l \ Y l−1 , then µ(Y l /Y l−1 ) ≥ µ(X k /X k−1 ).
The claimed property follows from this: given a ∈ Q, we have X (a) = X k for the
index k satisfying µ(X k /X k−1 ) ≥ a > µ(X k+1 /X k ). Choosing l minimal such that f (X k ) ⊂ Y l , we then have µ(Y l /Y l−1 ) ≥ µ(X k /X k−1 ) ≥ a, and thus Y l ⊂ Y (a) by definition. In case k = 0 there is nothing to show. For k = 1, suppose f (X 1 ) ⊂ Y l \ Y l−1 .
Then f induces a non-zero map between the semistable representations X 1 and
Y l /Y l−1 , showing µ(X 1 ) ≤ µ(Y l /Y l−1 ) as claimed. For general k, suppose that f (X k ) ⊂ Y l \ Y l−1 , and consider the diagram 0 → X k−1 α → X k → X k /X k−1 → 0 f ↓ 0 → Y l−1 → Y l β → Y l /Y l−1 → 0 If βf α equals 0, the map f induces a non-zero map X k /X k−1 → Y l /Y l−1 between semistable representations, and thus µ(X k /X k−1 ) ≤ µ(Y l /Y l−1 ) as desired. If βf α is non-zero, we have f (X k−1 ) ⊂ Y l \ Y l−1 , and we can conclude by induction that µ(Y l /Y l−1 ) ≥ µ(X k−1 /X k−2 ) > µ(X k /X k−1 )
, which again gives the desired estimate.
We call the slopes µ(X 1 /X 0 ), . . . , µ(X s /X s−1 ) in the unique HN filtration of X the weights of X. Definition 4.9. Given a ∈ Q, define T a as the class of all representations X all of whose weights are ≥ a, and define F a as the class of all representations X all of whose weights are < a. Proof. Assume X ∈ T a and Y ∈ F a . In the Q-indexed HN filtration, we thus have X (b) = X for all a ≤ b, and Y (b) = 0 for all a < b. By functoriality, any morphism f : X → Y is already zero, proving Hom(T a , F a ) = 0. Now assume Hom(X, F a ) = 0 for some representation X. Suppose X has a weight strictly less than a, then certainly the slope of the (semistable) top HN factor X/X s−1 is strictly less than a, too, thus it belongs to F a . But the projection map X → X/X s−1 is non-zero, a contradiction. Thus, X belongs to T a . Finally, assume Hom(T a , Y ) = 0 for some representation Y . If Y has a weight ≥ a, then certainly the slope of its (semistable) scss
Y 1 is ≥ a. Thus Y 1 belongs to T a . But the inclusion Y 1 → Y is non-zero, a contradiction. Thus, Y belongs to F a .
The inclusion properties of the various torsion and free classes follows from the definitions.
Further geometric properties of moduli spaces
5.1. The choice of Θ. In general, the choice of a "suitable" stability Θ for a given quiver, or a given dimension vector, is a very subtle problem. We will not consider this in general, but only work out the possible choices in some examples.
First note that there are two operations on functionals Θ as above which do not change the classes of (semi-)stable representations:
• first, we can multiply Θ by a non-negative integer;
• second, we can add an integer multiple of the functional dim to Θ.
This shows immediately that all choices of Θ for the multiple-loop quiver L m are equivalent to the choice Θ = 0. In this case, we know that all representations are semistable (of slope 0), and that the stable representations are precisely the simple ones. Thus in this case, the only relevant moduli spaces are the moduli
M st d (L m ) of d-dimensional simples.
For generalized Kronecker quivers, the above operations on stability conditions allow reduction to three choices, namely Θ 1 (d, e) = 0, Θ 2 (d, e) = e and Θ 3 (d, e) = d. The first two only lead to trivial notions of stability: for Θ 1 = 0, all representations are semistable, and the stables are precisely the simples S 1 , S 2 . For Θ 2 , assume that M is semistable of dimension vector (d, e). If d = 0 = e, consideration of a subrepresentation S 2 ⊂ M yields a contradiction. Thus, the only semistables are direct sums of copies of either S 1 or S 2 , and these simple are the only stables. Thus, the only non-trivial choice of stability is Θ 3 . In this case, a representation M is semistable if and only if for all non-zero subspaces
U of M i , we have dim m k=1 M α k (U ) ≥ dim M j dim M i dim U,
and it is stable if this inequality is strict for all proper non-zero subspaces.
Stable representations and Schur representations.
From Lemma 4.2, we know that every stable representation is a Schur representation. We will now show in an example (for a generalized Kronecker quiver), that in general there exists no choice Θ of stability making all Schur representations of a fixed dimension vector stable. It is also not possible in general to choose a Θ for a given Schur representation M such that M becomes Θ-stable.
Continuing the example from section 3.2, we have Therefore, the notion of stability chooses one of the two representations M (0, 1), M (1, 0) as "more canonical", due to its submodule structure.
As another example, we consider the 4-subspace quiver S 4 with dimension vector d = i 1 + i 2 + i 3 + i 4 + 2) and stability Θ = j * . A representation is given by four vectors v 1 , v 2 , v 3 , v 4 in k 2 , which we will assume to be non-zero. Then this representation is stable of no two of the vectors are proportional, and it is semistable if no three of them are proportional. It is indecomposable if it is semistable, and the four vectors cannot be grouped into two pairs of proportional ones. The stable representations therefore admit a normal form (e 1 , e 2 , e 1 + e 2 , e 1 + λe 2 ) for
λ = 0, 1. The moduli space M sst d (S 4 ) is isomorphic to a projective line P 1 via the map R sst d (Q) → P 1 given by (v 1 , v 2 , v 3 , v 4 ) → (det[v 1 |v 2 ] det[v 3 |v 4 ] : det[v 1 |v 4 ] det[v 3 |v 2 ])
(see section 5.5 for more details). In this way, we realize M st
d (K 4 ) is the open subset P 1 \ {0, 1, ∞}.
But the moduli space cannot distinguish between all the indecomposables: for example, all of the three isomorphism classes (e 1 , e 1 , e 2 , e 2 ), (e 1 , e 1 , e 2 , e 1 + e 2 ), (e 1 , e 2 , e 1 + e 2 , e 1 + e 2 ) are sent to the point (0 : 1) of P 1 . The first of these is decomposable into a direct sum U 1 ⊕ U 2 of three-dimensional representations. The second and the third are indecomposable, the second being a non-trivial extension of U 1 by U 2 , the third being a non-trivial extension of U 2 by U 1 .
As already suggested above, one may ask whether for a given Schur representation, there always exists a stability making this representation stable. The above example shows that this is not possible in general! The point of view of the present paper is not to worry about the different choices of stability (and even not about the question whether a particular moduli space is non-empty), but to try to formulate results which hold for arbitrary choices of stability.
Existence of stable
representations. An obvious question in relation to the choice of Θ is the following: under which conditions is M sst d (Q) (resp. M st d (Q)) non-empty? For the semistable representations, the Harder-Narasimhan filtration yields a recursive criterion, see [52].
Proposition 5.2. M sst d (Q)
is non-empty if and only if there exists no non-trivial decomposition d = d 1 + . . . + d s fulfilling the following three conditions:
• M sst d k (Q) = ∅ for all k = 1, . . . , s, • µ(d 1 ) > . . . > µ(d s ), • d k , d l = 0 for all k < l.
Criteria for non-emptyness of both M st d (Q) and M sst d (Q) can be formulated using the concept of generic subrepresentations [61]; these criteria are also highly recursive. We write e ֒→ d if the set of representations of dimension vector d admitting a subrepresentation of dimension vector e is dense in R d (Q). Whether this condition holds for two given dimension vectors can be determined recursively:
Universal bundles.
It is a general philosophy in moduli theory that moduli spaces should, most desirably, be equipped with so-called universal (or tautological) bundles.
As an elementary example, we consider the tautological bundle on the Grassmannian Gr k (V ) parametrizing k-dimensional subspaces of a vector space V , i.e. we can label the points of Gr k (V ) by k-dimensional subspaces U ⊂ V . There exists a vector bundle π : T → Gr k (V ), which is a subbundle of the trivial bundle p 1 : Gr k (V ) × V → Gr k (V ) with the following property:
π −1 (U ) ⊂ {U } × V con- sists of all ({U }, v) such that v ∈ U .
Similarly, in the context of quiver moduli, we ask for the following: let M st d (Q) be a moduli space of stable representations of Q of dimension vector d. We want to define vector bundles π i : V i → M st d (Q) of rank d i for all i ∈ I and vector bundle maps V α : V i → V j for all arrows α : i → j in Q such that the following holds:
) i ) = i∈I det(g i ) −si .
It is likely that no universal bundle on M d (Q) exists in case d is not coprime, but there is no proof of this yet.
Coordinates.
We now turn to the question of coordinates for quiver moduli. By definition, we have
M sst d (Q) = Proj( n≥0 k[R d ] G d ,χ n Θ ),
thus knowledge of generating (semi-)invariants provides coordinates for the moduli in the following sense:
Let R = n≥0 R n be such a semi-invariant ring. Then R 0 is finitely generated (being the invariant ring for the action of G d on R d ) by, say, f 1 , . . . , f s . Consider R as an R 0 -algebra. This is again finitely generated, since the ring of semi-invariants (more precisely, the underlying non-graded ring) can be viewed as the ring of invariants for the smaller group Ker(χ Θ ). Since the Proj-construction is not sensitive to "thinning" R, i.e. replacing R by R (k) = n≥0 R kn for k ≥ 1, we can choose generators g 0 , . . . , g t , homogeneous of some degree k ≥ 1, for R (k) . Then Proj(R) admits an embedding into A s × P t dual to the surjection k[x 1 , . . . , x s ][y 0 , . . . , y t ] → R attaching f i to x i and g j to y j . This procedure can be carried out in principle for R the ring of semi-invariants of the action of G d on R d with respect to a character χ Θ . We start with the ring of invariants:
Theorem 5.7 (Le Bruyn-Procesi). The ring of invariants for the action of G d on R d is generated by traces along oriented cycles, i.e. for a cycle ω = α u . . . α 1 given by To formulate a similar statement for semi-invariants, we have to introduce some additional notation. We start with the case of quivers without oriented cycles. Given representations M and N of Q, we can compute Hom(M, N ) and Ext 1 (M, N ) as the kernel and cokernel, respectively, of the map
i 1 α1 → i 2 α2 → . . . αu−1 → i u αu → i 1 in Q,d M,N : i∈I Hom k (M i , N i )→ α:i→j Hom k (M i , N j ) given by d M,N ((f i ) i ) = (N α f i − f j M α ) (α:i→j) .
In
c : R d (Q) × R d (Q) → k,
which is in fact a semi-invariant function for the action of G d × G e . We can also fix the representation M and vary the representation N to obtain a semi-invariant function c M on R e (Q). The following is proved in [63]:
Theorem 5.8. The functions c M for representations M such that dimM, d = 0 generate the ring of semi-invariants on R d .
In case Q has oriented cycles, we need a more general version unifying the above two theorems. Choose an arbitrary map v : P → Q between finitely generated projective representations P = i∈I P ai i and Q = Note again that the full semi-invariant ring, which is described by the above theorems, is not of the type we considered in the definition of moduli spaces, i.e. not associated to a single stability function Θ. The geometric object that is described by the Proj of this ring is the quotient of the stable points in R d by the action of the smaller group i∈I SL(M i ). It parametrizes quiver representations M together with fixed volume forms of all vector spaces M i , under isomorphisms preserving the volume forms.
5.
6. An example -subspace quivers. We consider an example to illustrate the above strategy for computing coordinates for quiver moduli and to show the difference between the full ring of semi-invariants and the ring of semi-invariants associated to a fixed character. Consider the m-subspace quiver S m and dimension vector (1, . . . , 1, 2). A representation is given as an m-tuple (v 1 , . . . , v m ) of vectors in k 2 , on which the group GL 2 (k) × (k * ) m acts via
(g, x 1 , . . . , x m ) * (v k ) k = ( 1 x k gv k ) k .
It is easy to see that the full ring of semi-invariants R is generated by the functions
D kl = det[v k |v l ] for 1 ≤ k < l ≤ m.
These functions fulfill the Plücker relations
D ik D jl = D ij D kl + D ik D jl
for all i < j < k < l. Thus, the Proj of this ring is nothing but the Grassmannian of 2-planes Gr 2 (k m ).
The function D kl is a semi-invariant for the character
χ(g, x 1 , . . . , x m ) = det(g) x k x l .
Let us consider the "most symmetric" stability Θ = (0, . . . , 0, −1). Now a mono- for some k ≥ 1. It turns out a minimal system of generators becomes rather large even for small values of n.
mial i<j D mij ij belongs to k[R d ] G
We consider the particular case n = 5. One can see directly that the semi-invariant ring is generated by the following functions:
c = D 12 D 23 D 34 D 45 D 51 , x i = D i,i+1 D i,i+4 D i+4,i D 2 i+2,i+3 for i ∈ Z 5 .
The Plücker relations give the following (defining) relations between the generators:
x i x i+1 = c 2 + cx i+3 for i ∈ Z 5 .
Consequently, the moduli space M The case m = 7 was worked out in [2], using results of [31]: the above methods yields an embedding of the 4-dimensional moduli space into P 35 , determined by 58 defining equations.
From these examples we can see that, even in simple examples, coordinatization of the moduli spaces leads to difficult explicit calculations in commutative algebra.
Another question is whether, even if we have explicit generators and defining relations, this is helpful for studying the moduli spaces, since it is difficult to extract global geometric information from defining equations.
Cohomology and cell decompositions
One of the possible directions towards a study of the global geometry of quiver moduli pursued by the author in [19,46,52,54,55,56,57,58] is the determination of Betti numbers of quiver moduli. We will first consider the question why knowledge of the Betti numbers should be interesting for such a study.
Whenever we use cohomology of varieties, we will work over the base field k = C of the complex numbers, and we will consider all quasiprojective varieties with their C-topology, induced from the natural C-topology on complex projective spaces. Then we consider singular cohomology (or singular cohomology with compact support) with coefficients in Q, disregarding all potential torsion phenomena. We then denote by b i (X) = dim Q H i (X, Q) the i-th Betti number of X (for arbitrary base fields k as before, ℓ-adic cohomology should be considered; standard comparison theorems guarantee the compatibility of these two approaches).
Cell decompositions.
Definition 6.1. A variety X is said to admit a cell decomposition if there exists a filtration
X = X 0 ⊃ X 1 ⊃ . . . ⊃ X t = ∅
by closed subvarieties, such that the successive complements X j−1 \ X j for j = 1, . . . , t are isomorphic to affine spaces A dj .
This notion is not to be confused with the topological notion of cell decomposition, for example in the context of CW-complexes. In the literature (for example [13]) one also finds a variant of this notion, where the successive complements are only required to be isomorphic to disjoint unions of affine spaces; further refinement of such a filtration leads to one in the above sense.
Examples of such varieties are provided by affine space itself, by projective spaces (where X i consists of all points (x 0 : . . . : x n ) in projective n-space such that the first i coordinates are zero, for i = 0, . . . , n − 1), Grassmannians, etc.. If X admits a cell decomposition, then all odd cohomology of X vanishes, and the 2i-th Betti number is given as the number of i-dimensional cells, i.e. the number of indices j such that X j−1 \ X j ≃ A i .
M st d (Q) = n k=1 Y k , where Y k ≃ A d k .
The restriction of each vector bundle V i to each Y k is a vector bundle over an affine space, and thus trivial. This means that we can find isomorphisms
φ ik : Y k × M i → V i | Y k .
For any arrow α : i → j we can then consider the composite map
f α = φ −1 jk • V α | Y k • φ ik : Y k × M i → Y k × M j .
For any point x in A d k ≃ Y k , the restriction f α (x) of f α to the fibres over x is a linear map from M i to M j by definition. Thus, the tuple (f α (x)) α defines a quiver representation M (x). The collection of all M (x) for x ∈ Y k thus give a normal form for all the quiver representations belonging to Y k .
We see that existence of a cell decomposition of the moduli space M st d (Q) gives an explicit parametrization of all isomorphism classes of stable representations of dimension vector d, together with explicit normal forms. This is of course very desirable in view of Problem 2.1.
One may conjecture that such cell decompositions exist for the moduli of stable representations whenever d is coprime for Θ. Instances of this conjecture will be proved later for certain (very special) dimension vectors of generalized Kronecker quivers (see section 7.3), and for the framed versions of moduli spaces in section 10.3.
Negative examples and discussion.
An interesting testing case for the above conjecture is provided by the surface considered in section 5.6: it is not clear whether this projective rational surface admits a cell decomposition.
Turning to the non-coprime case, one cannot hope for cell decompositions to exist in general. Consider again the example of the 4-subspace quiver S 4 and dimension vector (1, 1, 1, 1, 2). We have seen in section 5.2 that the moduli of stables is isomorpic to P 1 minus three points in this case. This space definitely has (twodimensional) first cohomology, so it cannot admit a cell decomposition (since this implies vanishing of the odd Betti numbers). Another example is provided by twodimensional simple representations of the two-loop quiver L 2 .
One might suspect that the problem is caused by the missing semistables (or semisimples). This is, however, not the case. Considering the dimension vector (2, 2) for the five-arrow Kronecker quiver, we get the counting polynomial t 13 + t 12 + 3t 11 + 2t 10 + 3t 9 + t 8 + t 7 − t 6 + t 3 + t 2 + t + 1 (see section 8) for M sst d (Q), thus there cannot exist a cell decomposition, as will be explained in section 8.1 (a polynomial counting points over finite fields of a variety admitting a cell decompositions necessarily has nonnegative coefficients).
Despite these negative results, there are several possible variants of the problem: the first possibility is to ask for a torus decomposition, i.e. we relax the defining condition of a cell decomposition and ask the successive complements to be isomorphic to tori instead. We will see in section 8.3 that this is supported by the conjecture 8.5.
Another option is to look at variants of quiver moduli, and to ask for these spaces to admit a cell decomposition. For moduli of simple representations, we will consider the noncommutative Hilbert schemes in section 10.3, and cell decompositions will be constructed. In general, the smooth models of section 10 provide candidatesagain one might conjecture that they always admit cell decompositions.
The most desirable variant of the original moduli spaces we would like to have is a desingularization of the moduli of semistables, or, in other words, a "compactification" of the moduli of stables. By this we mean a smooth variety X admitting a projective birational morphism to M sst d (Q) (then X is projective over M ssimp d (Q), and contains M st d (Q) as an open subset). Existence of cell decompositions for such desingularisation poses a difficult problem: suppose that X has a cell decomposition. In particular, we have X \ X 1 ≃ A d1 , i.e. there exists an open subset which is isomorphic to affine space -in other words, the variety X is rational. Since X maps to M sst d (Q) birationally, this would imply that the latter is rational, too. But it is shown in [62] that all quiver moduli are birational to moduli of simple representations of the multiple-loop quiver, and in this case rationality is a longstanding open problem, see [42].
One other possible relaxation of the notion of cell decomposition is therefore to ask for an orbifold decomposition, i.e. the successive complements should look like quotients of affine space (or a torus) by a finite group action.
There is one notable exception to this problem of construction of smooth compactifications. Namely, the moduli space M ssimp 2 (L m ) has been desingularized in [64]. This desingularization is analysed in detail in [51]. In particular, it is shown there that the desingularization admits a cell decomposition. Moreover, the obstruction to a generalization of the construction of [64] to higher dimensions is studied in detail, refining [44].
One of the problems in constructing cell decompositions is that there are only few general techniques for doing so. One is the Bialynicki-Birula method, to be discussed in section 7.3.
Betti numbers and prediction of cell decompositions.
One of the main motivations for computing and studying Betti numbers of quiver moduli can be seen in the following line of reasoning, based on the above discussion: suppose the Betti numbers of the moduli space in question are computed, and that they admit some combinatorial interpretation (for example numbers of certain types of trees in the case of Hilbert schemes in section 10.3). Then this gives a prediction for a combinatorial parametrization of the cells in a cell decomposition, and sometimes actually a construction of the cells.
6.5. Betti numbers for quiver moduli in the coprime case. We review the main results of [52]. Definition 6.2. Given Q, d and Θ as before, we define the following rational function in q:
P d (q) = d * (−1) s−1 q − P k≤l d l ,d k s k=1 i∈I di j=1 (1 − q −j ) −1 ∈ Q(q),
where the sum ranges over all tuples d * = (d 1 , . . . , d s ) of dimension vectors such that
• d = d 1 + . . . + d s • d k = 0 for all k = 1, . . . , s, • µ(d 1 + . . . + d k ) > µ(d) for all k < s.
We will see in section 9.3 how the definition of this function is motivated (basically, we have
P d (q) = |R sst d (Q)(F q )| |G d (F q )|
for any finite field F q ).
Theorem 6.3. If d is coprime for Θ, then (q−1)·P d (q) = i dim Q H i (M d , Q)q i/2 .
In particular, this theorem reproves a result of [38] that there is no odd cohomology of M d (Q) in the coprime case: namely, the left hand side of the formula in the theorem is a rational function in q, so the right hand side is so, too, and all potential contributations to half-powers of q -coming from the odd cohomologyhave to vanish.
A drawback of this formula is that, although we know a priori that the result is a polynomial in q, all summands are only rational functions in q, with denominators being products of terms of the form (1 − q i ). In particular, we cannot specialize the formula to q = 1, which would be very interesting because then the Poincare polynomial specializes to the Euler characteristic.
We also do not get a "positive" formula in the sense that one can see directly from the summands that the coefficients of the resulting polynomial have to be nonnegative integers. In contrast, the formula for P d (q) involves signs. We will see positive formulas for the Betti numbers of quiver moduli in special cases, namely in section 7.2 for generalized Kronecker quivers, and in section 10.3 for Hilbert schemes.
Nevertheless, the above formula gives rise to a fast algorithm for computing the Betti numbers (which was further optimized in [67] to compute the Euler characteristic of moduli of generalized Kronecker quivers). This serves as an important tool for computer experiments which motivated many of the developments described below.
6.6. Asymptotic aspects. One can argue that, in studying quiver moduli qualitatively (i.e. studying common features enjoyed by all quiver moduli, in contrast to determination of special features of particular ones) one should not consider a fixed dimension vector d, but consider either all of them at the same time (see results on generating series over all d such that µ(d) = d in Theorems 8.3 and 10.2), or consider the behaviour of the moduli for large d. The latter case is what is considered in this section.
We first consider a very simple instance of this principle: in case d is coprime for Θ, we will see in section 9.3 that the Poincare polynomial of singular cohomology is given by
(q − 1) · |R sst d (Q)(F q )| |G d |(F q )| . Since R sst d (Q)
is open in the affine space R sst d , we know that the counting polynomial is of degree dim R d (Q). For large enough coprime d, the number of non-semistable representations should be "small" compared to the number of all representations, so that
(q − 1) |R d (Q)| |G d |
should be a good approximation to the Poincare polynomial. This number equals (by a direct calculation)
q 1− d,d (1 − q −1 ) i∈I di k=1 (1 − q −k ) −1 .
A very surprising route towards predictions of the asymptotic behaviour of quiver moduli opens up in connection to methods from string theory, see e.g. [7,16,17]. The idea is, very roughly, to view e.g. moduli of representations of generalized Kronecker quivers as moduli spaces of the possible states of strings between branes. These should form a microscopic model for the behaviour of macroscopic physical systems like e.g. certain types of black holes. Known or expected properties of such physical systems then yield predictions on the microscopic system (i.e. the quiver moduli) "in the large", i.e. for large values of the dimension vector.
M. Douglas made the following conjecture for generalized Kronecker quivers K m : for large dimension vectors (d, e) (d and e coprime), the logarithm of the Euler characteristic log χ(M (d,e) (Q)) should depend continuously on the ratio e/d. A more precise formulation can be given as follows:
there should exist a continuous function f : R ≥0 → R with the following property: for any r ∈ R ≥0 and any ε > 0, there exist δ > 0 and N ∈ N such that for coprime (d, e) with d + e > N and |e/d − r| < δ, we have
|f (r) − log χ(M (d,e) (K m )) d | < ε.
This conjecture is extremely surprising mathematically, since there are no general geometric or representation-theoretic techniques to relate moduli spaces M d (Q) and M e (Q) for "close" coprime dimension vectors d and e. Nevertheless, computer experiments [67] using the above mentioned algorithm for computation of Betti numbers give substantial evidence for this conjecture. A posteriori, it turns out that, if such a function f exists, it already is uniquely determined up to a constant as
f (r) = C · r(m − r) − 1.
This can be seen using natural identifications of moduli spaces (which are special to moduli for generalized Kronecker quivers). Namely, the natural duality, resp. the reflection functors, yield isomorphisms
M st (d,e) (K m ) ≃ M st (e,d) (K m ), resp. M st d,e (K m ) ≃ M st (md−e,d) (K m ).
These identifications translate into functional equations for the function f (if it exists), which already determine it up to a scalar factor.
Using localization techniques (see section 7), it is possible to obtain exponential lower bounds for the Euler characteristic, thus proving part of the above conjecture; see [58,68].
A slight reformulation of the above (conjectural) formula for the asymptotic behaviour yields a conjecture for arbitrary quivers:
For every quiver Q, there exists a constant C Q such that for large coprime d, we have log
χ(M d (Q)) ∼ C Q − d, d .
It is a very interesting problem to make this more precise. If this conjecture is true in some form, it has the surprising consequence that the Euler characteristic of a quiver moduli "in the large" is already determined by it dimension 1 − d, d !
In one instance, the exponential behaviour of the Euler characteristic can indeed be proved: we consider the moduli space Hilb d,1 (L m ) (see section 10.3 for the definition), or, in other words, the moduli space M st (1,d) (Q) for the quiver Q given by vertices I = {i, j} and arrows
Q 1 = {(α : i → j), (β 1 , . . . , β m : j → j)}.
In this case, the parametrization of a cell decomposition by m-ary trees (as a special case of the combinatorial notions of section 10.3) yields the following [55]:
χ(Hilb d,1 (L m )) ∼ C · d −3/2 · (m m /(m − 1) (m−1) ) d .
In this case, it is even possible to describe the asymptotic behaviour of the individual Betti numbers [55]: for each d ∈ N, define a discrete random variable X d by
P(X d = k) = 1 χ(Hilb d,1 (L m )) · dim H (m−1)d(d−1)−2k (Hilb d,1 (L m ), Q).
Then the sequence of random variables 8/(m(m − 1)) · d −3/2 · X d admits a continuous limit distributation, the so-called Airy distribution [22].
Localization
The localization principle in topology states that a lot of topological information on a space X can be retrieved from the set of fixed points X T under the action of a torus T on X. For example, χ(X) = χ(X T ) for any action of a torus on a quasi-projective variety. See [12,13,20,28].
7.1. Localization for quiver moduli. We consider the torus T Q = (C * ) Q1 , i.e. one copy of C * for each arrow α in Q (in some situations, it is also interesting to consider arbitrary subtori, or for example C * embedded diagonally into T Q , see section 7.3).
The torus T Q acts on the path algebra CQ via rescaling of the generators α ∈ CQ corresponding to the arrows α. By functoriality, T Q acts on the category of representations and also on all moduli of representations. More precisely, T Q acts on R d (Q) (written as a right action) via
(M α ) α (t α ) α = (t α M α ) α .
This By definition, the second projection p 2 : G → T Q is surjective. On the other hand, it is injective since the stabilizer of M in P G d is trivial. Thus, we can invert the second projection, providing us with a map ϕ : T Q → P G d , such that
ϕ(t)M = M t for all t ∈ T Q .
We can lift ϕ to G d , again denoted by ϕ. Denote by ϕ i :
T Q → GL(M i ) the i-component of i ∈ I.
The defining condition of ϕ thus tells us that
ϕ j (t)M α ϕ i (t) −1 = t α M α
for all α : i → j in Q 1 and all t = (t α ) α ∈ T Q . The map ϕ i can be viewed as a representation of T Q on M i , which we can decompose into weight spaces, denoting by X(T Q ) the character group of T Q :
M i = λ∈X(TQ) M i,λ , where M i,λ = {m ∈ M i : ϕ i (t)m = λ(t)m for all t ∈ T Q }.
The character group X(T Q ) has a basis e α with e α (t) = t α , for α ∈ Q 1 . The above equation now yields
M α (M i,λ ) ⊂ M j,λ+eα for all α : i → j and all λ ∈ X(T Q ).
This means that M is automatically a kind of graded representation, which we can view as a representation of a covering quiver, defined as follows:
let Q be the quiver with set of vertices
Q 0 = Q 0 × X(T Q )
and arrows
Q 1 = {(α, λ) : (i, λ) → (j, λ + e α ), (α : i → j) ∈ Q 1 , λ ∈ X(T Q )}.
This covering quiver carries a natural action of X(T Q ) via translation. Then M can be viewed as a representation of Q of some dimension vector d lifting d, i.e. such that
π( d) := i,λ d (i,λ) = d i for all i ∈ I.
This representation is again stable, for the stabiliy Θ on Q defined by Θ( d) = Θ(π( d)) :
by rigidity of the HN-filtration under automorphisms, the HN-filtration is stable under all translation symmetries, thus the filtration descends to a HN filtration of the original representation M , which is necessarily trivial by stability of M . Conversely, stable representations of Q project to stable representations of Q. From this, we finally get:
Proposition 7.1. If d is Θ-coprime, the set of fixed points of T Q on M st d (Q) admits a description M st d (Q) TQ ≃ b d M st b d ( Q),
the union over all translation classes of dimension vectors d for Q which project to d.
Note that this result is trivial if the quiver Q is a tree, but it yields something non-trivial in case of generalized Kronecker quivers, in case Q has oriented cycles, etc.. For example, in the case of generalized Kronecker quivers, moduli of bipartite quivers appear, as in the following section.
Localization for generalized Kronecker quivers.
Consider the threearrow Kronecker quiver K 3 . Up to translation, we can assume the support of d to be contained in the connected component C of K 3 containing the vertex (i, 0). This component has the form of a hexagonal lattice:
. . .
• • ↑ ↑ • • ւ ց ւ ց • • • ↑ ↑ ↑ • (i, 0) • ւ ց ւ ց ւ ց • • • • ↑ ↑ • • ւ ց ւ ց • • • . . .
As a particular example, we consider the dimension vector d = (2, 3) for the m-arrow Kronecker quiver. In this case, it is easy to work out the stable representations of the covering quiver whose dimension vectors project to d. Namely, we find indecomposable representations supported on a subquiver of type A 5 with alternating orientation, and indecomposable representations supported on a subquiver of type D 4 , in both cases corresponding to the maximal roots of the respective Dynkin types. Additionally, we have to chose a labelling of the arrows, considered up to natural symmetry. Thus, we arrive at the following dimension vectors for the covering quiver: 1
i → 1 j ↓ 1 k → 1 l ↓ 1 1 i ↑ 2 j → 1 k ↓ 1
In the first case, the indices i, j, k, l ∈ {1, . . . , m} fulfill i = j = k = l, and they are considered up to the symmetry (i, j, k, l) ↔ (l, k, j, i). In the second case, the indices i, j, k ∈ {1, . . . , m} are pairwise different, and they are considered up to the natural S 3 -action. We conclude that the fixed point set consists of
m(m − 1) 3 2 + m(m − 1)(m − 2) 6 = m(m − 1)(3m 2 − 5m + 1) 6
isolated fixed points, so this number is precisely the Euler characteristic of the moduli space M (2,3) (K m ). This can also be obtained directly from Theorem 6.3, but the advantage here is that we get a positive formula a priori (see the discussion following Theorem 6.3).
For more general coprime dimension vectors d, this approach is the central tool in [68] for obtaining exponential lower bounds for χ(M st d (K m )), by constructing "enough" stable representations for the covering quiver.
Cell decompositions and the Bialynicki-Birula method.
Suppose the rank 1 torus C * acts on a smooth projective variety X. The following is proved in [8,9]: Theorem 7.2. Let C be a connected component of the fixed point set X C * , and let A(C) be the set of all x ∈ X such that lim t→0 xt ∈ C. Then both C and A(C) are smooth, and A(C) is locally closed in X. Associating to x ∈ A(C) the limit lim t→0 xt defines a morphism π : A(C) → C, which turns A(C) into a Zariski locally trivial affine bundle. There exists a descending filtration by closed subvarieties X = X 0 ⊃ X 1 ⊃ . . . ⊃ X t = ∅ such that the successive complements X i−1 \ X i are precisely the A(C).
In particular, in the case where X C * is finite, the sets A(x) for x ∈ X C * yield a cell decomposition of X!
We have seen in section 7.2 that there are finitely many fixed points of T Km acting on the moduli space M st (2,3) (K m ). Choosing a sufficiently general embedding of C * into T Km , these are precisely the C * -fixed points, and the above theorem proves that M st (2,3) (K m ) admits a cell decomposition. See [38] for details. In the particular case m = 3, the resulting cell decomposition has the following form: Proposition 7.3. If X is a stable representation of the 3-arrow Kronecker quiver of dimension vector (2,3), then X is isomorphic to exactly one of the following triples of 3 × 2-matrices ( * indicating an arbitrary entry):
1 1 1 1 or 1 1 * 1 1 or 1 * * 1 1 1 or 1 1 * * 1 1 or 1 1 * 1 * 1 or 1 1 * * * 1 1 or * * 1 * 1 1 1 or 1 * * 1 1 * 1 or 1 * * 1 * * 1 1 or * 1 * * * 1 1 1 or * 1 * * 1 * 1 1 or * * * * * 1 1 1 1 or * 1 * * * 1 * * 1 1
Arithmetic approach
8.1. Cell decompositions and counting points over finite fields. In the same spirit as the study of Betti numbers, counting rational points over finite fields can give predictions for the structure of potential cell decompositions. We present here the basic idea of this approach (without reference to schemes or other concepts from arithmetic algebraic geometry).
Suppose X is a quasiprojective variety, embedded as a locally closed subset of a projective space P N (C). Thus X is given by certain polynomial equalities and inequalities in the coordinates, i.e. for certain homogeneous polynomials P i , Q j . Suppose the coefficients appearing in these defining polynomials are all contained in some ring of algebraic numbers R (thus a finite extension of Z). For any prime p ⊂ R, the factor R/p is a finite field k. We can then consider the defining conditions of X modulo p and thus get a locally closed subset X(k) of P n (k) (which is a finite set), and we can study its cardinality |X(k)|, and in particular how it depends on k.
Suppose that X is a variety admitting a cell decomposition
X = X 0 ⊃ X 1 ⊃ . . . ⊃ X t = ∅ with X i−1 \ X i ≃ A di ,
such that all steps X i of the filtration are also defined over R. Then
|X(k)| = i (|X i−1 (k)| \ |X i (k)|) = i |(X i−1 \ X i )(k)| = i |A di (k)| = i |k| di .
Thus, in this case, we can define a polynomial with nonnegative integer coefficients
P X (t) = i q di ∈ N[t]
such that P X (|k|) = |X(k)|.
Like in the setting of Betti numbers, we can get a prediction for the nature of a cell decomposition from this counting polynomial. Note also that the property of admitting such a counting polynomial is very special among all varieties. One standard example for varieties without counting polynomial are elliptic curves. Also note the following elementary example: consider a unit circle X, defined by X = {(x, y) : x 2 + y 2 = 1}. It is defined over Z, and using stereographic projection, it is a simple exercise to see that |X(k)| equals |k| or |k| + 1, depending on whether −1 is a square in k or not. On the other hand, the change of variables u = x + iy, v = x − iy transforms X to {(u, v) : uv = 1}, a hyperbola with |X(k)| = |k| − 1 for any finite field k. Thus, the number of points over finite fields, and in particular the property of admitting a counting polynomial, depends on the chosen embedding of X into projective space.
If X admits a cell decomposition, we see from the above that P X (t) also equals i dim H 2i (X, Q)t i . This holds in much bigger generality for the class of so-called (cohomologically) pure varieties. Deligne's solution of the Weil conjectures [15] states in particular that any smooth projective variety is pure. Thus, for a smooth projective variety admitting a counting polynomial, we automatically know the Betti numbers. This is the method with which Theorem 6.3 was obtained.
For general quasi-projective varieties, one can at least compute the Euler characteristic from a counting polynomial as above, see e.g. [56]: Lemma 8.1. Suppose there exists a rational function P X (t) ∈ Q(t) such that P X (|k|) = |X(k)| for almost all reductions k of the ring R over which X is defined. Then P X (t) ∈ Z[t] is actually a polynomial with integer coefficients, and P X (1) = χ c (X), the Euler characteristic of X in singular cohomology with compact support.
We say that a variety X (or, more precisely, a chosen model of X over R) has the polynomial counting property if a polynomial as in the lemma exists.
8.2.
Counting points of quiver moduli over finite fields. For quiver moduli, there is a canonical choice for a model over Z, since we can define quiver representations over the integers, and since we have natural embeddings of quiver moduli into projective spaces by the results of section 5.5. It is proved in [56] that quiver moduli fit into the above discussion nicely: The proof uses Hall algebras in an essential way, see the sketch in section 9.3. In fact, in [56] only the case of M st d (Q) is considered, but the case of M sst d (Q) follows easily using the Luna stratification described in section 3.5.
The corresponding counting polynomials P M st d (Q) (q) and P M sst d (Q) (q) count isomorphism classes of absolutely stable, resp. semistable, representations of Q of dimension vector d (here a representation is called absolutely stable if it remains stable under base extension to an algebraic closure of the finite field k -see Remark 4.5 for a short discussion of scalar extensions of (semi-)stable representations).
To state an explicit formula for these polynomials, we have to introduce some notation. We consider the formal power series ring F = Q(q)[[t i : i ∈ I]] and define monomials t d = i∈I t di i for a dimension vector d ∈ NI. Besides the usual commutative multiplication, we also consider the twisted multiplication
t d • t e = q − d,e t d+e
on F . Denote by ψ k for k ≥ 1 the operator on F defined by ψ k (q) = q k and ψ k (t d ) = t kd .
We combine these operators into
Ψ(f ) := k≥1 1 k ψ k (f ),
which by [11] has an inverse
Ψ −1 (f ) = k≥1 µ(k) k ψ k (f )
involving the number theoretic Moebius function µ(k). Bases on this, one defines mutually inverse operators Exp(f ) = exp(Ψ(f )) and Log(f ) = Ψ −1 (log(f )) on F . Using these concepts, an explicit formula for the counting polynomials is proved in [46]:
Theorem 8.3.
For any Q, Θ, d as above and any µ ∈ Q, we have the following formulas which make key use of the rational functions P d (q) introduced in Definition 6.2:
( d∈NIµ P d (q)t d ) • Exp( 1 1 − q d∈NIµ P M st d (Q) (q)t d ) = 1 and d∈NIµ P M sst d (Q) (q)t d = Exp( d∈NIµ P M st d (Q) (q)t d ).
Let us consider the special case Θ = 0. Then the first formula of the above theorem can be made more explicit:
Corollary 8.4. We have ( d∈NI q − d,d i∈I di k=1 (1 − q −k ) −1 t d ) • Exp( 1 1 − q d∈NI P M simp d (Q) (q)t d ) = 1.
Here are some examples of the counting polynomials in the special case of the m-loop quiver L m , denoting a d (q) = P M simp d (Lm) (q):
a 1 (q) = q m , a 2 (q) = q 2m (q m − 1)(q m−1 − 1) q 2 − 1 , a 3 (q) = q 3m+1 (q m − 1)(q 2m−2 − 1)(q 2m−2 (q m + 1) − q m−2 (q + 1) 2 + q + 1) (q 3 − 1)(q 2 − 1) .
Based on computer experiments for the m-loop quiver and dimensions up to 12, the following conjecture is made in [46]: Even more optimistically, this conjecture suggests that there should be a decomposition into tori of any such moduli space, compare the discussion in section 6.3.
Moduli of simple representations.
It turns out that the first two terms in the Taylor expansion around q = 1 can be computed. The constant term is zero except in case d = e i for some i ∈ I: the action of the torus C * , diagonally embedded into T Q , has no fixed points. We can factor by this action to obtain a projectivization PM simp To state a formula for the linear term, we need some notation. We consider oriented cycles in the quiver Q, written as ω = α s . . . α 1 for arrows α 1 , . . . , α s in Q. We have the notion of cyclic equivalence of cycles as the equivalence relation generated by α 1 . . . α s ∼ α 2 . . . α s α 1 .
We call a cycle ω primitive if it is not cyclically equivalent to a proper power of another cycle, i.e. ω ∼ (ω ′ ) n for all n > 1. A notion of dimension vector dimω of a cycle ω can be defined by setting (dimω) i to equal the number of times ω passes through the vertex i ∈ I. Then the following formula is proved in [57]: For any indivisible element ν ∈ NQ 1 (i.e. gcd(ν α : α ∈ Q 1 ) = 1), define a covering quiver Q ν of Q as follows: the vertices are given by ( Q ν ) 0 = Q 0 × ZQ 1 /Zν, and the arrows are given as
Theorem( Q ν ) 1 = {(α, λ) : (i, λ) → (j, λ + α) : (α : i → j) ∈ Q 1 , λ ∈ ZQ 1 /Zν}.
The group ZQ 1 acts on Q ν by translation, and we consider dimension vectors d for Q ν up to translational equivalence. Each d projects to a dimension vector for Q. Since the Euler characteristic (in singular cohomology with compact support) is invariant under taking torus fixed points, and is additive with respect to disjoint unions, we can apply this theorem repeatedly. It turns out that, after finitely many localizations, the only resulting covering quivers contributing with non-zero Euler characteristic are cyclic quivers with dimension vector equal to 1 at any vertex; they contribute with an Euler characteristic equal to 1 since the projectivized space of simples is just a single point. These covering quivers are in fact parametrized by the cyclic equivalence classes of primitive cycles, yielding the claimed formula.
Although the Euler characteristic is computed in the proof as a purely combinatorial number, it admits various algebraic interpretations, most notably in relation to the Hochschild homology of the path algebra kQ. Namely, the zero-th Hochschild homology HH 0 (kQ) equals kQ/[kQ, kQ], the path algebra modulo additive commutators, an object which plays a central role in some approaches to noncommutative algebraic geometry (see e.g. [26]). Now HH 0 (kQ) inherits a NI-grading from kQ, and the degree d-part has a basis consisting of cyclic equivalence classes of cycles of dimension vector d. A similar result holds for HH 1 (kQ) (all other Hochschild homology being zero).
The Euler characteristic also admits a representation-theoretic interpretation: suppose a primitive cycle
ω : i 0 α1 → i 1 α2 → . . . αs → i s = i 0 of dimension vector d in Q is given. For each vertex i ∈ Q 0 , define K i = {k = 0, . . . , s − 1 : i k = i}.
Consider the d i -dimensional vector space M i with basis elements b k for k ∈ K i . For each arrow (α : i → j) ∈ Q 1 and each k ∈ K i , define
M (ω) α (b k ) = b k+1 , α = α k+1 , 0 , otherwise.
This defines a representation M (ω). It is easy to see that M (ω) ≃ M (ω ′ ) if and only if ω and ω ′ are cyclically equivalent, and that M (ω) is simple if and only if ω is primitive. One can hope that these representations enter as the 0-dimensional strata of a conjectural decomposition of PM simp d (Q) into tori, see the end of section 8.2.
It would be very interesting to have a description of all the individual Betti numbers in singular cohomology with compact support of PM simp d (Q), and to see how they relate to e.g. Hochschild homology. But such a description cannot be obtained in an obvious way via localization, since the projectivized space of simples is not projective. Again (compare the discussion in section 6.3), a smooth compactification is missing.
The knowledge so far about the counting polynomials P M simp d (Q) (q) suggests to view them in analogy to the polynomials i d (q) counting isomorphism classes of absolutely indecomposable representations of a quiver without oriented cycles, which are the basis for the Kac conjectures [34]. These state that i d (0) equals the multiplicity of the root d in the root system associated to Q (compare section 2.3), and that i d (q) ∈ N[q]. The first conjecture is proved in the case of indivisible d in [14]; a proof for arbitrary d is announced in [30].
Compare this to the results and conjectures above: we have a known number
P M simp d (Q) (q) q−1
| q=1 with combinatorial and algebraic interpretations, and we conjec-
ture that P M simp d (Q) (q) ∈ N[q − 1].
This suggests some deep similarities between the counting of indecomposables and the counting of simples.
It would also be very interesting to have a better understanding of the Euler characteristics χ c (M st d (Q)) for non-trivial Θ and non-coprime d. In principle, the above formulas allow to determine this number by evaluation of the counting polynomials at q = 1, but again, more explicit (combinatorial) formulas are desirable.
The role of Hall algebras
Several of the theorems above on Betti numbers and numbers of points over finite fields of quiver moduli can be proved using calculations in the Hall algebra of a quiver as introduced in [59]. where F X M,N denotes the number of subrepresentations U of X which are isomorphic to N , with quotient X/U isomorphic to M . This number is obviously finite. Also note that the sum in the definition of the multiplication is finite, since F X M,N = 0 implies dimX = dimM + dimN , and there are only finitely many (isomorphism classes of) representations of fixed dimension vector.
The above multiplication defines an associative Q-algebra structure on H q (Q) with unit 1 = [0]. This algebra is naturally NI-graded by the dimension vector.
We will also consider a completed (with respect to the maximal ideal spanned by non-zero representations) version of the Hall algebra, thus
H q ((Q)) = [M] Q[M ],
with the same multiplication as before. This version has the advantage that certain "generating series", like e.g. [M] [M ], can be considered in it.
9.2. Hall algebras and quantum groups. The Hall algebra is usually considered in relation to quantum groups: let Q be a quiver without oriented cycles, and define C q (Q) (the composition algebra) as the subalgebra of H q (Q) generated by the basis elements [S i ] corresponding to the simple representations S i for i ∈ I. Let C be the matrix representing the symmetric bilinear form (, ), to which we can associate a Kac-Moody algebra g, see [35]. Its enveloping algebra U(g) admits a quantum deformation, the quantized enveloping algebra U q (g) [32,45]. We will only consider its positive part U + q (g) (induced from the triangular decomposition of the Lie algebra g), which can be defined as the Q(q)-algebra with (Chevalley) generators E i for i ∈ I and defining relations (the q-Serre relations)
k+l=1−cij 1 − c ij k E k i E j E l i = 0 for all i = j in I.
There is a twisted version of the Hall algebra, which we denote by H q (Q) tw ; it is defined in the same way as H q (Q), but the multiplication is twisted by a power of q, namely
[M ] · [N ] = q − dimM,dimN [X] F X M,N [X].
The following is proved in [27]:
Theorem 9.1. The composition subalgebra C q (Q) tw of H q (Q) tw is isomorphic to the specialization of U + q (g) at q = |k|. We will now consider some special elements in H q ((Q)) and their evaluations.
Consider e d = dimM=d [M ],
the sum over all isomorphim classes of representations of dimension vector d.
We have
e d = dimM=d 1 Aut(M ) t d = dimM=d |G d M | |G d | t d = |R d (Q)| |G d | t d ,e sst d = |R sst d (Q)| |G d | .
By the results of section 4 on the Harder-Narasimhan filtration, every representation M admits a unique Harder-Narasimhan filtration
0 = M 0 ⊂ M 1 ⊂ . . . ⊂ M s = M.
Let d i be the dimension vector of the subquotient M i /M i−1 , for i = 1 . . . s. All the subquotients being semistable, and the HN-filtration being unique, we see that [M ] appears with coefficient equal to 1 in the product e sst ds · . . . · e sst d2 · e sst d1 .
Existence of the HN filtration yields the following identity: We can thus determine any e sst d inductively, the induction starting at dimension vectors for which every representation is semisimple (this is equivalent to Θ being constant on the support of d). Applying the evaluation map , this gives Corollary 9.4. We have
Lemma|R sst d (Q)| |G d | = |R d (Q)| |G d | − * q − P k<l d l ,d k s k=1 |R sst d k (Q)| |G d k | ,
the sum running over all decompositions as above of length s ≥ 2.
General arithmetic considerations prove that, in case d is Θ-coprime, we have
|M sst d (Q)| = (q − 1) |R sst d (Q)| |G d | (essentially since M sst d (Q)
is the quotient of R sst d (Q) by the group P G d , which acts freely in the coprime case). This gives an explicit (recursive) formula for the number of rational points of M sst d (Q). The result Theorem 6.3 is obtained from this by applying Deligne's solution of the Weil conjectures (see section 8.1) to pass from points over finite fields to Betti numbers, and by giving an explicit resolution of the recursion.
As a second example of the use of Hall algebras, used in [56], consider the element e µ = d∈NIµ e sst d ∈ H q ((Q)). What is surprising about this lemma is that the inverse is a sum over polystable representations only. Applying the evaluation map, we get Corollary 9.6. We have
d∈NIµ |R sst d (Q)| |G d | = m:S→N [S] (−1) mS |End(S)| ( m S 2 ) |Aut(S mS )| t P S mS dimS ,
where the sum runs over all maps (with finite support) from the set S of isomorphism classes of stable representations of slope µ to the set of nonnegative integers.
This identity forms the basis for the proof of Theorems 8.2 and 8.3: roughly, we can single out the summands corresponding to stable representations (i.e. the function m has precisely one non-zero value, equal to 1) to get a recursive formula for the number of isomorphism classes of stables in terms of the rational functions P d (q). Some arithmetic considerations allow passage to absolutely stable representations. Using Lemma 8.1, the theorem follows.
As a third application, we prove the cohomology formula Theorem 10.2 for the smooth models of section 10 using Hall algebras. For some n ∈ NI, consider the finitely generated projective representation P (n) = i∈I P ni i . Besides the element e µ ∈ H q ((Q)), consider the following elements:
Smooth models and Hilbert schemes
In this final section, we will consider a variant of quiver moduli (in some respect analogous to the quiver varieties of [49]) which enjoys several of the desirable properties which the original moduli lack in general: they admit universal bundles, they are always smooth and projective (over the moduli of semisimple representations), their Betti numbers can be calculated and the Poincare polynomial equals the counting polynomial for points over finite fields. In special cases we can even construct a cell decomposition and thus give a normal form. The drawback is that these moduli do not parametrize just isomorphism classes of representations, but equivalence classes of representations together with an additional structure. The material of this section is contained in [19].
10.1. Definition of smooth models. Let Q, d, Θ be a quiver, a dimension vector and a stability as before. Choose another dimension vector d ∈ NI and consider extended quiver data Q, d, Θ defined as follows: the vertices of Q are given by Q 0 = Q 0 ∪ {∞}, and the arrows of Q are those of Q, together with n i arrows from the additional vertex ∞ to any i ∈ I. We extend d to a dimension vector d for Q by defining d ∞ = 1, and we define a stability Θ for Q by setting Θ i = N Θ i for i ∈ I and Θ ∞ = Θ(d) + 1, for some sufficiently large integer N ∈ N.
It is now easy to see that (because of the additional entry 1 in d) the dimension vector d is always Θ-coprime, so that the resulting moduli space of (semi-)stable representation M In other words, this moduli space, which we call a smooth model (for M sst d (Q)), parametrizes semistable representations M together with a map from a fixed projective to M which "avoids all subrepresentations contradicting stability of M ".
The map forgetting the extra datum of the map f induces a projective morphism π : M Θ d,n (Q) → M sst d (Q). The fibres of this map can be described explicitely using the Luna stratification of section 3.5. In particular, the generic fibre -the fibre over the stable locus M st d (Q) -is isomorphic to projective space P n·d−1 , where n · d = i∈I n i d i .
In the case where d is Θ-coprime, the smooth model stays very close to the original moduli space: it is isomorphic to the projectivization of the bundle i∈I V ni i . In all other cases, the smooth models M Θ d,i (Q) can therefore be viewed as "projectivizations of non-existing universal bundles".
10.2. Cohomology of smooth models. Since the smooth models are a particular case of moduli in the coprime case, we know from the discussion in section 6.5 that the odd Betti numbers vanish, and that the even Betti numbers can be computed using Theorem 6.3. We will make this formula more explicit, using again the rational functions P d (q) of Definition 6.2. As noted in section 9.3, the result uses the identity of Lemma 9.7 in the Hall algebra of Q and the passage from counting points over finite fields to Betti numbers, as in section 8.1.
Hilbert schemes.
We consider the special case Θ = 0 and denote the smooth model M 0 d,n (Q) by Hilb d,n (Q), which we call a Hilbert scheme for Q. It parametrizes (arbitrary) representations M of dimension vector d, together with a surjective map from P (n) to M . In other words, it parametrizes subrepresentations U ⊂ P (n) of finitely generated projective representations such that dimP (n) /U = d.
In this case, there are much more explicit results on the structure of that space. For example, in the case of quivers without oriented cycles, Hilb d,n (Q) can be described as an iterated Grassmann bundle, see [54].
For general quivers, we can give an explicit (and non-recursive) criterion for nonemptyness of Hilb d,n (Q) (compare the discussion of non-emptyness in section 5. (1) n i ≥ d, i for all i ∈ I, (2) for every i ∈ supp(d) there exists j ∈ supp(n) and a path from j to i in supp(d).
We also have a positive combinatorial formula for the Betti numbers (compare the discussion following Theorem 6.3), based on certain multipartitions: Let Λ d be the set of tuples (λ i = (λ i 1 ≥ λ i 2 ≥ . . . ≥ λ i di ≥ 0)) i∈I of partitions of length d i , which we call multipartitions of length d. The weight of such a multipartition λ is defined as
|λ| = i∈I di k=1 λ i k .
We define a subset S d,n consisting of multipartitions λ of length d as fulfilling the following condition:
for every 0 ≤ e < d, there exists a vertex i ∈ I such that λ i di−ei < n i − e, i . 10.4. Cell decompositions for Hilbert schemes. In fact, all Hilb d,n (Q) admit cell decompositions, which we will now construct (the special case of the multiple loop quivers was considered before in [55]). We need some combinatorial notation.
For each vertex i ∈ I, define a covering quiver (in fact, a tree) Q i of Q as follows: the vertices of Q i are parametrized by the paths ω on Q starting in i. The arrows in Q i are given by α : ω → (αω) for arrows (α : j → k) ∈ Q 1 and paths ω in Q starting in i and ending in j. Obviously Q i has a unique source corresponding to the empty path at i. We have a natural projection from Q i to Q associating to a vertex ω of Q i the terminal vertex of the path ω. A full subquiver T of Q i is called a tree if it is closed under predecessors. To such a tree T , we can associate a dimension vector d(T ) for Q, where d(T ) j is defined as the number of vertices in T whose corresponding paths ω have terminal vertex j.
For n ∈ NI, we define Q n as the disjoint union of n i copies of each Q i , for i ∈ I. The vertices of Q n are labelled by triples (i, j, ω), which means that the path ω starting in i is placed in the j-th copy of T i . An n-forest T * is a full subquiver of Q n which is closed under predecessors; in other words, it is a tuple (T ij ) i∈I, k=1,...,ni of trees T ij in Q i . The dimension vector d(T * ) of an n-forest T * is defined as i,j d(T ij ). Theorem 10.5. Each Hilb d,n (Q) admits a cell decomposition, whose cells are parametrized by n-forests of dimension vector d.
To construct the cells, we need a total ordering on n-forests. First, choose an arbitary total ordering on I. For each pair of vertices i, j ∈ I, choose a total ordering on the arrows from i to j. Then, define a total ordering on Q 1 as follows: (α : i → j) ≤ (β : k → l) if one of the following conditions holds:
• i = k, j = l and α ≤ β in the ordering chosen on the arrows from i to j,
• i = k and j < l,
• i < k. Now we define a total ordering on the vertices of Q 1 : let ω = (α s . . . α 1 ) and ω ′ = (β t . . . β 1 ) be two paths in Q starting in i. Then ω ≤ ω ′ if α k < β k for the minimal index k such that α k = β k ; if no such k exists, we set ω ≤ ω ′ if s ≤ t.
Finally, we define a total ordering on the vertices of Q n : we define (i, j, ω) ≤ (i ′ , j ′ , ω ′ ) if one of the following conditions holds:
• i < i ′ in the total ordering on I,
• i = i ′ and j < j ′ ,
• i = i ′ , j = j ′ and ω ≤ ω ′ in the total ordering on vertices of Q i .
Define the corona C(T * ) of T * as the set of all vertices (i, j, ω) of Q n which are not elements of T * , but whose (unique) immediate predeccessor in Q n is an element of T * .
Choose a basis v ij for each vector space V i . For a representation M and a path ω = α s . . . α 1 in Q, write M ω = M αs • . . . • M α1 . Given an n-forest T * of dimension vector d, let Z T * be the set of all points (M, f ) such that the following conditions hold:
• the collection of elements b (i,j,ω) = M ω (f i (v i,j )) for (i, j, ω) ∈ T * (i.e. ω ∈ T i,j for all i ∈ I and j = 1, . . . , n i ) forms a basis of M = i∈I M i ,
• for each (i, j, ω) ∈ C(T * ), the element M ω (f i (v i,j )) belongs to the span of all suitable b (i ′ ,j ′ ,ω ′ ) for (i ′ , j ′ , ω ′ ) ∈ T * such that (i ′ , j ′ , ω ′ ) < (i, j, ω).
Theorem 10.6. There exists a filtration Hilb d,n (Q) = X 0 ⊃ X 1 ⊃ . . . ⊃ X t = ∅ such that the successive complements X q−1 \ X q are precisely the sets Z T * defined above, where T * runs over all n-forests of dimension vector d. Consequently, Hilb d,n (Q) admits a cell decomposition, whose cells are parametrized by the nforests of dimension vector d.
As an immediate corollary, we can describe χ(Hilb d,n (Q)) as the number of nforests of dimension vector d. Combinatorial considerations allow to describe the generating functions of Euler characteristics as solutions to algebraic equations in the formal power series ring Q[[I]] as follows:
Corollary 10.7. For n ∈ NI, define F n (t) = d∈NI χ c (Hilb d,n (Q))t d .
Then we have F n (t) = i∈I F i (t) ni and F i (t) = 1 + t i · α:i→j F j (t) for all i ∈ I.
One might conjecture that the generating functions d∈NIµ χ c (M Θ d,n (Q))t d ∈ Q[[I]] for arbitrary smooth models are always algebraic.
d, e ∈ ZI. We denote by dim d = i∈I d i the total dimension of d. Standard examples of quivers in this paper will be: • the m-loop quiver L m with I = {i} and Q 1 = {(α 1 , . . . , α m : i → i)}, • the m-arrow generalized Kronecker quiver K m with I = {i, j} and Q 1 = {(α 1 , . . . , α m : i → j)}, • the m-subspace quiver S m with I = {i 1 , . . . , i m , j} and Q 1 = {(α k : i k → j) : k = 1, . . . , m}.
Problem 2. 1 .
1Classify the representations in mod kQ up to isomorphism, and give normal forms for the representations.
(d, e) = d, e − e, d the symmetrization of the Euler form of Q. On ZI, we define reflections s i for i ∈ I by s i (d) = d − (d, i)i, and we define the Weyl group W (Q) as the subgroup of GL(ZI) generated by the s i for i ∈ I. The fundamental domain F (Q) is defined as the set of all non-zero dimension vectors 0 = d ∈ NI with connected support (i.e. the full subquiver with set of vertices supp(d) = {i ∈ I, d i = 0} is connected), such that (d, i) ≤ 0 for all i ∈ I. The set of real roots ∆ re (Q) = W (Q)I ⊂ ZI is defined as the set of all Weyl group translates of coordinate vectors, and the set of (positive) imaginary roots ∆ im (Q) = W (Q)F (Q) is defined as the set of all Weyl group translates of elements of the fundamental domain. Finally, we define ∆(Q) as the union of both sets of roots.Theorem 2.2. There exists an indecomposable representation of Q of dimension vector d ∈ NI if and only if d is a root, i.e. d ∈ ∆(Q). In case d ∈ ∆ re (Q), there exists a unique indecomposable (up to isomorphism) of dimension vector d. In case d ∈ ∆ im (Q), the number of parameters of the set of indecomposable representations is 1 − d, d .
Lemma 2. 3 .
3Almost all representations of K 3 of dimension vector d (i.e. a Zariski-open set in the space M n (k) 3 of all representations) are isomorphic to one of the representations M (λ * , a * * ).
M
(1, t) = M (1, 0) in U.
Theorem 3.3. V st is an open (but possibly empty) subset of V , and the restriction π : V st → π(V st ) =: V st /G
χ n the subring of semi-invariants for all powers of the character χ. This is naturally an N-graded subring of k[V ] with k[V ] G as the subring of degree 0 elements. An element v ∈ V is called χ-semistable if there exists a function f ∈ k[V ] G,χ n for some n ≥ 1 such that f (v) = 0. Denote by V χ−sst the (open) subset of χsemistable points. An element v is called χ-stable if the following conditions are satisfied: v is χ-semistable, its orbit Gv is closed in V χ−sst , and its stabilizer in G is zero-dimensional. Denote by V χ−st the (again open) subset of stable points.
It will be denoted by M ssimp d (Q) and called the moduli space of semisimple representations. The stabilizer of M in G d is nothing else than the automorphism group Aut(M ), thus its stabilizer in P G d is zero-dimensional (in fact trivial) if and only if M is a Schur representation. Thus, we see that the open subset R d (Q) st consists precisely of the simple representations of dimension vector d, and that the quotient variety R d (Q) st (Q)/P G d = M simp d (Q) is a moduli space for (isomorphism classes of) simple representations.
Corollary 3 . 7 .
37The variety M Θ−sst d (Q) parametrizes isomorphism classes of Θpolystable representations of Q of dimension vector d.
Lemma 4 . 10 .
410For each a ∈ Q, the pair (T a , F a ) defines a torsion pair in mod kQ. For a < b, we have T a ⊃ T b and F a ⊂ F b .
Lemma 5. 1 .
1The representation M (λ, µ) for (λ, µ) = 0 is semistable if and only if it is stable if and only if λ = 0. Proof. The representation M (0, 1) admits a subrepresentation of dimension vector (1, 1), contradicting (semi-)stability. If λ = 0, the only possible dimension vectors of subrepresentations of M (λ, µ) are easily worked
Theorem 5.3. e ֒→ d if and only if e ′ , d − e ≥ 0 for all e ′ ֒→ e.Based on this notion, we have the following criterion from[61]:Theorem 5.4. We have (1) M sst d (Q) = ∅ if and only if µ(e) ≤ µ(d)for all e ֒→ d,(2) M st d (Q) = ∅ if and only if µ(e) < µ(d) for all e = d such that 0 = e ֒→ d.A "less recursive" criterion is formulated in[1].Theorem 5.5. Assume d = s k=1 m k d k can be written as a positive combination of dimension vectors d k such that M st d k (Q) = ∅ for all k. Then M st d (Q) = ∅ if and only if (m 1 , . . . , m s ) is the dimension vector of a simple representation of the quiver with vertices i 1 , . . . , i s and δ k,l − d k , d l arrows between each pair of vertices i k , i l . This reduces the problem to the question of existence of simple representations, which is solved in [43]: Theorem 5.6. We have M simp d (Q) = ∅ if and only if supp(d) is a quiver of type A n with cyclic orientation and d i = 1 for all i ∈ supp(d), or supp(d) is not of the above type and d, i ≤ 0 ≥ i, d for all i ∈ supp(d).
consider the fibres π −1 i (M ) for some point M ∈ M st d (Q). Then the representation of Q induced on the vector spaces π −1 i (M ) by the maps V α is isomorphic to M . The idea behind the construction of this universal representation (see[37]) is quite obvious: we consider the trivial vector bundles R st d (Q) × M i → M i and the vector bundle maps (M, m i ) → (M, M α (m i )) for α : i → j. These become G d -bundles via the standard action of G d on each M i . We want these bundles to descend to bundles on the quotients M std (Q) = R st d (Q)/G d .This works only if the induced action of the stabilizer of a point on the fibres of the bundle is trivial, which is not true: consider the action of scalar matrices in G d , which are the stabilizers of stable representations. The way out of this problem is to twist the G d -action on R st d (Q) × M i by a character χ, which necessarily has to take the value λ −1 on a scalar matrix λ ∈ G d . Such a character exists if and only if the dimension vector d is coprime in the sense that gcd(d i : i ∈ I) = 1: we can then choose a tuple s i of integers such that i∈I s i d i = 1 and define χ((g i
we consider the function tr ω assigning to a representation M = (M α ) α the trace tr(M αu · . . . · M α1 ). Taking enough traces along oriented cycles, we thus get an embedding of M (s)simp d (Q) into an affine space.
case dimM, dimN = dim Hom(M, N ) − dim Ext 1 (M, N ) = 0, we thus have a map d M,N between vector spaces of the same dimension, and we can consider its determinant c(M, N ) = det d M,N . Varying M and N in their respective spaces of representations, we get a polynomial function
(
P i denoting the projective indecomposables associated to the vertex i ∈ I), such that i∈I (a i − b i )d i = 0. Then the induced map Hom(v, M ) : Hom(Q, M ) → Hom(P, M ) is a map between vector spaces of the same dimension, and again we can consider its determinant c v (M ) = det Hom(v, M ). This defines a semi-invariant function on R d (Q). Again by [63], we have: Theorem 5.9. The ring of semi-invariant functions on R d (Q) is generated by the functions c v .
d χΘ if and only if for any i = 1, . . . , n, we have j<i m ji + j>i m ij = 2k
5 ) is the surface in P 5 with coordinates (x 1 : x 2 : x 3 : x 4 : x 5 : c) determined by the five equations above.
6. 2 .
2The importance of cell decompositions for quiver moduli. Now assume that M st d (Q) is a quiver moduli admitting a universal representation V, and assume that M st d (Q) admits a cell decomposition. Then we can write
action naturally commutes with the (left) G d -action on R d . The action fixes (semi-)stable representations, since it does not change the possible dimension vectors of subrepresentations. Thus, the T Q -action on R sst d (Q) descends to an action on M (s)st d (Q). We will now derive a description of the fixed point set M st d (Q) TQ in the case where d is coprime for Θ. Let M = (M α ) α be a T Q -fixed point in M d (Q). Thus M is a stable representation, and in particular it has trivial endomorphism ring. Consider the group G = {(g, t) ∈ P G d × T Q : gM = M t}.
X
= {(x 0 : . . . : x N ) : P i (x 0 , . . . , x N ) = 0, Q j (x 0 , . . . , x N ) = 0}
Theorem 8. 2 .
2For arbitrary quivers Q, stabilities Θ and dimension vectors d, both M st d (Q) and M sst d (Q) have the polynomial counting property.
Conjecture 8 . 5 .
85When written as a polynomial in the variable q − 1, all polynomials P M simp d (Q) (q) have nonnegative coefficients, i.e. P M simp d (Q) (q) ∈ N[q − 1].
) (but note that PM simp d (Q) is not a projective variety since M simp d (Q) is not necessarily affine).
8. 6 .
6The Euler characteristic (in cohomology with compact support) of PM simp d (Q), or, equivalently, the constant term in the Taylor expansion of P M simp d (Q) (q) around q = 1, is given as the number of cyclic equivalence classes of primitive cycles in Q of dimension vector d.The proof of this theorem works via the localization principle, in a refined form so that it applies to the action of T Q on PM simp d (Q). The additional difficulty is that P G d does not act freely on the projectivization of the subset R simp d (Q) of R d (Q) consisting of the simple representation, so that the argument of section 7.1 does not apply. Instead, this refined form of localization leads to the following description of the fixed point components:
Theorem 8 . 7 .
87The fixed point set PM simp d (Q) TQ is isomorphic to the disjoint union of moduli spaces of the same type PM simp b d ( Q ν ), where ν ranges over the indivisible elements of ZQ 1 , and d ranges over the equivalence classes of dimension vectors for Q ν projecting to d.
9. 1 .
1Definition of Hall algebras. Let k be a finite field with q elements. Let H q (Q) be a Q-vector space with basis elements [M ] indexed by the isomorphism classes of k-representations of Q. Define a multiplication on H q (Q) by [M ] · [N ] := [X]
γ
M [M ], where γ M is zero if M is not polystable, and γ M = [S] (−1) mS |End(S)| ( m S 2 )if M = S S mS as a direct sum of stable representations of slope µ.
0 (P (n) , M )|[M ], where Hom 0 (Z, M ) denotes the set of all maps f : Z → M with the following property: if f (Z) ⊂ U ⊂ M for U ∈ mod µ (Q), then U = M . Then the following identity holds, see [19]: Lemma 9.7. We have e µ h µ,n = e µ,n in H q ((Q)). Application of the evaluation map immediately yields the formula of Theorem 10.2.
e d ( Q) is smooth. We denote this moduli space by M Θ d,n (Q). Theorem 10.1. The moduli space M Θ d,n (Q) is smooth, and projective over the moduli of semisimple M ssimp d (Q). It parametrizes pairs consisting of a semistable representation M of Q of dimension vector d, together with a map f : P (n) → M from the finitely generated projective representation P (n) = i∈I P ni i to M with the property: if U is a proper subrepresentation of M containing Im(f ), then µ(U ) < µ(M ). These pairs are parametrized up to isomorphisms respecting the additional data, i.e. (M, f ) and (M ′ , f ′ ) are equivalent if there exists an isomorphism ϕ : M → M ′ such that f ′ = ϕ • f .
Theorem 10. 2 .
2In the skew formal power series ring Q q [[I]], we have the following identity:d∈NIµ i dim H i ((M Θ d,n (Q), Q)q i/2 t d = ( d∈NIµ P d (q)t d ) −1 · ( d∈NIµ q n·d P d (q)t d ).
.
Let Q be an arbitrary quiver, and let d and n be dimension vectors. We have Hilb d,n (Q) = ∅ if and only if the following two conditions are fulfilled:
Theorem 10. 4 .
4We have the following formula for the Betti numbers of the Hilbert scheme Hilb d,n (Q):i dim H i (Hilb d,n (Q), Q)q i/2 = q n·d− d,d λ∈S d,n q −|λ| .
9.3. Applications of Hall algebras to quiver moduli. The Hall algebra admits an evaluation homomorphism to a skew polynomial ring: as in section 8.2, consider the ring Q q [I] which has basis elements t d for d ∈ NI and multiplicationt d · t e = q − d,e t d+e .We have a natural skew formal power series version Q q [[I]] of Q q [I]. Then we can define the evaluation morphism as in [56]: Lemma 9.2. The map sending [M ] to |Aut(M)| · t dimM induces Q-algebra morphisms : H q (Q) → Q q [I] and : H q ((Q)) → Q q [[I]], respectively.1
since the cardinality of the orbit G d M of M in R d equals the order of the group G d , divided by the order of the stabilizer, which by definition equals the automorphism group of M . the sum over all isomorphism classes of semistable representations of dimension vector d. Similarly to the above, we haveNext, consider
e sst
d =
dimM=d
M semistable
[M ],
9.3. We have e d = * e sst ds · . . . e sst d1 , the sum running over all decompositions d 1 + . . . + d s = d of d into non-zero dimension vectors such that µ(d 1 ) > . . . > µ(d s ).
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Markus Reineke, C -Mathematik Fachbereich, D -42097 Wuppertal. Germany E-mail20Bergische Universität WuppertalMarkus Reineke, Fachbereich C -Mathematik, Bergische Universität Wuppertal, Gaußstr. 20, D -42097 Wuppertal, Germany E-mail: [email protected]
| []
|
[
"The Decentralized Structure of Collective Attention on the Web",
"The Decentralized Structure of Collective Attention on the Web"
]
| [
"Lingfei Wu \nDepartment of Media and Communication\nCity University of Hong Kong\nHong KongChina\n",
"Jiang Zhang \nDepartment of Systems Science\nSchool of Management\nBeijing Normal University\nBeijingChina\n"
]
| [
"Department of Media and Communication\nCity University of Hong Kong\nHong KongChina",
"Department of Systems Science\nSchool of Management\nBeijing Normal University\nBeijingChina"
]
| []
| Background : The collective browsing behavior of users gives rise to a flow network transporting attention between websites. By analyzing the structure of this network we uncovered a non-trivial scaling regularity concerning the impact of websites.Methodology: We constructed three clickstreams networks, whose nodes were websites and edges were formed by the users switching between sites. We developed an indicator C i as a measure of the impact of site i and investigated its correlation with the traffic of the site (A i ) both on the three networks and across the language communities within the networks.Conclusions: We found that the impact of websites increased slower than their traffic. Specifically, there existed a relationship C i ∼ A i γ (γ < 1). We suggested that this scaling relationship characterized the decentralized structure of the clickstream circulation: the World Wide Web is a system that favors small sites in reassigning the collective attention of users. | null | [
"https://arxiv.org/pdf/1110.6097v2.pdf"
]
| 14,933,688 | 1110.6097 | 8873da3faa8f3c1b517a8ffdd3c311af3d7de8d8 |
The Decentralized Structure of Collective Attention on the Web
6 Nov 2012
Lingfei Wu
Department of Media and Communication
City University of Hong Kong
Hong KongChina
Jiang Zhang
Department of Systems Science
School of Management
Beijing Normal University
BeijingChina
The Decentralized Structure of Collective Attention on the Web
6 Nov 20121
Background : The collective browsing behavior of users gives rise to a flow network transporting attention between websites. By analyzing the structure of this network we uncovered a non-trivial scaling regularity concerning the impact of websites.Methodology: We constructed three clickstreams networks, whose nodes were websites and edges were formed by the users switching between sites. We developed an indicator C i as a measure of the impact of site i and investigated its correlation with the traffic of the site (A i ) both on the three networks and across the language communities within the networks.Conclusions: We found that the impact of websites increased slower than their traffic. Specifically, there existed a relationship C i ∼ A i γ (γ < 1). We suggested that this scaling relationship characterized the decentralized structure of the clickstream circulation: the World Wide Web is a system that favors small sites in reassigning the collective attention of users.
Introduction
The explosive growth of the World Wide Web in the past two decades presents an urgent challenge for developing a quantitative, predictive theory of the interaction between the Web and users. Previous studies analyzed the structure of the hyperlink network [1][2][3][4][5] and also the individual browsing records [6][7][8][9] in order to investigate the collective surfing behavior. While these studies paved way for the following research, they have limitations restricted by the data. Firstly, as pointed out in [10], hyperlinks are too simple to represent the rich interactions between sites as a result of users' various online activities. From bookmarks and default home pages to historical viewing records, there are many different ways in which clickstreams are generated between sites of no hyperlink connections [10]. Secondly, although individual surfing records has been extensively investigated [7][8][9]11], there is still a lack of research studying collective browsing behavior from a network perspective [12].
There are generally two different opinions concerning the surfing dynamics. One is the "rich-get-richer" paradigm, which suggests that user navigation strengthens the inequality of traffic among sites [13][14][15]. The other is the "egalitarian" paradigm arguing that the surfing activities of users actually makes the Web a level-play place where new sites have a greater chance of acquiring popularity [16]. In the current study we investigated clickstreams formed by a large number of users to examine these two effects. We collected data from Google (www.google.com) and Alexa (www.alexa.com) and constructed three website-level clickstream networks at different time points [12,16,17]. In each of the networks, the nodes were websites and the edges showed the daily percentage of global users who visited two websites successively. We defined A i as the traffic of site i and C i the impact of the site on the rest of sites in clickstream circulation. If C i increases faster than A i , the "rich-get-richer" effect is supported; otherwise, the "egalitarian" effect is supported. It turned out that C i scales sublinearly with A i as C i ∼ A i γ (γ < 1). This scaling pattern was observed to be universal, existing both in the three clickstream networks and across the language communities within the networks. We suggested that this pattern, as an evidence of the "egalitarian" effect, resulted from the decentralized structure of clickstream networks. That is, compared to large sites, small sites had a disproportionately larger impact in the circulation of clickstreams.
We would like to point out that the presented approach of clickstream network analysis is not only interesting at its own right, but also provides a new method investigating various online activities. For example, traditional studies on news diffusion focused on the diffusion of news among users [18,19], but from the perspective of this approach, we can also understand the diffusion process in a "reversed" way, that is, the allocation and transmission of users' attention among news [20]. Therefore, the rise and decay of news is the result of the competition among them for users' collective attention [20]. Obviously, News can also be tags [21], videos [22] or any other type of information resources when clickstream network is applied to analyze a specific type of online activity.
Materials and Methods
Data collection
We at first selected three lists of top 1000 sites at different time points. Two of them were selected from Google statistics (http://www.google.com/adplanner/static/top1000/) and the rest one was selected from Alexa reports (http://www.alexa.com)(please refer to Supplemental Materials for the detailed information of these lists). We then downloaded from Alexa the clickstreams between the sites on the lists. From the downloaded data we constructed three clickstream networks (which are called w1, w2, and w3 hereafter), in which a directed, weighted edge from nodes i to j indicated the daily percentage of the global Web users who visited i and j successively. It should be noted that as Alexa only reports a maximum of ten top inbound and outbound clickstreams for each site, our dataset does not neccessarily include all the clickstreams between the studied sites. We actually constructed and studied the "backbone networks" of the clickstreams on the Web [23]. That is, we extracted the top clickstreams connecting the largest sites on the Web. Essential statistics of the three networks, including degree distribution, distribution of weights, and weighted degree distribution, are shown in the Supplementary Materials. In Fig.1 we plot w2 as an example of the clickstream networks, in which we also show the language-based communities of sites to be introduced in the section of Results.
The definition of A i , C i , and γ
To show how to calculate A i , C i , and γ exactly, let us take a look at an example clickstream network provided by Fig.2a. In this network nodes are websites, edges are clickstreams, and the weights of edges reflect the number of distinct users that hop between sites. We balance the network (dashed lines) by adding two artificial nodes, "source" and "sink", to make sure that at each node the sum of inbound and outbound streams are equal [25]. Then we normalize the matrix form of the balanced network F ′ by row to obtain the transition matrix M as given by Eq.1. An element m ij of M denotes the probability that a random user visits website i and j successively. Note that there are only n + 1 rows (columns) in M for the row (column) corresponding to "sink" should be removed in order to derive Eq.5.
m ij = f ′ ij n+1 k=1 f ′ ik , ∀i, j = 0, 1, · · ·, n(1)
Now we define A i and C i as follows Figure 1. The visualization of w2. Small white circles represent websites, big gray circles correspond to language communities, and edges show the clickstreams. Websites of the same language are placed together within a big gray circle and are assigned the same color of edges, with their traffic being reflected by the size of small circles. In calculating the coordinates of the websites, we designed a new algorithm called "two-level spring embedding algorithm", which used spring embedder for two times. Firstly we aggregated w2 into a community-level network, in which nodes were communities and edges were the clickstreams transported between them. We visualized this network using a spring embedding algorithm [24] but only plotted the nodes, which are exactly the gray circles in the figure, whose size is proportional to the total amount of clickstream within a community. Secondly we applied the said spring embedding algorithm algorithm on each of the communities and rescaled the coordinates of websites in order to place them in the gray circles. We found that with the help of the "two-level spring embedding algorithm" we could show the communities particulary clear while remaining the topological structure of the entire network.
A i = n+1 k=1 f ′ ik , ∀i = 1, 2, · · ·n(2)C i = G i n k=1 u ik , ∀i = 1, 2, · · ·n(3)
In Eq.3, u ij is the element of
U = 1 I − M = I + M + M 2 + · · · + M ∞(4)
and G i is defined as:
G i = n j=1 f ′ 0j u ji u ii .(5)
Where f ′ 0j is the balanced flow from "source" to j. From Eq.4 and Eq.5 we know that u ij calculates the fraction of the total flow from i to j along all possible pathes over the total traffic of i, and thus G i is total flow transported from "source" to i [26] summed over all possible pathes (excluding the flow on the self-loop of i). Using the data of the example network, Figure.3 gives a summary of the above mentioned calculations, which prepares data for the testing of the scaling relationship:
C i ∼ A i γ .(6)
In sum, A i stands for the traffic of an arbitrary site i in the balanced clickstream network. C i reflects the circulated clickstreams moderated by i, in specifically, the total number of users who have visited this site and still remain in the network. Therefore, we can treat C i as a measure of the total (both direct and indirect) impact of site i on the rest of sites in clickstream circulation [27,28]. γ measures the increase of impact with traffic, averaged over all sites. We can interpret γ as the level at which large sites dominate the circulation of clickstreams [29]. For example, suppose we have two clickstream networks of the same traffic distribution but are different in γ, A i = {1, 2, 3, 4, 5}, γ ′ = 1/2, and γ ′′ = 2. We can derive that 2)) ≈ 27% circulated clickstreams. Similarly, we can also derive that in the latter network the largest node controls 45% circulated clickstreams, leading us to the conclusion that the latter network is more centralized. To conclude, γ < 1 implies a "democratic" flow structure in which the impact of websites are evenly distributed, whereas γ > 1 is the signature of a "oligarchic" flow structure in which a small group of "hubs" controls the entire network.
C ′ i = {1
As suggested in [28,30,31], a scaling relationship between A i and C i , to be existed, allows one to regress Log(C i ) on Log(A i ) and obtain γ as the slope of ordinary least square (OLS) regression. For example, Figure.1b plots log(C i ) against log(A i ) in the example network, in which the data point corresponding to node 2 (A 2 = 60; C 2 = 125) is colored in red.
Results
A scaling pattern that reveals the decentralized structure of clickstream networks The data points from three networks are plotted in different colors and styles: blue squares for w1, red circles for w2, and green triangles for w3. The values of γ (black line) were 0.95, 0.92, and 0.96, respectively. Please refer to Table.1 for more information concerning the fitting of the three networks.
The traffic A i of a site is proportional to the probability that a random user chooses it as the entrance of the virtual world [5], thus is widely used to indict the website popularity [6,7]. However, if we want to study the long-distance, complex interactions between sites, the investigation on the distribution of traffic [7][8][9] is not enough. We should probe into the transportation of traffic between sites, that is, the flow of clickstreams [12,17]. In the current study, to examine the "rich-get-richer" effect against the "egalitarian" effect, we defined the impact C i of a arbitrary site as the amount of the circulated clickstreams controlled by the site and investigated its relationship with the traffic A i on three clickstream networks (please refer to the section of Data and Method for the calculation of A i and C i ).
As shown by Figure.4, we found a scaling relationship C i ∼ A i γ , in which γ was estimated to be in the range of 0.92 ∼ 0.96 (Table.1). This finding suggested that the impact of websites increases slower than its traffic, which is an evidence of the decentralized structure of the clickstream networks [28,30].
The scaling pattern across language communities
In the last section, we ignored the differences between users in investigating the scaling property of the clickstream networks. However, this is a naive assumption concerning the different preferences of users in the Web browsing [6,7]. Among the various demographic and psychological factors that contribute to the preferences [16,32], we chose to control the linguistic variance in the further investigation of the scaling property. Specifically, we divided the clickstream networks into language-based website communities and then observed the scaling pattern across the communities.
With the help of the AlchemyAPI (http://www.alchemyapi.com/), which turned out to be very efficient in identifying the languages used by sites, we detected 16 language communities from w1, 17 from w2, and 50 from w3. In Table.2 we presented the result of w2 as an example (the result of the rest two networks were given in the Supplementary Materials). Please note that the communities listed in Table.2 is less than those given by Figure.1, because there were several communities within which the clickstreams were too few to support an effective estimation of γ. As suggested by Table.2 and Figure.5, in most of the communities there existed the relationship C i ∼ A i γ (γ < 1), and the value of γ seem to be invariant of community size. It means that these communities share the common decentralized structure with the entire network. This finding also implies that, despite a variety of demographic and psychological that shapes the preferences of users [16,32], there are universal regularities in collective surfing behavior [32].
We noted that a majority of clickstreams across communities occurred between the English community and non-English communities ( Figure.1). This is because users generally use no more than two languages (the mother language and English) in surfing the Web. As a consequence, the clickstream network formed a "wheel-like" structure. Whether this structure contributes to the discussed scaling property is an interesting question worth further investigation. The daily clickstreams is derived by summing up the number of unique users over all edges in the clickstream network.
The robustness of the scaling pattern
The necessity of the work presented in this section is twofold. Firstly, to overcome the limitations of the currently used data sets. As Alexa only provides the top ten inbound and outbound clickstreams for each site, we have to ignore the rest of the potential clickstreams in the data analysis. If the discussed scaling pattern is sensitive to this missing of clickstreams, our conclusion would probably be biased and thus can not be generalized to a larger scale of observation. Secondly, by testing the robustness of the scaling pattern against network reconstructions we may obtain the understanding towards the mechanism leading to the observed pattern.
In the current study, we investigated the robustness of the scaling relationship against two types of network reconstructions, the selective removal of clickstreams and the reshuffling of edges and weights. In both reconstructions we used four statistics to characterize the scaling pattern, including γ, R 2 , ρ, and D. γ and R 2 are the best fitted parameter and the explained variance of the OLS regression, respectively. ρ is the Pearson correlation coefficient between log(C i /A i ) and log(A i ) [31]. The reason for introducing ρ is because the scaling exponent γ was always close to 1 in the current study, to focus on the non-linear nature of the data we removed the linear dependence between A i and C i by calculating C i /A i and then observed the direction and strength of the correlation. If C i is irrelevant of A i or has a trivial, linear relationship with A i , ρ will approximate 0; if there exists a relationship C i ∼ A i γ (γ < 1), which also reads as log(C i /A i ) ∼ (γ − 1)log(A i )(γ − 1 < 0), ρ will deviate from 0 negatively. D is the Kolmogorov-Smirnov statistic that quantifies the distance between two empirical distribution functions [33,34]. It can be used to determine whether two data sets come from populations with the same distribution, which is called the KS test [33]. We calculated D between the distributions of the empirical and predicted values of C i (the latter was A i γ , in which γ was obtained in the OLS regression) and compared it with 0.035, the expected value of D corresponding to a confidence level equals 0.1 (as suggested by [34]) and a sample size equals 1200 [33]. The value of D smaller than 0.035 was treated as a "good" result since we could not reject the null hypothesis that the empirical and predicted values of C i were from the same distribution, in other words, the prediction of C i by A i γ was validated. Please note that we were using the most rigorous criterion of the KS test in setting the sample size as 1200, because the number of nodes in the three clickstream networks and their reconstructed versions was actually smaller than this number and thus allowed a larger expected value of D [33]. Finally, we would like to stress that among the four statistics, the KS statistic was the only one based on formal statistical tests [33] thus should be treated as the most important criterion.
We called the first reconstruction "backbone network analysis" [23], in which we gradually removed edges of small weights from a network and observed the change of the statistics. Specifically, we defined 0 ≤ α < 1 as the portion of the edges to be kept. For a given α, we removed from every node 1 − α incoming and outgoing edges of the least weights. As shown by Figure.6, the scaling pattern was very stable against the removal of edges. In particular, the values of γ and R 2 did not change much while the networks lost as much as 70% edges as α decreased from 1 to 0.2. More importantly, during this process the scaling relationship was validated by the KS test. Actually, we can even try to forecast the scaling pattern given the condition that more clickstreams were retrieved: the value of γ would probably be smaller and the fitting is likely to be better (to be indicated by the smaller D and the larger R 2 ). In sum, our finding of γ < 1 in the clickstream networks is possible to be generalized into the larger scale of observation. The second reconstruction was called "reshuffling analysis". In this reconstruction we reshuffled the clickstream networks (in the ways given by Table.3) to examine the contributions of the linking structure and weights in forming the scaling relationship. Please note that the "randomly shuffled links" was different from the "randomly connected links" in Table.3, because the former kept the long-tail degree distribution of the original network (as shown in the Supplementary Materials) whereas the latter lead to a binomial degree distribution. In particular, in generating randomly connected links we selected w pairs of numbers (with replacement) from n unique numbers randomly, in which w and n were the number of links and nodes of the network to be reconstructed, respectively. The readers who are familiar with the theories of complex networks would find that we were actually creating Erdos-Renyi random graphs, whose degree distribution is binomial [1]. Similarly, the "randomly shuffled weights" was different from the "uniformly distributed weights", for we permuted the order of weights and kept their long-tail distribution in the former combination, but created new weights uniformly distributed (between the minimum and maximum values of the original weights) in the latter.
For each of the combinations listed in Table.3, we ran 100 times of simulations and recorded the mean and standard deviation of the aforementioned four statistics. After that, we plotted γ vs. R 2 and ρ vs. D as given by Figure.7, in which the center of the disks indicated the means and the radius in the corresponding direction reflected the standard deviations. We plotted the results of the combinations in different colors and edge styles and marked the results of the three original networks by "+" (w1), " * " (w2), and "×" (w3). It turned out that across the three clickstream networks, the original networks always had the smallest D and largest R 2 . In examining the scaling pattern by the KS test, we ruled out three combinations whose values of D were greater than 0.035, including f , g, and h. From previous discussions on "randomly connected links" we know that these types of reconstructed networks shared the same binomial degree distribution [1]. Therefore, we can naively conclude that the change of the degree distribution from long-tail to binomial blurred the scaling pattern, whereas the change of the weight distribution did not. In other words, the scaling pattern was determined by the topological structure of the clickstream networks.
Compared with f , g, and h, the rest of combinations were close to the original networks in terms of D or other statistics. If we continuously lower the level of confidence in the KS test (e.g., to 0.05), it is possible to ruled out more combinations. However, we would like to suggest that, before a comprehensive understanding of the mechanism leading to the scaling pattern is achieved, such a tuning is very trivial and does not provide much insight. Therefore, in the current study we would rather stop at the conclusion that topological structure matters in the forming of scaling patterns and leave the contribution of other factors, e.g., the distribution of weights, as a open question.
Discussion
We studied collective browsing behavior from a flow network perspective. We defined C i as a measure of the impact of websites i on other sites through users' collective, continuous surfing activities and found it scaled to website traffic A i with an exponent smaller than 1. This pattern unrevealed the decentralized structure of the three clickstream networks. Further, we found that this scaling pattern appeared universally across language-based communities within the clickstream networks and that the value of γ was independent of communities size. Finally, we examined the stability of the scaling pattern against the reconstructions of the clickstream networks. It turned out that the scaling relationship was robust against the selective removal of edges but sensitive to the permutation on the linking structure.
Our finding has relevant theoretical and practical consequences. Although the "rich-get-richer" paradigm has been widely accepted as a mechanism of hyperlink formations since Barabasi et al. [13], we should not simply assume that this paradigm also suits the dynamics of collective surfing behavior [14,15]. It is already pointed out in [16] that the traffic of websites scaled to its number of inbound links with an exponent approaches 0.8. In this work we found the sublinear relationship between the impact and the traffic. Put these findings together, we can conclude that the survival probability of small sites in the Web ecological system is higher than what was suggested by their in-degree [13] or the Page Rank values based on hyperlink structure [5,16]. Moreover, we would like to emphasize that it is only by studying empirical clickstream networks can the rich interactions between sites be comprehensively understood.
The found scaling relationship provide a quantitative prediction of the impact of a website from its traffic. Online advertising usually measures the impact of websites by their traffic [35,36], but our study offers a more precise calculation of the impact of sites based on their role in the circulation of clickstreams. This approach has potential application in the estimation of the value of sites and also the planning of online marketing campaigns.
Figure 2 .
2An example clickstream network (a) and the fittings of γ in the example network (b). The red node denotes the values of A 2 and C 2 .
Figure 3 .
3Summary of the steps in deriving matrix U .
Figure 4 .
4The scaling relationship between A i and C i in the three clickstream networks.
Figure 5 .
5The change of γ with community size N . The data points of the three networks are plotted in blue squares (w1), red circles (w2), and green triangles (w3), respectively. The y axis is plotted in the linear scale and the x axis is plotted in the base-e log scale.
Figure 6 .
6The change of the number of nodes, the number of links, γ, R 2 , correlation, and KS statistic with the increase of α. The data points from three networks are plotted in different colors: blue squares for w1, red circles for w2, and green triangles for w3. The dashed, black line in the last figure shows the critical value of KS statistics as 0.035 given the condition of 1200 sample size and 0.1 level of confidence.
Figure 7 .
7The mean and standard deviation of statistics of interested of the eight combinations in the reshuffling. The dashed, black line in the left figure shows the critical value of KS statistics as 0.035 given the condition of 1200 sample size and 0.1 level of confidence.
, 1.4, 1.7, 2, 2.2} and C ′′ i = {1, 4, 9, 16, 25}. The impact of the largest node is 2.2 in the former network, meaning that it controls (2.2/(1 + 1.4 + 1.7 + 2.
Table 1 .
1The statistics of three studied clickstream networks.Network N sites N edges Daily clickstreams
γ
R 2 of γ
w1
979
11906
5.45 × 10 9
0.95
0.98
w2
956
11529
1.38 × 10 10
0.92
0.95
w3
1189
17061
6.06 × 10 9
0.96
0.99
Note: The daily clickstreams is obtained by summing up the number of unique users over all edges in a
clickstream network.
Table 2 .
2The scaling exponent across language communities in w2.Community N sites N edges Daily clickstreams
γ
R 2 of γ
English
516
6188
8.64 × 10 9
0.94
0.94
Chinese
214
2130
3.18 × 10 9
0.94
0.77
Japanese
63
481
4.83 × 10 8
0.86
0.88
Portuguese
31
115
4.48 × 10 7
0.91
0.83
French
25
57
1.12 × 10 7
0.84
0.57
Russian
21
94
8.50 × 10 7
0.97
0.94
German
15
64
1.78 × 10 7
0.91
0.76
Korean
11
53
6.11 × 10 7
0.98
0.84
Polish
10
43
1.72 × 10 7
1.05
0.91
Vietnamese
7
25
8.70 × 10 6
0.86
0.61
Thai
3
6
1.67 × 10 6
0.31
0.71
Table 3 .
3The combinations in the reshuffling.Original weights Randomly shuffled weights Uniformly distributed weights
Original links
w1 /w2 /w3
a
b
Randomly shuffled links
c
d
e
Randomly connected links
f
g
h
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| []
|
[
"Searches for Pulsar-like Candidates from Unidentified Objects in the Third Catalog of Hard Fermi-LAT (3FHL) sources with Machine Learning Techniques",
"Searches for Pulsar-like Candidates from Unidentified Objects in the Third Catalog of Hard Fermi-LAT (3FHL) sources with Machine Learning Techniques"
]
| [
"C Y Hui \nDepartment of Astronomy and Space Science\nChungnam National University\n34134DaejeonKorea\n",
"Jongsu Lee \nDepartment of Space Science and Geology\nChungnam National University\n34134DaejeonKorea\n",
"K L Li \nDepartment of Astronomy and Space Science\nChungnam National University\n34134DaejeonKorea\n\nDepartment of Physics\nUNIST\n44919UlsanKorea\n\nInstitute of Astronomy\nNational Tsing Hua University\n30013HsinchuTaiwan\n",
"Sangin Kim \nDepartment of Space Science and Geology\nChungnam National University\n34134DaejeonKorea\n",
"Kwangmin Oh \nDepartment of Space Science and Geology\nChungnam National University\n34134DaejeonKorea\n",
"Shengda Luo \nFaculty of Information Technology\nMacau University of Science and Technology\nAvenida Wai LongTaipa, Macau\n",
"Alex P Leung \nFaculty of Information Technology\nMacau University of Science and Technology\nAvenida Wai LongTaipa, Macau\n",
"A K H Kong \nInstitute of Astronomy\nNational Tsing Hua University\n30013HsinchuTaiwan\n",
"J Takata \nInstitute of Particle physics and Astronomy\nHuazhong University of Science and Technology\nChina\n",
"K S Cheng \nDepartment of Physics\nUniversity of Hong Kong\nPokfulam RoadHong Kong\n"
]
| [
"Department of Astronomy and Space Science\nChungnam National University\n34134DaejeonKorea",
"Department of Space Science and Geology\nChungnam National University\n34134DaejeonKorea",
"Department of Astronomy and Space Science\nChungnam National University\n34134DaejeonKorea",
"Department of Physics\nUNIST\n44919UlsanKorea",
"Institute of Astronomy\nNational Tsing Hua University\n30013HsinchuTaiwan",
"Department of Space Science and Geology\nChungnam National University\n34134DaejeonKorea",
"Department of Space Science and Geology\nChungnam National University\n34134DaejeonKorea",
"Faculty of Information Technology\nMacau University of Science and Technology\nAvenida Wai LongTaipa, Macau",
"Faculty of Information Technology\nMacau University of Science and Technology\nAvenida Wai LongTaipa, Macau",
"Institute of Astronomy\nNational Tsing Hua University\n30013HsinchuTaiwan",
"Institute of Particle physics and Astronomy\nHuazhong University of Science and Technology\nChina",
"Department of Physics\nUniversity of Hong Kong\nPokfulam RoadHong Kong"
]
| [
"MNRAS"
]
| We report the results of searching pulsar-like candidates from the unidentified objects in the 3 rd Catalog of Hard Fermi-LAT sources (3FHL). Using a machine-learning based classification scheme with a nominal accuracy of ∼ 98%, we have selected 27 pulsarlike objects from 200 unidentified 3FHL sources for an identification campaign. Using archival data, X-ray sources are found within the γ−ray error ellipses of 10 3FHL pulsar-like candidates. Within the error circles of the much better constrained X-ray positions, we have also searched for the optical/infrared counterparts and examined their spectral energy distributions. Among our short-listed candidates, the most secure identification is the association of 3FHL J1823.3-1339 and its X-ray counterpart with the globular cluster Mercer 5. The γ−rays from the source can be contributed by a population of millisecond pulsars residing in the cluster. This makes Mercer 5 as one of the slowly growing hard γ−ray population of globular clusters with emission > 10 GeV. Very recently, another candidate picked by our classification scheme, 3FHL J1405.1-6118, has been identified as a new γ−ray binary with an orbital period of 13.7 days. Our X-ray analysis with a short Chandra observation has found a possible periodic signal candidate of ∼ 1.4 hrs and a putative extended X-ray tail of ∼ 20 arcsec long. Spectral energy distribution of its optical/infrared counterpart conforms with a blackbody of T bb ∼ 40000 K and R bb ∼ 12R at a distance of 7.7 kpc. This is consistent with its identification as an early O star as found by infrared spectroscopy. | 10.1093/mnras/staa1113 | [
"https://arxiv.org/pdf/2004.10945v2.pdf"
]
| 216,080,741 | 2004.10945 | 92b237b1f25ad761620fd498d80f8683045cbb42 |
Searches for Pulsar-like Candidates from Unidentified Objects in the Third Catalog of Hard Fermi-LAT (3FHL) sources with Machine Learning Techniques
2015
C Y Hui
Department of Astronomy and Space Science
Chungnam National University
34134DaejeonKorea
Jongsu Lee
Department of Space Science and Geology
Chungnam National University
34134DaejeonKorea
K L Li
Department of Astronomy and Space Science
Chungnam National University
34134DaejeonKorea
Department of Physics
UNIST
44919UlsanKorea
Institute of Astronomy
National Tsing Hua University
30013HsinchuTaiwan
Sangin Kim
Department of Space Science and Geology
Chungnam National University
34134DaejeonKorea
Kwangmin Oh
Department of Space Science and Geology
Chungnam National University
34134DaejeonKorea
Shengda Luo
Faculty of Information Technology
Macau University of Science and Technology
Avenida Wai LongTaipa, Macau
Alex P Leung
Faculty of Information Technology
Macau University of Science and Technology
Avenida Wai LongTaipa, Macau
A K H Kong
Institute of Astronomy
National Tsing Hua University
30013HsinchuTaiwan
J Takata
Institute of Particle physics and Astronomy
Huazhong University of Science and Technology
China
K S Cheng
Department of Physics
University of Hong Kong
Pokfulam RoadHong Kong
Searches for Pulsar-like Candidates from Unidentified Objects in the Third Catalog of Hard Fermi-LAT (3FHL) sources with Machine Learning Techniques
MNRAS
0002015Accepted XXX. Received YYY; in original form ZZZPreprint 1 May 2020 Compiled using MNRAS L A T E X style file v3.0gamma-rays: stars -X-rays: stars -X-rays: binaries -pulsars: general
We report the results of searching pulsar-like candidates from the unidentified objects in the 3 rd Catalog of Hard Fermi-LAT sources (3FHL). Using a machine-learning based classification scheme with a nominal accuracy of ∼ 98%, we have selected 27 pulsarlike objects from 200 unidentified 3FHL sources for an identification campaign. Using archival data, X-ray sources are found within the γ−ray error ellipses of 10 3FHL pulsar-like candidates. Within the error circles of the much better constrained X-ray positions, we have also searched for the optical/infrared counterparts and examined their spectral energy distributions. Among our short-listed candidates, the most secure identification is the association of 3FHL J1823.3-1339 and its X-ray counterpart with the globular cluster Mercer 5. The γ−rays from the source can be contributed by a population of millisecond pulsars residing in the cluster. This makes Mercer 5 as one of the slowly growing hard γ−ray population of globular clusters with emission > 10 GeV. Very recently, another candidate picked by our classification scheme, 3FHL J1405.1-6118, has been identified as a new γ−ray binary with an orbital period of 13.7 days. Our X-ray analysis with a short Chandra observation has found a possible periodic signal candidate of ∼ 1.4 hrs and a putative extended X-ray tail of ∼ 20 arcsec long. Spectral energy distribution of its optical/infrared counterpart conforms with a blackbody of T bb ∼ 40000 K and R bb ∼ 12R at a distance of 7.7 kpc. This is consistent with its identification as an early O star as found by infrared spectroscopy.
INTRODUCTION
Fermi Gamma-ray Space Telescope has brought us into a new era of high energy astronomy by significantly expanding the population of γ−ray sources. In particular for pulsars, thanks to the much improved sensitivity of the Large Area Telescope (LAT) on board Fermi, our understandings of their high energy properties have been advanced considerably in the last decade (see Hui (2018) for a recent review). Currently, there are 234 γ−ray pulsars have been detected, E-mail: [email protected] which is > 30 times of their population before the launch of Fermi. Not only enlarging the population, Fermi LAT also has uncovered previously unknown classes of γ−ray pulsars (Abdo et al. 2013) such as millisecond pulsars (MSPs). Furthermore, other γ−ray phenomena related to pulsars have also been found. For example, γ−ray emission were discovered from a number of globular clusters (Abdo et al. 2009;Kong et al. 2010;Tam et al. 2011), which can be originated from the collective contribution of the magnetospheric radiation from MSPs in the cluster (Abdo et al. 2010) and/or from the inverse Compton scattering between the relativistic pulsar wind outflow and the local soft photon field Hui et al. 2011). Also, flares in X-ray , GeV and TeV regimes from the γ−ray binaries, which contains a pulsar and a OB companion, were detected before/after the periastron passage (e.g. Tam et al. (2018)). These flares are suggested to be resulted from the intrabinary shocks .
In the previous Fermi LAT point source catalogs obtained from the full band all-sky survey (> 100 MeV), there are approximately one-third of the sources have their nature unidentified (e.g. 2FGL; Nolan et al. (2012); 3FGL: (Acero et al. 2015)). The locations of these unidentified Fermi objects provide us with a "treasure map" for searching interesting objects with multiwavelength observations. By imposing a suitable set of classification criteria, one can select some promising candidates from these unidentified sources for searching the counterparts within their γ−ray positional error ellipses. For example, by choosing the unidentified objects that have low γ−ray flux variability for discriminating them from the AGN-like sources (i.e. small variability indices) and with curved spectral shape similar to the pulsars (i.e. large curvature significances), one can obtain a list of pulsar candidates for follow-up identifications (e.g. Kong et al. 2012;Hui et al. 2015;Saz Parkinson et al. 2016). A significant fractions of MSPs were discovered by this method (Clark 2017).
Apart from the full-band γ−ray source catalogs, lists of sources in the hard γ−ray bands have also been compiled. In the third Catalog of Hard Fermi LAT sources (3FHL) (Ajello et al. 2017), it contains 1556 objects detected in the energy range of 10 GeV to 2 TeV. 136 of them have their nature identified and 1220 "associated" sources have been classified primarily by the positional coincidence with sources of known nature. Among these 1356 sources, 59 sources are labeled as pulsars and the rest includes mostly AGNs. The remaining 200 sources do not have any association/identification in the 3FHL catalog.
A recent systematic investigation have been carried out for pinpointing the nature of these unidentified 3FHL objects (Kaur et al. 2019). They have selected 110 sources from 200 unidentified 3FHL sources which have their fields covered by archival Swift-XRT data for their analysis. Among them, 52 sources have a single X-ray sources detected in their 95% γ−ray error ellipses and have been selected for further analysis. By cross-matching the X-ray positions with catalogs of different wavelengths, Kaur et al. (2019) have classifed 36 of these sources as AGN candidates.
While their work is successful in identifying a number of AGN candidates, their approach is not very efficient as they have to analyze a large number of sources without any pre-screening. A lot of effort have been spent on analyzing the data of the sources that are unlikely to be their target-ofinterest (i.e. AGN). A more efficient approach is to select the promising candidates first with machine learning algorithms and then look into the archival data and/or carry follow-up observations afterward. This is the approach we adopted in our investigation.
In this work, we present a systematic searches for pulsar-like candidates from the unassociated/unidentified 3FHL objects with machine learning techniques and performed a follow-up multiwavelength identification campaign. While the population of pulsars with energies > 100 MeV has been significantly expanded, the population in the very high energy regime (VHE > 100 GeV) remains to be rather small. So far only three pulsars have their pulsed emission detected at energies > 50 GeV (cf. (Hui 2018) for a review). Besides their magnetospheric radiation, interaction of the pulsar emission and/or wind particles with their surroundings can also produce VHE photons such as those in γ-ray binaries and globular clusters. The hard γ−ray pulsar-like candidates investigated in this work have the potential for enlarging VHE pulsar population and the related phenomena.
PSR-LIKE CANDIDATE SELECTION WITH MACHINE LEARNING TECHNIQUES
Using the 3FHL sources with identified/associated nature for training and testing a classifier, we can perform a binary classification of 3FHL sources between pulsars (PSR) and non-pulsars (NON PSR) by employing machine learning techniques. Among 65 features in the catalog, sixteen features were removed in the our preprocessing stage. Eleven features are manually removed as we believe they are not useful in determining the source nature, such as their 3FHL names and alternative names. We have also set a threshold of removing any feature with more than 10% of null values, and five more features are therefore automatically removed.
We use 1356 identified/associated sources as our sample for the feature selection and building prediction models. Among them, 1231 γ−ray sources are identified/associated with extragalactic objects, which include starburst galaxy, BL Lac, flat-spectrum radio quasar type of blazar, nonblazar active galaxy, narrow-line seyfert 1, radio galaxy and blazar candidate of uncertain type. On the other hand, there are 125 γ−ray sources reside in our Galaxy. These Galactic 3FHL sources include pulsars, pulsar wind nebula, supernova remnant, high mass binary, binary, globular clusters and star formnation regions. Since we are interested in looking for the pulsar-like candidates, we perform a one-againstall classification. Instead of using the original labels in the catalog for identifying their nature, we add a column to divide them into two classes. For all the sources identified as (or associated with) pulsars, we put them in the class of "PSR". Otherwise, we label them as "NON PSR".
In the previous work of selecting pulsar candidates from the unidentified Fermi objects, γ−ray flux variability is an important feature for us to distinguish the pulsar-like sources from the AGN-like sources (e.g. Hui et al. 2015). However, there is no feature for indicating variability in 3FHL catalog. Instead of relying on our current knowledge for differentiating the γ−ray properties between pulsars and the other γ−ray sources, we employ an automatic feature selection algorithm (Leung et al. 2017;Luo et al. 2020) for picking the features which can help discriminate a source is PSR-like or not. We achieve this by adopting a scheme of Recursive Feature Elimination (RFE). RFE is a backward selection method with unimportant features are sequentially eliminated during a recursive process (Leung et al. 2017). With this machine-learning based technique, attributes and patterns of the data that are overlooked by human investigators can be highlighted.
After the preprocessing stage, an optimal set of fea-tures can be automatically selected by using RFE. For each iteration in the stage of RFE, we evaluate the performance of a random forest classifier by computing the root-meansquared error (RMSE). The classification performance is evaluated by plotting the RMSE with the corresponding number of features. The performance profile produced by the RFE for the 3FHL catalog is shown in Figure 1. While the minimum of the profile is attained by using 30 out of all 49 features, it appears to be rather flat for the number of features 15. The minimum corrsponds to 30 features (i.e. solid blue symbol in Figure 1) can be a result of local fluctuations.
In view of this, we chose to suffice a little bit of performance by accepting an upper margin of error of 5% in the RMSE value to trade for a simpler model with better interpretability. A simpler model is often easier to understand and more robust (cf. Luo et al. 2020).
With this imposed scheme, a minimal set of 17 features is selected. The selected features are summarized in Table 1 which are ranked by their importance scores.
In Figure 2, we show the two-dimensional projections of the feature space for the highly ranked features: Flux_Density_Error, Powerlaw_Index and Pivot_Energy. The known PSR and NON PSR sources in 3FHL catalog are plotted as red dots and blue dots respectively. These chosen features suggest that the hardness of γ−rays is a key factor for differentiating PSR and NON PSR sources. This can be readily shown by the distributions the Powerlaw_Index/Spectral_Index (Figure 2). The harder a source is, the smaller these features will be. On the other hand, Pivot_Energy is defined as the energy at which the error on differential photon flux is minimal (Ajello et al. 2017). A softer γ−ray source has a smaller Pivot_Energy, and therefore it anti-correlates with the Powerlaw_Index/Spectral_Index. For the feature Flux_Density_Error, it is the error on differential photon flux at Pivot_Energy (Ajello et al. 2017). For the hard sources, which have larger Pivot_Energy, their differential fluxes at Pivot_Energy tend to be smaller. Since Flux_Density_Error generally scales with the differential flux (see Luo et al. 2020), this feature naturally anticorrelates with Pivot_Energy.
One surprising result is that the feature Curve_Significance, which many previous studies have relied on selecting pulsar candidates (Kong et al. 2012;Hui et al. 2015;Saz Parkinson et al. 2016), does not appear to be a highly ranked defining characteristic for pulsars in 3FHL catalog. It has been found that the γ−ray spectra of most of the pulsars are characterized by a power-law with an exponential cut-off at energies 5 GeV Abdo et al. 2013). As all the pulsars included the in 3FHL catalog are detected in the energy range of 10 GeV to 2 TeV, which beyonds the typical range of the spectral cut-off of most pulsars, their less curved spectra can be a selection effect. This may explain why Curve_Significance is not among the top-ranked features for discriminating pulsars from the others in hard γ−ray band.
Using the features in Table 1 to build the prediction model, we compare the performances of different classifiers. Seven prediction models are built with the following machine learning methods: Random Forest (RF), Generalized Additive Models (GAM), Logistic Regression (LR), Boosted Figure 1. The performance profile of PSR/NON PSR classification in the 3FHL catalog. The optimal performance is achieved by using thirty features (solid symbol). Allowing a tolerance of 1.05 as the margin of error in the RMSE value, a minimal set of 17 features are selected for building the model which is highlighted by the circle.
Features
Importance Scores Logistic Regression (Boost LR), Support Vector Machines (SVM), Decision Trees (DT) and Logistic Trees (LT). For each of these tested classifiers, the data of labeled sources are randomly divided into training/test sets with a ratio of 70%/30%. During the training stage, some parameters of various classifiers are tuned for optimizing their performances with the training data set as the input. Such parameters are automatically optimized by using a 10-fold cross-validation empirically. For quantifying the performance of each model, we compute the overall accuracy which is defined as the ratio of the correct classification in the test set. A comparison of the overall accuracies of different classifiers is summarized in Table 2. Among all the tested classifiers, an optimal overall accuracy of 98.03% is achieved with RF. Using a scheme of nested cross-validation (Luo et al. 2020), we found that the standard deviations of all the quote accuracies in Table 2 are 1%.
To further characterize the model performance with RF, we computed the receiver operating characteristic (ROC) curves for the PSR/NON PSR classification task with both training set and test set. ROC curve is a plot of sensitivity (i.e. probability of detection) against specificity (i.e. 1probability of false alarm). A good model should minimize the false alarm and avoid missing any detection, and hence its ROC curve would be pushed toward the top-left corner. The training and test ROC curves of RF classifier are shown in Figure 3. The Area Under the Curve (AUC) of an ROC curve provides another measure for the classification performance The larger the AUC, the better the performance. An AUC of 98.2% is obtained for the test ROC.
Using the prediction model with the best threshold obtained from the test ROC curve in Figure 3, we run the PSR/NON PSR classification on the 200 unidentified/unassociated sources. 27 of them have been classified as PSR by our model. We summarize their properties in Table 3, which includes their names in 3FHL catalog, γ−ray positions and errors, the corresponding name in 3FGL catalog (if there is any), as well as the confidence score of belonging to PSR class assigned by our model. The confidence score for a given source provides a gauge for the reliability of the class assignment as predicted by the model, which should not be interpreted as the probability of the source as a PSR. The distributions of these 27 selected PSR candidates in the projected feature spaces are shown by the black trian- Figure 3. The training and test ROC curves produced by random forest classifier using 3FHL catalog gles in Figure 2. Except for two outliers (3FHL J1915.2-1323 and 3FHL J0737.5+6534) with low PSR confidence scores, other candidates are clustered in the regime occupied by the known pulsars in 3FHL catalog. Kaur et al. (2019) have reported 36 unidentified 3FHL sources which most likely belong to AGNs family. In comparing their list (Table 4 in their paper) with our PSR candidate list, only one source 3FHL J0541.1-4855, which has a relatively low PSR confidence score of 0.166, is overlapped. This provides further confidence for our method and the PSR candidates selected by this scheme.
DATA ANALYSIS
3.1 Searching for X-ray/optical sources within the γ−ray error ellipses
We have searched for X-ray counterparts associated with our short-listed 3FHL sources by using archival X-ray spectral imaging data. We attempted to detect the X-ray sources within the γ−ray error ellipses of these candidates with a wavelet detection algorithm. Only the X-ray sources detected at a significance larger than 4σ are considered as genuine in our work. Among 27 PSR-like candidates in Table 3, ten of them have X-ray sources found within their 95% confidence γ-ray error ellipses. The results are summarized in Table 4. X-ray images of the fields of these 10 PSR-like candidates are shown in Figure 4. We found that these selected candidates have been observed either by Chandra, XMM-Newton or Swift. If Chandra data is available for a PSR-like candidate, we solely used its data to determine the positions of the X-ray counterparts as Chandra can provide the best positional accuracy among all X-ray telescopes. For the cases there is no archival Chandra data but with XMM-Newton available, the positions of the X-ray counterparts are determined by the MOS cameras (merged MOS1/2 data) because their pixel size provide a full sampling of the point spread function of the mirror. For Swift XRT observations, we noticed that their exposures are typically a few ks which are unconstraining for our searches of relatively faint sources potentially associated with pulsars. Therefore, the observations by Swift XRT will be ignored in this work.
By assuming an absorbed power-law with a photon index of Γ x = 2 and the column absorption n H adopted at the value of the Galactic HI column density in the directions towards these X-ray sources (Kalberla et al. 2005), with the aid of PIMMS (ver. 4.9a), we systematically computed the absorption-corrected X-ray fluxes F x for all the X-ray sources in an energy range of 0.3 − 10 keV by using their count rates. And hence, we obtained their X-ray to γ−ray flux ratios F x /F γ with F γ as the energy flux in 10 GeV to 2 TeV as obtained from 3FHL catalog. F x and F x /F γ are summarized in column 8 and column 9 in Table 4.
In Figure 5, we compare the distributions of log F x /F γ of these X-ray sources with those of the known pulsars in the same energy ranges. The range of log F x /F γ spanned by these X-ray sources is bracketed by those of the known pulsars, except for two sources J18007 X8 and J17472 X8 which have the lowest log F x /F γ . We have also examined the temporal variability of these X-ray sources. We first search for the short-term variability within each observation window by using the Gregory-Loredo variability algorithm (Gregory & Loredo 1992). By testing whether the arrival times of these sources are uniformly distributed, only J18007 X1 has a probability of > 90% as a variable source.
Apart from the short-term variabilities, a number of X-ray sources have been observed more than once. These multi-epoch X-ray data allows us to further examine their long-term flux variability. We compare the difference of the fluxes with their errors combined by quadrature, i.e.:
|F obs1 − F obs2 | / σ 1 obs1 + σ 2 obs1 .
The largest difference found for each source are summarized in the column 10 in Table 4. We consider a source to have long-term variability if the maximal difference of its flux in two observations is larger than 4σ. Four sources, J17472 X5, J0737 X1, J0737 X2 and J0737 X5 are found to be significantly variable. For those with non-detection in certain epoch(s), we have placed lower bounds on their long-term variabilities instead.
Since the X-ray data provide much better constraints on the positions of the potential counterparts, we are able to search for the possible optical/infra-red counterparts of these X-ray sources. We searched the following optical and
F X /F γ Variability (h m s) (d m s) (arcsec) (σ) (10 −3 cts/s) (10 −14 erg cm −2 s −1 ) (10 −3 ) (S / L)3FHLN / > 2.3σ
Note. The chance coincidence can be seen next to the 3FHL name. * means that the coordinate and significance are extracted from. For whole analysis, we used the wavdetect task in CIAO. The counts rate was calculated from the net counts for *'s instrument with exposure time. MNRAS 000, 1-19 (2015) Figure 5. Comparison for the distributions of log F x /F γ of known pulsars in 3FHL and the sources in Table 4. The fluxes in Xray and γ−ray are evaluated in 0.3-10 keV and 10 GeV-2 TeV respectively.
(near-)infrared source catalogs for the counterpart to the Xray sources (search radius = 1 ): Pan-STARRS DR2 (PS1; Chambers et al. 2016), GLIMPSE (Spitzer Science 2009), the Spitzer point-source catalog of seven nearby galaxies (Khan et al. 2015), VISTA Variables in the Via Lactea (VVV; Saito et al. 2012;Minniti et al. 2017), WISE allsky catalog (Cutri & et al. 2012), and Gaia DR2 (Gaia Collaboration et al. 2018). Most of the 3FHL sources are close to the Galactic plane with heavy extinction, and several (near-)infrared catalogs were thus used. For every optical counterpart, we made an extinction corrected spectral energy distribution (SED) and fit it with the blackbody model. Dereddening was done using the extinction curve of Fitzpatrick (1999) with an extinction value (A v ) inferred from the hydrogen column density in the X-ray analysis (i.e., N H /Av = 2.21 × 10 21 ; Güver &Özel 2009). Table 5 shows the SED fitting results. Except for those have Gaia distance measurements (Bailer-Jones et al. 2018), a distance of 1 kpc is assumed for the calculations of the blackbody radii (the radius is proportional to the distance). As mentioned, most of the sources are highly absorbed and therefore their SEDs are largely affected by the dereddening. Given that the extinctions adopted are full Galactic values, the SEDs of some nearby sources (e.g., J16263 X2) could be over-corrected and appear to be much bluer than they should be. Under-correction is also possible if a source is with high intrinsic absorption.
Detailed Analysis of Individual PSR-like Candidates
The details of the X-ray observations and data analyses of these ten PSR candidates are given in the followings:
3FHL J1748.6-2816
Both Chandra (Obs ID: 2269) and XMM-Newton (Obs ID: 0694641401) have observed the field of 3FHL J1748.6-2816 on 2001 July 16 and 2012 September 30 for the effective exposures of 18 ks and 32 ks respectively. In both observations, only one X-ray source is detected within the γ−ray positional error ellipse which is denoted as J17486 X1 (see Figure 4). Searching in SIMBAD, we found the nature of this source remains to be unidentified. For estimating its absorptioncorrected X-ray flux as given in Table 4, we adopted the count rate from the Chandra observation and assumed a column absorption of n H = 1.3 × 10 22 cm −2 at the same level as the Galactic HI column density in the corresponding direction (Kalberla et al. 2005). Besides J17486 X1, there are other sources are detected serendipitously in the whole field-of-view (FoV) covered by the cameras in both observations. We have considered the possibility that one or more sources lie within the error ellipse by chance. We counted the number of X-ray sources detected in the whole FoV and computed the source density. Based on this, we estimated the number of chance coincidences λ expected within the γ-ray error ellipse. Assuming a Poisson distribution, the probability of finding one or more chance coincidences of X-ray sources is given by:
P (n ≥ 1) = ∞ n=1 λ n e −λ n! = 1 − e −λ(1)
For 3FHL J1748.6-2816, we found that P (n ≥ 1) ∼ 40% and ∼ 34% in Chandra and XMM-Newton observations respectively. J17846 X1 does not show any X-ray flux variability neither in individual observations nor between two observations at different epoch. Optical/IR counterpart of J17846 X1 has been identified. A blackbody fit to its extinction-corrected SED yields a temperature of T bb ∼ 1.2 × 10 4 K and an emitting region with a radius of R bb ∼ 1.3d kpc R (cf . Table 5), where d kpc is the distance at unit of 1 kpc.
In this work, a detailed X-ray spectral fitting will be carried out for those sources with more than 50 net counts detected. The results are summarized in Table 6. Since the net counts of J17486 X1 collected from both observations is ∼ 140 cts, we have extracted its spectrum and fitted with both absorbed power-law model and absorbed blackbody model. Both models result in a similar goodness-offit. The best-fit power-law yields a column absorption of n H = 1.3 +0.7 −0.5 × 10 22 cm −2 , a photon index of Γ x = 3.1 +0.9 −0.7 and an absorption-corrected of F x ∼ 2.1 × 10 −13 erg cm −2 s −1 in 0.3-10 keV. The best-fitted Γ ∼ 3 appears to be quite steep which indicate the X-ray emission is rather soft. Considering a purely thermal emission scenario, the best-fit blackbody yields a temperature of kT = 0.6 ± 0.1 keV with an absorption-corrected of F x ∼ 2.7 × 10 −14 erg cm −2 s −1 in 0.3-10 keV. The normalization of the blackbody implies an X-ray emission region with a radius of ∼ 13.4d kpc m.
3FHL J1839.4-0553
The γ−ray error ellipse of 3FHL J1839.4-0553 has been covered by two Chandra observations with ACIS-I on 2008 March 9 (Obs. ID: 7493) and 2007 November 5 (Obs. ID. (a) Unless a Gaia distance is found, d = 1 kpc is assumed. (b) Observed magnitudes of the shortest wavelength (in the brackets) that can be found in the aforementioned catalogs.
(c) Distance and the extinction are adopted from Corbet et al. (2019). (d) Only two data points in the SED and therefore no uncertainty can be obtained. Table 5. Results of blackbody fits to the optical/IR SED of the possible counterparts associated with the X-ray sources found in the error ellipses of PSR-like 3FHL sources. Table 6. X-ray spectral properties of X-ray sources with more than fifty net counts collected from the archival data. The results from both power-law fits and blackbody fits are summarized. The quoted uncertainties are 1σ for one parameter of interest. 7630) for an effective exposure of 20 ks and 28 ks respectively. Within the 95% γ−ray error ellipse, there are two X-ray sources J18394 X1 and J18394 X2 (see Figure 4). J18394 X1 has been detected by both observations. Therefore, we are able to estimate its long-term variability which is only at 1.4σ level. On the other hand, J18394 X2 is out of the FoV in one Chandra observation (Obs. ID. 7630). The nature of both X-ray sources is not known. Their absorptioncorrected X-ray fluxes as given in Table 4 is estimated by assuming a column absorption of n H = 1.8×10 22 cm −2 , which is consistent with the total Galactic HI absorption at that direction, with the count rates observed by Obs. ID: 7493. Taking all the X-ray sources detected in the entire FoV in the observation, P (n ≥ 1) is found to be ∼ 65%. Searching for their counterparts in other wavelengths with the archival data does not yield any positive result.
J18394 X1 has ∼ 58 net counts collected from both observations and therefore we have further examined its X-ray spectrum (see Table 6). Its X-ray spectrum appears to be rather flat in the energy range of 0.5-8 keV. Both powerlaw and blackbody fits suggest the column absorption can be lower than that inferred from the Galactic HI absorption. The best-fit power-law yields n H < 2.6 × 10 22 cm −2 , Γ x = 0.02 +0.77 −0.54 and F x ∼ 6.4 × 10 −14 erg cm −2 s −1 in 0.3-10 keV. On the other hand, the best-fit blackbody yields n H < 1.8 × 10 22 cm −2 , kT = 2.6 +2.9 −0.9 keV and F x ∼ 5.3 × 10 −14 erg cm −2 s −1 in 0.3-10 keV.
3FHL J1823.3-1339
3FHL J1823.3-1339 has been observed by XMM-Newton on 23 March 2002 (Obs. ID: 0040140201) for an effective exposure of ∼ 11 ks. Only one single X-ray source, J18233 X1, is detected within its γ−ray error ellipse. Its X-ray image as observed by MOS1/2 camera onboard XMM-Newton is displayed in Figure 4. Using the serendipitous X-ray sources detected in the FoV of MOS camera, the probability of finding an X-ray source within the error ellipse of 3FHL J1823.3-1339 is found to be P (n ≥ 1) ∼ 12%, which is the lowest among all the selected 3FHL PSR-like candidates in this work. We noticed that the feature is apparently extended with an angular size of ∼ 30 arcsec and has the peak emission located at RA (J2000)=18 h 23 m 19 s Dec (J2000)=−13 • 40 02 . This extended X-ray feature is identified for the first time. A close-up view of J18233 X1 is shown in the right panel of Figure 6.
Searching for the nature of this extended source in SIM-BAD, we found that it is possibly associated with a poorlystudied globular cluster Mercer 5, which is discovered in the GLIMPSE Survey (Mercer et al. 2005). It is highly obscured in optical regime as it resides in a region of high visual extinction, A V ∼ 8.5 − 12.5 mag (Longmore et al. 2011), which suggests an X-ray absorption at the level of (1.2 − 2.8) × 10 22 cm −2 (Güver &Özel 2009). In left panel of Figure 6, we compare the X-ray morphology of J18233 X1 with the K s band 2MASS image of Mercer 5. by overlaying the X-ray contours on the infrared image. The distribution of the stars in Mercer 5 is comparable with the morphology of J18233 X1. The peak of the X-ray emission coincides with the region with highest stellar density.
The net counts of J18233 X1 collected from all EPIC cameras on XMM-Newton (MOS1/2 + PN) is 322 cts. This enables us to carry out a detailed analysis. In examining its X-ray spectrum, we found that it can be welldescribed by an absorbed power-law model with a goodnessof-fit of χ 2 = 38.55 for 40 d.o.f.. The observed X-ray spectra of J18233 X1 and the best-fitted power-law model are displayed in Figure 7. The X-ray emission of J18233 X1 is quite hard. The best-fit yields a column absorption of n H = 1.9 +0.8 −0.6 × 10 22 cm −2 , a photon index of Γ x = 1.1 ± 0.3 and an absorption-corrected flux in 0.3-10 keV of F x ∼ 3 × 10 −13 erg cm −2 s −1 . The X-ray column absorption inferred from the spectral fit is consistent with that deduced from the n H − A v correlation. This suggests that J18233 X1 and Mercer 5 are very likely to be located at the same distance from us.
We have also attempted to search for X-ray periodicity from J18233 X1. However, we do not find any significant periodic signal from the existing data.
We further investigated if 3FHL J1823.3-1339 can also be the γ−ray counterpart of Mercer 5. 3FHL J1823.3-1339 is also identified in the 3FGL catalog with designation 3FGL 1823.2-1339 (Acero et al. 2015). In 0.1-100 GeV, its energy flux is f γ = (9.3 ± 0.8) × 10 −11 erg cm −2 s −1 . At a distance of d ∼ 5.5 kpc as estimated by Gaia (DR2) (Baumgardt et al. 2019), this corresponds to a γ−ray luminosity of L γ ∼ 3.4 × 10 35 erg/s. On the other hand, the metallicity of Mercer 5 is estimated to be [Fe/H]∼-0.86 (Peñaloza et al. 2015). Using its metalicity, we can estimate the expected L γ from a globular cluster by using the empirical relation: log L γ = (0.6 ± 0.2)[Fe/H]+(35.6±0.2) (Hui et al. 2011). This implies that the γ−ray luminosity of Mercer 5 is expected at the order of ∼ 10 35 erg/s. This is consistent with the observed luminosity within the tolerance of the uncertainties of fitted parameters and L γ . This suggests the γ−rays from 3FHL J1823.3-1339/3FGL 1823.2-1339 are likely from Mercer 5.
3FHL J1748.1-2903
3FHL J1748.1-2903 has been observed by two Chandra observations with ACIS-I CCD array on 2006 October 31 (Obs. ID. 7158) and 2017 July 13 (Obs. ID. 19448) with effective exposures of ∼ 14 ks and ∼ 45 ks respectively. Using these data, two sources namely J17481 X1 and J17481 X2 are detected within the γ−ray error ellipse. A merged image is shown in Figure 4. However, J17481 X1 can only be detected in the 2017 observation and J17481 X2 can only be detected in the 2006 observation. Based on the limiting flux in the corresponding epoch of non-detection, we placed the limits on the long-term variability for J17481 X1 and J17481 X2 as > 2σ and > 3σ respectively. Both of them are potentially variable X-ray sources. The absorptioncorrected X-ray fluxes of J17481 X1 J17481 X2 tabulated in Table 4 is estimated with their count rates in the corresponding observation and a total Galactic HI column density of n H = 1.1 × 10 22 cm −2 . No optical/IR counterpart were found for these two sources. The net counts for both sources are < 50 cts and therefore no further analysis will be proceeded.
3FHL J1857.0+0059
3FHL J1857.0+0059 has been observed by XMM-Newton (Obs. ID. 0784040201) on 2016 October 13 for an effective exposure of ∼ 37 ks. Within its γ−ray ellipse, only one Xray source J18570 X1 is detected in this data (cf. Figure 4). The nature of J18570 X1 remains unidentified in SIMBAD. The P (n ≥ 1) inferred from this observation is ∼ 40%. The absorption-corrected X-ray flux of J1857 X1 as given in Table 4 is estimated by assuming a total Galactic HI column density of n H = 1.1 × 10 22 cm −2 . Searching for its multiwavelength counterpart does not yield any result.
On the other hand, we noted that a pulsar PSR J1857+0057 is lying within the 3FHL error circle. The angular separation between PSR J1857+0057 and J18570 X1 is ∼ 5.5 arcmin. Therefore, there is no association between these two objects. PSR J1857+0057 has a spin-down power of E = 4.7 × 10 31 erg/s (Manchester et al. 2005). At a distance of d ∼ 2.5 kpc as inferred by the dispersion measure of this pulsar, 3FHL J1857.0+0059/3FGL J1857.2+0059 has a luminosity of L γ ∼ 3.6 × 10 34 erg/s at energies > 100 MeV which is three orders of magnitude larger than E. Therefore, we concluded that PSR J1857+0057 cannot be associated with this γ−ray source.
There are ∼ 85 net counts collected from J18570 X1 in this XMM-Newton observation. In examining its Xray spectrum, we found that a best-fit power-law yields Right panel: X-ray image of J18233 X1 in 0.3-10 keV with data from MOS1 and MOS2 onboard XMM-Newton combined. This is the only X-ray source lies within the γ−ray error ellipse of 3FHL J1823.3-1339. An apparently extended X-ray feature is discovered at the location of Mercer 5. We overlay the X-ray contours on the infrared image for comparing the morphology at different wavelengths. Top is north and left is east. 3.2.6 3FHL J1800.7-2357 3FHL J1800.7-2357 has been observed by Chandra (Obs. ID. 10997) on 2010 July 30 for an effective exposure of ∼ 80 ks. 3FHL J1800.7-2357/3FGL J1800.8-2402 resides in a region where the supernova remnant (SNR) W28 interacts with a number of surrounding molecular cloud (MC) (Aharonian et al. 2008). A complex of TeV emission is found in this field (see Figure 8). 3FHL J1800.7-2357 apparently coincides with a protrusion at the eastern edge of HESS J1800-240B. This leads us to speculate whether this feature is indeed a distinct source or a part of the γ−rays from the SNR-MC interactions. For investigating this issue, VHE observation facility with improved spatial resolution is required (e.g. CTA).
Cui et al. (2018) have reported an updated analysis of this field with 9 years Fermi LAT data. In a 10-200 GeV sky map, they have found features which spatially match HESSJ1800-240A, HESSJ1800-240B and HESSJ1800-240C. HESSJ1800-240B is the brightest among them (Figure 1 in . The GeV spectra of both HESSJ1800-240B and HESSJ1800-240C show flux discontinuities which suggests there can be several emission components contribute to the γ−rays detected at their locations. While the emission below ∼ 1 GeV can come from the a nearby source with unknown origin, argue that the γ−rays with energies 1 GeV from all three spatial components of HESSJ1800-240 have a hadronic origin which is dominated by the interactions with the local sea of Galactic cosmic rays.
On the other hand, HESSJ1800-240B is potentially associated with a massive star formation region G5.89-0.39 (Hampton et al. 2016). This suggests finding young neutron stars or pulsars in this region is not unreasonable. However, the high density of X-ray sources in this region makes the probability of having more than one chance coincidence within the error ellipse of 3FHL J1800.7-2357, P (n ≥ 1), almost close to 100%. Eight X-ray sources are detected within the γ−ray error ellipse (Figure 4). Total Galactic HI column density of n H = 1.2 × 10 22 cm −2 and the count rates of these sources obtained in this observation are adopted for estimating their F x (cf. Table 4). Based on Gregory-Loredo variability algorithm, J18007 X1 is the only X-ray source that in this investigation that shows possible variability in a single observation with a probability > 90%. Its X-ray light curve is shown in Figure 9. Since the net counts collected from J18007 X1 is < 50 cts, we do not carry out any further analysis of this source.
We have also identified the optical/IR counterparts of J18007 X2, J18007 X3, J18007 X5, J18007 X6 and J18007 X8. By fitting the blackbody model to their SED, temperatures in the range of T ∼ 1300 − 21100 K are yielded (Table 5). For J18007 X5 and J18007 X8, their counterparts can also be found in Gaia DR2. Parallax measurements suggest J18007 X5 and J18007 X8 are located at the distance of 0.9 kpc and 2.5 kpc respectively. Adopting these distances, the blackbody radii of the optical/IR counterparts of J18007 X5 and J18007 X8 are found to be 0.2R and 2.4R respectively. tive exposure of ∼ 13 ks. Two X-ray sources, J14051 X1 and J14051 X2, are detected within the γ−ray error ellipse. The probability of chance coincidence is estimated to be P (n ≥ 1) ∼ 34%. Searching in SIMBAD, we found that both X-ray sources are unclassified. Their F x as given in Table 4 are estimated with their detected count rates and a total Galactic HI column density of n H = 1.9 × 10 22 cm −2 . J14051 X1 is among the brightest X-ray sources discovered in this work, which is detected at a S/N ratio of ∼ 24σ. ∼ 69 net counts from this source have been collected by ACIS-S in this observation and this allows us to perform a detailed analysis.
We found its X-ray spectrum can be well-described by an absorbed power-law model with a photon index of Γ x = 2.7 +1.4 −1.1 (Figure 10). It yields a goodness-of-fit of χ 2 = 13.97 for 21 d.o.f.. The best-fit column absorption is found to be n H = 1.5 +0.8 −0.5 × 10 23 cm −2 which is much larger than the total Galactic HI column density along the direction toward this source. Adopting this best-fit model, the absorptioncorrected X-ray flux becomes F x = 2.4 × 10 −12 erg cm −2 s −1 in 0.3-10 keV. On the other hand, its spectrum can also be fitted by an absorbed blackbody model and results in a comparable goodness-of-fit ( χ 2 = 13.71 for 21 d.o.f.). It yields an n H = 9.2 +5.1 −3.4 ×10 22 cm −2 and a temperature of kT = 1.3 +0.4 −0.3 keV. The best-fit normalization implies a thermal emission region with a radius of ∼ 10.6 d kpc m where d kpc is the distance to the source in unit of kpc. The best-fit blackbody model implies an absorption-corrected X-ray flux to be F x ∼ 3.0 × 10 −13 erg cm −2 s −1 in 0.3-10 keV.
We have also examined its X-ray temporal properties. Although the variability analysis with the Gregory-Loredo algorithm does not indicate any significant variability, a visual inspection of its barycentric-corrected light curve suggest a possible underlying structure. In particular, there appears to have two peaks separated by ∼ 5000 s which suggests a possible periodicity. Searches for the periodic signal around this value by epoch-folding yield a candidate signal at P ∼ 5090 s with χ 2 = 12.2 for 7 d.o.f.. The X-ray light curve of J14051 X1 folded at this putative period is shown in Figure 11. Although the statistical significance of this folded light curve for being different from a uniform distribution is low (pre-trial p−value ∼ 10%), its apparently sinusoidal nature makes it as a promising candidate for further investigation.
Visual examination of the X-ray image of J14051 X1 suggests it is possibly extended. A close-up view of J14051 X1 is shown in Figure 12. The source appears to be slightly elongated along the northwest-southeast orientation. Also, it apparently extends towards southwest. In order to further investigate its spatial nature, we compute its brightness profiles along the aforementioned orientations with sampling regions illustrated by the upper panels in Figure 13. In the lower-right panel of Figure 13, we show the brightness profile of J14051 X1 along the northwestsoutheast orientation. The source appears to have an extent of ∼ 10 arcsec towards northwest. As J14051 X1 has an Figure 12. A smoothed image of the field around J14051 X1 as observed by Chandra ACIS-S3 in 0.3-8 keV. The black cross illustrates the X-ray position of J14051 X1 as given in Table 4. The source is apparently elongated along the NW-SE orientation and it also appears to be extended towards SW.
off-axis angle of ∼ 4.1 arcmin in this observation, the apparent elongation can be a result of distorted point spread function (PSF). To examine this, we have used the Chandra Ray Tracer (ChaRT) to simulate the PSF. The adopted inputs for simulating the PSF are the energy spectrum of J14051 X1 with the same exposure, roll, and off-axis angle as in the ACIS-S3 observation. Then we computed the brightness profile of the simulated data with the same set of sampling regions in the upper-right panel of Figure 13. The result is displayed as the dotted line in the lower-right panel of Figure 13, which matches the observed profile pretty well. Hence, we conclude that the elongation of J14051 X1 along the northwest-southeast orientation is due to the degraded angular resolution as a result of large off-axis angle.
On the other hand, the brightness profile for the southwestern extended feature is shown in the lower-left panel of Figure 13. The feature appears to have an extension of ∼ 20 arcsec towards southwest before it falls to the background. We have also compared the observed profile with the simulated PSF. In this direction, the profile of the simulated PSF falls to the background within the the bin corresponds to the peak in the observed profile. Therefore, this ∼ 20 arcsec extent cannot be accounted by the distorted PSF. The signal-to-noise ratio of this feature is ∼ 4σ. We have examined the Digitized Sky Survey optical image for the region of this feature. We do not find any optical counterpart to account for this putative extended X-ray feature.
We have also identified the IR counterpart of J14051 X1. Using the extinction of A v = 8.5 as inferred from the Galactic HI column density, we constructed the extinction-corrected SED. Fitting a blackbody to this SED yields a temperature of T bb ∼ 2013 K and a emitting area with a radius of R bb ∼ 6.6R d kpc with both T bb and R bb as free parameters.
3FHL J1626.3-4915
Part of the positional error ellipse of 3FHL J1626.3-4915 has been covered by a Chandra ACIS-I observation (Obs. ID. 13287) on 2012 June 16 for an effective exposure of ∼ 10 ks. It is also partially covered by an XMM-Newton observation (Obs ID. 0403280201) on 2007 February 14. However, this observation was seriously contaminated by high background. After removing these contaminated time intervals, an effective exposure of ∼ 6 ks is remained in the XMM-Newton data. In the Chandra observation, two sources, J16263 X1 and J16263 X2, are detected. The brighter one J16263 X1 can also be detected in the short XMM-Newton exposure. The difference of its flux in these two frames is only ∼ 2σ. For J16263 X2, it is below the detection threshold in the XMM-Newton observation. This places a limit of > 3.7σ on its longterm variability. Another source J16263 X3, which is not covered by the FoV of the Chandra observation, is detected by XMM-Newton. From the sources serendipitously detected in these observations, the probability of chance coincidence is found to be P (n ≥ 1) ∼ 99% and P (n ≥ 1) ∼ 61% in the Chandra and XMM-Newton observation respectively. For computing the F x of all three detected sources as given in Table 4, we adopt a column absorption of n H = 1.9 × 10 22 cm −2 based on the HI estimate and their net count rates.
∼ 87 net counts are collected from J16263 X1 alto-gether from Chandra and XMM-Newton data. This allows us to carry out a more detailed analysis. We found that its X-ray spectrum can be fitted equally well with both absorbed power-law ( χ 2 = 16.31 for 23 d.o.f.) and absorbed blackbody models ( χ 2 = 15.78 for 23 d.o.f.). The bestfit power-law model yields a column absorption of n H = 6.5 +10.3 −6.5 × 10 21 cm −2 , a photon index of Γ x = 2.0 +0.7 −0.6 and an unabsorbed flux F x = 2.7 +3.8 −0.9 × 10 −13 in 0.3-10 keV. For the best-fit blackbody model, it yields n H < 3.4 × 10 21 cm −2 , a temperature kT = 0.9 ± 0.1 keV, emitting area with a radius of R = 4.5 +2.1 −4.5 d k pc km, and an unabsorbed flux F x = (1.2 ± 0.2) × 10 −13 in 0.3-10 keV. The goodness-of-fit for both models are comparable (see Table 6) For J16263 X3, there are ∼ 130 net counts collected by XMM-Newton. By fitting its spectrum with an absorbed power-law, we obtain the best-fit results of n H = 8.4 +2.4 −2.0 × 10 21 cm −2 , Γ x = 4.0 +0.7 −0.6 and an unabsorbed flux F x 1.9 × 10 −12 in 0.3-10 keV. On the other hand, the best-fit blackbody model yields n H = 1.6 +1.9 −1.5 × 10 21 cm −2 , kT = 0.4 ± 0.1 keV, an emitting area with a radius of R = 68 +40 −21 d kpc m, and an unabsorbed flux F x ∼ 1.2 × 10 −13 in 0.3-10 keV. Although the power-law model yields a better goodness-of-fit ( χ 2 = 10.51 for 23 d.o.f.), its photon index is too steep to account for any reasonable non-thermal emission scenario.
We have identified the optical/IR counterparts of J16263 X1 and J16263 X2. For J16263 X1, after correcting the extinction with A v = 8.9 by assuming the total Galactic HI column density, its optical/IR counterpart can be described by a blackbody of T bb ∼ 2000 K and R bb ∼ 1.1R d kpc . However, its SED only has two data points and do not allow us to properly constrain its blackbody parameters and compute the uncertainties. For J16263 X2, we found that the adopted A v = 8.9 might over-correct the extinction and results in an unphysical high blackbody temperature. On the other hand, a possible counterpart of J16263 X2 is identified by Gaia and place an estimate on its distance to be 3.6 kpc. Using the Chandra observation in 2010, eight X-ray sources were detected within the γ−ray ellipse. In the 2000 Chandra observation, only J17472 X1, J17472 X2, J17472 X4, J17472 X5 and J17472 X6 can be detected. In comparing these two frames, J17472 X5 is found to be variable at the level of 4.3σ. On the other hand, the nondetections of J17472 X3, J17472 X7 and J17472 X8 place limits on their variabilities to be > 1.9σ, > 1.1σ and > 0.3σ respectively. In all the XMM-Newton observations, only J17472 X1 can be detected. Its flux as measured by these observations are all consistent with that obtained in Chandra observation.
Their unabsorbed F x given in Table 4 are estimated by their count rate detected in the 2010 Chandra observation with the total Galactic HI column density n H = 1.2 × 10 22 cm −2 in that direction. Since 3FHL J1747.2-2822 lies along the direction towards the Galactic centre, the spatial density of the X-ray sources is rather high. Using the serendipitous X-ray sources detected in these data, P (n ≥ 1) is estimated to be as high as ∼ 100% in all observations. Among the detected X-ray sources, J17472 X1 is the brightest and is detected by all observations. However, we found that it coincides with the giant molecular cloud Sgr B2. The association is further confirmed by the detection of the iron line at 6.4 keV which is likely to be originated from the interaction between the hard X-rays from the Sgr A* and the cloud (Dogiel et al. 2015). Therefore, J17472 X1 is not the main interest for this work and will not be further concerned. For the other seven X-ray sources, their net counts are all less than 50 and no further analysis will be performed.
We have also identified an optical/IR counterpart of J17472 X8. Using an extinction of A v = 5.7 as inferred from the total Galactic HI column density, a blackbody fitting to the extinction-corrected SED yields a temperature of T bb = 1900 ± 200 K and an emitting area with a radius of R bb = 11 ± 2d kpc R . . Eight X-ray sources have been found within its γ−ray error ellipse by using the longest Chandra observation at 2004 December. Based on their count rates obtained from this observation and the adopted column absorption of n H = 4.5 × 10 20 cm −2 which is consistent with the total Galactic HI column density in that direction, we estimated their unabsorbed F x in Table 4. J07375 X1 is the brightest X-ray sources among them. Its flux is found to be significantly variable at a level as high as ∼ 12σ. The shortest timescale of its X-ray variability found in this study is ∼ 4 months. J07375 X2 and J07375 X5 also exhibit long-term X-ray varibility at the level up to 8σ and 4σ among these observations. J07375 X1 and J07375 X2 have 1096 and 454 net counts collected from all these archival data respectively, and therefore we have carried out a more detailed analysis. Their photon statistics from each observation are high enough to allow us performing multi-epoch spectral analysis. By fixing n H at 4.5 × 10 20 cm −2 , the best-fit parameters in each epoch of J07375 X1 and J07375 X2 are summarized in Table 7 and Table 8 respectively. Evidences for spectral variabilities are found from both sources.
Among eight X-ray sources, six of them have optical/IR counterparts identified. The results of blackbody fitting to their SEDs are summarized in Table 5. Their inferred low temperatures and small radii suggest they can possibly be late-type stars. For J07375 X3, counterpart has also been found by Gaia which suggests a distance of 1.6 kpc. Table 8. X-ray spectral properties of J07375 X2 at different epochs.
SUMMARY & DISCUSSIONS
With an optimal set of features selected by RFE algorithm (see Table 1 & Figure 1), a supervised classification model is built from a training set of labeled PSR/NON PSR 3FHL objects. Using this model, we have selected 27 PSR-like objects with a nominal accuracy of ∼ 98% from the unknown 3FHL sources for identification campaign (see Table 3). Utilizing the archival X-ray data, we have found X-ray counterparts from 10 3FHL PSR-like candidates (see Table 4 & Figure 4). These identifications allows us to systematically constrain the positions of the potential X-ray counterparts to arcsecond accuracies, estimate the X-ray to γ−ray flux ratios F x /F γ and temporal variabilities. Except for J18007 X8 and J17472 X8, the F x /F γ for all the other X-ray sources conform with that for the known pulsars detected in the energy range of 10 GeV to 2 TeV. For the sources with their Xrays found to be significantly varying in a given observation window and/or across different epochs, their flux variabilities make them less likely to be a typical pulsar which has rather stable X-ray emission. On the other hand, we cannot exclude the possibility of these variable X-ray sources as γ−ray binaries. Also, their X-ray positions enable us to search for the optical/IR counterparts and estimate the surface temperatures and sizes of the possible companion stars by assuming a blackbody model (see Table 6). For those have more than 50 net counts collected from the archival X-ray data, we have carried out more detailed analysis. Among them, J18233 X1 which is associated with 3FHL J1823.3-1339 is one of the most interesting source. They are very likely to be the X-ray and γ−ray counterparts of the globular cluster Mercer 5 (cf. Figure 6). The association between J18233 X1 and Mercer 5 is supported by the consistency between the column absorption obtained from the X-ray spectral fitting and that deduced from the optical extinction. On the other hand, the association between 3FHL J1823.3-1339 and Mercer 5 is suggested by the agreement between its γ−ray luminosity L γ at the distance of the globular cluster and the general trend of L γ −[Fe/H] as observed in the γ-ray globular cluster population (Hui et al. 2011).
Because of the frequent stellar encounters, globular clusters are efficient in producing compact binaries, including millisecond pulsars (MSPs), through dynamical interactions (Pooley et al. 2003;Hui et al. 2010). It is a general consensus that the γ−ray emission from a globular cluster is originated from its MSPs. Therefore, we speculate that Mercer 5 is hosting a MSP population awaited to be discovered. Pulsar searches targeted at this cluster are encouraged to examine this assertion.
There are two different scenarios in explaining the γ−ray emission mechanism of a globular cluster. While their γrays can be collectively contributed by the magnetospheric radiation from the MSPs (Abdo et al. 2010), it is possible that the inverse Compton scattering (ICS) between the relativistic pulsar wind and the ambient soft photons can result in the observed γ-rays . The ICS scenario is suggested by the correlation between L γ and the energy densities of the ambient soft photon fields (Hui et al. 2011). Such scattering can boost the soft photons to an energy > 10 GeV . As the γ-ray spectrum of a pulsar typically has an exponential cut-off at a few GeV, their magnetospheric radiation is unlikely to have significant contribution in the hard γ-ray band. Therefore, the globular clusters detected at energies > 10 GeV can help us to constrain the parameters of the ICS model . Although there are 30 γ−ray globular clusters have been identified in the Fermi LAT 8 years point source catalog (Fermi LAT collaboration 2019), only two of them, 47 Tuc and Terzan 5, are included in the 3FHL catalog. On the other hand, Figure A3 in de Menezes et al. (2019) shows that 2MS-GC01, NGC6440 and NGC2808 seem to have emission above 10 GeV. The survey with the upcoming CTA holds the potential in further expanding the population of hard γ-ray globular clusters.
Apart from MSPs, globular clusters also host different classes of compact X-ray binaries (e.g. low-mass X-ray binaries, cataclysmic variables). Ascribing to the relatively poor spatial resolution of XMM-Newton, the X-ray counterpart of Mercer 5, J18233 X1, identified in this work is likely resulted from a blend of unresolved X-ray point source population. Its X-ray spectrum can be well-described by a power-law model with Γ x ∼ 1.1 (Figure 7) which is apparently harder than the faint unresolved X-ray populations found in the other clusters (Hui et al. 2009). A spectral imaging analysis with high spatial resolution by Chandra is necessary to resolve and classify the X-ray binaries in Mercer 5.
Another interesting identification in our campaign is 3FHL J1405.1-6118 and its promising X-ray counterpart J14051 X1. The X-ray spectrum of J14051 X1 can be described by a power-law of Γ x ∼ 2.7 with a large column absorption n H ∼ 1.5 × 10 23 cm −2 (Figure 10). Such large X-ray absorptions are commonly seen in the high-mass Xray binaries (HMXBs) (Paul et al. 2017). Very recently, a γ-ray periodic modulation of P b ∼ 13.7 days in 0.2-500 GeV has been discovered which makes 3FHL J1405.1-6118 (= 4FGL J1405.1-6119) the third γ−ray binary found from the initial discovery of periodic modulation of the LAT light curve (Corbet et al. 2019). X-ray modulation of J14051 X1 at the same period has also been found by Swift XRT (cf. Figure 4 in Corbet et al. (2019)). Taking the phase zero at MJD 56498.7, Chandra exposure used in our work corresponds the orbital phase of ∼ 0.07−0.08 which is not included in Corbet et al. (2019). Using our best-fit spectral model and with the aid of PIMMS, the flux observed by Chandra translates into a Swift XRT count rate of ∼ 10 −3 cts/s which is consistent with that in the phase interval of ∼ 0.03 − 0.25 as reported in the Table 1 of Corbet et al. (2019).
For the optical/IR counterpart of 3FHL J1405.1-6118, the blackbody fitting to its SED yields a temperature of T ∼ 2000 K and a radius of R ∼ 7R in the case that we adopt A v = 8.5, d = 1 kpc and with both T bb and R bb as free parameters. On the other hand, based on the near-IR spectroscopy, Corbet et al. (2019) identify the counterpart as an O6 III star. To examine whether this inference can be consistent with our photometric result, we redo the blackbody fitting with T bb fixed at 40000 K which is typical for O stars and adopt the mean derived extinction of A v = 31.6 (E (B − V) = 10.2) reported by Corbet et al. (2019). At a distance of d = 7.7 kpc (Corbet et al. 2019), it yields a radius of R bb = 12.9 ± 0.8R which is consistent with the expected size for an O6 III star.
While Corbet et al. (2019) have found the orbital period of ∼ 13.7 days, the X-ray light curve observed by Chandra suggests a periodicity candidate at P ∼ 1.4 hrs ( Figure 11). For the HMXBs with the X-ray pulses from the neutron stars detected, their spin periods span a range from ∼ 0.03 s to ∼ 4 hrs (Liu et al. 2006). Therefore, this signal can possibly be originated from the neutron star rotation. Deeper follow-up observations are strongly encouraged to examine this putative signal.
Apart from the periodic signal candidate, this short Chandra observation also reveals a putative extended X-ray feature associated with J14051 X1 at a significance of ∼ 4σ (Figures 12 & 13). A deeper observation is required to confirm its spatial nature with higher signal-to-noise ratio and examine if there is any spectral variation across it. Evidences of such X-ray features have been found from a number of γray binaries, including PSR B1259-63/LS 2883 (Pavlov et al. 2015), LS I+61 303 (Paredes et al. 2007), andLS 5039 (Durant et al. 2011). Except for PSR B1259-63/LS 2883, the nature of the compact objects for the other γ−ray binaries remain unknown. For PSR B1259-63/LS 2883, its extended X-ray feature can be resulted from synchrotron radiation emitted by the relativistic particles accelerated at the shock between the pulsar wind and the massive star outflow (Tavani & Arons 1997). On the other hand, if the γ-ray binary is powered by a microquasar, the extended X-ray nebula can be originated from the relativistic particles produced by the Blandford-Znajek process (Blandford & Znajek 1977) or from an MHD jet. Pulsar searches of 3FHL J1405.1-6118 can help to discriminate these two competing scenarios.
Figure 4 .
4X-ray sources (solid circles) found within the 95% γ−ray positional error ellipses (dashed ellipses)of our selected PSR-like 3FHL sources. Top is north and left is east in all images.
Figure 6 .
6Left panel: K s band image of a field centered at Mercer 5 obtained by 2MASS. A central concentration of stars can be noted.
Figure 7 .
7The X-ray spectra of J18233 X1 which is positionally coincident with the globular cluster Mercer 5 as observed by XMM-Newton MOS1/2 + PN cameras and simultaneously fitted to an absorbed power-law(upper panel) and contribution to the fitting residuals (lower panel). n H = 3.0 +1.8 −1.3 × 10 22 cm −2 , Γ x = 1.3 +0.7 −0.6 and F x ∼ 5.8 × 10 −14 erg cm −2 s −1 in 0.3-10 keV. And a best-fit blackbody yields n H = 1.3 +1.1 −0.7 × 10 22 cm −2 , kT = 1.6 +0.6 −0.4 keV and F x ∼ 3.6 × 10 −14 erg cm −2 s −1 in 0.3-10 keV.
Figure 8 .
8A γ−ray excess map at energies > 0.1 TeV of the W28 field as obtained by H.E.S.S.(Aharonian et al. 2008). The VHE sources HESS J1801-233 and the complex of HESS J1800-240 (regions A, B & C) can be clearly seen. The location and the angular size of the supernova remnant W28 is illustrated by the dotted yellow circle. The dotted white ellipse is the 95% confidence γ−ray positional uncertainty of 3FHL J1800.7-2357.
3.2.7 3FHL J1405.1-6118 3FHL J1405.1-6118 has been observed by Chandra ACIS-S on 2013 September 19 (Obs. ID. 14888) for an effec-
Figure 9 .
9The X-ray light curve of J18007 X1 with a bin size of 7500 s as observed by Chandra in 0.3-7 keV.
Figure 10 .
10The X-ray spectrum of J14051 X1 as observed by Chandra ACIS-S with the best-fitted absorbed power-law(upper panel) and contribution to the fitting residuals (lower panel).
Figure 11 .
11X-ray light curve of J14051 X1 in 0.5-7 keV as obtained by Chandra ACIS-S. It is folded at the period of P = 1.4 hrs.
Figure 13 .
13The brightness profile of the putative SW extended feature associated with J14051 X1 (lower-left panel) as sampled from regions in the Chandra ACIS-S3 raw image (upper-left panel). The brightness profile of J14051 X1 along the NW-SE orientation as sampled from the regions illustrated in the upper-right panel is displayed in the lower-right panel. The dotted line in this plot represents the expected profile from a point-like source. The average background level and its 1σ deviation are indicated by horizontal lines that were calculated by sampling from the source-free regions.
located along the line-of-sight towards the Galactic centre. The 95% γ−ray positional error ellipse of 3FHL J1747.2-2822 has been covered by two Chandra observations: Obs.IDs.: 944 (2000 March 29; 100 ks), 11795 (2010 July 29; 98 ks) and five XMM-Newton observations Obs.IDs: 0802410101 (2018 April 2; 99 ks), 0694641401 (2012 September 30; 34 ks), 0694641301 (2012 September 26; 47 ks), 0694640601 (2012 September 6; 41 ks), and 0203930101 (2004 September 4; 33 ks).
Table 3. 27 PSR candidates selected from 3FHL catalog. θ 95 are their γ−ray positional uncertainty at 95% confidence level.3FHL name
RA (J2000) Dec (J2000)
θ 95
3FGL name
PSR confidence score
h m s
d m s
degree
3FHL J1748.6-2816
17 48 38.4
-28 16 41
0.035
3FGL J1748.3-2815c
0.858
3FHL J1839.4-0553
18 39 24.3
-05 53 46
0.040
3FGL J1839.3-0552
0.852
3FHL J1823.3-1339
18 23 21.7
-13 39 46
0.031
3FGL J1823.2-1339
0.84
3FHL J1748.1-2903
17 48 08.9
-29 03 42
0.039
3FGL J1747.7-2904
0.79
3FHL J1139.2-6248
11 39 16.7
-62 48 09
0.044
3FGL J1139.0-6244
0.768
3FHL J1857.0+0059
18 57 05.7
+00 59 23
0.058
3FGL J1857.2+0059
0.668
3FHL J1753.8-2537
17 53 48.1
-25 37 46
0.026
3FGL J1754.0-2538
0.66
3FHL J1802.3-3043
18 02 23.7
-30 43 20
0.049
3FGL J1802.4-3043
0.658
3FHL J1907.0+0713
19 07 00.6
+07 13 43
0.046
...
0.602
3FHL J1800.7-2357
18 00 44.1
-23 57 12
0.051
3FGL J1800.8-2402
0.59
3FHL J1603.3-6011
16 03 22.9
-60 11 59
0.048
3FGL J1603.7-6011
0.522
3FHL J1855.5+0142
18 55 35.8
+01 42 55
0.047
...
0.516
3FHL J1405.1-6118
14 05 06.2
-61 18 06
0.034
3FGL J1405.1-6119
0.466
3FHL J1306.3-6042
13 06 22.5
-60 42 39
0.028
3FGL J1306.4-6043
0.45
3FHL J1626.3-4915
16 26 23.9
-49 15 23
0.076
3FGL J1626.2-4911
0.418
3FHL J1112.5-6054
11 12 33.5
-60 54 40
0.072
3FGL J1111.9-6058
0.4
3FHL J1747.2-2822
17 47 17.7
-28 22 00
0.033
...
0.372
3FHL J1824.3-0621
18 24 18.0
-06 21 05
0.045
3FGL J1824.3-0620
0.364
3FHL J0725.6-5008
07 25 39.1
-50 08 25
0.043
3FGL J0725.4-5007
0.334
3FHL J0541.1-4855
05 41 10.7
-48 55 43
0.072
...
0.166
3FHL J1915.2-1323
19 15 16.4
-13 23 30
0.051
...
0.162
3FHL J0737.5+6534
07 37 35.3
+65 34 43
0.033
...
0.158
3FHL J1657.6-4656
16 57 37.1
-46 56 54
0.095
3FGL J1657.6-4653
0.144
3FHL J1803.1-6709
18 03 10.7
-67 09 49
0.053
3FGL J1803.3-6706
0.142
3FHL J1200.3+0201
12 00 22.7
+02 01 44
0.061
3FGL J1200.4+0202
0.132
3FHL J0110.9+4346
01 10 56.5
+43 46 54
0.082
...
0.124
3FHL J0115.4-2916
01 15 24.2
-29 16 57
0.055
...
0.118
Table 4 .
4Properties of X-ray Sources within γ-ray Error Ellipses (95% Confidence) of Selected 3FHL Unidentified ObjectsSource
R.A. (J2000) Decl. (J2000)
σ pos
Signifi.
Inst.
Counts rate
F unabs
0.3−10k eV
Table 7 .
7X-ray spectral properties of J07375 X1 at different epochs.3.2.10 3FHL J0737.5+65343FHL J0737.5+6534 is located very far away from the Galactic plane with b ∼ 29 • . Its γ−ray positional error ellipse has been covered by three Chandra observations with Obs.IDs:
© 2015 The Authors
C. Y. Hui et al.
MNRAS 000, 1-19(2015)
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| []
|
[
"RADIO PROPERTIES OF YOUNG STELLAR OBJECTS IN THE CORE OF THE SERPENS SOUTH INFRARED DARK CLOUD",
"RADIO PROPERTIES OF YOUNG STELLAR OBJECTS IN THE CORE OF THE SERPENS SOUTH INFRARED DARK CLOUD"
]
| [
"Nicholas S Kern ",
"Jared A Keown ",
"John J Tobin ",
"Adrian Mead ",
"Robert A Gutermuth "
]
| []
| []
| We present deep radio continuum observations of the star-forming core of the Serpens South Infrared Dark Cloud with the Karl G. Jansky Very Large Array (VLA). Observations were conducted in two bands centered at 7.25 GHz (4.14 cm) and 4.75 GHz (6.31 cm) with a σ rms of 8.5 and 11.1 µJy/beam, respectively. We also use 2MASS, Spitzer and Herschel data to put our radio observations in the context of young stellar populations characterized by near and far infrared observations. Within a 5' x 5' region of interest around the central cluster, we detect roughly eighteen radio sources, seven of which we determine are protostellar in nature due to their radio spectral indices and their association with infrared sources. We find evidence for a previously undetected embedded Class 0 protostar and reaffirm Class 0 protostellar classifications determined by previous millimeter wavelength continuum studies. We use our infrared data to derive mid-infrared luminosities for three of our protostellar sources and find relative agreement between the known YSO radio luminosity vs bolometric luminosity correlation. Lastly, we marginally detect an additional six radio sources at the 2-3σ level that lie within two arcseconds of infrared YSO candidates, providing motivation for higher sensitivity studies to clarify the nature of these sources and further probe embedded and/or low luminosity YSOs in Serpens South. | 10.3847/0004-6256/151/2/42 | [
"https://arxiv.org/pdf/1511.09082v1.pdf"
]
| 118,423,501 | 1511.09082 | c9e6ca0250fd8eb92c68110379b9f0be655a4340 |
RADIO PROPERTIES OF YOUNG STELLAR OBJECTS IN THE CORE OF THE SERPENS SOUTH INFRARED DARK CLOUD
29 Nov 2015
Nicholas S Kern
Jared A Keown
John J Tobin
Adrian Mead
Robert A Gutermuth
RADIO PROPERTIES OF YOUNG STELLAR OBJECTS IN THE CORE OF THE SERPENS SOUTH INFRARED DARK CLOUD
29 Nov 2015Submitted to AJ Submitted to AJPreprint typeset using L A T E X style emulateapj v. 5/2/11
We present deep radio continuum observations of the star-forming core of the Serpens South Infrared Dark Cloud with the Karl G. Jansky Very Large Array (VLA). Observations were conducted in two bands centered at 7.25 GHz (4.14 cm) and 4.75 GHz (6.31 cm) with a σ rms of 8.5 and 11.1 µJy/beam, respectively. We also use 2MASS, Spitzer and Herschel data to put our radio observations in the context of young stellar populations characterized by near and far infrared observations. Within a 5' x 5' region of interest around the central cluster, we detect roughly eighteen radio sources, seven of which we determine are protostellar in nature due to their radio spectral indices and their association with infrared sources. We find evidence for a previously undetected embedded Class 0 protostar and reaffirm Class 0 protostellar classifications determined by previous millimeter wavelength continuum studies. We use our infrared data to derive mid-infrared luminosities for three of our protostellar sources and find relative agreement between the known YSO radio luminosity vs bolometric luminosity correlation. Lastly, we marginally detect an additional six radio sources at the 2-3σ level that lie within two arcseconds of infrared YSO candidates, providing motivation for higher sensitivity studies to clarify the nature of these sources and further probe embedded and/or low luminosity YSOs in Serpens South.
INTRODUCTION
Serpens South is a young stellar cluster that is a part of the broader Aquila Rift complex of dark clouds. Discovered in 2008 by Gutermuth et al. (2008) as a part of the Spitzer Space Telescope's Gould Belt Legacy Survey, Serpens South has been found to harbor an unusually high ratio of Class I to Class II young stellar objects (YSO), where a Class I YSO is representative of the late mass accretion phase onto a central protostar and Class II of a classical T Tauri pre-main sequence star (see Greene et al. 1994). This suggests that Serpens South is in a very early phase of cluster formation and makes it one of the most active sites of star formation within 1 kpc. Since its discovery, it has become the center of a wide range of scholarship. This has consisted of near, mid and far infrared mappings with Spitzer and Herschel tracing heated dust around protostars (Gutermuth et al. 2008;Bontemps et al. 2010), millimeter mappings tracing cold dust (Maury et al. 2011), near infrared polarimetry revealing the importance of global magnetic fields in the cluster's formation history ), molecular outflows studies (e.g. Nakamura et al. 2011;Teixeira et al. 2012;Plunkett et al. 2015a,b), and a wealth of spectral line surveys probing filamentary infall (e.g. Kirk et al. 2013;Friesen et al. 2013;Tanaka et al. 2013;Fernández-López et al. 2014;Nakamura & Li 2014). In spite of the wealth of optical, infrared and submillimeter data currently available, Serpens South has received little attention in the radio continuum. Presently, only one radio continuum study of Serpens South has been conducted by Ortiz-León et al. (2015), who did not detect any radio sources associated with known YSOs in the central core of Serpens South.
Because the core of Serpens South has been shown to harbor a high density of YSOs in the earliest phases of their development, it is an interesting region to search for highly embedded YSOs and allows for a large number of possible protostellar radio detections with only one telescope pointing. The Karl G. Jansky Very Large Array (VLA) is an ideal instrument for this aim. It has been a proven tool for detecting radio emission around YSOs since the early 1990s and its recently upgraded capabilities make it an even more powerful tool for this purpose (e.g., Curiel et al. 1989;Anglada et al. 1998;Beltrán et al. 2001;Shirley et al. 2007;Dzib et al. 2013). The goal of this paper is to for the first time shed light on the radio properties of the clustered protostars in the core of Serpens South.
The study of radio emission around YSOs is an important asset to star formation studies because radio emission can penetrate the high column densities that obscure YSOs at optical and sometimes infrared wavelengths. In the case of highly embedded and very young objects, radio emission can provide evidence for the presence of a central source (Andre et al. 2000). Recent studies at millimeter wavelengths have also shown that embedded, low-luminosity protostars can go undetected and overlooked in near infrared imaging of starless cores (e.g. Schnee et al. 2012), making long-wavelength studies of star forming regions essential to fully understanding prestellar populations. Additionally, the shape of a YSOs spectral energy distribution (SED) at radio wavelengths can provide information about collimated out-flows, magnetic field activity and other high-energy processes around a protostar (Feigelson & Montmerle 1999).
The distances to Serpens South, W40 and the Serpens Main cloud are not agreed upon in the literature. When Serpens South was discovered in 2008, Gutermuth et al. (2008) adopted a distance of 260 pc ± 37 pc based on evidence that its LSR velocities matched LSR velocities of the Serpens Main cloud, which was then thought to be a part of the larger Aquila Rift complex estimated to lie at 260 pc (Straižys et al. 2003). However, VLBA parallax measurements conducted in 2010 have established the distance to the Serpens Main cloud as 429 ± 2 pc (Dzib et al. 2011). Gutermuth et al. (2008) also argued that Serpens South lies in front of W40, claiming that its cold filaments are seen in absorption against emission from W40. If we are to follow the initial LSR velocity analysis by Gutermuth et al. (2008), we would equate Serpens South to Serpens Main and say Serpens South lies at approximately 429 pc, while W40 lies further away. Indeed, previous radio and x-ray studies adopt a distance of 600 pc to W40, although they admit the distance is poorly constrained (Kuhn et al. 2010;Rodríguez et al. 2010). Here, we adopt a distance of 429 pc to Serpens South, similar to other recent studies of the region (Plunkett et al. 2015a;Ortiz-León et al. 2015).
OBSERVATIONS AND DATA
We observed Serpens South with one pointing on July 2, 2013, for 1 hour with the VLA in its C array configuration under project number DEM0009. In order to derive a more accurate spectral index, we configured our C band observation to have subbands centered at 4.75 GHz (6.31 cm) and 7.25 GHz (4.14 cm), each with bandwidths of 1.024 GHz. At 4.75 and 7.25 GHz, our images had primary beam diameters of 9.5 and 6.2 arcminutes respectively. Our observation focused on a 5 arcmin x 5 arcmin region around Serpens South's central filament, with a phase center positioned at α(J2000) = 18 h 30 m 05.00 s , δ(J2000) = −02 • 02 ′ 30.0 ′′ ( Figure 1). During our hour-long observation, we switched from Serpens South to J1804 + 0101 every 10 minutes for complex gain calibrations, giving us a total of 45 minutes on source. At the time of our observation two antennas were not functioning properly, leaving us with a total of twenty five antennas.
We manually flagged, calibrated and imaged our data with standard procedures using Common Astronomy Software Applications (CASA) 4.1.0. We also thoroughly inspected the data and manually flagged for obvious radio frequency interference (RFI). We used J1331+305 (3C286) as a flux and bandpass calibrator, and J1804+0101 as a gain and phase calibrator (S 4.75GHz = 0.70 ± 0.02 Jy, S 7.25GHz = 0.66 ± 0.02 Jy). We deconvolved the Stokes I images with the Cotton-Schwab algorithm (Schwab 1984) using the CLEAN method (Högbom 1974;Clark 1980). We experimented with natural, robust and uniform weighting, and found the best compromise between noise level and source resolution using robust weighting (Briggs 1995), with the robust parameter set to 0.5. The synthesized beam sizes and RMS values for our two images are detailed in Table 1. We performed a primary beam correction on our images by dividing the CLEANed images by the modeled flux re- 3.1 x 2.5 13.4 8.5 a 4.75 and 7.25 GHz is equivalent to 6.31 and 4.14 cm respectively. b Using robust weighting in the CLEAN method sponse of the antennas.
We cross-referenced the positions of our radio sources with the 2MASS catalogue and ran extractions over the four IRAC bands of Spitzer, the 24 µm MIPS band on Spitzer, and the 70 µm PACS band on Herschel. We use this wide range of infrared data to help determine if a radio source is associated with a YSO or if it is extragalactic. We used a 2 arcsecond maximum matching tolerance for these extractions. Apertures and adopted corrections were consistent with Gutermuth et al. (2009). The final errors on our infrared fluxes consist of the zero flux offset error used to convert magnitudes to fluxes (2% contribution) and an assumed 10% error on our final magnitudes, which encompasses the absolute flux calibration. We were able to produce at least one infrared source extraction on six of our radio sources, whose infrared spectral energy distributions are shown in Figure 2.
METHODOLOGY
In choosing our sources, we restrict ourselves to a circular region centered on our phase center extending out to 50% of the primary beam response, which equates to roughly 5 arcminutes in diameter. We identify our sources based on a visual inspection of the data, judging their appearance and strength in both our 7.25 and 4.75 GHz images and considering their spatial proximity to infrared associations. Similar to other radio studies, we consider a detection firm if it has a clear infrared association and has a radio flux of at least 4σ in either frequency band. In addition, we consider all objects with radio fluxes > 5σ as sources. Given our beam size relative to the size of our field of view, there is less than a 1% chance that even one random Gaussian noise fluctuation would produce a signal in excess of 5σ in any one of our synthesized beams. This left us with a total of 18 radio sources in our field of view. We note that we detect an additional ∼6 sources at the 2-3σ level that lie within 2 arcseconds of an infrared YSO candidate, however, we do not include them in our final source list. Further observations with higher sensitivity will likely be able to determine whether these sources are spurious, extragalactic or protostellar in nature.
After applying primary beam corrections, we used CASA's IMFIT function to fit 2D gaussians and derive integrated flux densities for each of our radio sources. Flux errors were calculated by adding three sources of error in quadrature: the flux error output from IMFIT, an assumed 5% systematic absolute error on the flux calibration, and the percent uncertainty due random pointing errors in the individual antennas following the prescription outlined in Dzib et al. (2014). Source positions were determined by using the centers of IMFIT's 2D Gaussian fits to the higher resolution 7.25 GHz image. (Gutermuth et al. in prep.). The green box indicates our 5 arcmin by 5 arcmin region of interest with the VLA, while the cyan box shows the extent of the zoomed-in figure to the right. Right: VLA 7.25 GHz radio continuum of the core of Serpens South. The circles represent Spitzer Class I and II YSOs.
It is unlikely that all of our radio sources will be associated with YSOs; there will be a level of contamination from background galaxies that emit in the radio. We can calculate the number of random background sources we would expect to find in our radio images. We use the formulation found in Shirley et al. (2007) and Dzib et al. (2013) who draw from radio studies done by Fomalont et al. (1991). The density of random background radio sources above a flux limit of S µJy at 6 cm (4.9 GHz) is given by
n(> S) = 0.42 ± 0.05 S 30 µJy −1.18±0.19 arcmin −2 .
Therefore, the number density of sources with flux S greater than 50 µJy at 6 cm is 0.23 arcmin −2 . This leads to a 0.1% chance that a background source falls inside a 4.8 x 3.9 arcsec synthesized beam centered on a radio source. It also gives us on average ∼ 6 ± 1.5 background sources within our 5' x 5' region of interest above a 5σ level. There are about 7 to 9 sources that we identify as extragalactic, which agrees with this analysis within two standard deviations at worst.
Radio and Infrared Spectral Indices
Radio emission from a YSO can have two components: a thermal component coming from free-free bremsstrahlung of an ionized region and a non-thermal component generally in the form of gyrosynchrotron emission created by magnetic fields in the stellar corona. While high mass stars have strong enough internal luminosities to support compact HII regions, low mass stars generally do not. Thermal emission from ionized regions around low-mass YSOs is typically thought to be driven by collimated outflows that shock the surrounding material (Anglada et al. 1998). Because outflows are common among less-developed YSOs, thermal radio emission is found predominately around the earliest protostellar phases, Class 0 and I. Non-thermal emission is generally found around more developed pre-main sequence T Tauri stars (Class II & III) because their stellar coronae are exposed. There are a few cases, however, where non-thermal emission has been found around young protostars, which could in part be due to geometric effects or envelope-clearing by a companion star (Dzib et al. 2010). This suggests that young YSOs may in fact emit copious amounts of non-thermal emission that is then absorbed by thermal emission coming from larger size scales around the protostar.
At radio wavelengths, the spectral index is defined as the difference of the natural log of the radio flux at two different radio wavelengths, divided by the difference of the natural log of those wavelengths:
α radio = ln(S λ1 /S λ2 ) ln(λ 2 /λ 1 ) ,(1)
where λ 2 > λ 1 (Shirley et al. 2007). Note that S λ is defined as a flux density and not a flux (e.g. λ · S λ or ν · S ν ). Flat radio spectral indices (α > −0.1) are generally indicative of optically thin thermal free-free emission from an ionized plasma, while rising indices (α ∼ 2.0) are representative of the optically thick case (Ghavamian & Hartigan 1998). Steeply falling radio spectral indices (α ∼ −2.0) are common from sources of non-thermal radio emission (e.g. AGN). Because of this, radio spectral indices are commonly used to help discriminate between thermal and non-thermal emission processes; however, this can be complicated due to errors on measured fluxes and the blending of multiple emission components. Other ways to deduce the emission mechanism of a radio source is to measure the variability of its radio flux over time and to probe for polarization, neither of which we can do in this present study.
In the infrared, four main categories of YSOs exist based on the shape of their spectrum: Class I, Flat-Spectrum, Class II and Class III YSOs. These classifications roughly correspond to unique development phases in chronological order; however, it is good to note that a direct connection between infrared class and unique evolutionary stage can be confused by possible misclassifications based on inclination effects (Crapsi et al. 2008). Greene et al. (1994) presented YSO classifications based on infrared spectral indices, where the infrared spectral index α IR is defined as
α IR = ln ((λ 2 · S λ2 )/(λ 1 · S λ1 )) ln(λ 2 /λ 1 ) :(2)Class I 0.3 ≤α IR (3) Flat Spectrum −0.3 ≤α IR < 0.3 (4) Class II −1.6 ≤α IR < −0.3 (5) Class III α IR < −1.6,(6)
with λ 2 > λ 1 . We derive our infrared spectral indices between the IRAC 8 µm band and the MIPS 24 µm band. Our infrared spectral indices have a standard error of ±0.2 based on an error analysis that combines in quadrature the zero flux offset error used to convert magnitudes to fluxes (2% contribution) and an assumed 10% error in our extracted magnitudes. There are five sources where we have successful source extractions at 8 and 24 µm. Figure 2 shows these SEDs and our derived infrared spectral indices.
RESULTS: CLASSIFICATIONS AND RADIO PROPERTIES
We detail the properties of our eighteen radio sources in Table 2, which includes their integrated and peak radio fluxes, derived spectral indices and classifications based on our radio and infrared data. Other detections of our sources in previous works are also discussed here, which come primarily from four studies: a near-infrared spectral line study of molecular hydrogen jets and outflows done by Teixeira et al. (2012) (referred to as T12), a 1.2 millimeter continuum survey done by Maury et al. (2011) (referred to as M11), a 2.7 millimeter spectral line study of molecular outflows done by Plunkett et al. (2015a) (referred to as P15), and a centimeter radio continuum study with the VLA done by Ortiz-León et al. (2015) (referred to as OL15). The major differences between our study and the VLA study of OL15 are (1) they had higher angular resolution by over a factor of ten, and (2) they had lower point source sensitivity by over a factor of two. Their lower sensitivity did not enable them to make any firm radio detections of any YSO candidates in our field of view. For radio sources we report as having radio fluxes within their detectable limits, their nondetection could be due to either their small beam size over-resolving the emission structure, time variability in the source's emission, or both. For the case of the former, this would suggest that there is a significant source of thermal emission originating at or above size scales of hundreds of AU: the physical distance corresponding to the angular separation of OL15's 0.3 arcsecond synthesized beam assuming a distance of 429 pc to the cloud.
Protostellar Sources
Seven of our eighteen radio sources, VLA 7,11,12,13,14,16 and 17, are likely to be protostellar in nature. All except VLA 14 lie within 2 arcseconds of an infrared/submillimeter-identified Class 0, I or II YSO. All sources have either flat or rising radio spectral indices indicative of optically thin or thick thermal emission expected from a young YSO. Figure 3 shows our 7.25 GHz radio contours against various infrared and submillimeter images of the core of Serpens South. Figure 4 shows a zoom-in on our 7.25 GHz and 4.75 GHz radio contours for most of these sources along with Spitzer Class I & II YSOs (Gutermuth et al. in prep).
VLA 7 is spatially associated with a Class II YSO in the infrared that has a calculated α IR of -0.7. It has a radio spectral index of ∼1.3 indicative of a mix of optically thin and thick thermal emission.
VLA 11 is associated with an infrared source with α IR =0.7, which makes it a Class I protostar. In the IRAC bands, VLA 11's SED seems to be turning over, but its MIPS and PACS emission clearly shows it has a rising SED at mid-infrared wavelengths.
VLA 11 was also found to be driving a jet. T12 found a molecular hydrogen emission-line object (MHO) feature that they conclude is driven by the infrared source associated with our VLA 11 source, which they call "P2." We note that while T12 claims this YSO to be Class 0 by citing the Herschel study of Bontemps et al. (2010), there was no officially published catalogue of Herschel protostars from Serpens South at the time. Currently, only a catalogue of dense cores and YSO candidates without classifications exists for Herschel data in the Aquila region (Konyves et al. 2015).
M11 images a millimeter peak, called "MM16," centered East of VLA 11 but encompasses VLA 11 within its FWHM fit. While they suggest MM16 is a blending of multiple sources, we definitively show that there are indeed at least two sources associated with the their millimeter peak. P15 also showed this with higher resolution mm continuum imaging, detecting both VLA 11, which they call "CARMA-5", and the source at the center of M11's MM16.
VLA 12 is not detected short-ward of 24 microns, meaning it is a highly embedded source. It is, however, the peak structure of diffuse 1.2 mm emission and was therefore classified as a Class 0 source (Maury et al. 2011). This classification used a 25 arcsecond FWHM fit, which also encompassed the infrared sources associated with VLA 13 and 14 (see Figure 4). Here, we reaffirm VLA 12's Class 0 classification with the detection of thermal radio emission indicative of outflows and the presence of a central YSO (Andre et al. 2000). Recent ALMA data also finds VLA 12 to be a driver of a highly collimated, episodic molecular outflow (Plunkett et al. 2015b), running 4 • East of the North-South vertical direction. VLA 12 is also detected by P15 at 2.7 mm, who resolve VLA 12's emission from its nearby companions VLA 13 and VLA 14 (referred to as CARMA-7). At the 3σ level, VLA 12's radio emission has elongated structure towards the North-West (Figure 4), which aligns with its continuum emission at 1 mm (Plunkett et al. 2015b).
VLA 13 is associated with an infrared source that has a full set of infrared extractions except for the IRAC 1 band at 3.6 µm (Figure 2). The lack of a flux at the IRAC 1 band, as noted by P15, would prevent it from being classified as a YSO candidate by Evans et al. (2009). However, because we (1) have fluxes associated with VLA 13 from the IRAC 2 band to the PACS 70µm band and (2) have detected thermal radio emission, we conclude that VLA 13 is likely an embedded Class 0/I protostar. It shows comparatively strong radio emission suggesting it could be driving a relatively powerful outflow, perhaps being the most dominant driver of outflows in the densely populated region of Serpens South's core. VLA 14 is the only YSO candidate we detect that has yet to be definitively detected by any previous study. In far-infrared images from 24 to 250 µm it is clear that, in projection, VLA 14 lies within a gaseous envelope (Figure 3). However, because of the low resolution of the far infrared images that blends the emission of VLA 13 with VLA 14, we do not retrieve any source extraction within 2 arcseconds of VLA 14's radio peak. Even in the higher resolution MIPS 24 µm image, VLA 14 is not seen as an individual source but rather an extension of the emission structure from VLA 13 (Figure 3). This is also seen in P15's mm continuum emission structure. In the near-infrared, VLA 14 is too deeply embedded to be seen. This is likely why previous studies missed this source and it implies that if a central source exists it is likely young and/or a low luminosity object. There are three main reasons why we conclude that VLA 14 is very likely an embedded Class 0 protostar: (1) it has a steeply rising radio spectral index indicative of optically thick thermal radio emission, which is also suggestive of outflows and therefore a central source, (2) it is a spatially resolved structure from VLA 13 in both our 7.25 and 4.75 GHz radio images, and (3) VLA 16 is associated with an infrared source that has a steeply rising infrared spectral index and is therefore classified as a Class I YSO, although just looking at its SED at IRAC wavelengths it may be confused as a Flat-Spectrum or Class II source. It has a relatively flat radio spectral index suggesting optically thin thermal radio emission.
VLA 17 's infrared SED suggests it is a Flat-Spectrum source that may be transitioning to a Class II source, due to the fact that its 70 µm flux is significantly lower than its 24 µm flux. It has comparatively strong radio flux with a slightly rising radio spectral index, which, similar to VLA 13, suggests that it could be a dominant contributor to large-scale outflow from the core. VLA 17 is also detected by the CARMA study, which they label "CARMA-1," and is also claimed to be a driver of a MHO feature referred to as "Y1" by Teixeira et al. (2012). In both images, the blue solid contours represent the 7.25 GHz radio signal and start at the 3σ level and increase to 4, 5, 6, 8, 10, 15, 20, 30, and 50σ. The blue circles are Spitzer identified Class I sources (Gutermuth et al. in prep.). Green circles are the FWHM fits of the 1.2 mm emission peaks from Maury et al. (2011). Magenta circles are fits to the 2.7 mm emission from Plunkett et al. (2015a). Projected size scale assumes a distance of 429 pc to Serpens South. 70µm and have negative radio spectral indices indicative of non-thermal synchrotron emission typical of a galaxy's radio spectrum. VLA 2, 3 and 4 are interestingly aligned along an axis ( Figure 5). One possible explanation is that these three sources actually form one source: one side of a radio jet emanating from the central galaxy (VLA 2) and ending in a radio lobe (VLA 4). These strong sources of radio emission were not detected by the high resolution radio survey done by OL15. The most probable reason why OL15 did not detect these strong sources is because they are extended sources, meaning that the high resolution (∼0.3 " ) synthesized beam of OL15 resolved out the radio flux to a level below their point source sensitivity. While VLA 1 and 5 have no infrared, far-infrared or millimeter associations and are isolated from the Serpens South filament, they have flat and rising radio spectral indices suggesting optically thin or thick free-free emission respectively. Although galaxies can produce freefree emission from large HII regions, it is generally a sub-dominant component compared to its synchrotron output when an AGN is present. Moreover, in Dzib et al. (2013)'s study of 190 radio sources in the Ophiuchus star forming cloud, they did not find a single extragalactic source with a radio spectral index larger than ∼0. However, we should note that VLA 1 and 5 are relatively far from our phase center (θ ∼ 2.3 ′ ) meaning that VLA pointing errors could be larger than accounted for in our error analysis. We tentatively classify these sources as extragalactic due to their lack of any association at shorter wavelengths.
Extragalactic Sources
Unclear Sources
We are unsure as to the nature of VLA 10 ( Figure 4). We treat the whole elongated structure as one source. At our higher resolution 7.25 GHz image, we can resolve two distinct emission components that are blended at our lower resolution 6.3 cm image. Although our measured fluxes are uncertain due to source blending at 4.75 GHz, VLA 10's peak flux derived spectral index is strongly negative suggesting non-thermal emission and an extragalactic classification. However, VLA 10's Southern structure is also within 2 arcseconds of a Spitzer identified Class I source. This raises the possibility that the Northern structure is a background source while the Southern structure is a YSO radio source. Another possibility is that VLA 10 is representative of post-shock free-free emission from an outflow emanating from either the Southern VLA 10 structure, the Northern VLA 10 structure, or from sources VLA 11, VLA 9, or another source. Indeed, recent millimeter studies tracing 12 CO outflow emission shows a low-velocity redshifted outflow structure that aligns spatially and morphologically with VLA 10's radio emission (Plunkett et al. 2015a).
Further evidence that would support a nonextragalactic classification of both the South and North structures of VLA 10 is the fact that the dust emission seen at and longward of 70 µm neatly fits around VLA 10's elongated morphology, as seen in Figure 3-(e). Lastly, its elongation towards the North-West direction aligns with the direction of the large-scale red-shifted molecular outflow seen in P15. Higher resolution, multi-epoch observations should be able to spatially separate the individual emission components and better determine the nature behind their radio emission.
Similar to VLA 10, the nature of VLA 9 is unclear. Although it has no strong infrared associations, it does lie within 5 arcseconds of a weak infrared source. It is placed in a region of high YSO surface density, falls in front of diffuse, far-infrared emission and has a relatively flat spectral index: increasing the chances it is a YSO candidate. Its 4.75 GHz emission blends with VLA 10 at the 3σ level. While this could be due to blending of VLA 9 and 10's emission structure due to the larger synthesized beam at 4.75 GHz, it does raise the possibility that VLA 9 is tied to the structure powering VLA 10's radio emission (Figure 4).
VLA 18 is spatially distinct from the filament and has a flat radio spectral index of about 0 ± 0.1, consistent with optically thin thermal radio emission driven by protostellar outflows. It has three extremely weak infrared associations at the IRAC infrared bands from 3.6 to 8.0 µm, the 24 µm MIPS band, and a weak and diffuse association at 100 and 160 µm from Herschel images. However, while these associations are visible upon manual inspection of the data, the only ones strong and compact enough to enable successful source extractions are the IRAC band associations. VLA 18 also lies ∼3 arcseconds from a Herschel -identified starless core (see Konyves et al. 2015), which further supports a YSO classification. Currently, we do not have enough evidence to firmly support a protostellar classification due to the lack of far infrared associations, however, higher resolution millimeter and radio studies could clarify the nature of this source.
DISCUSSION
The Serpens South region is filled with YSOs identified in the infrared. Spitzer images reveal upwards of 91 YSOs in the entire Serpens South cloud, 59% of which are Class I protostars (Gutermuth et al. 2008). At the core, this fraction rises to 77%, suggesting a recent onset of star formation in this concentrated area (within about 2 × 10 5 years). Gutermuth et al. (2008) define the core of Serpens South within a circular region centered at α=18.30.03 and δ=-02.01.58.2 and extending out by 2.5 arcminutes, which is similar in both position and size to our radio observation. Within this spatially defined core lies a strong peak in sub-millimeter emission, corresponding to the highly clustered radio sources of VLA 11 to VLA 16 (see Figure 3-f).
The three Class 0 sources in the central cluster lie within a projected separation of only 10 arcseconds, which corresponds to 0.04 pc or roughly 4,100 AU, although they may lie further away in 3D coordinates. The fact that we detect three clustered Class 0 protostars suggests that the onset of star formation in the central cluster is very recent, on timescales of the Class 0 phase duration of ∼ 10 5 years (Dunham et al. 2015).
Out of the roughly 35 Class I protostars in our field of view, we detect radio flux from only six of them. In addition, the radio fluxes of our YSO sources are for the most part weak, on the order of or less than one hundred micro-Janksy. Ortiz-León et al. (2015) suggested that, assuming a spherical wind model and using Rodriguez et al. (1989) as a guide, young YSOs in Serpens South are expected to emit about 0.34 mJy of thermal radio flux given a distance of 429 pc. If that distance is increased to 700 pc, that expectation falls to 0.12 mJy, which is out of their detection range but still well within ours, possibly explaining their lack of any YSO detections in the core of Serpens South. The fact that we only pick up a handful of detections with over a factor of two increase in sensitivity over Ortiz-León et al. (2015) suggests that the distance to Serpens South is likely not what is causing our lack of YSO radio detections (assuming a spherical wind model), otherwise we would have likely detected many more Class I sources in the radio.
At high resolution scales, the collimated outflows that are the energy sources of thermal radio emission can be resolved (e.g. Rodríguez et al. 2003). Although our spatial resolution is not high enough to probe the inner regions of outflows, even at 3 arcsecond resolution scales we recover some source morphology for VLA 11 in our 7.25 GHz map; two-component Gaussian fitting with the IMFIT task on our 7.25 GHz image returns an estimated deconvolved beam size of 3.1±0.2 arcsec × 0.8±0.3 arcsec with a position angle of 11±2 degrees for VLA 11. This morphology also is in the same direction of MHO features that Teixeira et al. (2012) concluded were being driven by VLA 11. Further high resolution outflow studies may be able to clarify this connection.
Because thermal radio emission is driven by outflows, the strength of YSOs' relative radio flux can be used as a rough proxy for relative outflow strength. VLA 13 has the strongest radio flux out of our YSO candidates and it is centrally located within the core of Serpens South, where there is strong 12 CO emission tracing molecular outflows. We therefore speculate that it could be driving a relatively strong outflow and could be a strong contributor to Serpens South's large scale molecular outflow seen in the millimeter regime (see Nakamura et al. 2011;Plunkett et al. 2015a).
5.1. The S radio and L bol Relationship A correlation between centimeter flux and bolometric luminosity is known to exist for low and intermediate mass protostars. This correlation is related to another relationship, the outflow force versus bolometric luminosity relationship, which shows that higher luminosity YSOs drive more powerful outflows (Wu et al. 2004). Because thermal radio emission is driven by outflows and stronger outflows produce more thermal emission (Ghavamian & Hartigan 1998), the centimeter flux versus bolometric luminosity relation is naturally an extension of the former relationship. Anglada (1995) was one of the first to relate a YSO's centimeter continuum emission to its bolometric luminosity. Since then, this relationship has been updated and improved upon with protostars at lower and lower bolometric luminosities (e.g., Anglada et al. 1998;Beltrán et al. 2001;Shirley et al. 2007;AMI Consortium et al. 2011a). The intrinsic scatter of this relationship is likely affected by emission variability.
The bolometric luminosity of a protostar is dominated by emission from its envelope at sub-millimeter wavelengths. However, in the core of Serpens South, it is not possible to construct sub-millimeter SEDs for individual protostars because they are blended in the Herschel images. To get an estimate of their bolometric luminosities, we have relied on protostellar luminosity studies that have developed a relationship that correlates a YSO's mid-infrared luminosity to its bolometric luminosity (see section 3.2 of Kryukova et al. 2012). The mid-infrared flux is defined as a sum of 2MASS J,H, and K fluxes, IRAC 1,2,3 and 4 fluxes and the MIPS 24µm flux with coefficients determined by Equation (6) of Kryukova et al. (2012). We use our infrared source extractions to create mid-infrared fluxes for three of our protostellar sources and estimate their bolometric luminosities using Equation (5) from Dunham et al. (2013). We re-calculated our protostars' infrared spectral index, α IR , using the IRAC 3.6 µm band and the MIPS 24 µm band to be consistent with the derivation found in the former study. Figure 6 shows three of our protostellar sources overlaid on the 3.6 cm luminosity and 6.0 cm luminosity vs. bolometric luminosity correlations provided by Shirley et al. (2007). Using this sample, Shirley et al. (2007) report correlations of log(L 3.6 /L 1 mJy kpc 2 ) = − (2.24 ± 0.03) + (0.71 ± 0.01) log(L bol /L ⊙ ), log(L 6.0 /L 1 mJy kpc 2 ) = − (2.51 ± 0.03)
+ (0.87 ± 0.02) log(L bol /L ⊙ )
for the 3.6 cm and 6.0 cm emission components respectively. Because the detection and characterization of low luminosity protostars is still an ongoing process, it is hard to tell for sure what the trend of this relationship is at the low-luminosity end where our protostars reside. However, deep radio studies by AMI Consortium et al. (2011b) suggest that this relationship may tentatively extend linearly from log bolometric luminosities of 1 down to -1. To compare this relationship to our sample of protostars, we used our derived radio spectral indices to extrapolate our 4.1 cm and 6.3 cm radio fluxes to 3.6 and 6.0 radio fluxes. The radio luminosities use an assumed distance of 429 pc to Serpens South. Errors on our derived bolometric luminosity account for errors in our infrared fluxes, infrared spectral indices, and errors in the scaling relationships provided by Kryukova et al. (2012) and Dunham et al. (2013). Within our errors, we Figure 2 from Shirley et al. (2007) showing the 3.6 cm vs. bolometric luminosity correlation (left) and 6.0 cm vs. bolometric luminosity (right) for protostars. Three of our protostellar sources are plotted with bolometric luminosities derived from a correlation that utilizes the protostars' mid-infrared luminosity (see Kryukova et al. 2012;Dunham et al. 2013). find that our data roughly agrees with the relationship as it currently stands. Although visually our data might seem to create some tension at the low-luminosity end, this is complicated by the fact that this S radio vs L bol relationship does not presently include all radio detections of protostars, has an intrinsic scatter, can be affected by factors such as emission variability and has yet to be fully probed at the low-luminosity end. Our derived mid-infrared luminosities could be affected by the fact that we do not have 2MASS J, H, K band extractions for some of our protostars (Figure 2). To test how sensitive our derived luminosities are to this lack of data, we artificially added JHK fluxes that roughly matched an extrapolation of the protostars' SED and found that the newly derived bolometric luminosities change by at most 0.03 in log space: much smaller than the errors already associated with the bolometric luminosity derivation. We therefore conclude that we are largely unaffected by the fact that we do not have J, H, K fluxes for determining our protostars' mid-infrared luminosity.
CONCLUSIONS
We have presented deep, interferometric radio observations of centimeter continuum emission from the core of Serpens South with the Karl G. Jansky Very Large Array. We detect radio emission from a number of YSO candidates identified in the infrared and millimeter and from one previously unidentified source. For Class 0 protostars identified in the millimeter but not detected in the infrared, we provide definitive evidence for a compact central source through our detection of thermal radio emission, indicative of outflows expected during the protostellar phase. Our 3 arcsecond resolution also allows us to separate and resolve individual YSO sources that were previously confused by lower spatial resolution studies. Out of the 40+ infrared YSO candidates in our field of view, we detect radio emission at or above a level of 50 µJy/beam for only six. We also detect emission at a weaker level for another six that lie within 2 arcseconds of infrared YSO candidates with IR detections in some but not all of the IRAC 1, 2, 3, 4 and MIPS 24µm band, providing motivation for a higher sensitivity follow-up study to detect more embedded objects in Serpens South. Our detection of YSOs overlooked by infrared studies reinforce the notion that longer wavelength studies at millimeter and radio wavelengths are necessary to construct accurate demographics of prestellar populations. Finally, the fact that Ortiz-León et al. (2015) did not make any firm detections of our YSO candidates suggests that either (1) the sources have significant time variability in their emission and/or (2) that the high-angular resolution of Ortiz-León et al. (2015) (∼ 0.3arcsec) over-resolved the emission structure from these YSOs. The latter, if true, would suggest that there is a significant source of thermal emission originating at or above size scales of hundreds of AU.
Although we cannot probe the inner regions of the collimated outflows that drive YSO thermal radio emission, our detections provide future high resolution studies a starting point for where thermal radio emission exists and its approximate strength. However, in fitting twocomponent Gaussians to our radio sources with CASA'a IMFIT task, we recover a deconvolved beam size for VLA 11 in our 7.25 GHz image that agrees with a previous work that found a jet emanating from VLA 11 in roughly the same direction (Teixeira et al. 2012). Serpens South has also been found to drive large scale molecular outflows Plunkett et al. 2015a). Our strongest source of thermal radio emission is VLA 13, which lies almost directly in the center of both Serpens South's core and the peak of its extended sub-millimeter emission. We therefore speculate that VLA 13 could be a dominate driver of molecular outflow in the region.
We cross reference infrared catalogues and run source extractions over near and far infrared images from 2 µm to 70 µm to build infrared SEDs, which we use to both classify our protostellar sources and derive mid-infrared & bolometric luminosities. We find relative agreement with the known radio vs. bolometric luminosity relationship of protostars for three of our sources given our errors.
Although the core of Serpens South may seem to be relatively radio quiet judging from the high-resolution radio survey of Ortiz-León et al. (2015), we find a host of interesting radio sources that show clear signs of being protostellar in nature that have for the first time shed light on the radio properties of YSOs in the core of the Serpens South infrared dark cloud. Future radio studies of Serpens South would benefit from multi-epoch, polarization-capable studies that have an angular resolution of ∼1 arcsecond and a point source sensitivity of less than 10 µJy/beam in order to detect a larger number of young stellar objects.
The observations for this paper were conducted as part of the NRAO REU program funded by the National Science Foundation. N.S.K., J.A.K., and A.M. were summer students at the NRAO when the observations utilized by this study were taken. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. J.J.T. acknowledges support provided by NASA through Hubble Fellowship grant #HST-HF-51300.0 A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. This work is supported by grant 639.041.439 from the Netherlands Organisation for Scientific Research (NWO).
Figure 2 .
2SEDs of all protostellar candidates that have infrared associations. Infrared bands are 2MASS H and K from 1.6 -2.2 µm, IRAC 1, 2, 3 and 4 from 3.6 -8.0 µm, MIPS 24µm and PACS 70µm. Our derived infrared spectral index from 8 to 24 µm is shown for sources with successful fits in those bands and have errors of ±0.2.
it is set against strong and extended submillimeter emission indicative of an extended envelope: the same millimeter emission from Maury et al. (2011) (SerpS-MM18) that blends VLA 14 with both VLA 12 and 13 and the submillimeter emission from Plunkett et al. (2015a) (CARMA-6) that blends VLA 14 with VLA 13.
VLA 2, 3, 4, 6, and 15 are very likely extragalactic in nature. They have no visible associations from 1 µm to
Figure 3 .
37.25 GHz (4.14 cm) radio contours plotted over infrared and far-infrared images ranging from 4.5 to 250 µm. Radio contours start at the 4σ level, with the deconvolved beam size shown at lower-left.(a) Spitzer IRAC 4.5 µm image. (b) Spitzer IRAC 5.8 µm image. (c) Spitzer IRAC 8.0 µm image. (d) Spitzer MIPS 24 µm image. (e) Herschel PACS 70 µm image. (f ) Herschel SPIRE 250 µm image. Note how the radio sources in the central cluster are not visible in the near infrared images, but start to become visible long-ward of 24 microns.
Figure 4 .
4Left: Zoom-in of VLA 12 and surrounding sources. Right: Zoom-in of the elongated VLA 10 source.
Figure 5 .
5Zoom-in on sources VLA 2, 3 and 4. Blue solid and red dashed contours correspond to 7.25 and 4.75 GHz radio images respectively, which start at their respective 3σ levels. This system of extragalactic sources could be in fact one central galaxy with powerful radio lobes.
Figure 6 .
6Figure 6. Figure 2 from Shirley et al. (2007) showing the 3.6 cm vs. bolometric luminosity correlation (left) and 6.0 cm vs. bolometric luminosity (right) for protostars. Three of our protostellar sources are plotted with bolometric luminosities derived from a correlation that utilizes the protostars' mid-infrared luminosity (see Kryukova et al. 2012; Dunham et al. 2013). Figure reproduced from Shirley et al. (2007) with permission.
Figure reproduced from Shirley et al. (2007) with permission.
Table 1
1Observation SummaryFrequency a
Beam Size b
Position Angle
Image RMS
(GHz)
(arcsec x arcsec)
(degrees)
(µJy beam −1 )
4.75
4.8 x 3.8
14.8
11.1
7.25
Table 2
2VLA Centimeter Continuum Sources Centers of 2D Gaussian fits for sources in 7.25 GHz map using CASA's IMFIT procedure. b Integrated flux values and errors using IMFIT from images deconvolved with Briggs weighting, robust=0.5(Briggs 1995). c Peak flux values calculated by taking the brightest pixel at the source's center. d Spectral index of integrated flux from 7.25 to 4.75 GHz, see subsection 3.1 for details on calculation. e Spectral index of peak flux from 7.25 to 4.75 GHz.Source
R.A. a
Decl. a
S int
4.75GHz
b
S peak
4.75GHz
c
S int
7.25GHz
S peak
7.25GHz
α int
radio
d
α peak
radio
e
Class
Other Names
(J2000)
(J2000)
(µJy)
(µJy beam −1 )
(µJy)
(µJy beam −1 )
VLA 1
18 30 09.68 -02 00 32.8
749 ± 46
718
844 ± 81
794
0.28 ± 0.11
0.24 ± 0.12
Extragal.
J183009.68-020032.7 i
VLA 2
18 30 04.79 -02 00 34.1
177 ± 12
138
87 ± 10
72
-1.67 ± 0.14 -1.55 ± 0.18
Extragal.
-
VLA 3
18 30 02.55 -02 00 48.5
140 ± 8
99
116 ± 14
87
-0.46 ± 0.14
-0.3 ± 0.18
Extragal.
-
VLA 4
18 30 01.53 -02 00 51.6 1663 ± 101
1178
946 ± 84
515
-1.33 ± 0.11
-1.96 ± 0.1
Extragal.
-
VLA 5
18 30 14.79 -02 01 27.3
106 ± 9
95
167 ± 20
173
1.07 ± 0.15
1.41 ± 0.18 Extragal.?
-
VLA 6
18 30 06.62 -02 01 28.9
79 ± 8
61
33 ± 5
56
-2.04 ± 0.18 -0.22 ± 0.25
Extragal.
-
VLA 7
18 30 05.81 -02 01 44.9
48 ± 4
40
84 ± 9
63
1.29 ± 0.15
1.08 ± 0.32
Class II
-
VLA 8
18 30 06.89 -02 02 11.5
39 ± 4
46
50 ± 6
50
0.55 ± 0.17
0.18 ± 0.3
Extragal.?
-
VLA 9
18 30 02.80 -02 02 25.3
66 ± 3
50
60 ± 3
44
-0.22 ± 0.08
-0.31 ± 0.3
Unknown
-
VLA 10
18 30 02.95 -02 02 32.2
287 ± 16
119
260 ± 22
73
-0.23 ± 0.1
-1.15 ± 0.17 Unknown
-
VLA 11
18 30 03.36 -02 02 45.8
38 ± 3
36
73 ± 5
58
1.5 ± 0.11
1.14 ± 0.35
Class I
P2 g ,CARMA-5 h
VLA 12
18 30 04.11 -02 03 02.1
30 ± 3
24
79 ± 4
58
2.31 ± 0.12
2.06 ± 0.49
Class 0
MM18 f ,CARMA-7 h
VLA 13
18 30 03.54 -02 03 08.4
186 ± 9
178
231 ± 13
227
0.51 ± 0.08
0.58 ± 0.11
Class 0/I
P3 g ,CARMA-6 h
VLA 14
18 30 03.84 -02 03 12.2
31 ± 1
37
83 ± 5
61
2.29 ± 0.09
1.19 ± 0.34
Class 0
CARMA-6 h
VLA 15
18 30 04.60 -02 03 14.1
68 ± 3
52
49 ± 4
51
-0.74 ± 0.11 -0.08 ± 0.28
Extragal.
-
VLA 16
18 30 03.25 -02 03 26.6
63 ± 3
58
55 ± 3
65
-0.28 ± 0.09 0.27 ± 0.24
Class I
-
VLA 17
18 30 01.32 -02 03 42.9
146 ± 8
135
161 ± 12
167
0.22 ± 0.1
0.5 ± 0.13
Flat/II
Y1 g
VLA 18
18 30 09.40 -02 04 08.8
170 ± 10
182
167 ± 18
134
-0.04 ± 0.12 -0.71 ± 0.14 Extragal.?
-
a f From Maury et al. (2011)
g From Teixeira et al. (2012)
h From Plunkett et al. (2015a)
i From Ortiz-León et al. (2015)
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| []
|
[
"Level Lines of the Gaussian Free Field with general boundary data",
"Level Lines of the Gaussian Free Field with general boundary data"
]
| [
"Ellen Powell ",
"Hao Wu "
]
| []
| []
| We study the level lines of a Gaussian free field in a planar domain with general boundary data F. We show that the level lines exist as continuous curves under the assumption that F is regulated (i.e., admits finite left and right limits at every point), and satisfies certain inequalities. Moreover, these level lines are a.s. determined by the field. This allows us to define and study a generalization of the SLE 4 (ρ) process, now with a continuum of force points. A crucial ingredient is a monotonicity property in terms of the boundary data which strengthens a result of Miller and Sheffield and is also of independent interest.Note that this definition is the same for any two functions F 1 and F 2 which are equal almost everywhere, since the harmonic extensions of such functions are necessarily equal. From Definition 1.1, we can see that the so-called level lines of the GFF have an intriguing property that distinguishes them from level lines of an ordinary smooth function. Namely, once one conditions on a level line, the conditional expectation of the field on one side of the curve differs by 2λ from the value on the other side. In a sense, a level line is more like a "level cliff" where there is a prescribed jump between the two sides of the curve.• (W (n) ) n∈N is tight in the metrisable space of continuous functions on [0, ∞) with the topology of uniform convergence on compact subsets of [0, ∞). | 10.1214/16-aihp789 | [
"https://arxiv.org/pdf/1509.02462v4.pdf"
]
| 119,279,569 | 1509.02462 | 90e7f30c32993956f0320be0784369fd637128c0 |
Level Lines of the Gaussian Free Field with general boundary data
Ellen Powell
Hao Wu
Level Lines of the Gaussian Free Field with general boundary data
We study the level lines of a Gaussian free field in a planar domain with general boundary data F. We show that the level lines exist as continuous curves under the assumption that F is regulated (i.e., admits finite left and right limits at every point), and satisfies certain inequalities. Moreover, these level lines are a.s. determined by the field. This allows us to define and study a generalization of the SLE 4 (ρ) process, now with a continuum of force points. A crucial ingredient is a monotonicity property in terms of the boundary data which strengthens a result of Miller and Sheffield and is also of independent interest.Note that this definition is the same for any two functions F 1 and F 2 which are equal almost everywhere, since the harmonic extensions of such functions are necessarily equal. From Definition 1.1, we can see that the so-called level lines of the GFF have an intriguing property that distinguishes them from level lines of an ordinary smooth function. Namely, once one conditions on a level line, the conditional expectation of the field on one side of the curve differs by 2λ from the value on the other side. In a sense, a level line is more like a "level cliff" where there is a prescribed jump between the two sides of the curve.• (W (n) ) n∈N is tight in the metrisable space of continuous functions on [0, ∞) with the topology of uniform convergence on compact subsets of [0, ∞).
Introduction
The relationship between Schramm-Loewner Evolution (SLE) and the two-dimensional Gaussian free field (GFF) is at the heart of recent breakthroughs in Liouville quantum gravity, imaginary geometry and more generally, random conformal geometry. Starting with the seminal papers of [Dub09], [SS13], [SS09], one key idea is to make sense of SLE-type curves as a level lines of an underlying Gaussian free field h in a domain, which we take to be the upper half plane H without loss of generality in the rest of the paper. When the field h is given the boundary values λ := π/2 on R + and −λ on R − , the corresponding level line is a chordal SLE 4 curve. A considerable extension of that theory is described in [MS16a], which introduced the notion of flow lines and counter flow lines of the GFF. In this case it turns out that the curves are given by SLE κ processes with κ ∈ (0, 4) and κ ∈ (4, ∞) respectively.
It is also natural to wonder for which sort of boundary data the notion of level line makes sense. In [MS16a] and [WW16], the hypothesis on the boundary data is extended from the above to any arbitrary piecewise constant function on the real line. The goal of this paper will be to relax this assumption. Assuming solely that the boundary data F is a regulated function, i.e., the left and right limits
F(t + ) = lim h→0+ F(t + h); F(t − ) = lim h→0− F(t + h)
(1.1) exist and are finite for all t ∈ R, and that for some c > 0
F(x) ≤ λ − c, x < 0; F(x) ≥ −λ + c, x ≥ 0 (1.2)
which roughly corresponds to the non existence of a continuation threshold, we can show that the corresponding level line is well defined almost surely as a continuous transient curve. Moreover, it is almost surely determined by the field. This also allows us, for a zero boundary GFF h, to consider the set of level lines of different heights. By this we mean the level lines of h + F, where F ranges over (the bounded harmonic extensions of) all regulated functions on R. Strengthening the results of [MS16a], [WW16], we are able to prove a general monotonicity principle for the level lines, which is both a key tool in our existence proof, and an interesting result in its own right. This is deeply intertwined with the reversibility property of the level lines, which we are also able to prove in general; see Theorems 1.4 and 1.5.
A further point of interest is that we obtain some continuity in the level lines as a consequence of our proof. That is, if we take a sequence of piecewise constant functions F n converging monotonically uniformly to some F, then the level lines of height F n for a zero boundary GFF converge almost surely to the level line of height F. This convergence is with respect to Hausdorff distance, after conformally mapping everything to the unit disc.
We remark that our hypothesis on the boundary data is satisfied by a wide range of functions, including the special class of functions of bounded variation. Any such function can be described almost everywhere as the integral of a finite Radon measure ρ, and this connection allows us to deduce that the marginal law of a level line with such boundary data is given by what we call an SLE 4 (ρ) process. This is the natural analogue of an SLE 4 (ρ) process, where the vector ρ is replaced by a measure. Our results therefore demonstrate the existence of such processes, as well as establishing some further properties.
We first recall the definition of what it means for a curve, and more generally a Loewner chain, to be a level line. If we have a Loewner chain (K t ,t ≥ 0) in H, with associated sequence of conformal maps g t : H \ K t → H, we will often want to describe the image under g t of a point x on the real line. To do this, for any x ≤ 0 we define a process V L t (x) by setting it equal to g t (x) if x / ∈ K t and if x ∈ K t , taking it to be the image of the leftmost point of R ∩ K t under g t . We define a process V R t (x) for x ≥ 0 analogously. The process V L t (x) for x ∈ R − , or V R t (x) for x ∈ R + , is what we define to be the image of x under g t .
Definition 1.1 ( [MS16a,WW16]). Suppose that F is L 1 with respect to harmonic measure on R viewed from some point in H and that h is a zero boundary GFF in H. If (K t ,t ≥ 0) is a Loewner chain and (g t ,t ≥ 0) is the corresponding sequence of conformal maps, set f t = g t − W t , and let V R t (x) (resp. V L t (x)) be the image of x ≥ 0 (resp. x ≤ 0) under g t . Let η 0 t be the bounded harmonic function on H with boundary values (see Figure 1.1)
F( f −1 t (x)), if x ≥ V R t (0 + ) −W t , λ , if 0 ≤ x < V R t (0 + ) −W t , −λ , if V L t (0 − ) −W t ≤ x < 0, F( f −1 t (x)), if x < V L t (0 − ) −W t . Define, for z ∈ H \ K t ,
η t (z) = η 0 t ( f t (z)). We say that K is a level line of h + F if there exists a coupling (h, K) such that the following domain Markov property holds: for any finite K-stopping time τ, given K τ , the conditional law of (h + F)| H\K τ is equal to the law of h • f τ + η τ . Fig. 1.1: The left hand side shows the boundary values of the harmonic function η τ in H \ K τ . This is the image under f −1 τ of the harmonic function η 0 τ in H, whose boundary values are shown on the right hand side.
f τ λ −λ 0 λ F F • f −1 τ −λ F 0 V R τ (0 + )−W τ K τ V L τ (0 − )−W τ F • f −1 τ
More generally, we say that a Loewner chain (K t ,t ≥ 0) is a level line of a GFF h in a domain D from a ∈ ∂ D to b ∈ ∂ D if (ϕ(K t ),t ≥ 0) is a level line of ϕ(h) as in Definition 1.1, where ϕ is a conformal map from D → H sending a to 0 and b to ∞.
Theorem 1.2.
[Coupling] Assume the same notations as in Definition 1.1. Suppose that the function F is regulated and satisfies (1.2) for some c > 0. Then there exists a coupling satisfying the conditions in Definition 1.1. Moreover, in this coupling, the Loewner chain K is almost surely generated by a continuous and transient curve γ with almost surely continuous driving function.
The inequality on F in Theorem 1.2 guarantees that the corresponding level line will reach its target point ∞ before "dying" at some continuation threshold. Indeed, the level line of a GFF with piecewise constant boundary data is only defined until the first time that it hits a section of R + where the boundary data is less than −λ or a section of R − where it is greater than λ . In our case, if we allowed F to approach −λ (resp. λ ) at some point in R + (resp. in R − ), then our current framework would not control the behaviour of the level line around this point (see discussion below.) Thus, we do not treat this situation here.
Theorem 1.3.
[Determination] If (h, γ) are coupled as in Theorem 1.2, then γ is almost surely determined by h. Moreover, the curve γ is almost surely simple. We call γ the level line of h + F.
With this in hand, we can consider the collection of level lines determined by a given field. The following two theorems describe the interactions between the curves; corresponding to what one might expect from the level lines of a smooth function.
Theorem 1.4.
[Monotonicity] Suppose that F, G are functions satisfying the conditions in Theorem 1.2, and that F(x) ≥ G(x) for x ∈ R. Suppose that h is a zero boundary GFF on H and γ F (resp. γ G ) is the level line of h + F (resp. h + G). Then γ F lies to the left of γ G almost surely.
Theorem 1.5.
[Reversibility] Suppose that h is a GFF on H whose boundary value satisfies the conditions in Theorem 1.2. Let γ be the level line of h from 0 to ∞ and γ be the level line of −h from ∞ to 0. Then the two paths γ and γ (viewed as sets) are equal almost surely. Now we will explain the relevance of Conditions (1.1) and (1.2), which we need for our approach to work. Although one can make sense of what it means to be a level line of h+F for any F in L 1 (as in Definition 1.1), before this work the existence of the coupling was only known for piecewise constant boundary data. The assumption that the boundary data F is regulated corresponds precisely to the fact that F can be uniformly approximated by piecewise constant functions. Indeed, our argument will use an approximation of F by such functions, and a limit of the corresponding level lines. Thus with our current approach we are unable to say anything about functions which are not regulated. However, since Definition 1.1 still makes sense for a wider class of functions, it is an interesting question to determine the most general restrictions under which a coupling exists. For example, if one takes a GFF with boundary data which is −λ in a neighbourhood to the left of 0 and λ in a neighbourhood to the right of 0 then one can allow much rougher boundary data away from these neighbourhoods (for example, even Neumann boundary conditions, see [KI13]), and construct a weaker form of "local coupling" with an SLE variant. Whether these types of coupling can be extended to a strong coupling as in Definition 1.1, where the curve is also determined by the field, or whether the condition near 0 can be relaxed is currently unknown.
Concerning Condition (1.2); the key to the proof of Theorem 1.2 is the continuity and transience of the approximating level lines (with piecewise constant boundary data). This allows us to use the results of [KS16] (see details in Section 2.2) to obtain a continuous limiting curve. If Condition (1.2) failed, the approximating level lines would only be defined up to a continuation threshold, and we would not be able to obtain such a limit. The continuity of the limiting curve is absolutely crucial to the proofs of Theorems 1.3 to 1.5. In fact, if the existence and the continuity of level lines were obtained for other boundary data, one could use similar proofs to get the corresponding theorems. However, whether continuity still holds in this set up is also a difficult open problem. Although it is natural to conjecture that for general regulated boundary data the level line will exist as a continuous curve until hitting a point on the boundary where Condition (1.2) fails, a "continuation threshold" as in [MS16a], [WW16], it is unclear whether or not the continuity will break down around this point.
Finally, we identify the law of the level lines. It is proved in [MS16a,WW16] that the level lines of GFF with piecewise constant boundary data are SLE 4 (ρ) processes where ρ is a vector. In our context, when the boundary data is of bounded variation, the level lines turn out to be SLE 4 (ρ) processes, where ρ is now a Radon measure. With the help of the GFF, we are able to obtain the existence, the continuity, and the reversibility of such processes, properties which are far from clear by the definition of the process through Loewner evolution.
Theorem 1.6. Assume the same notations as in Theorem 1.2. Suppose further that F is of bounded variation. Then in the coupling (h, γ) given by Theorem 1.2, the marginal law of γ is that of an SLE 4 (ρ L ; ρ R ) process (see Section 2.5) where ρ R (resp. ρ L ) is a finite Radon measure on R + (resp. on R − ) and
F(x) = λ (1 + ρ R ([0, x])), x ≥ 0; F(x) = −λ (1 + ρ L ((x, 0])), x < 0
almost everywhere. In particular, we have the following properties of the SLE 4 (ρ L ; ρ R ) process. Suppose that there exists c > 0 such that
ρ L ((x, 0]) ≥ −2 + c, x < 0, ρ R ([0, x]) ≥ −2 + c, x > 0.
Then
(1) There exists a law on continuous curves from 0 to ∞ in H with almost surely continuous driving functions, for which the associated Loewner chain is an arSLE 4 (ρ L ; ρ R ) process.
(2) The above continuous curve is almost surely simple and transient.
(3) The time reversal of the above SLE 4 (ρ L ; ρ R ) process has the same law as SLE 4 (ρ L ;ρ R ) ,wherẽ
ρ R ([x, ∞]) = ρ L ((x, 0]), x < 0;ρ L ((x, ∞]) = ρ R ([0, x]), x > 0.
Remark 1.7. Although Theorem 1.6 gives us existence of SLE 4 (ρ L ; ρ R ) processes, we do not derive uniqueness in law. That is, we have not excluded the possibility that there exists another law on Loewner chains satisfying the definition of an SLE 4 (ρ L ; ρ R ) process.
Remark 1.8. Item (3) is the so-called reversibility of SLE. The reversibility was derived previously for SLE κ in [Zha08b], for SLE κ (ρ) where ρ is a vector in [Zha08a,?,MS16b,WW13]. In Theorem 1.6, we derive the reversibility of SLE 4 (ρ) where ρ is a Radon measure.
Outline. The structure of the paper is as follows. In Section 2, we discuss briefly the necessary background theory, and collect some results that will be important to us. We also define the class of SLE κ (ρ) process and generalize some of the theory from [MS16a], [WW16] which will help us in the sequel. In Sections 3 and 4, we set up a general framework for the level lines of a GFF, under the assumption that they exist and are given by continuous transient curves. In particular, we show that they are monotonic in the boundary data, and describe where they can and cannot hit the boundary. Sections 5 and 6 address the existence of continuous transient curves which can be coupled as level lines of a GFF, provided the boundary data satisfies the conditions of Theorem 1.2. The proof of this is via an approximation argument; using a general theory for the weak convergence of curves, as set out in [KS16]. The key point in the proof is the monotonicity obtained in Section 4. In Section 7 we prove Theorems 1.3 to 1.5 using the ideas from Sections 3 and 4. Finally, we complete the proof of Theorem 1.6 in Section 8.
Preliminaries
Regulated functions and functions of bounded variation
We say that a function F on R is regulated if it admits finite left and right limits
F(t + ) = lim h→0+ F(t + h); F(t − ) = lim h→0− F(t + h)
at every point t ∈ R, including ∞. Equivalently, see [Die69,Secion 7.6], F is regulated if it can be uniformly approximated on R by piecewise constant functions which change value only finitely many times. It is this formulation of the definition that will be useful to us in the sequel. Another type of function which is of particular interest in the current paper is the class of functions of bounded variation. Let us consider the connection (2.3) between pairs of Radon measures (ρ L ; ρ R ) and functions F on the real line. We saw above that piecewise constant functions correspond to purely atomic measures. In general, finite Radon measures are in one-to-one correspondence with right-continuous functions of bounded variation.
The space of functions of bounded variation are those F which satisfy
sup a<b sup ∑ i |F(x i ) − F(x i−1 )| : {x i } a finite partition of [a, b] < ∞.
For a proof of this equivalence, see [Fol99,Theorem 3.29]. Note that these functions are clearly regulated. So, provided they satisfy the correct bounds on R − and R + , functions of bounded variation meet the conditions of Theorem 1.2. Furthermore, if a bounded variation function is also absolutely continuous, then the corresponding measures (ρ L ; ρ R ) are absolutely continuous with respect to Lebesgue measure, and writing
ρ L (dx) = f L (x) dx, ρ R (dx) = f R (x) dx
we have that the function is differentiable almost everywhere with derivative equal to f L (x) on R − and f R (x) on R + .
A result on the convergence of curves
To show existence of the level line of a GFF with general boundary data as given in Theorem 1.2, we will attempt to approximate it by level lines of the field with piecewise constant boundary data. For this, a result from [KS16] on the weak convergence of curves, satisfying certain conditions on crossing probabilities, will be crucial.
In order to state the result, we need to define what we mean by crossings of topological quadrilaterals. Definition 2.2. We will often consider topological quadrilaterals in H which lie on the boundary in the sense that S 1 , S 3 ⊂ R and S 0 , S 2 ⊂ H. If we have such a quadrilateral, then we say that a curve γ :
[T 0 , T 1 ] → C crosses Q if there is a subinterval [t 0 ,t 1 ] ⊂ [T 0 , T 1 ], such that γ(t 0 ,t 1 ) ⊂ V but γ[t 0 ,t 1 ] intersects both S 0 and S 2 .
Essentially, the condition that will be required for weak convergence will be the following:
Condition 2.3. For any simple curve γ on H we say that Q is a topological quadrilateral in H τ := H \ γ[0, τ] if it is the image of the square (0, 1) 2 under a homeomorphism ψ. We define the sides of Q: S 0 , S 1 , S 2 , S 3 , to be the images of
{0} × (0, 1), (0, 1) × {0}, {1} × (0, 1), (0, 1) × {1}
under ψ. We consider Q such that the opposite sides S 1 , S 3 are contained in ∂ H t and define a crossing of Q to be a curve in H t which connects the two opposite sides S 0 and S 2 . Finally, we say that Q is avoidable if it doesn't disconnect γ(τ) and ∞ inside H t . A family Σ of probability measures on simple curves from 0 to ∞ in H is said to satisfy a conformal bound on an unforced crossing if there exists a constant M > 0 such that for any P ∈ Σ, for any stopping time τ, and any avoidable quadrilateral Q of H τ whose modulus m(Q) is greater than M,
P (γ[τ, ∞) crosses Q | γ[0, τ]) ≤ 1/2. γ[0, τ ] ∞ 0 Q S 2 S 0 S 1 S 3 (a) An avoidable quadrilateral of H τ = H \ γ[0, τ]. γ[0, τ ] ∞ 0 Q S 2 S 1 S 0 S 3 (b) An unavoidable quadrilateral of H τ = H \ γ[0, τ].
Fig. 2.1
Now we may state the result.
Proposition 2.4. Suppose that (W (n) ) n∈N is a sequence of driving processes of random Loewner chains that are generated by continuous simple random cuves (γ (n) ) n∈N in H, satisfying Condition 2.3. Suppose that the (γ (n) ) n∈N are parameterized by half plane capacity. Then
• (W (n) ) n∈N is tight in the metrisable space of continuous functions on [0, ∞) with the topology of uniform convergence on compact subsets of [0, ∞).
• (γ (n) ) n∈N is tight in the metrisable space of continuous functions on [0, ∞) with the topology of uniform convergence on the compact subsets of [0, ∞).
Moreover, if the sequence converges weakly in either of the topologies above, then it also converges weakly in the other and the limits agree in the sense that the law of the limiting random curve is the same as the that of the random curve generated under the law of the limiting driving process. In particular, any subsequential limit of the sequence of curves a.s. generates a Loewner chain with continuous driving function.
Proof. This may be found in [KS16] cf. Theorem 1.5 and Corollary 1.7.
In fact, we will need to apply this theorem when the curves (γ (n) ) n∈N correspond to certain SLE 4 (ρ L ; ρ R ) processes. In this case they may hit the real line, and so are not necessarily contained in H, as required by the Proposition. However, as discussed before the proof of Theorem 1.10 in [KS16], the result extends to curves such as ours, and so we may apply it without concern.
The zero boundary Gaussian free field
In this section we will describe the zero boundary Gaussian free field (GFF) in an arbitrary domain D C. We will always assume that the domain has harmonically non-trivial boundary, meaning that a Brownian motion started from a point in the interior will hit the boundary almost surely.
We start with the Green's function G D in D, which is the unique function in D such that
• ∆G D (z, ·) = 2πδ z (·) for each z ∈ D, and
• G D (z, w) = 0 if z or w is in ∂ D.
Explicitly,
G D (z, w) = − log |z − w| −G z (w)
whereG z (w) is the harmonic extension of w → − log |z − w| from ∂ D to D. The Green's function is conformally invariant in the sense that for any conformal map φ on D, and z, w ∈ D, we have
G D (z, w) = G φ (D) (φ (z), φ (w)).
Roughly speaking, the GFF will be the random Gaussian "function" on D with cov(h(z), h(w)) = G D (z, w). However, it can only be made sense of rigorously as a random distribution on D. For H s (D) the space of smooth compactly supported functions on D, we let (·, ·) denote the normal L 2 inner product on H s (D). We may also endow H s (D) with the Dirichlet inner product defined by
( f , g) ∇ = 1 2π D ∇ f (z) · ∇g(z)d 2 z
and we denote its Hilbert space completion under Dirichlet inner product by H(D). For {φ n } n≥0 an orthonormal basis of H(D), we define the zero boundary GFF h to be the random sum h := ∑ n α n φ n , where the α n 's are i.i.d. Gaussians with mean 0 and variance 1. This almost surely diverges in H(D), but makes sense as a distribution. That is, the limit ∑ n α n (φ n , p) := (h, p) almost surely exists for each p ∈ H s (D), and p → (h, p) is almost surely a continuous linear functional on H s (D). Note that for any f ∈ H s (D) we have that −∆ f = p is also in H s (D) and so can define
(h, f ) ∇ := 1 2π (h, p).
Then (h, f ) ∇ is a Gaussian with mean 0 and variance
1 4π 2 ∑ n (φ n , p) 2 = ∑ n (φ n , f ) 2 ∇ = ( f , f ) ∇ .
In fact, this characterizes the Gaussian free field. Furthermore, noticing that for p ∈ H s (D)
∆ −1 p := 1 2π D G D (·, w)p(w) dw
is a smooth function in D whose Laplacian is p and vanishes on ∂ D, we see that for any f , g, p, q ∈ H s (D) where h 1 and h 2 are independent, h 1 is harmonic in W , and h 2 is a zero boundary GFF in W .
cov((h, f ) ∇ , (h, g) ∇ ) = ( f , g) ∇ , cov((h, p), (h, q)) = D×D p(z)G D (z, w)q(w)d 2 zd 2 w.
This tells us that, given h| D\W , the conditional law of h| W is that of a zero boundary GFF in W , plus the harmonic extension of h| D\W to W .
Suppose that F is L 1 with respect to harmonic measure on R viewed from some point (hence every point) in H; we also denote its bounded harmonic extension to H by F. Then the GFF with mean F is defined to be the sum, h + F, of a zero boundary GFF and F. Proposition 2.6. Suppose that D 1 and D 2 are two simply connected domains with non empty intersection, and h i is a zero boundary GFF on D i for i = 1, 2. Let F i be harmonic on D i , i = 1, 2 and U ⊂ D 1 ∩ D 2 be a simply connected open domain. Then
(1) If dist(U, ∂ D i ) > 0 for i = 1, 2, then the laws of (h 1 + F 1 )| U and (h 2 + F 2 )| U
are mutually absolutely continuous.
(2) Suppose there is a neighbourhoodŪ ⊂ U such that D 1 ∩ U = D 2 ∩ U and that F 1 − F 2 tends to 0 along sequences approaching points in ∂ D i ∩U . Then the laws of
(h 1 + F 1 )| U and (h 2 + F 2 )| U are mutually absolutely continuous. Proof. [MS16a, Proposition 3.2].
Local sets for the GFF
The theory of local sets for the Gaussian free field was first introduced by Schramm and Sheffield in [SS13], and we quote several of their results here. For D a simply connected domain and A a random closed subset ofD, we let
A δ := {z ∈ D : d(z, A) ≤ δ }
and A δ be the smallest σ -algebra for which A and the restriction of h to the interior of A δ are measurable. Setting A = ∩ δ ∈Q + A δ we obtain a σ -algebra which is intuitively the smallest such making A, and h restricted to some infinitesimal neighbourhood of A, measurable. With this in mind, we will often refer to A as (A, h| A ).
Definition 2.7. Suppose that (h, A) is a coupling of a GFF in D and a random closed subset A ⊂ D. Then we say that A is a local set for h if either of the following equivalent statements hold:
(1) For any deterministic open subset U ⊂ D we have that, given the orthogonal projection of h onto h ⊥ (U), the event {A ∩ U = / 0} is independent of the orthogonal projection of h onto H(U). This means that the conditional probability of {A ∩ U = / 0} given h is a measurable function of the orthogonal projection of h onto H ⊥ (U).
(2) Given A , the conditional law of h is that of h 1 + h 2 , for h a zero boundary GFF on D \ A and h 1 an Ameasurable random distribution which is almost surely harmonic on D \ A.
In this case, we let C A be the conditional expectation of h given (A, h| A ), corresponding to h 1 in Item (2).
The interactions between local sets display some nice properties, which we will describe in the following propositions.
Proposition 2.8. Suppose that A 1 , A 2 are local sets for a GFF h, which are conditionally independent given h. Then A = A 1 ∪ A 2 is also local for h and moreover, given (A 1 , A 2 , A, h| A ), the conditional law of h is given by C A plus an instance of the zero boundary GFF in D \ A.
Proof. [SS13, Lemma 3.10].
Proposition 2.9. Let A 1 , A 2 be connected local sets which are conditionally independent and A = A 1 ∪ A 2 . Then C A − C A 2 is almost surely a harmonic function in D \ A which tends to zero along any sequence converging to a limit in • a connected component of A 2 \ A 1 which is larger than a singleton, or
• a connected component of A 1 ∩ A 2 which is larger than a singleton, if the limit is at a positive distance from
either A 2 \ A 1 , or A 1 \ A 2 .
Proof. [SS13, Lemma 3.11] and [MS16a, Proposition 3.6].
Proposition 2.10. Let A 1 , A 2 be connected local sets which are conditionally independent and
A = A 1 ∪A 2 . Suppose that C is a σ (A 1 )-measurable connected component of D \ A 1 such that {C ∩ A 2 = / 0} almost surely. Then C A | C = C A 1 | C almost surely, given A 1 . Proof. [MS16a, Proposition 3.7].
Proposition 2.11. Let h be a GFF and (Z(t),t ≥ 0) a family of closed sets such that Z(τ) is local for every Zstopping time τ. Suppose futher that for a fixed z ∈ D, CR(z, D \ Z(t)) is almost surly continuous and monotonic in t. Then, if we reparameterise time by
log CR(z, D \ Z(0)) − log CR(z, D \ Z(t)), the process C Z(t) (z) − C Z(0) (z)
has a modification which is Brownian motion until the first time that Z(t) accumulates at z. In particular, C Z(t) (z) has a modification which is almost surely continuous in t.
Proof. This is proved in [MS16a, Proposition 6.5]. Since we need the argument in the proof later, we briefly recall the proof here. For s ≥ 0, set
τ(s) := inf{t ≥ 0 : log CR(z, D \ Z(0)) − log CR(z, D \ Z(t)) = s}.
We need only show that the increments of the process C Z(τ(t)) (z) are independent, and stationary with Gaussian distribution. By [MS16a, Lemma 6.4], we know that for any s < t, the conditional law of
C Z(τ(t)) (z) − C Z(τ(s)) (z),
given (Z(τ(s)), h| Z(τ(s)) ), is a Gaussian with mean 0 and variance
log CR(z, D \ Z(τ(s))) − log CR(z, D \ Z(τ(t))) = t − s.
This means it must also be independent of (Z(τ(s)), h| Z(τ(s)) ), and so of C Z(τ(s)) (z). This completes the proof.
SLE κ (ρ) processes
We call a compact set K ⊂ H an H-hull if H := H \ K is simply connected. For any such hull one can show that there exists a unique conformal map φ from H → H which is normalized at ∞ in the sense that
φ (z) = z + 2a z + o( 1 z ), as z → ∞,
for some constant a which we call the half-plane capacity of K. For a continuous real-valued function (W t ,t ≥ 0) with W 0 = 0 we can define the solution g t (z) to the chordal Loewner equation
∂ t g t (z) = 2 g t (z) −W t , g 0 (z) = z.
This is well defined for each z ∈ H until the first time, τ(z), that g t (z) −W t hits 0. Setting K t = {z ∈ H : τ(z) ≤ t} and H t = H \ K t we find that g t is the conformal map from H t to H normalized at ∞, and the half-plane capacity of K t is equal to 2t. We call the family (K t ,t ≥ 0) the Loewner chain driven by (W t ,t ≥ 0). One class of Loewner chains that we will be particularly interested in are those generated by continuous curves; that is, those for which there exists a continuous curve γ such that K t is the hull generated by γ[0,t] for all t. Chordal SLE κ is the Loewner chain driven by W t = √ κB t , where B t is a standard one-dimensional Brownian motion. It is characterised by the special properties of conformal invariance and the domain Markov property. Specifically, (µ −1 K µ 2 t ,t ≥ 0) has the same law as (K t ,t ≥ 0) for any µ > 0, and for any stopping time τ, the law of ( f τ (K t+τ ),t ≥ 0) is the same as that of K. Here f τ := g τ −W τ .
It is known that SLE κ is almost surely generated by a continuous curve for all κ. In the special case κ ∈ [0, 4], it has also been shown that the curve is almost surely simple. Moreover we know that lim t→∞ γ(t) = ∞ almost surely; a property we refer to as transience. These facts were all proved in [RS05].
Definition 2.12. Let ρ L and ρ R be finite Radon measures on R − = (−∞, 0] and R + = [0, ∞) respectively, and
(B t ,t ≥ 0) be a standard one-dimensional Brownian motion. We say that W t , (V L t (x)) x∈R − , (V R t (x)) x∈R + t≥0 de- scribe an SLE κ (ρ L ; ρ R )
process, if they are adapted to the filtration of B and the following hold:
(1) The processes W t , B t , V L t (x) x∈R − and V R t (x) x∈R + satisfy the following SDE on time intervals where W t does not collide with any of the V L,R t (x):
dW t = √ κdB t + R − ρ L (dx) W t −V L t (x) dt + R + ρ R (dx) W t −V R t (x) dt (2.1) and dV L t (x) = 2dt V L t (x) −W t , x ∈ R − ; dV R t (x) = 2dt V R t (x) −W t , x ∈ R + . (2.2)
(2) We have instantaneous reflection of W t off the V L,R t (x), ie. it is almost surely the case that for Lebesgue almost all times t we have that W t = V L,R t (x) for each x ∈ R. The SLE κ (ρ L ; ρ R ) process is then defined to be the Loewner chain driven by W .
Remark 2.13. Note that it is not immediate from the definition that such a process exists. Indeed, we will only show the existence for κ = 4 and a specific subset of (ρ L ; ρ R ).
We define the continuation threshold of the process to be the be the infinum of values of t for which
either ρ L {x ∈ R − : V L t (x) = W t } ≤ −2 or ρ R {x ∈ R + : V R t (x) = W t } ≤ −2.
Observe that the case ρ L ≡ 0, ρ R ≡ 0 corresponds simply to SLE κ . Another special case is when the Radon measures are purely atomic. If this occurs we instead consider (ρ L ; ρ R ) to be a pair of vectors
ρ L = (ρ L l , · · · , ρ L 1 ), ρ R = (ρ R 1 , · · · , ρ R r ) with associated force points x L = (x L l < · · · < x L 1 ≤ 0), x R = (0 ≤ x R 1 < · · · x R r )
in the obvious way. In this case, it is proved in [MS16a, Theorem 2.2] that a slightly stronger version of Definition 2.12 determines a unique law on SLE κ (ρ L ; ρ R ) processes, defined for all time up until the continuation threshold. The additional condition they impose is that W t , B t , V L t (x) x∈R − and V R t (x) x∈R + in fact must satisfy (2.1) and (2.2) at all times. This ensures the uniqueness in law of these processes.
Through their connection with the GFF, which we will discuss in the next section, it was shown in [MS16a] that SLE κ (ρ L ; ρ R ) processes are almost surely generated by continuous curves up to and including the continuation threshold. Moreover, on the event that the continuation threshold is not hit before the curves reach ∞, the curves are almost surely transient. One can also show that the curves are absolutely continuous with respect to SLE κ as long as they are away from the boundary.
Level lines of the GFF with piecewise constant boundary data
As discussed in the introduction, the theory of level lines and flow lines of a GFF with piecewise constant boundary data has been studied previously in a number of works, including [Dub09], [MS16a], [SS13] and [WW16]. We collect in this section some results that will be useful in our article.
Suppose that F is a bounded harmonic function in H whose boundary value is piecewise constant on R and changes only finitely many times. Then F can be described almost everywhere in terms of a pair of purely atomic finite Radon measures (ρ L ; ρ R ), corresponding to vectors (ρ L ; ρ R ), via the relation
F(x) = λ (1 + ρ R ([0, x])), x ≥ 0; F(x) = −λ (1 + ρ L ((x, 0])), x < 0. (2.3)
When κ = 4, which corresponds to level lines of the GFF, the following results are known for any (ρ L ; ρ R ): (see [WW16, Theorems 1.1.1 and 1.1.2])
• There exists a coupling (K, h) where K is an SLE 4 (ρ L ; ρ R ) process and h is a zero boundary GFF, such that K is a level line of h + F.
• If h is a zero boundary GFF and K an SLE 4 (ρ L ; ρ R ) process, coupled such that K is a level line of h + F, then K is almost surely determined by h.
This allows us, for any such F and an instance of the zero boundary GFF h in H, to define the level line, γ, of h + F. It has been shown in [WW16, Theorem 1.1.3] that γ is in fact almost surely continuous up to and including the continuation threshold, and it is transient when the continuation threshold is not hit.
More generally, for any simply connected domain D and x, y in ∂ D, we say that γ is the level line of a GFF h in D started at x and targeted at y, if φ (γ) is the level line of h • φ −1 , where φ is any conformal map from D to H which sends x to 0 and y to ∞.
One nice property of the level lines is what we call monotonicity. Suppose that h is a GFF with piecewise constant boundary values, changing only finitely many times. For u ∈ R, we define the level line of h with height u to be the level line of h + u, and denote it by γ u . Then, for any u 1 ≥ u 2 , the level line γ u 1 lies to the left of γ u 2 almost surely, see [WW16, Theorem 1.1.4].
Another property of the level lines is their reversibility. Suppose that h is a GFF with piecewise constant boundary values changing only finitely many times. Let γ be the level line of h from 0 to ∞ and γ be the level line of −h from ∞ to 0. Then, on the event that neither hit their continuation thresholds before reaching their target points, we have γ = γ almost surely as sets. This implies the reversibility of the SLE 4 (ρ R ; ρ R ) process: conditioned on the event that the continuation threshold is not hit, the time reversal of the process is another SLE 4 (ρ L ; ρ R ) process, now from ∞ to 0 in H with appropriate weights and force points, conditioned not to hit its continuation threshold. See [WW16, Theorem 1.1.6]. Finally, we include a list of results from [WW16] that will be useful for the later proofs.
Lemma 2.14. Suppose that h is a zero-boundary GFF and F is the bounded harmonic extension of the piecewise constant boundary data which changes finitely many times. Let γ be the level line of h + F. We already know that γ is almost surely continuous up to and including the continuation threshold.
(1) [WW16, Theorem 1.1.3] The curve γ is almost surely simple and is continuous up to and including the continuation threshold.
(2) [WW16, Remark 2.5.15] For any open interval I of (−∞, 0) ∪ (0, ∞), assume that
either F(x) ≥ λ , ∀x ∈ I, or F(x) ≤ −λ , ∀x ∈ I.
Then almost surely γ ∩ I = / 0.
First generalizations to the GFF with general boundary data
In this section, we generalize some results concerning level lines with piecewise constant boundary data to general boundary data. In fact, the ideas in the proof for Lemma 2.16 when the boundary condition is piecewise constant [SS13, Lemmas 2.4-2.6] work for general boundary data with proper adjustment. In order to be self-contained, we still give a complete proof here.
Lemma 2.15. Suppose that (K t ,t ≥ 0) is a Loewner chain driven by a continuous process (W t ,t ≥ 0). Denote by (g t ,t ≥ 0) the corresponding sequence of conformal maps and f t = g t −W t the centered conformal maps. For any
fixed z ∈ H, define C t (z) = log CR(z, H) − log CR(z, H \ K t ).
Then, we have that
dC t (z) = 4ℑ( f t (z)) 2 | f t (z)| 4 dt. Proof. The conformal radius CR(z, H \ K t ) is equal to 2/|φ t (z)| for φ t any conformal map from H \ K t to H which sends z to i, an example of which is given by m t • f t , where m t : H → H is the Möbius transformation defined by m t (w) = ℑ( f t (z))w ℜ( f t (z)) 2 + ℑ( f t (z)) 2 − ℜ( f t (z))w .
This gives us that
C t (z) −C 0 (z) = − log 2 + ℜ(log m t ( f t (z))) + ℜ(log g t (z)).
However, in this case we can calculate m t ( f t (z)) explicitly, and find that −ℜ(log m t ( f t (z))) = log ℑ f t (z). Since we also know that
dg t (z) = −2g t (z) f t (z) 2 dt, dℑ( f t (z)) = −2ℑ( f t (z)) | f t (z)| 4 dt,
we can compute
dC t (z) = 4ℑ( f t (z)) 2 | f t (z)| 4 dt,
and this implies the result.
Lemma 2.16. Assume the same notations as in Definition 1.1. Suppose that the Loewner chain K is almost surely generated by a random continuous curve γ on H from 0 to ∞ whose driving function W is almost surely continuous. For z ∈ H and t ≥ 0, set τ(t) = inf{s : log CR(z, H) − log CR(z, H \ K s ) = t}.
Then the pair (h, K) can be coupled as in Definition 1.1 if and only if (η τ(t) (z),t ≥ 0) is a Brownian motion with respect to the filtration generated by (W τ(t) ,t ≥ 0) for any z ∈ H.
Proof. If (h, K) is coupled as in Definition 1.1, by Proposition 2.11, we know that (η τ(t) (z),t ≥ 0) is a Brownian motion. Moreover, by the proof of Proposition 2.11, we see that, for s < t, the variable η τ(t) (z) − η τ(s) (z) is independent of K τ(s) and has the law of Gaussian with mean zero and variance t − s. This implies that (η τ(t) (z),t ≥ 0) is a Brownian motion with respect to the filtration generated by (W τ(t) ,t ≥ 0). For the converse, assume that, for each z ∈ H, the process (η τ(t) (z),t ≥ 0) is a Brownian motion with respect the filtration generated by (W τ(t) ,t ≥ 0). We will begin by showing that there exists a Brownian motion (B t ,t ≥ 0) (with respect to the filtration of (W t ,t ≥ 0)) such that, for all z, we have
dη t (z) = ℑ 2 f t (z) dB t . (2.4)
Define U t (z) = η t (z) + arg( f t (z)).
(2.5)
We have the following observations.
• By the definition of η t (·), we know that U t (·) is the bounded harmonic function on H \ K t with the boundary values given by F + 2λ on R − \ K t , λ along the boundary of K t , and F on R + \ K t . Therefore, for fixed z, process (U t (z),t ≥ 0) is of bounded variation and is measurable with respect to the filtration generated by (W t ,t ≥ 0).
• By the assumption, for fixed z, the process (η t (z),t ≥ 0) is a Brownian motion when parameterized by
C t (z) = log CR(z, H) − log CR(z, H \ K t ).
Thus, by Lemma 2.15, we see that
d η t (z) = dC t (z) = 4 ℑ 1 f t (z) 2 dt.
Moreover, since (η τ(t) (z),t ≥ 0) is a Brownian motion with respect to the filtration of (W τ(t) ,t ≥ 0), we know that, for any s < t, the variable η t (z) − η s (z) is independent of (W u , u ≤ s) (we implicitly use the fact that K is a Loewner chain generated by a continuous curve with continuous driving function), thus the process (η t (z),t ≥ 0) is a local martingale with respect to the filtration of (W t ,t ≥ 0).
Combining these two facts, we know that arg( f t (z)) = arg(g t (z) − W t ) is a semimartingale, and hence W t is a semimartingale at least up to the first time that z is swallowed by K t . Note that
d arg( f t (z)) = ℑ −1 f t (z) dW t + ℑ 2dt ( f t (z)) 2 − d W t 2( f t (z)) 2 ,
and therefore, we have d W t = 4dt for all such t. Note however that the process W does not depend on z, and since we can always choose z far away as we want, we can argue that (W u , 0 ≤ u ≤ t) is a semimartingale up to time t for any t > 0, with d W t = 4dt. Thus, there exists a Brownian motion (B t ,t ≥ 0) and a process of bounded variation
(V t ,t ≥ 0) such that W t = 2B t − V t .
We emphasize that the processes B and V do not depend on z. Plugging in Equation (2.5), we have that
dη t (z) = ℑ 2 f t (z) dB t , dU t (z) = ℑ 1 f t (z) dV t ,(2.d η t (z), η t (w) = ℑ 1 f t (z) ℑ 1 f t (w) dt.
We know that, for each z, η t (z) is a continuous martingale. We can also extend the definition of η t (z) by setting it equal to its limit as s ↑ τ(z) at all times after τ(z). We further define for z, w ∈ H and t ≤ τ(z) ∧ τ(w),
G t (z, w) := G H ( f t (z), f t (w))
where we again extend this to all times after τ(z)∧τ(w), by setting it constant and equal to its limit as t ↑ τ(z)∧τ(w).
Observe that, in each connected component of H \ γ[0,t], the function η t is the bounded harmonic function with boundary values shown in Figure 2.2. We also know that G t (z, w) is non-decreasing in t for any fixed z, w.
Putting all of the above together, we can deduce by stochastic calculus that for any p ∈ H s (H), (η t , p) is a continuous martingale with
d (η t , p) = −dE t (p), where E t (p) := p(z)p(w)G t (z, w) d 2 zd 2 w.
Now we are ready to show that the pair (h, K) is coupled as in Definition 1.1. Since for each z, w ∈ H and nonnegative p ∈ H s (H) we have that η t (z) is a martingale and G t (z, w), E t (p) are non-decreasing, it must be that all the limits η ∞ (z), G ∞ (z, w) and E ∞ (p) exist. We leth be equal to η ∞ − η 0 plus a sum of independent zero boundary GFF's; one in each connected component of H \ γ. To show that (K,h) are coupled in the correct way we must verify that the marginal law ofh is that of a zero boundary GFF in H, and that (K,h) satisfies the correct domain Markov property. This amounts to showing that for each non-negative p ∈ H s (H): • (h, p) is a Gaussian with mean 0 and variance E 0 (p).
0 F F λ −λ F λ −λ −λ γ[0, t]
• For any K-stopping time τ, the conditional law of (h + η 0 )| H\K τ , p given K τ is a Gaussian with mean (η τ , p) and variance E τ (p).
To see the first point, for any µ > 0 we calculate
E[exp(−µ(h, p))] = E[E[exp(−µ(h, p))|K]] = E exp −µ(η ∞ − η 0 , p) − µ 2 2 E ∞ (p) = E exp −µ(η ∞ − η 0 , p) + µ 2 2 (E 0 (p) − E ∞ (p) exp − µ 2 2 E 0 (p) = exp − µ 2 2 E 0 (p) ,
where the last line follows from the fact that (η t , p) is a continuous bounded martingale with mean η 0 (p) and quadratic variation E 0 (p) − E ∞ (p). The second point follows similarly, replacing the initial expectation with a conditional one.
Non-boundary intersecting regime
All the conclusions in Sections 3 and 4 are proved in [MS16a,WW16] for level lines with piecewise constant boundary data and constant height difference. Although many of the ideas from these papers are fundamental to our proofs, there are several places where they fail for general boundary data. Therefore, we treat the general case here and give complete proofs in the next two sections.
Lemma 3.1. Suppose that γ is a random continuous curve from 0 to some γ-stopping time T with almost surely continuous driving function. Assume that γ is coupled with a zero boundary GFF h as a level line of h
+ F up to time T where F(x) ≥ −λ , ∀x < 0; F(x) ≥ λ , ∀x ≥ 0.
Then almost surely γ[0, T ] ∩ (0, ∞) = / 0.
Proof. Assume the same notations as in Definition 1.1. First, for any z ∈ H, define U t (z) in the same way in Equation (2.5), and we will explain that the process (U t (z), 0 ≤ t ≤ T ) is non-increasing. By the definition of η t (z), we know that U t (·) − λ is the harmonic function on H \ K t with the boundary values given by F + λ ≥ 0 on R − \ K t , zero along the boundary of K t , and F − λ ≥ 0 on R + \ K t . This harmonic function is non-increasing in t, and thus U t (z) is non-increasing. Next, we will show that the process Z t := V R t (0 + ) −W t cannot hit zero. By the proof of Lemma 2.16, we know that there exist a Brownian motion B and a process of bounded variation V such that W = 2B −V and Equation (2.6) holds. Since U t (z) in non-increasing in t, the process V t is non-decreasing in t up to the time that z is swallowed. However, the process V does not depend on z, and since we can always choose z far away, we have that (V u , 0 ≤ u ≤ t) is non-decreasing in u for any t > 0. Then we have, for all t,
dZ t ≥ −2dB t + 2dt Z t .
We can compare Z t /2 with a Bessel process of dimension 2, and so may conclude that Z t cannot hit 0. This implies that the curve cannot hit (0, ∞).
Remark 3.2. The proof of Lemma 3.1 also applies to the case when
F(x) ≤ −λ , x < 0; F(x) ≤ λ , x ≥ 0
by symmetry. In this case we see that for γ satisfying the same conditions as in Lemma 3.1, we have Then, almost surely, the curve (γ(t), 0 ≤ t ≤ T ) does not hit the boundary except the two end points.
γ[0, T ] ∩ (−∞, 0) = / 0 almost surely. 0 C ≥ λ γ[0, T ] λ+F −G ≥ λ −λ+F −G ≥ −λ −λ I = (a, b) (a) The boundary value of −h − G given γ[0, T ε ]. 0 C ≥ λ γ[0, T ] λ+F −G ≥ λ −λ+F −G ≥ −λ −λ I = (a, b) γ G (b) γ G cannot
Proof. It is sufficient to show that, for any 0 < a < b < ∞, the curve γ does not hit the interval I = (a, b). We prove by contradiction. Suppose that γ does hit I with positive probability, and on this event, define T ε to be the first time that γ gets within ε of I. Since F is bounded, suppose that F ≥ −C for some finite C ≥ λ . Let G be the bounded harmonic extension of the function which is equal to −C on R − and is equal to λ on R + . Note that F ≥ G. Let γ G be the level line of −h − G from ∞ to 0. By Lemma 2.14(1) and (2), we know that γ G is almost surely continuous and transient; and that γ G almost surely does not hit I.
Leth be h restricted to the unbounded connected component of H \ γ[0, T ε ], then conditionally on γ[0, T ε ], the field −h − G is a GFF with boundary data as shown in Figure 3.1(a). Moreover, given γ[0, T ε ], the curve γ G is coupled withh so that it is a level line of −h − G up until the first time that γ G hits γ[0, T ε ] (by Propositions 2.8 to 2.10). Since F − G is positive on H, we see from Lemma 3.1 that γ G cannot hit the left side of γ[0, T ε ] or (−∞, 0] before hitting the right side of γ[0, T ε ] or the tip γ(T ε ), see Figure 3.1(b). In any case, this implies that γ G has to get within ε of I. Since this holds for any ε > 0 on the event that γ hits I and γ G is continuous, we can conclude that γ G hits I with positive probability, contradiction.
Lemma 3.4. Assume the same notations as in Lemma 3.3. Then γ is almost surely simple.
Proof. First, we argue that, for any γ-stopping time τ, we have γ[0, τ) ∩ γ(τ, T ) = / 0 almost surely. Given γ[0, τ], denote byh the restriction of h + F to H \ γ[0, τ] (since γ does not hit the boundary, this set only has one connected component). By the domain Markov property in Definition 1.1, we know that, given γ[0, τ], the curve γ| [τ,T ) is coupled withh as its level line. Note that the boundary value ofh + F is F ≤ −λ on R − , is −λ along the left side of γ[0, τ], is λ along the right side of γ[0, τ], and is F ≥ λ along R + . By Lemma 3.3, we know that γ(τ, T ) cannot hit γ(0, τ).
Next, we show that γ is almost surely simple. For any q > 0, define A q to be the event that γ(0, q) ∩ γ(q, T ) = / 0. If γ has double point, then A q happens for some positive rational q, since γ is continuous. However, by the above argument, we know that ∪ q∈Q + A q has zero probability. Therefore, γ is almost surely simple.
Proposition 3.5. Suppose that h is a zero boundary GFF and that F is bounded and satisfies
F(x) ≤ −λ , ∀x < 0; F(x) ≥ λ , ∀x ≥ 0.
Suppose that γ (resp. γ ) is a random continuous transient curve from 0 to ∞ (resp. from ∞ to 0) with almost surely continuous driving function.
Assume that γ is coupled with h as a level line of h + F, that γ is coupled with h as a level line of −h − F, and that the triple (h, γ, γ ) are coupled so that γ and γ are conditionally independent given h. Then almost surely γ equals γ. In particular, this implies that γ is almost surely determined by h.
Proof. First, we argue that, for any γ -stopping time τ , given γ [0, τ ], the curve γ almost surely first exits H \ γ [0, τ ] at γ (τ ). Denote byh the restriction of h to H \ γ [0, τ ]. Given γ [0, τ ], the curve γ is coupled with h as a level line ofh + F. The boundary value ofh + F is F ≤ −λ on R − , is −λ along the left side of γ [0, τ ], is λ along the right side of γ [0, τ ], and is F ≥ λ on R + . Thus, by Lemma 3.3, we know that γ must exit H \ γ [0, τ ] at γ (τ ).
Next, we show that γ and γ are equal. Since γ hits γ [0, τ ] for the first time at γ (τ ) for any γ -stopping time τ , we know that γ hits a dense countable set of points along γ in reverse chronological order. By symmetry, γ hits a dense countable set of points along γ. Since both γ and γ are continuous simple curves, the two curves (viewed as sets) are equal.
Monotonicity
Lemma 4.1. Suppose that h is a zero boundary GFF and that F is bounded. Suppose that γ is a random continuous curve from 0 to some γ-stopping time T with almost surely continuous driving function. Assume that γ is coupled with h as a level line of h + F up to time T .
(1) Then the curve (γ(t), 0 ≤ t ≤ T ) almost surely does not intersect any open interval I of (0, ∞) such that F(x) ≥ λ ∀x ∈ I.
Symmetrically, it does not intersect any open interval of
(−∞, 0) where F(x) ≤ −λ .
(2) In addition, if (γ(t), 0 ≤ t ≤ T ) is almost surely simple, then it does not hit any open interval I of (−∞, 0)
where F(x) ≥ λ . Symmetrically, it does not intersect any open interval of (0, ∞) where F(x) ≤ −λ .
Proof of Lemma 4.1, Item (1). We first show the conclusion when I = (a, b) for 0 < a < b and F(x) ≥ λ , ∀x ∈ I. Pickã,b such that a <ã <b < b. It is sufficient to show that, for any suchã,b, the curve (γ(t), 0 ≤ t ≤ T ) does not hit the intervalĨ = [ã,b]. We prove by contradiction. Suppose that the curve (γ(t), 0 ≤ t ≤ T ) hitsĨ with positive probability. Since F is bounded, we have that F ≥ −C for some C ≥ λ . Let G be the bounded harmonic extension of the function which is equal to −C on R − ∪ (0, a) ∪ (b, ∞) and λ on (a, b). Note that F ≥ G. Let γ G be the level line of −h − G from b to a. Note that since G is piecewise constant we know by Lemma 2.14(1) that the curve γ G is continuous from b to a, and the boundary data also means, by Lemma 2.14(2), that it does not hitĨ. This means we can repeat the same argument as in the proof of Lemma 3.3 to show that γ G hitsĨ with positive probability and obtain a contradiction.
Proof of Lemma 4.1, Item (2). Now, let I = (a, b) for a < b < 0, and suppose that F(x) ≥ λ , ∀x ∈ I and (γ(t), 0 ≤ t ≤ T ) is almost surely simple. It will be sufficient for us to prove that γ does not hitĨ = [ã,b] for any a <ã <b < b. First note that if γ hits [−∞, a] before hitting I, since γ grows towards ∞, it can never hitĨ thereafter and we are done.
If not, let ϕ be the Möbius transform of H that sends the triplet (b, 0, ∞) to (∞, 0, 1). Then (ϕ(γ(t)), 0 ≤ t ≤ T ) is a continuous curve, coupled with a zero-boundary GFFh as a level line ofh + F • ϕ −1 until the first time it hits [1, ∞]. By Item (1), we know that (ϕ(γ(t)), 0 ≤ t ≤ T ) cannot hit the interval (ϕ(a), ∞) before this time. Thus (γ(t), 0 ≤ t ≤ T ) cannot hitĨ without first hitting the point b. Let τ be the time at which γ hits b, setting τ = T if this never happens, and τ be the first time at which γ hitsĨ, again setting τ = T if necessary. By the previous reasoning, if we do not have {τ < τ < T } then we are done, so assume this occurs with positive probability. On this event, since γ is a continuous curve with continuous Loewner driving function, we see that {τ ≤ t ≤ τ : γ(t) ∈ R} has Lebesgue measure 0, and so there exists a time τ < σ < τ with γ(σ ) / ∈ R. Let w be the left-most point in (γ(t), 0 ≤ t ≤ σ ) ∩ (−∞, 0). Then applying the same argument as above, now to (γ(t), σ ≤ t ≤ T ) in the domain (a, b) replaced by (a, w), we see that (γ(t), σ ≤ t ≤ T ) must first hit w before it can hitĨ. This is a contradiction to the simplicity of γ. (2) If there exists X ∈ (0, ∞) and c > 0 such that
H \ (γ(t), 0 ≤ t ≤ σ ) withF(x) ≥ λ , ∀x ∈ (−X, ∞); F(x) ≥ −λ + c, ∀x ∈ (X, ∞)
and in addition the curve (γ(t), 0 ≤ t ≤ T ) is almost surely simple, then (γ(t), 0 ≤ t ≤ T ) almost surely does not hit ∞. Symmetrically, if there exists X ∈ (0, ∞) and c > 0 such that
F(x) ≤ −λ , ∀x ∈ (X, ∞); F(x) ≤ λ − c, ∀x ∈ (−X, ∞)
and the curve (γ(t), 0 ≤ t ≤ T ) is almost surely simple, then (γ(t), 0 ≤ t ≤ T ) almost surely does not hit ∞
We point out that Item (2) is not a consequence of Item (1) in Lemma 4.2. In fact, if F is piecewise constant and F(x) ∈ (−λ + c, λ − c) on (−∞, −X) ∪ (X, ∞) for c ∈ (0, λ ), then the level line of h + F is transient, and hence hits ∞ almost surely.
Proof of Lemma 4.2, Item (1). We may assume that F ≥ −λ + c on (a, b) where 0 < a < x 0 < b and again prove by contradiction. Suppose that the curve (γ(t), on (a, b). Note that F ≥ G. Let γ G be the level line of −h − G from b to a. Note that γ G is continuous and does not the point {x 0 } by Lemma 2.14(3). Thus we can repeat the same argument as in the proof of Lemma 3.3 and show that γ G hits {x 0 } with positive probability, which is a contradiction.
0 ≤ t ≤ T ) does hit {x 0 } with some positive probability. Since F is bounded, suppose that F ≥ −C for some C ≥ λ . Let G be the function which is equal to −C on R − ∪ (0, a) ∪ (b, ∞), and is (−λ + c) ∧ λ
Proof of Lemma 4.2, Item (2). We prove by contradiction. Suppose that γ does hit ∞ with positive probability. Since F is bounded, suppose that F ≥ −C for some C ≥ λ . Let G be the function which is equal to −C on (−X, 0) ∪ (0, X) and is (−λ + c) ∧ λ on (X, ∞) ∪ (−∞, −X). Note that F ≥ G. Let γ G be the level line of −h − G from −X to X. Since G is piecewise constant, we know the curve γ G is continuous and does not hit the point ∞. By Lemma 4.1(2), since γ is almost surely simple, we know that γ cannot hit (−X, ∞) before it hits ∞. Thus, we can repeat the same argument as in the proof of Lemma 3.3 and show that γ G hits ∞ with positive probability, contradiction. See more details in Figure 4.1.
0 C ≥ λ γ F λ+F −G ≥ λ (λ−c) ∨ (−λ) X −X ∞
(a) Suppose that γ F hits ∞ with positive probability. Let T ε be the first time that it enters {z : |z| > 1/ε}.
Since F ≥ λ on (−∞, −X), the curve γ F can never hit (−∞, −X). Given γ F [0, T ε ], the boundary data of the field −h − G is shown in this figure. 0 C ≥ λ γ F λ+F −G ≥ λ X −X ∞ γ G (b)
By the choice of G, we see that γ G cannot hit the union of (−X, 0) and the left side of γ F [0, T ε ] before hitting γ F [0, T ε ]. Therefore, γ G has to enter {z : |z| > 1/ε}. This holds for all ε > 0, thus γ G hits ∞ with positive chance, contradiction.
F(x) ≥ G(x), ∀x ∈ R.
Suppose that γ F (resp. γ G ) is a random continuous transient curve from 0 to ∞ (resp. from ∞ to 0) with almost surely continuous driving function. Assume that γ F is coupled with a zero boundary GFF h as a level line of h + F from 0 to ∞ and that γ G is coupled with h as a level line of −h − G from ∞ to 0, and that the triple (h, γ F , γ G ) is coupled so that γ F and γ G are conditionally independent given h. Then almost surely γ F stays to the left of γ G .
Proof. Note that, by Lemma 4.3, γ F is almost surely simple. It is sufficient to show that, for any γ G -stopping time τ , the point γ G (τ ) is to the right of γ F .
Leth be h restricted to H \ γ G [0, τ ]. Then we know that, given γ G [0, τ ], the conditional law ofh + F is that of a GFF with boundary data as shown in Figure 4.2(a). Moreover, γ F is coupled as a level line ofh + F up until the first time it hits γ G [0, τ ]. Given γ G [0, τ ], let τ be the first time that γ F hits γ G [0, τ ].
Consider the set γ G [0, τ ], there are two possibilities for the intersection γ G [0, τ ] ∩ (0, ∞): Case (a), the intersection is nonempty, and in this case we denote by x G the last point in the intersection; Case (b), the intersection is empty and in this case we set x G = +∞ ie. to the right of γ G [0, τ ].
Since the boundary data on the right hand side of γ G [0, τ ] is greater than λ and the boundary data is bounded away from −λ in a neighborhood of x G , by Lemma 4.1 we know that γ F cannot hit the right hand side of γ G [0, τ ] before hitting the left side of γ G [0, τ ] or exiting H at ∞, approaching from the left. We also know that γ F cannot hit {x G } before this time, by Lemma 4.2(1) in Case (a) and by Lemma 4.2(2) in Case (b). Therefore γ F cannot hit the union of the right hand side of γ G [0, τ ] and {x G } (i.e. the blue section in Figure 4.2(a)) of the boundary before Fig. 4.2: The curve γ F cannot hit the blue section in the figure. hitting the left hand side of γ G [0, τ ] or exiting H at ∞, approaching from the left. In the latter case, we are done. In the former case, γ F first hits γ G [0, τ ] from its left hand side at time τ. If γ G (τ ) is strictly to the left of γ F , then it must be the case that after time τ, γ F wraps around γ G [0, τ ] and then hits the right hand side of γ G [0, τ ] or exits at x G . Let τ δ be the first time after τ that γ F is in the right connected component of
0 F F −λ+F −G λ+F −G x G γ G [0, τ ] (a) 0 F F −λ+F −G λ+F −G x G γ G [0, τ ] γ F (b)H \ (γ F [0, τ] ∪ γ G [0, τ ]) and dist(γ F (t), γ G [0, τ]) ≥ δ , setting τ δ = ∞ if this never happens. If γ G (τ )
is strictly to the left of γ F (so in particular not on the curve γ F ) with positive probability then we know that {τ δ < ∞} occurs with strictly positive probability. However, given γ G [0, τ ] ∪ γ F [0, τ δ ], the conditional law of h + F is that of a GFF with boundary values as shown in Figure 4.2(b), and (γ F (t),t ≥ τ δ ) is a level line of this field (by Propositions 2.8 to 2.10.) Therefore, by Lemmas 4.1 and 4.2 again, we know that it cannot hit the right hand side of γ G [0, τ ] or exit at x G , and hence cannot reach ∞. Thus we obtain a contradiction.
Lemma 4.5. Suppose that F is bounded and satisfies Condition (1.2). Suppose that γ F (resp. γ F ) is a random continuous transient curve from 0 to ∞ (resp. from ∞ to 0) with almost surely continuous driving functions.
Assume that γ F is coupled with a zero boundary GFF h as a level line of h + F from 0 to ∞, that γ F is coupled with h as a level line of −h − F from ∞ to 0, and that the triple (h, γ F , γ F ) is coupled so that γ F and γ F are conditionally independent given h. Then almost surely γ F = γ F . In particular, γ F is almost surely determined by h.
Proof. By Lemma 4.4, we know that γ F almost surely stays to the left of γ F and (by the same arguments) that γ F almost surely stays to the right of γ F . Combining with the fact that γ F , γ F are simple by Lemma 4.3, we know that almost surely γ F = γ F . Since γ F and γ F are coupled with h so that they are conditionally independent given h, γ F = γ F implies that γ F must be almost surely determined by h.
Lemma 4.6. Suppose that F and G are bounded, and satisfy Condition (1.2). Suppose further that
F(x) ≥ G(x), x ∈ R.
Suppose that γ F , γ G (resp. γ G ) are random continuous transient curves from 0 to ∞ (resp. from ∞ to 0) with almost surely continuous driving functions.
Assume that γ F (resp. γ G ) is coupled with a zero boundary GFF h as a level line of h + F (resp. h + G), that γ G is coupled with h as a level line of −h − G from ∞ to 0, and that (h, γ F , γ G , γ G ) is coupled so that γ F , γ G and γ G are conditionally independent given h. Then almost surely γ F stays to the left of γ G .
Proof. We have the following observations.
• By Lemma 4.4, we know that γ F stays to the left of γ G .
• By Lemma 4.5, we know that γ G = γ G .
Combining these two facts, we see that γ F stays to the left of γ G .
Corollary 4.7. Suppose that F and G are piecewise constant functions changing value only finitely many times and that they satisfy Condition (1.2). Suppose further that
F(x) ≥ G(x), x ∈ R.
Let γ F (resp. γ G ) be the level line of h + F (resp. h + G) for h a zero boundary GFF as in Section 2.6. Then almost surely γ F stays to the left of γ G .
Proof. From the results in Section 2.6, we have the existence, the continuity and transience of γ F and γ G , and also γ G which is the level line of −h − G from ∞ to 0. Moreover, we know that each of γ F , γ G and γ G is almost surely determined by h. By Lemma 4.6, we know that γ F stays to the left of γ G almost surely.
Estimates on crossing probabilities
In this section, we will consider SLE 4 (ρ L ; ρ R ) processes for vectors
ρ L = (ρ L l , · · · , ρ L 1 ), ρ R = (ρ R 1 , · · · , ρ R r ),
with associated force points
x L = (x L l < · · · < x L 1 ≤ 0), x R = (0 ≤ x R 1 < · · · x R r ),
such that for some c > 0,C < ∞,
− 2 + c λ ≤ j ∑ i=1 ρ L i ≤ −1 + C λ , 1 ≤ j ≤ l, −2 + c λ ≤ k ∑ i=1 ρ R i ≤ −1 + C λ , 1 ≤ k ≤ r. (5.1)
We will show that if (γ (n) ) n∈N are a family SLE 4 (ρ L ; ρ R ) processes as above (with the same c,C), then they satisfy Condition 2.3. Here, we know that the processes are generated by continuous curves, due to the results of [MS16a,WW16].
Note that these processes correspond to level lines of (h + F n ) n∈N for h a zero boundary GFF, where Condition (5.1) means that the F n 's are uniformly bounded (lying in (−C,C)) and satisfy, for all n ≥ 0,
F n (x) ≤ λ − c, x < 0; F n (x) ≥ −λ + c, x ≥ 0.
These are the same conditions we require on F in Theorem 1.2. Therefore, the tactic will be to approximate such an F by piecewise constant functions F n on R, and show that the laws of the corresponding SLE 4 (ρ L,n ; ρ R,n ) processes converge weakly using Proposition 2.4. This limiting law will be our candidate for the level line of h + F.
Lemma 5.1. Suppose that (γ (n) ) n∈N are a family of SLE 4 (ρ L , ρ R ) processes satisfying Condition (5.1) for some c > 0,C < ∞ and all n. Then they satisfy Condition 2.3.
Proof. Recall, we would like to show that our family satisfies a conformal bound on an unforced crossing. That is, that there exists a constant M > 0, such that for any of our processes γ (n) , any stopping time τ and any avoidable quadrilateral of H τ = H \ K (n) τ whose modulus m(Q) is greater than M,
P γ (n) [τ, ∞) crosses Q | γ (n) [0, τ] ≤ 1/2. Here (K (n)
For n ≥ 0, the law of γ (n) is that of an SLE 4 (ρ L,n ; ρ R,n ) process with force points located at (x L,n ; x R,n ). Denote its driving function by W (n) , its sequence of conformal mappings by g (n) , and set f (n) = g (n) −W (n) . By the results of [WW16], we know that γ (n) can be coupled with h a zero boundary GFF in H, as the level line of h + F n , for
F n (x) = −λ 1 + ∑ {i:x L,n i ≥x} ρ L,n i , x < 0 λ 1 + ∑ {i:x R,n i ≤x} ρ R,n i , x ≥ 0.
Moreover, for any stopping time τ, we know by the domain Markov property that, conditionally on γ (n) [0, τ], the curve evolves from time τ onwards as a level line of a GFF with boundary conditions η (n) τ in the remaining domain. Here η (n) is defined corresponding to F n as in Definition 1.1. The important thing to notice is that, as a result of the condition (5.1), we have
η (n) τ • ( f (n) τ ) −1 ≥ −C, on (−∞, 0); η (n) τ • ( f (n) τ ) −1 ≥ −λ + c on [0, ∞)
for any τ and n. Therefore, if we set
G 1 (x) := −C, x < 0 −λ + c, x ≥ 0 we have that G 1 ≤ η (n) τ • ( f (n) τ ) −1 , on R. Similarly if we set G 2 (x) := λ − c, x < 0 C, x ≥ 0 then G 2 ≥ η (n) τ • ( f (n)
τ ) −1 , on R. Now, consider an avoidable topological quadrilateral Q of H τ . The avoidability assumption means that, when we map it to H via f Suppose we are in the first case. We would like to bound above the probability of γ (n) [τ, ∞) crossing Q, where Q has modulus greater than M for some positive M. Equivalently, we must bound the probability of f (n) τ (γ (n) [τ, ∞)) crossing Q , noting by conformal invariance that Q also has modulus greater than M. If Q = (V , (S k ) 1≤k≤4 ), we let Q = (V , S 0 , S 2 ) be the doubly connected domain where V is the interior of the closure of V ∪ V * (V * the reflection of V in the real line) and S 0 , S 2 are it's inner and outer boundary. Following the arguments in the proof of [KS16, Theorem 1.10], we let x = min(R ∩ S 0 ) > 0 and r = max{|z − x| : z ∈ S 0 } > 0. We see that Q is a doubly connected domain separating x and a point on ∂ B(x, r) from 0 and ∞ (see Figure 5.1.) However, [Ahl73, Thoerem 4.7] tells us that among all such domains, the one with the largest modulus (here defined as the extremal length of the curve family connecting S 0 and S 2 in V , which satisfies m(Q ) = m(Q )/2) is the domain formed by removing (−∞, 0] ∪ [x, x + r] from the complex plane. This modulus is also calculated explicitly in [Ahl73] and so we may deduce that exp(2πm(Q )) ≤ 16 x r + 1 .
Since
m(Q ) = m(Q) 2 ≥ M 2 ,
this means that r ≤ νx for
ν = 1 16 exp(πM) − 1 −1 . (5.2)
Note that ν can be made as small as we like by choosing M large. It is also clear that for f
B(x, r) Q = Q ∪ Q x 0 Q Q S 2 S 2 S 0 S 0 S 3 S 3 S 1 S 1ρ L = −1 + C λ ; ρ R = −2 + c λ (5.3)
(left and right force points at the origin), intersecting it. Therefore, we have
P γ (n) [τ, ∞) crosses Q | γ (n) [0, τ] ≤ P f (n) τ (γ (n) [τ, ∞)) crosses Q | γ (n) [0, τ] ≤ P f (n) τ (γ (n) [τ, ∞)) intersects B(x, r) | γ (n) [0, τ] ≤ P SLE 4 (ρ L ; ρ R ) intersects B(x, r) (ρ L , ρ R are defined in Equation (5.3))
= P SLE 4 (ρ L ; ρ R ) intersects B(1, r/x) (by scaling invariance)
≤ P SLE 4 (ρ L ; ρ R ) intersects B(1, ν) . (ν is defined in Equation (5.2))
Since we know that the SLE 4 (ρ L ; ρ R ) process with left and right force points at 0 almost surely does not hit the point 1 (in fact, there is exact estimate on this event, see for instance [MW16, Theorem 1.8]), we see that by choosing M large enough, and so ν small enough, we can make the right hand side less than 1/2. Thus there exists an M such that the left hand side is bounded above uniformly by 1/2 whenever m(Q) ≥ M.
For the second case, when the boundary arcs S 1 , S 3 of Q both lie on the negative real line, we may use symmetrical arguments, replacing G 1 by G 2 .
Corollary 5.2. Suppose that (γ (n) ) n∈N are a family of SLE 4 (ρ L ; ρ R ) processes satisfying Condition (5.1) for all n. Suppose further than they are all parameterised by half plane capacity and that (W (n) ) n∈N are the corresponding family of driving functions. Then Moreover, if the sequence converges weakly in either of the topologies above, then it also converges weakly in the other and the limits agree in the sense that the law of the limiting random curve is the same as the that of the random curve generated under the law of the limiting driving process.
f (n) τ 0 ∞ 0 ∞ γ (n) [0, τ ] f (n) τ (γ (n) [τ, ∞]) B(x, r) −λ λ −C ≤ F n F n ≥ −λ+c −λ λ ≥ −C ≥ −λ+c
Proof. This is a direct consequence of Proposition 2.4 and the remarks in Theorem 1.10 of [KS16].
6 Existence of the coupling-proof of Theorem 1.2
In this section we will show existence of the coupling described by Theorem 1.2. Recall we would like to prove that for F on R which is regulated, so can be approximated uniformly by piecewise constant functions changing value only finitely many times, and which satisfies Condition (1.2), there exists a coupling of a Loewner chain K with a zero boundary GFF h, such that K is a level line of h + F. Moreover, we will show that K is almost surely generated by a continuous and transient curve γ.
To do this we will take a sequence of piecewise constant functions F n (changing value only finitely many times), which uniformly approximate F, and consider the level lines, denoted by γ (n) , of h + F n for a zero boundary GFF h. Observe that we can choose the F n so that the level lines are a family of SLE 4 (ρ L ; ρ R ) processes satisfying the conditions of Corollary 5.2. Thus, the tightness given by the corollary will allow us to extract a subsequential limit.
Proposition 6.1. Let F satisfy the conditions of Theorem 1.2. Suppose that (F n ) n∈N are piecewise constant functions on R, changing value only finitely many times. Let h be a zero boundary GFF and γ (n) be the level line of h + F n for each n. Suppose further that they are all parameterized by half plane capacity and that (W (n) ) n∈N are the corresponding family of driving functions.
Then, if the (F n ) converge uniformly to F on R, we have that:
(1) There exists a subsequence of the γ (n) which converges weakly in the space of continuous functions on [0, ∞) with the topology of uniform convergence on compact subsets of [0, ∞).
(2) The limiting law describes a continuous curve from 0 to ∞ in H which generates a Loewner chain with a.s. continuous driving function.
(3) The limiting curve can be coupled with a zero boundary GFF h, as a level line of h + F.
Remark 6.2. We will later see that this limiting law does not depend on the choice of approximation, as any continuous curve which can be coupled with a zero boundary GFF as a level line of h + F must have a unique law: see Remark 7.1. In particular, this tells us that we actually have convergence of the whole sequence in distribution.
Proof of Theorem 1.2. Theorem 1.2 is a direct consequence of Proposition 6.1.
Proof of Proposition 6.1, Items (1), (2). Note that the weak convergence directly follows from Corollary 5.2, as does the fact that the limiting law corresponds to a continuous curve generating a Loewner chain with almost surely continuous driving function.
Definition 6.3. Suppose that F is L 1 with respect to harmonic measure on R and that γ is a continuous curve with continuous Loewner driving function. We set η 0 t in the same way as in Definition 1.1. Then for any z ∈ H we can define, for t less than the first time that γ swallows z,
η t (F, γ, z) = η 0 t ( f t (z))
as in Definition (1.1), emphasising the dependence on F and γ. Let
C t (γ, z) = log CR(z, H) − log CR(z, H \ K t ), η t (F, γ, z) = η τ(t) (F, γ, z), where τ(t) := inf{s ≥ 0 : C s (γ, z) = t}.
Finally, define (W t (γ, z)) t≥0 for γ to be the driving function of γ reparameterised by C t (γ, z).
To prove Proposition 6.1, Item (3), i.e. to see that the limiting curve can be coupled as a level line in the way we want, we will use Lemma 2.16. This tells us that if we defineη t (F, γ, z) as above for our limiting curve γ, we need only show that for each z ∈ H, the process (η t (F, γ, z),t ≥ 0) is a Brownian motion with respect to the filtration generated by (W t (γ, z),t ≥ 0). Lemma 6.4. Let (γ (n k ) ) be a subsequence of the random curves in Proposition 6.1, parameterised by half plane capacity, which converge weakly to some γ in the space of continuous functions on [0, ∞) with the topology of uniform convergence on compacts. Then for every z ∈ H,
W (γ (n k ) , z),η(F, γ (n k ) , z) d − → W (γ, z),η(F, γ, z)
in C([0, ∞); R) ×C([0, ∞); R) with respect to the product topology of uniform convergence on compacts.
We postpone the proof of Lemma 6.4 and first tell the readers how we obtain Proposition 6.1 from Lemma 6.4. Lemma 6.5. Let (γ (n k ) ) be a subsequence of the random curves in Proposition 6.1, parameterised by half plane capacity, which converge weakly to some γ in the space of continuous functions on [0, ∞) with the topology of uniform convergence on compacts. Then for every z ∈ H,
W (γ (n k ) , z),η(F n k , γ (n k ) , z) d − → W (γ, z),η(F, γ, z)
in C([0, ∞); R) ×C([0, ∞); R) with respect to the product topology of uniform convergence on compacts.
Proof. By Lemma 6.4, we have that W (γ (n k ) , z),η(F, γ (n k ) , z) d − → W (γ, z),η(F, γ, z) (6.1) with respect to the product topology of uniform convergence on compacts. It is also clear that, for all t, z and any curve γ
η t (F n k , γ , z) −η t (F, γ , z) ≤ sup x∈R |F(x) − F n k (x)|.
Indeed,η t (F n k , γ , ·) andη t (F, γ , ·) are by definition harmonic extensions of functions whose boundary values differ by at most the right hand side. Since sup x∈R |F(x) − F n k (x)| → 0 by assumption, we may conclude that, for any T > 0, almost surely as k → ∞,
sup t∈[0,T ] η t (F n k , γ (n k ) , z) −η t (F, γ (n k ) , z) → 0. (6.2)
Combining Equations (6.1) and (6.2), we obtain the conclusion.
Proof of Proposition 6.1, Item (3). Fix z ∈ H. Since γ (n k ) is coupled as a level line of h + F n k we know by Lemma 2.16 that η t (F n k , γ (n k ) , z),t ≥ 0 is a Brownian motion for each k, with respect to the filtration of W t (γ (n k ) , z),t ≥ 0 . Therefore, by the weak convergence in Lemma 6.5, we have that if γ is the limiting law of the γ (n k ) 's, the process η t (F, γ, z) must also have the law of Brownian motion, with respect to the filtration of W t (γ, z),t ≥ 0 . Applying Lemma 2.16 again proves the proposition.
Proof of Lemma 6.4. Fix z ∈ H. We will show that the laws of (W (γ (n k ) , z),η(F, γ (n k ) , z)) converge weakly in k to the law of (W (γ, z),η(F, γ, z)). To do this, we begin by showing that this family of laws is tight in C([0, ∞); R) × C([0, ∞); R) with respect to the product topology of uniform convergence on compacts. This allows us to extract a further subsequence along which the (W (γ (n k ) , z),η(F, γ (n k ) , z))'s converge. We then argue that the limit of this subsequence must be equal to that of (W (γ, z),η(F, γ, z)), so in fact our whole original subsequence converged, and the limit is (W (γ, z),η(F, γ, z)). Note that the proof of this lemma would be trivial if (W (·, z),η(F, ·, z)) was a continuous function on the set of curves, however, this is not quite the case. It is essentially a continuous function when restricted to a set in which the γ (n k ) 's lie with high probability. By the proof of [KS16, Theorem 1.5], we know that for every M > 0 we can find a subset E of the space of continuous curves in H such that inf
k P(γ (n k ) ∈ E) ≥ 1 − 1 M (6.3)
when the (γ (n k ) ) are parameterised by half plane capacity, and
• E is relatively compact with respect to the topology of uniform convergence on compacts,
• curves in E correspond to Loewner chains with continuous driving functions parameterised by half plane capacity, and
• if a sequence of curves in E converges with respect to uniform convergence on compacts, then their driving functions also converge uniformly on compacts along a further subsequence, and the limits agree.
For the construction of such an E, see Section 3.5 of [KS16], in particular the definition (60) and the discussion in the closing paragraphs. See also the opening paragraph of Section 3.6. We argue that the set {(W (γ , z),η(F, γ , z)) : γ ∈ E} is a relatively compact subset of C([0, ∞); R)×C([0, ∞); R) with respect to the product topology of uniform convergence on compacts. Thus by (6.3) the laws of the W (γ (n k ) , z),η(F, γ (n k ) , z) are tight in this topology. It is sufficient to verify the following claim: if γ n → γ is any convergent sequence of curves in E, whose driving functions also converge uniformly on compacts, then for any T > 0, as n → ∞, W t (γ n , z) −W t (γ , z) → 0. (6.5)
• W n t → W t uniformly on [0, S].
• g n t → g t uniformly on {(t, z) ∈ [0, S] × H : d(z, K t ) > δ } for any δ > 0. Same reason as above.
Combining these three facts, we have that the quantity |η t (F, γ n , z) − η t (F, γ , z)| converges uniformly to 0 on t ∈ [0, S], implying Equation (6.8).
Combining Equations (6.6), (6.7) and (6.8), we obtain Equation (6.4) by noting that
sup t∈[0,T ] η τ n (t) (F, γ n , z) − η τ(t) (F, γ , z) ≤ sup t∈[0,T ] η τ n (t) (F, γ n , z) − η τ n (t) (F, γ , z) + sup t∈[0,T ] η τ n (t) (F, γ , z) − η τ(t) (F, γ , z) .
We obtain Equation (6.5) by the same method as above, which is much simpler in this case, and so we omit the details.
Finally, we show that if (γ (n k ) ) k∈N converges weakly, and there exists a further subsequence along which (W (γ (n k ) , z),η(F, γ (n k ) , z)) converges, then the limit must be (W (γ, z),η(F, γ, z)). To do this, for any M ∈ N take E relatively compact such that (6.3) holds, and note that by the above claim we have that
A E := (γ ,W (γ , z),η(F, γ , z)) : γ ∈ E is relatively compact in C([0, ∞); C) ×C([0, ∞); R) ×C([0, ∞); R)
, and its closure is equal to
(γ ,W (γ , z),η(F, γ , z)) : γ ∈ E .
This means that the joint laws of (γ (n k ) ,W (γ (n k ) , z),η(F, γ (n k ) , z)) are also tight, and thus we can extract an even further subsequence along which we have joint convergence. If P * is the law of this joint limit then,
P * A E ≥ inf k P γ (n k ) ∈ E ≥ 1 − 1 M
and so we see that the probability of our marginal laws agreeing in the sense we want must be greater than 1 − 1 M . Since this holds for every M, agreement must hold almost surely, and as these marginal laws are equal to the limiting laws of the individually convergent sequences, the result follows.
Proof of Theorems 1.3 to 1.5
Proof of Theorem 1.4. Suppose that γ F and γ G are continuous transient curves from 0 to ∞ in H, coupled with a zero-boundary GFF h as level lines of h+F and h+G respectively. Suppose further that γ G is a continuous transient curve from ∞ to 0 and is coupled with h as a level line of −h − G from ∞ to 0, such that the four objects h, γ F , γ G , γ G are coupled with γ F , γ G , γ G are conditionally independent given h. From Theorem 1.2, we have the existence of γ F , γ G and γ G . By Lemma 4.6, we know that γ F stays to the left of γ G almost surely.
Proof of Theorems 1.3 and 1.5. Suppose that γ F is a continuous transient curve which is coupled with h as a level line of h + F from 0 to ∞, as in Theorem 1.2. Let γ F be a continuous curve coupled with h as a level line of −h − F from ∞ to 0, such that γ F and γ F are conditionally independent given h. The existence of γ F is given by Theorem 1.2. Lemma 4.5 then tells us that γ F = γ F almost surely. In particular, γ F is almost surely determined by h.
Remark 7.1. By applying Theorem 1.3, we see that if γ is the weak limit of any sequence of level lines as in Proposition 6.1, then γ can be coupled as the level line of a GFF and is moreover determined by the GFF in this coupling. Thus, the law of γ is uniquely determined. In particular, it does not depend on the sequence of approximating level lines.
Lemma 7.2. Let F be as in Theorem 1.2. Suppose that F n ↓ F approximate F uniformly on the real line, where the F n are decreasing, and are piecewise constant with value changing only finitely many times.
Let h be a zero boundary GFF in H, γ n be the level line of h + F n for each n, and γ be the level line of h + F. Denote by H n the open sets corresponding to the strict right hand sides of γ n . By monotonicity these are almost surely decreasing. Define H = ∩ n H n .
Then ∂ H coincides with γ almost surely. In other words, the sequence of curves γ n converges to γ almost surely.
Proof. First, we show that ∂ H has the same law as γ F . We use a conformal mapping to take everything to the unit disc, as it will be more convenient to work in a space where our sets are compact. We endow H with the metric it inherits from the unit disc U via the map ϕ(z) = (z − i)/(z + i). Namely, let d * (·, ·) denote the metric on H given by d * (z, w) = |ϕ(z) − ϕ(w)|.
We write H for the completion of H with respect to d * . For compact sets A, B ⊂ H, we have the d * -induced
Hausdorff distance d H * (A, B) = inf{ε > 0 : A ⊂ B (ε) , B ⊂ A (ε) },
where A (ε) denotes the open ε-neighborhood of A with respect to the metric d * . Note that d H * makes the set of all non-empty compact subsets of H (with metric d * ) into a compact metric space. We have the following observations.
• The sets H n form an almost surely decreasing sequence of compact subsets of H, which therefore converge to H with respect to d H * . This implies that γ n = ∂ H n almost surely converges to ∂ H with respect to d H * .
• By the assumptions on F n , we know that the laws of γ n fall in to the framework of Proposition 6.1. This means that we can extract a subsequence which converges weakly in the space of continuous functions on [0, ∞) with respect to uniform convergence on compacts. Moreover, the limiting curve can be coupled with a zero boundary GFFh as the level line ofh + F. Furthermore, the subsequence converges weakly, to the same limit, in the space of curves from [0, 1] → H with respect to the topology of uniform convergence modulo reparameterisation, where the metric on H is given by d * . This requires a slight extension of Proposition 2.4, which was stated here, but is nonetheless still true, by the extended version given in [KS16, Corollary 1.7]. By continuity, we therefore have that along this subsequence the curves also converge weakly to the same limit with respect to d H * . Thus ∂ H has the law of a continuous curve which can be coupled with a zero boundary GFFh as the level line ofh + F.
• By Theorem 1.3, we know that the law on continuous curves which can be coupled with a GFFh as a level line ofh + F, is unique.
Combining these three facts, we may conclude that ∂ H has the same law as γ F .
Next, we show that ∂ H coincides with γ almost surely. We have the following observations.
• By the above analysis, we know that ∂ H has the same law as γ.
• By Theorem 1.4, we know that ∂ H lies to the left of γ almost surely.
Combining these two facts, we obtain that ∂ H coincides with γ almost surely.
Proof of Theorem 1.6 and concluding remarks
In this section, we prove Theorem 1.6: the key ingredient being the proof of Lemma 8.1. This lemma is proved in [MS16a,WW16] for SLE 4 (ρ) process when ρ is a vector. The proof given in these papers will work with minor modifications for the case when ρ is a Radon measure but, to be self-contained, we still give a complete proof here.
Lemma 8.1. Suppose we are given a random continuous curve in H from 0 to ∞ whose Loewner driving function W is almost surely continuous. If (ρ L ; ρ R ) are a pair of finite Radon measures on R − , R + and F is the corresponding function of bounded variation, define (η t ,t ≥ 0) as in Definition 1.1. For z ∈ H and t ≥ 0, define τ(t) = inf{s : log CR(z, H) − log CR(z, H \ K s ) = t}.
Then W, (V L (x)) x∈R − , (V R (x)) x∈R + can be coupled with a standard Brownian motion to describe an SLE 4 (ρ L ; ρ R ) process if (η τ(t) (z),t ≥ 0) evolves as a Brownian motion with respect to the filtration generated by (W τ(t) ,t ≥ 0) for any z ∈ H.
Proof. Suppose that (η τ(t) (z),t ≥ 0) is a Brownian motion with respect to the filtration generated by (W τ(t) ,t ≥ 0) for each z ∈ H. This implies that (η t (z),t ≥ 0) is a local martingale with respect to the filtration generated by (W t ,t ≥ 0). Our first step will be to show that W t is a continuous semi-martingale. By the definition of η t (·), we know that, for each z ∈ H,
2η t (z) = − R − arg(g t (z) −V L t (x)) ρ L (dx) − arg(g t (z) −W t ) + (π − arg(g t (z) −W t )) + R + π − arg(g t (z) −V R t (x)) ρ R (dx). (8.1)
This follows from the integration by parts formula for functions of bounded variation, and the integral expression for the harmonic extension of a bounded function on the real line. Note here that the integrals are well defined, since for each fixed t, z the integrands are continuous, bounded functions in x, and ρ L , ρ R are assumed to be finite measures. Indeed, g t (z) and (V L,R t (x)) x∈R are adapted and differentiable, and we may also differentiate under the integral in (8.1) by finiteness of ρ L , ρ R . Therefore, we can deduce that all the terms in (8.1) apart from the only one, arg(g t (z) − W t ), involving W t , are semi-martingales. Since η t (z) is itself a local martingale, this means that arg(g t (z)−W t ) must also be a semi-martingale. Now, note that by Schwartz's formula, we can write log(g t (z)−W t ), up to a constant, as a linear functional (an integral against a test function) of arg(g t (z) − W t ). So log(g t (z) − W t ) is also a semi-martingale, and thus it's exponential, and consequently W t itself, must be a semi-martingale also. Hence we can write W t := M t −V t for M a local martingale and V of bounded variation.
Substituting this into the expression (8.1) we see that, on intervals where W t does not collide with the V L,R t , the drift of 2η t is equal to the imaginary part of
2 R − ρ L (dx)/ V L t (x) −W t g t (z) −W t dt + −2dV t (g t (z) −W t ) + d W t − 4dt (g t (z) −W t ) 2 + 2 R + ρ R (dx)/ V R t (x) −W t g t (z) −W t dt,
which of course must vanish. Therefore, multiplying by (g t (z) − W t ) 2 and evaluating at z such that g t (z) − W t is arbitrarily close to 0, we can deduce that d W t = 4dt. On subsequently removing the third term, we also find an expression for dV t , and can conclude that W t satisfies (2.1) in Definition 2.12 on intervals where W t does not collide with the V L,R t . All that remains is to show that we have instantaneous reflection of W t off the V L,R t (x). It suffices to show that the number of times the curve γ hits the real line has Lebesgue measure 0. However, this is always the case for a continuous curve with continuous driving function, which we know for example by [MS16a, Lemma 2.5].
Remark 8.2. We believe that Lemma 8.1 could be made into an if and only if statement if we strengthened Definition 2.12 of an SLE κ (ρ L ; ρ R ) process to also require that, almost surely,
V L t (x) = x + t 0 2ds V L s (x) −W s , x ∈ R − ; V R t (x) = x + t 0 2ds V R s (x) −W s , x ∈ R + and (8.2) W t = √ κB t + t 0 ds R − ρ L (dx) W s −V L s (x) + t 0 ds R + ρ R (dx) W s −V R s (x) ,(8.3)
Definition 2. 1 .
1A topological quadrilateral Q = (V ; S k , k = 0, 1, 2, 3) consists of a domain V , along with four boundary arcs S 0 , S 1 , S 2 , S 3 , which can be mapped homeomorphically to a square in such a way that the boundary arcs are in counterclockwise order and correspond to the edges of the square. For any topological quadrilateral, there exists a unique positive L and a conformal map from Q onto the rectangle [0, L] × [0, 1], such that the boundary arcs are mapped to the edges of the quadrilateral and, in particular, S 0 is mapped to {0} × [0, 1]. We call this unique L the modulus of Q, denoted by m(Q).
( 3 )
3[WW16, Proposition 2.5.11] For any point x 0 ∈ (0, ∞), assume that there exists c > 0 such that F ≥ −λ + c in a neighborhood of {x 0 }, then almost surely γ does not hit {x 0 }. Symmetrically, for x 0 ∈ (−∞, 0), assume that there exists c > 0 such that F ≤ λ − c in a neighborhood of {x 0 }, then almost surely γ does not hit {x 0 }.
Fig. 2. 2 :
2The function η t (·) is a harmonic function in each connected component of H \ γ[0,t] with boundary values as above.
hit the left side of γ[0, T ε ] or (−∞, 0].
Fig. 3 . 1 :
31Explanation of the boundary values in the proof of Lemma 3.3. Lemma 3.3. Suppose that γ is a random continuous curve from 0 to some γ-stopping time T with almost surely continuous driving function. Assume that γ is coupled with a zero boundary GFF h as a level line of h + F up to time T where F(x) ≤ −λ , ∀x < 0; F(x) ≥ λ , ∀x ≥ 0.
Lemma 4 . 2 .
42Suppose that h is a zero boundary GFF and that F is bounded. Suppose that γ is a random continuous curve from 0 to some γ-stopping time T with almost surely continuous driving function. Assume that γ is coupled with h as a level line of h + F up to time T .(1) For any fixed point x 0 ∈ (0, ∞), if there exists c > 0 such that F ≥ −λ + c in a neighborhood of x 0 , then the curve (γ(t), 0 ≤ t ≤ T ) almost surely does not hit {x 0 }. Symmetrically, for any fixed point x 0 ∈ (−∞, 0), if there exists c > 0 such that F ≤ λ − c in a neighborhood of x 0 , then the curve (γ(t), 0 ≤ t ≤ T ) almost surely does not hit {x 0 }.
Fig. 4 . 1 :
41Explanation of the boundary values in the proof of Lemma 4.2, Item (2).
its image Q is a topological quadrilateral in H as in Definition 2.1 with S 1 , S 3 (the arcs touching the boundary) either both lying in [0, ∞), or both in (−∞, 0].
Fig. 5 . 1 :
51Since Q separates x and a point on ∂ B(x, r) from ∞, we obtain a lower bound on m(Q ) = m(Q)/2. Since m(Q) ≥ M this gives us an upper bound on r/x. Moreover, we know that for a curve to cross Q it must necessarily intersect B(x, r).
τ
(γ (n) [τ, ∞)) to cross Q , it must necessarily intersect B(x, r). However, the law of f(n) τ (γ (n) [τ, ∞)) is that of the level line ofh + η −1 , forh a zero boundary GFF in H. By the monotonicity result Corollary 4.7, we see that this level line lies to the left of the level line ofh + G 1 almost surely (see Figure 5.2.) Thus, the probability of f
τ
(γ (n) [0, τ]) intersecting B(x, r) is less than the probability of an SLE 4 (ρ L ; ρ R ) process with
Fig. 5 . 2 :
52The boundary values of h+F n given γ (n) [0, τ] are marked in the left panel. Thus f(n) τ (γ (n) [τ, ∞])is the level line of a zero boundary GFFh + boundary data as depicted in the right panel. By monotonicity, it must therefore lie to the left of the level line ofh + G 1 (marked in red.) Consequently, the probability that f
τ
(γ (n) [τ, ∞]) intersects B(x, r) is less than the probability that the red level line does.• (γ (n) ) n∈N is tight in the metrisable space of continuous functions on [0, ∞) with the topology of uniform convergence on the compact subsets of [0, ∞).
sup t∈[0,T ] η t (F, γ n , z) −η t (F, γ , z) → 0 (6.4)and supt∈[0,T ]
Lemma 4.3. Suppose that h is a zero boundary GFF and that F is bounded and satisfies Condition (1.2). Suppose that γ is a random continuous transient curve from 0 to ∞ with almost surely continuous driving function. Assume that γ is coupled with h as a level line of h + F. Then γ is almost surely simple.Proof. We can repeat the same argument as in the proof of Lemma 3.4 replacing Lemma 3.3 by Lemmas 4.1(1)
and 4.2(1).
Lemma 4.4. Suppose that F and G are bounded, F satisfies Condition (1.2), and that
Preliminaries
Non-boundary intersecting regime
t ,t ≥ 0) denotes the sequence of hulls generated by γ (n) .
• By Definition 6.3, we know that η t (F, γ , ·) (resp. η t (F, γ n , ·)) is the bounded harmonic function with boundary values equal F on R \ K t (resp. on R \ K n t ), −λ on the left side of K t (resp. K n t ), and λ on the right side of K t (resp. K n t ).
Proof of Theorem 1.6 and concluding remarks
Proof of Theorem 1.6 and concluding remarks
Acknowledgments. We thank Nathanaël Berestycki, Jason Miller, Steffen Rohde, and Scott Sheffield for helpful discussions. We thank Avelio Seplveda and Juhan Aru for precious comments on the previous version of this paper. The main part of this work was done while H. Wu was at MIT and H. Wu's work is funded by NSF DMS-1406411. E. Powell's is funded by a Cambridge Centre for Analysis EPSRC studentship.Relative compactness then follows because the choice of E means that any sequence of curves in E has a convergent subsequence along which the driving functions also converge.We will prove the above claim now. We let K t (resp. K n t ) be the hull generated by γ (resp. γ n ) in the capacity parameterisation and W t , g t (resp. W n t , g n t ) be the corresponding driving functions, and functions H \ K t (resp. H \ K n t ) to H, normalised at ∞. We define f t = g t −W t and f n t = g n t −W n t as usual, and consider these to be extended to the boundary, also writing f t (0 + ) for V R t (0 + ) −W t . Writeτ(t) := inf{s ≥ 0 : C s = t}, τ n (t) := inf{s ≥ 0 : C n s = t}. First, we will show that for any T > 0 before the first time that γ swallows z, as n → ∞,We have the following observations.• By Lemma 2.15, and since C 0 = C n 0 = 0, we haveand A.4])Combining these three facts, we obtain Equation (6.6).Second, we show that, for any T > 0 before γ swallows z, as n → ∞, |C t −C n t | → 0.Since supandη t (F, γ , z) is uniformly continuous on [0, S], we see that it must converge to 0.Third, we show that, for any T > 0 before γ swallows z, as n → ∞,We need only show that, on any time interval [0, S] such that S is strictly less than the time γ swallows z, the quantity |η t (F, γ n , z) − η t (F, γ , z)| converges uniformly to 0. We have the following observations.
That is, using the stronger definition, we could show that any such process can always be coupled with the Gaussian Free Field as generalized level line. This would also give us uniqueness in law for the SLE 4 (ρ L ; ρ R ) process among continuous curves. However. it seems that (8.2) and (8.3) are hard to verify assuming only thatThat is, using the stronger definition, we could show that any such process can always be coupled with the Gaussian Free Field as generalized level line. This would also give us uniqueness in law for the SLE 4 (ρ L ; ρ R ) process among continuous curves. However, it seems that (8.2) and (8.3) are hard to verify assuming only that
Combining Theorem 1.2 with Lemmas 2.16 and 8.1 in the case that F is of bounded variation, we know that in the coupling (h, γ) given by Theorem 1.2, the marginal law of γ is that of an SLE 4 (ρ L ; ρ R ) process. This gives us existence of the process. Moreover, we know the curve γ is almost surely continuous and. transient and also satisfies the reversibility property (3) of Theorem 1.6, by Theorem 1.5Proof of Theorem 1.6. Combining Theorem 1.2 with Lemmas 2.16 and 8.1 in the case that F is of bounded varia- tion, we know that in the coupling (h, γ) given by Theorem 1.2, the marginal law of γ is that of an SLE 4 (ρ L ; ρ R ) process. This gives us existence of the process. Moreover, we know the curve γ is almost surely continuous and transient and also satisfies the reversibility property (3) of Theorem 1.6, by Theorem 1.5.
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Reversibility of chordal SLE. Dapeng Zhan, [email protected] Annals of Probability. 364Ellen Powell Department of Pure Mathematics and Mathematical Statistics University of CambridgeDapeng Zhan. Reversibility of chordal SLE. The Annals of Probability, 36(4):1472-1494, 2008. Ellen Powell Department of Pure Mathematics and Mathematical Statistics University of Cambridge, Cambridge, England [email protected]
| []
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[
"Computational Flows in Arithmetic",
"Computational Flows in Arithmetic"
]
| [
"Amirhossein Akbar \nInstitute of Mathematics Academy of Sciences of the Czech Republic\n\n",
"Tabatabai * [email protected] \nInstitute of Mathematics Academy of Sciences of the Czech Republic\n\n"
]
| [
"Institute of Mathematics Academy of Sciences of the Czech Republic\n",
"Institute of Mathematics Academy of Sciences of the Czech Republic\n"
]
| []
| A computational flow is a pair consisting of a sequence of computational problems of a certain sort and a sequence of computational reductions among them. In this paper we will develop a theory for these computational flows and we will use it to make a sound and complete interpretation for bounded theories of arithmetic. This property helps us to decompose a first order arithmetical proof to a sequence of computational reductions by which we can extract the computational content of low complexity statements in some bounded theories of arithmetic such as I∆ 0 , T k n , I∆ 0 + EXP and PRA. In the last section, by generalizing term-length flows to ordinal-length flows, we will extend our investigation from bounded theories to strong unbounded ones such as IΣ n and PA + TI(α) and we will capture their total NP search problems as a consequence. | null | [
"https://arxiv.org/pdf/1711.01735v1.pdf"
]
| 2,586,791 | 1711.01735 | 357b0ad29fd8e6f4f7155a5da9cbee82e21089f3 |
Computational Flows in Arithmetic
6 Nov 2017 November 7, 2017
Amirhossein Akbar
Institute of Mathematics Academy of Sciences of the Czech Republic
Tabatabai * [email protected]
Institute of Mathematics Academy of Sciences of the Czech Republic
Computational Flows in Arithmetic
6 Nov 2017 November 7, 2017
A computational flow is a pair consisting of a sequence of computational problems of a certain sort and a sequence of computational reductions among them. In this paper we will develop a theory for these computational flows and we will use it to make a sound and complete interpretation for bounded theories of arithmetic. This property helps us to decompose a first order arithmetical proof to a sequence of computational reductions by which we can extract the computational content of low complexity statements in some bounded theories of arithmetic such as I∆ 0 , T k n , I∆ 0 + EXP and PRA. In the last section, by generalizing term-length flows to ordinal-length flows, we will extend our investigation from bounded theories to strong unbounded ones such as IΣ n and PA + TI(α) and we will capture their total NP search problems as a consequence.
Introduction
Intuitively speaking, proofs are information carriers and they transfer the informational content of the assumptions to the informational content of the conclusion. This open notion of content though admits different many interpretations in different many disciplines. The most trivial and the least informative one is the truth value of a sentence and it is pretty clear that this truth value is preserved by sound proofs. The other example, and the more useful interpretation, is the computational content of a sentence, which plays the main role in the realm of proof theory and theoretical computer science. The notion of the computational content also admits different kinds of interpretations, from the witnesses of existential quantifiers a la Herbrand to dialectica-type interpretation of higher order arithmetical statements via Gödel's type theory T . What we want to investigate in this paper is one of these computational interpretations and in the rest of this introduction we will try to explain it.
Let us explain the idea step by step. First of all, we will focus on our interpretation of the computational content of a sentence. The answer is simply the following: We will interpret a sentence as a computational problem and by its computational content we roughly mean any way that can solve the problem computationally. It is clear that this notion of content is vague and imprecise but note that what is important is not the content itself but how it flows. (Compare this situation to the cardinal arithmetic where the notion of a cardinal of an infinite set is secondary compared to the notion of equipotency.) Therefore, it is important to interpret the computational preservation of information and we have a very natural candidate for that: the computational reductions. Let us illuminate what we mean by an example. Consider the formula ∀y ≤ t(x)∃z ≤ s(x)A(x, y, z). What we mean by this sentence is the total search problem which reads y in the domain [t(x)] and finds z in the range [s(x)] such that A(x, y, z) holds. This is a computational problem and by its content we mean any kind of computational method to solve this search problem. Now consider the situation that we have another search problem ∀u ≤ m(x)∃v ≤ n(x)B (x, u, v). The question is how it is possible to transfer the content of the first one to the content of the second one. In other words, if we have a way to solve the first search problem, how can we find a way to solve the second one? One of the many ways to reduce the second one to the first one is the reduction technique which we can define in the case of our example as the following: A computational reduction from ∀u ≤ m(x)∃v ≤ n(x)B (x, u, v) to ∀y ≤ t(x)∃z ≤ s(x)A(x, y, z) is the pair of two functions f and g with a certain complexity such that f reads u and finds y = f (x, u) and g reads u, z and computes v = g(x, u, z) such that
A(x, f (x, u), z) → B(x, y, g(x, u, z)).
So far, we have explained our interpretation of sentences and the way that the content is preserved. Now it is time to find a natural interpretation for proofs as the information carriers. For this goal, we will translate a first order proof of a sequent Γ ⇒ ∆ to a sequence of simple provable computational reductions from Γ to ∆ which formalizes the concept of a flow of computational information and for this reason we will call these sequences just computational flows or simply flows. Therefore, what we have to do is to show that this flow interpretation is sound and complete with respect to some certain theories, i.e, we have to show that if there exists a proof for Γ ⇒ ∆ then there exists a flow from Γ to ∆ and vice versa. This is the main goal of the whole paper.
As the final part of this introduction, let us say something about the structure of the paper. First of all, we will develop the theory on the abstract scale to make everything more clear and general. However, to control the problems arising from this extreme abstraction, we will limit ourselves just to the languages of arithmetic, the theories of bounded arithmetic and to some weak unbounded theories. Secondly and using these flows, we will reprove some recent characterizations of search problems in the Buss' hierarchy of bounded arithmetic via game induction principle [6], [5] or higher PLS problems [3] and a characterization of NP search problems of Peano Arithmetic [1]. Then we will generalize these results to prove some new characterizations of low-complexity search problems from higher order bounded theories of arithmetic and stronger theories such as I∆ 0 + EXP to strong fragments of Peano arithmetic like IΣ n or even stronger theories like PA + TI(α) for ǫ 0 α.
The Theory of Flows
In this section we will present a general definition of a bounded theory of arithmetic and then we will use two different types of flows to decompose the proofs of these theories. First of all, let us fix a language which can be an arbitrary extension of a ring-type language for numbers: Definition 2.1. Let L be a first order language of arithmetic extending {0, 1, +, −, ·, ⌊ · · ⌋, ≤}. By R we mean the first order theory consisting of the axioms of commutative discrete ordered semirings (the usual axioms of commutative rings minus the existence of additive inverse plus the axioms to state that ≤ is a total discrete order such that < is compatible with addition and multiplication with non-zero elements), plus the following defining axioms for − and ⌊ · · ⌋: (x ≥ y → (x − y) + y = x) ∧ (x < y → x − y = 0), and ((y + 1) · ⌊ x y ⌋ ≤ x) ∧ (x − (y + 1) · ⌊ x y ⌋ < y + 1).
Note that to avoid division by zero and to have a total function symbol in the language, by ⌊ x y ⌋ we actually mean ⌊ x y+1 ⌋. Definition 2.2. Let B be a theory. A class of terms, T, is called a B-term set if:
(i) It is closed under all L R -basic term operations of the language L provably in B, i.e. for any basic operation f and any t( x) ∈ T, there exist
r( x) ∈ T such that B ⊢ r( x) = f (t( x)).
(ii) It is closed under substitution, i.e. if t( x, y) ∈ T and s is an arbitrary term (not necessarily in T ) then t( x, s) ∈ T provably in B, i.e. there exists r( x) ∈ T such that B ⊢ r( x) = t( x, s).
Moreover, if a term set T has a subset of monotone majorizing terms provably in B, it is called a B-term ideal. By a monotone majorizing subset we mean a set of terms X ⊆ T such that for any t(
x) ∈ T there exists s( x) ∈ X such that B ⊢ t( x) ≤ s( x) and for any r( x) ∈ X, B ⊢ x ≤ y → r( x) ≤ r( x).
Example 2.3. For any theory B and any language extending L R , there are two trivial B-term sets. T all consisting of all terms of the language and T cls consisting of all closed terms. To have a non-trivial example, consider the language of bounded arithmetic extending the language of R and define T p as the class of all terms majorized by a term in the form p(| x|) for some polynomial p provably in BASIC + R. The majorzing subset is the set of all terms in the form p(| x|).
Definition 2.4. (i) By an R-conjunction between A( x) and B( x) we mean a formula C( x) such that R ⊢ A( x) ∧ B( x) ↔ C( x).
(ii) By an R-disjunction between A( x) and B( x) we mean a formula C( x)
such that R ⊢ A( x) ∨ B( x) ↔ C( x). (iii) By an R-negation for A( x) we mean a formula C( x) such that R ⊢ ¬A( x) ↔ C( x).
(iv) By an R-bounded universal quantification for A( x, y) we mean a formula C( x) such that R ⊢ ∀y ≤ t( x)A( x, y) ↔ C( x).
(v) By an R-bounded existential quantification for A( x, y) we mean a formula C( x) such that R ⊢ ∃y ≤ t( x)A( x, y) ↔ C( x).
Using the general setting we have set so far we can also define a general definition of π and σ-classes. Definition 2.5. (i) A class of formulas Π is called a π-class of the language L if it includes all quantifier-free formulas of L, is closed under substitutions and subformulas, and is closed under an R-conjunction, an R-disjunction and an R-bounded universal quantifier. And finally, if ∃y ≤ t B(y) ∈ Π then B has an R-negation in Π and also ∀y ≤ t ¬B(y) has an R-negation in Π, i.e ¬∀y ≤ t ¬B(y) ∈ Π.
(ii) A class of formulas Σ is called a σ-class of the language L if it includes all quantifier-free formulas of L, is closed under substitutions and subformulas, is closed under an R-conjunction, an R-disjunction and an R-bounded existential quantifier. And finally, if ∀y ≤ t B(y) ∈ Σ then B has an R-negation in Σ and also ∃y ≤ t¬ B(y) has an R-negation in Σ, i.e ¬∃y ≤ t ¬B(y) ∈ Σ.
We can also define a bounded hierarchy:
Definition 2.6. Let Φ be a class that includes all quantifier-free formulas and is closed under all boolean operations. The hierarchy
{Σ k (Φ), Π k (Φ)} ∞ k=0
is defined as the following:
(i) Π 0 (Φ) = Σ 0 (Φ) is the class Φ, (ii) If B(x) ∈ Σ k (Φ) then ∃x ≤ t B(x) ∈ Σ k (Φ) and ∀x ≤ t B(x) ∈ Π k+1 (Φ) and (iii) If B(x) ∈ Π k (Φ) then ∀x ≤ t B(x) ∈ Π k (Φ) and ∀x ≤ t B(x) ∈ Σ k+1 (Φ).
Example 2.7. The most well-known examples of π and σ-classes are U k and E k classes in the language of Peano arithmetic, Π b k and Σ b k classes and Π b k andΣ b k classes in the language of bounded arithmetic. But there are also some other useful examples, like the classes based on doubly sharply bounded formulas following with alternating sharply bounded quantifiers in the language of bounded arithmetic plus the function # 3 .
We are ready to state the general definition of bounded arithmetic. Definition 2.8. Let A be a set of quantifier-free axioms, T be a A-term ideal and Φ be a class of bounded formulas closed under substitution and subformulas. By the first order bounded arithmetic, B(T, Φ, A) we mean the theory in the language L which consists of axioms A, and the (T, Φ)induction axiom, i.e. ,
A(0) ∧ ∀x(A(x) → A(x + 1)) → ∀xA(t(x)),
where A ∈ Φ and t ∈ T. Equivalently, we can define B(T, Φ, A) as a proof system of the following form:
Axioms:
A ⇒ A ⊥ ⇒ ⇒ A
Where in the rightmost rule, A ∈ A.
Structural Rules:
Γ ⇒ ∆ (wL) Γ, A ⇒ ∆ Γ ⇒ ∆ (wR) Γ ⇒ ∆, A Γ, A, A ⇒ ∆ (cL) Γ, A ⇒ ∆ Γ ⇒ ∆, A, A (cR) Γ ⇒ ∆, A Γ 0 ⇒ ∆ 0 , A Γ 1 , A ⇒ ∆ 1 (cut) Γ 0 , Γ 1 ⇒ ∆ 0 , ∆ 1
Propositional Rules:
Γ 0 , A ⇒ ∆ 0 Γ 1 , B ⇒ ∆ 1 ∨L Γ 0 , Γ 1 , A ∨ B ⇒ ∆ 0 , ∆ 1 Γ ⇒ ∆, A i ∨R (i = 0, 1) Γ ⇒ ∆, A 0 ∨ A 1 Γ, A i ⇒ ∆ ∧L (i = 0, 1) Γ, A 0 ∧ A 1 ⇒ ∆, C Γ 0 ⇒ ∆ 0 , A Γ 1 ⇒ ∆ 1 , B ∧R Γ 0 , Γ 1 ⇒ ∆ 0 , ∆ 1 , A ∧ B Γ 0 ⇒ A, ∆ 0 Γ 1 , B ⇒ ∆ 1 , C → L Γ 0 , Γ 1 , A → B ⇒ ∆ 0 , ∆ 1 , C Γ, A ⇒ B, ∆ → R Γ ⇒ ∆, A → B Γ ⇒ ∆, A ¬L Γ, ¬A ⇒ ∆ Γ, A ⇒ ∆ ¬R Γ ⇒ ∆, ¬A
Quantifier rules:
Γ, A(s) ⇒ ∆ And the following induction rule:
Induction: Γ, A(y) ⇒ ∆, A(y + 1) (Ind) Γ, A(0) ⇒ ∆, A(t)
For every A ∈ Φ and t ∈ T. Example 2.9. With our definition of bounded arithmetic, different kinds of theories can be considered as bounded theories of arithmetic, for instance I∆ 0 , S k i , T k i , I∆ 0 +EXP and PRA are just some of the well-known examples. The most important property of the sequent calculus of bounded theories of arithmetic is cut elimination:
Theorem 2.10. (Cut Elimination) If B(T, Φ, A) ⊢ Γ ⇒ ∆ then
there exists a free-cut free proof for the same sequent in the same system.
The following corollary is very useful:
Corollary 2.11. If Γ ∪ ∆ ⊆ Φ and B(T, Φ, A) ⊢ Γ ⇒ ∆ then
there exists a proof of the same sequent in the same sytem such that all formulas occurring in the proof is in Φ.
In the following we will define two different types of reductions as the building blocks of flows. These reductions are generalizations of the usual reductions in computablity theory, from many to one reductions between recursive languages to polytime reductions between total NP search problems. Definition 2.12. Let A( x) and B( x) be some formulas in Π k (Φ) and {F i } k i=1 be a sequence of terms. By recursion on k, we will define F = {F i } k i=1 as a deterministic Π k (Φ)-reduction from B( x) to A( x) and we will denote it by A( x) ≤ F,k d B( x) when:
(i) If A( x), B( x) are in Π 0 (Φ), we say that the empty sequence of functions is a deterministic reduction from B to A iff B ⊢ A( x) → B( x). (ii) If A = ∀ u ≤ p( x)C( x, u), B = ∀ v ≤ q( x)D( x, v) and F = {F i } k+1 i=1 is a sequence of terms, then A( x) ≤ F,k+1 d B( x) iff B ⊢ v ≤ q( x) → F k+1 ( x, v) ≤ p( x) and F k+1 ( x, v) ≤ p( x) → C( x, F k+1 ( x, v)) ≤F ,k d v ≤ q( x) → D( x, v) whereF = {F i } k i=1 . (iii) If A = ∃ u ≤ p( x)C( x, u), B = ∃ v ≤ q( x)D( x, v) and F = {F i } k+1 i=1 is a sequence of terms, then A( x) ≤ F,k+1 d B( x) iff B ⊢ u ≤ p( x) → F k+1 ( x, u) ≤ q( x) and y ≤ p( x) ∧ C( x, u) ≤F ,k d F k+1 ( x, u) ≤ q( x) ∧ D( x, F k+1 ( x, u)) whereF = {F i } k i=1 . We say B is (Π k (Φ), B)-deterministicly reducible to A and we write A ≤ (Π k (Φ),B) d B, when there exists a sequence of terms F such that A ≤ F,k d B.
Definition 2.13. Let B be a first order bounded arithmetic and A( x) and B( x) be some formulas in the language L. We say B is non-deterministically B-reducible to A( x) and we write A(
x) ≤ B n B( x) if B ⊢ A( x) → B( x)
. The natural question is that how this proof-theoretic based concept can be called a computational reduction and if so, why is it a non-deterministic reduction as opposed to the above-mentioned deterministic reduction? The answer is the following well-known Herbrand theorem: Theorem 2.14. (Herbrand Theorem) If B is a universal bounded arithmetic then the following are equivalent:
(i) A( x) ≤ B n B( x). (ii) There exists a Herbrand proof for A( x) → B( x) in B.
Generally speaking, we intend to decompose arithmetical proofs to a sequence of reductions, and the base theory for those reductions preferably are simple universal and possibly induction-free theories. Therefore, we can use the Herbrand theorem for each step of the reduction to witness the essentially existential quantifiers in A → B. This is actually what is happening in the deterministic reductions, but here the difference is the use of ∨-expansions in the Herbrand proof. Intuitively, these expansions allow us to use some constantly many terms to witness one existential quantifier as opposed to just one term in the case of deterministic reductions. Moreover, expansions make some room for interaction in providing the witnessing terms which makes the concrete witnesses extremely complicated. For these reasons, we call these reductions non-deterministic.
In the following examples we will illuminate the difference between deterministic and non-deterministic reductions and the importance and the naturalness of the latter.
Example 2.15. Let A(x, y) ∈ Π b
k be a formula and consider the sentences ∃y, z ≤ t (A(x, y)∨A(x, z)) and ∃w ≤ t A(x, w). Intuitively, the first formula is equivalent to ∃y ≤ t A(x, y) ∨ ∃z ≤ t A(x, z) which is equivalent to the the second formula ∃w ≤ t A(x, w). Therefore, it seems quite reasonable to assume that if we have the second one, we can reduce the first one to it. Moreover, since this equivalence is quite elementary and it is just on the level of pure first order logic, we expect the reduction to have the lowest possible complexity. Fortunately, for the non-deterministic reduction it is obviously the case. But let us try to understand how the computational aspect of this reduction works. To do so, we have to take a look at a proof of the statement ∃y, z ≤ t (A(x, y)∨A(x, z)) → ∃w ≤ t A(x, w). The most simple proof works as follows: Assume y and z such that A(x, y) ∨ A(x, z). Then there are two possibilities: If A(x, y) then pick w = y and if A(x, z) then pick w = z. In a more computational interpretation, if we define g(x, y, z) = y, h(x, y, z) = z we have
A(x, y) ∨ A(x, z) ≤ d A(x, g(x, y, z)) ∨ A(x, h(x, y, z)).
What does it mean? It simply means that to have a reduction from the second statement to the first one we need two different copies of ∃w ≤ tA(x, y); one to handle the case A(x, y) and the other to handle the case A(x, z). This is available in proof theory via the contraction rule and it is absent in the computational interpretations of reduction. To fill this gap we allow these different copies which can be considered as some kind of non-determinism.
Example 2.16. In this example we want to show that it is generally impossible to simulate the non-deterministic reductions by deterministic ones. For that reason, we use a special case of the last example. Assume A(x, y, z, t) = (y = 0 ∧ B(x, t)) ∨ (y = 1 ∧ ¬B(x, z)) where B(x, t) ∈ Π b 0 is an arbitrary formula and the language consists of all polynomial computable functions (L PV ). We want to show that there is no polynomial time computable reduction from
∃y, y ′ ≤ 1∃t, t ′ ≤ s∀z, z ′ ≤ s (A(x, y, z, t) ∨ A(x, y ′ , z ′ , t ′ )) to ∃u ≤ 1∃v ≤ s∀w ≤ s A(x, u, v, w)
even if we assume B = T h(N). Assume that there exists a polytime reduction, hence there exist a polytime function f such that:
∃t, t ′ ≤ s∀z, z ′ ≤ s (A(x, y, z, t) ∨ A(x ′ , y ′ , z ′ , t ′ )) implies ∃v ≤ s∀w ≤ s A(x, f (x, y, y ′ ), v, w).
Pick y = 0 and y ′ = 1. It is easy to see that the left side is true because either ∃t ≤ s B(x, t) or ∀z ≤ s ¬B(x, z) is true, hence the right side should be true, as well. But the truth of the right side means
(f (x, 0, 1) = 0 ∧ ∃v ≤ s B(x, v)) ∨ (f (x, 0, 1) = 1 ∧ ∀w ≤ s ¬B(x, w))
which means that we have a polytime decision procedure for the NP predicate ∃w ≤ s B(x, v) which implies NP = P.
Remark 2.17. The example 2.16 shows that pure logical deductions are far beyond the power of deterministic reductions. In other words, it is possible to prove B by A just by some elementary methods of logic but it does not mean that B can be reducible to A. Let us explain where the problem is. At the first glance, it seems that all logical rules are completely syntactical and amenable to low complexity reductions. It is correct everywhere except for one logical rule: The contraction rule which is more or less responsible for all kinds of computational explosions like the explosion of the lengths of the proofs after the elimination of cuts. Notice that the reason that we have the equivalence in the Example 2.15 is this contraction rule and it is easy to see that this rule is a source of some non-determinism. Therefore, it seems natural to use non-deterministic reductions to simulate computationally what is going on in the realm of proofs.
So far, we have defined the concept of reduction which can be considered as a way to transfer the computational content of the source to the content of the target. They are similar to simple syntactic rules in the general proof theory. Then what is the counterpart of the concept of a proof (which is basically a combination of those simple rules)? The answer is the notion of a flow; a sequence of reductions which allows us to transfer information and computational contents.
Definition 2.18. Let Π be a π-class, A( x), B( x) ∈ Π and T a term ideal. A non-deterministic (T, Π, B)-flow from A( x) to B( x) is a pair (t, H) where t( x) ∈
T is a term and H(u, x) ∈ Π is a formula such that the following statements are provable in B:
(i) H(0, x) ↔ A( x). (ii) H(t(x), x) ↔ B( x). (iii) ∀u < t(x) H(u, x) → H(u + 1, x). If there exists a non-deterministic (T, Π, B)-flow from A( x) to B( x) we will write A( x) ⊲ (T,Π,B) n B( x). Moreover, if Γ and ∆ are sequents of formulas in Π, by Γ ⊲ (T,Π,B) n ∆ we mean Γ ⊲ (T,Π,B) n ∆.
And also we have deterministic flows:
Definition 2.19. Let A( x), B( x) ∈ Π k (Φ). A (Π k (Φ), B)-deterministic flow from A( x) to B( x) is the following data: A term t( x), a formula H(u, x) ∈ Π k (Φ)
and sequences of terms E 0 , E 1 , G 0 , G 1 and F (u) such that the following statements are provable in B:
(i) H(0, x) ≡ (E 0 ,E 1 ) d A( x). (ii) H(t(x), x) ≡ (G 0 ,G 1 ) d B( x). (iii) ∀u < t(x)H(u, x) ≤ F (u) d H(u + 1, x). If there exists a deterministic (Π k (Φ), B)-flow from A( x) to B( x) we will write A( x) ⊲ (Π k (Φ),B) d B( x). Moreover, if Γ and ∆ are sequents of formulas in Π k (Φ), by Γ ⊲ (Π k (Φ),B) d ∆ we mean Γ ⊲ (Π k (Φ),B) d ∆.
In the following we will prove a sequence of lemmas to make a high-level calculus of deterministic and non-deterministic flows. Then we will use this calculus to show that this flow interpretation is sound and complete with respect to the corresponding bounded arithmetic. All lemmas are true both for deterministic and non-deterministic flows, but note that for the deterministic flows we mean the (T, Π k (Φ), B)-flow and for the non-deterministic case we mean the (Π, B)-flow all the time. Therefore, when we write A ∈ Π, for the deterministic case we mean A ∈ Π k (Φ) and when we write ⊲ we mean both deterministic and non-deterministic cases.
Lemma 2.20. (Conjunction Application) Let C( x) ∈ Π be a formula. If A( x) ⊲ B( x) then A( x) ∧ C( x) ⊲ B( x) ∧ C( x). Proof. For the deterministic case, since A( x) ⊲ d B( x), by Definition 2.19, there exists a term t( x), a formula H(u, x) ∈ Π k (Φ) and sequences of terms E 0 , E 1 , G 0 , G 1 and F (u) such that B ⊢ A( x) ≡ E 0 ,E 1 H(0, x), B ⊢ B( x) ≡ G 0 ,G 1 H(t( x), x), and B ⊢ ∀u < t( x) H(u, x) ≤ F (u) d H(u + 1, x). Now define t ′ = t, H ′ (u, x) = H(u, x)∧C( x) and E ′ 0 , E ′ 1 , G ′ 0 , G ′ 1 and F ′ (u)
as the corresponding sequences of terms extending their counterparts by using the quantifiers in C to witness themselves by the identity terms. It is clear
that the new data is a deterministic (Π k (Φ), B)-flow from A( x) ∧ C( x) to B( x) ∧ C( x).
For the non-deterministic case do the same, without the sequences of the terms and use the fact that if
B ⊢ H(u, x) → H(u + 1, x) then, B ⊢ H(u, x) ∧ C( x) → H(u + 1, x) ∧ C( x). Lemma 2.21. (Disjunction Application) Let C( x) ∈ Π be a formula. If A( x) ⊲ B( x) then A( x) ∨ C( x) ⊲ B( x) ∨ C( x). Proof. For the deterministic case, since A( x) ⊲ B( x) then by Definition 2.19, there exists a term t( x), a formula H(u, x) ∈ Π k (Φ) and sequences of terms E 0 , E 1 , G 0 , G 1 and F (u) such that the conditions in the Definition 2.19 is provable in B. Now define t ′ = t, H ′ (u, x) = H(u, x) ∨ C( x) and E ′ 0 , E ′ 1 , G ′ 0 , G ′ 1 and F ′ (u)
as the corresponding sequences of terms extending their counterparts by using the quantifiers in C to witness themselves by the identity terms. It is clear that the new data is a deterministic (Π k
(Φ), B)-flow from A( x) ∨ C( x) to B( x) ∨ C( x).
For the non-deterministic case do the same, without the sequences of the terms and use the fact that if
B ⊢ H(u, x) → H(u + 1, x) then, B ⊢ H(u, x) ∨ C( x) → H(u + 1, x) ∨ C( x). Lemma 2.22. (i) (Weak Gluing) If A( x) ⊲ B( x) and B( x) ⊲ C( x) then A( x) ⊲ C( x). (ii) (Strong Gluing) If s ∈ T and A(y, x)⊲A(y+1, x) then A(0, x)⊲A(s, x).
Proof. For (i) and for the deterministic case, since
A( x) ⊲ d B( x) there exists a term t( x), a formula H(u, x) ∈ Π k (Φ) and sequences of terms E 0 , E 1 , G 0 , G 1 and F (u) such that B proves the conditions in the Definition 2.19. On the other hand since B( x) ⊲ d C( x) we have the corresponding data for B( x) to C( x) which we show by t ′ ( x), H ′ (u, x), E ′ 0 , E ′ 1 , G ′ 0 , G ′ 1 and F ′ (u). Define s( x) = t( x) + t ′ ( x) + 1, I(u, x) = H(u, x) u ≤ t( x) B( x) u = t( x) + 1 H ′ (u − t( x) − 2, x) t( x) + 1 < u ≤ t( x) + t ′ ( x) + 1
and the sequence of terms in the same pointwise way. Then, it is easy to check that this new data is a deterministic (Π k (Φ), B)-flow from A( x) to C( x). For the non-deterministic case do the same without the sequences of the terms and notice that since T is closed under successor and addition and t, t ′ ∈ T, we have s ∈ T.
For (ii) and for the deterministic case, if we have
A(y, x) ⊲ d A(y + 1, x)
it is enough to glue all copies of the sequences of reductions for 0 ≤ y ≤ s, to have A(0, x) ⊲ d A(s, x). More precisely, assume that all reductions have the same length t ′ ( x) greater than t(s, x). This is an immediate consequence of the facts that we can find a monotone majorization for t(y, x) like r(y, x), and since y ≤ s we have t(y, x) ≤ r(y, x) ≤ r(s, x). Now it is enough to repeat the last formula in the flow to make the flow longer to reach the length t ′ ( x, z) = r(s, x) where z is a vector of variables in s. Now, define
t ′′ ( x, z) = s × (t ′ ( x) + 2), I(u, x) = H(u, y, x) y(t ′ + 2) < u < (y + 1)(t ′ + 2) A(y, x) u = y(t ′ + 2)
and
F (u) = F (u, y) y(t ′ + 2) < u < (y + 1)(t ′ + 2) − 1 E 0 (u, y) u = y(t ′ + 2) G 1 (u, y + 1) u = (y + 1)(t ′ + 2) − 1 and E ′ 0 = E ′ 1 = G ′ 0 = G ′ 1 = id. It is easy to see that this new sequence is a deterministic (Π k (Φ), B)-flow from A(0, x) to A(s, x).
For the non-deterministic case, notice that T is closed under substitution, sum and product and therefore, t ′′ ∈ T which makes (t ′′ , I) a non-deterministic (T, Π, B)-flow from A(0, x) to A(s, x).
Lemma 2.23. (Conjunction and Disjunction Rules)
(i) If Γ, A ⊲ ∆ or Γ, B ⊲ ∆ then Γ, A ∧ B ⊲ ∆. (ii) If Γ 0 ⊲ ∆ 0 , A and Γ 1 ⊲ ∆ 1 , B then Γ 0 , Γ 1 ⊲ ∆ 0 , ∆ 1 , A ∧ B. (iii) If Γ ⊲ ∆, A or Γ ⊲ ∆, B then Γ ⊲ ∆, A ∨ B. (iv) If Γ 0 , A ⊲ ∆ 0 and Γ 1 , B ⊲ ∆ 1 then Γ 0 , Γ 1 , A ∨ B ⊲ ∆ 0 , ∆ 1 .
Proof. (i) and (iii), for both deterministic and non-deterministic cases, are trivial. For (ii), in the deterministic case, if Γ 0 ⊲ ∆ 0 , A, then by conjunction
application with Γ 1 we have Γ 0 ∧ Γ 1 ⊲( ∆ 0 ∨A)∧ Γ 1 . Moreover, we have Γ 1 ⊲ ∆ 1 ∨ B and again by conjunction application Γ 1 ∧ ( ∆ 0 ∨ A) ⊲ ( ∆ 1 ∨ B) ∧ ( ∆ 0 ∨ A). Therefore by weak gluing Γ 0 ∧ Γ 1 ⊲ ( ∆ 1 ∨ B) ∧ ( ∆ 0 ∨ A).
But it is easy to see that
( ∆ 1 ∨ B) ∧ ( ∆ 0 ∨ A) ≤ d ∆ 1 ∨ ∆ 0 ∨ (A ∧ B)
.
Hence Γ 0 , Γ 1 ⊲ ∆ 0 , ∆ 1 , (A ∧ B). For (iv), if Γ 0 , A ⊲ ∆ 0 then by disjunction application with Γ 1 ∧ B we have ( Γ 0 ∧ A) ∨ ( Γ 1 ∧ B) ⊲ ∆ 0 ∨ ( Γ 1 ∧ B).
Moreover, we have Γ 1 ∧ B ⊲ ∆ 1 , hence again by disjunction application
( Γ 1 ∧ B) ∨ ∆ 0 ⊲ ∆ 0 ∨ ∆ 1 .
Hence, by weak gluing,
( Γ 0 ∧ A) ∨ ( Γ 1 ∧ B) ⊲ ∆ 0 ∨ ∆ 1 .
However, it is clear that
Γ 0 ∧ Γ 1 ∧ (A ∨ B) ≤ d ( Γ 0 ∧ A) ∨ ( Γ 1 ∧ B). Hence, Γ 0 , Γ 1 , (A ∨ B) ⊲ ∆ 0 , ∆ 1 .
The following lemma makes it possible to compute a characteristic function of any A ∈ Ψ k ∈ {Π k (Φ), Σ k (Φ)} by a Σ k+1 (Φ) deterministic flow of reductions. This is a very important tool to reduce the complexity of deciding a complex formula to just deciding one equality. We will see its use in full force in the case of handling the contraction rule.
Lemma 2.24. (Computability of Characteristic Functions) Suppose {Σ k (Φ), Π k (Φ)} ∞
k=0 is a hierarchy and B has characteristic terms for all φ ∈ Φ, then for any
Ψ ∈ {Π k (Φ), Σ k (Φ)} if A( x) ∈ Ψ then ⊲ (Σ k+1 ,B) d ∃i ≤ 1 [(i = 0 → A) ∧ (i = 1 → ¬A)].
Proof. We prove the theorem by using induction on the number of bounded prefix quantifiers of A. If A ∈ Π 0 (Φ), then there is nothing to prove because it is enough to put i = χ A which belongs to the terms of B by the assumption.
If A = ∀z ≤ t( x)B(z, x), then by IH we have ⊲ (Σ k+1 ,B) d ∃r ≤ 1 [(r = 1 → B( x, u + 1)) ∧ (r = 0 → ¬B( x, u + 1))].
Now, we want to prove that there exists a reduction from the conjunction of
G(u + 1) = ∃k ≤ 1 [(k = 1 → B( x, u + 1)) ∧ (k = 0 → ¬B( x, u + 1))] and H(u) = ∃i ≤ 1 [(i = 1 → ∀z ≤ u B( x, z)) ∧ (i = 0 → ∃z ≤ u ¬B( x, z))] to H(u+1) = ∃j ≤ 1 [(j = 1 → ∀z ≤ u+1 B( x, z))∧(j = 0 → ∃z ≤ u+1 ¬B( x, z))].
Witness j as the following:
j = 1 i = k = 1 0 o.w.
Then for the other quantifiers use the following scheme:
If i = k = 1, then witness ∀z ≤ u + 1 B( x, z) by ∀z ≤ u B( x, z) and B( x, u + 1). If i = 1 and k = 0, then to witness ∃z ≤ u + 1 ¬B( x, z) use ¬B( x, u + 1) and finally if i = k = 0, then use ∃z ≤ u ¬B( x, z) to witness ∃z ≤ u + 1 ¬B( x, z). Therefore G(u + 1) ∧ H(u) ⊲ d H(u + 1). By IH, ⊲ d G(u + 1). Hence, by conjunction application H(u) ⊲ d G(u + 1) ∧ H(u)
and then by gluing H(u) ⊲ d H(u + 1) and finally by strong gluing
H(0) ⊲ d H(t( x)). Since H(0) ≡ d G(0) and ⊲ d G(0), hence ⊲ d H(0) which means ⊲ d H(t( x)). Lemma 2.25. (Negation Rules) If Γ, ∆ ⊆ Π k+1 and A ∈ Π k ∪ Σ k then (i) If Γ, A ⊲ ∆ then Γ ⊲ ∆, ¬A. (ii) If Γ ⊲ ∆, A then Γ, ¬A ⊲ ∆.
Proof. Since we have conjunction and disjunction application, it is enough to prove that
(i) ⊤ ⊲ Π k+1 A( x) ∨ ¬A( x). (ii) A( x) ∧ ¬A( x) ⊲ Π k+1 ⊥.
The reason for this sufficiency is the following: For the deterministic case we will prove the existence of a Σ k+1 -flow. Then the claim will be clear using negation on all the elements of the flow. For (i), notice that
For (i), if we have Γ, A ⊲ ∆ then Γ ∧ A ⊲ ∆,∃i ≤ 1 [(i = 0 → A) ∧ (i = 1 → ¬A)] ≤ d A ∨ ¬A.
It is enough to witness A and ¬A in both sides with themselves. But since
⊲ (Σ k+1 ,B) d ∃i ≤ 1 [(i = 0 → A) ∧ (i = 1 → ¬A)], we can deduce ⊲ (Σ k+1 ,B) d A∨¬A.
For the deterministic case of (ii), notice that A ∧ ¬A ≤ d ⊥ because it is enough to witness the quantifiers of A by ¬A and vice versa.
As we observed in the Example 2.16, the main difference between the deterministic and non-deterministic reductions is the contraction rule that the non-deterministic reduction can handle by definition and the deterministic reduction obviously can not. In the following lemma, we will show that it is possible to simulate the contraction rule by deterministic reductions in the cost of extending one reduction to a sequence of them, i.e., a flow.
Lemma 2.26. (Structural rules) (i) If Γ, A, B, Σ ⊲ ∆ then Γ, B, A, Σ ⊲ ∆. (ii) If Γ ⊲ ∆, A, B, Σ then Γ ⊲ ∆, A, B, Σ. (iv) If Γ ⊲ ∆ then Γ, A ⊲ ∆. (v) If Γ ⊲ ∆ then Γ ⊲ ∆, A. (iii) If Γ, A, A ⊲ ∆ then Γ, A ⊲ ∆. (vi) If Γ ⊲ ∆, A, A then Γ ⊲ ∆, A.
Proof. The weakening and the exchange cases are trivial. For the contraction case notice that since we have conjunction and disjunction applications and also the gluing rule, it is enough to prove the following claim:
Claim. If Ψ ∈ {Σ k (Φ), Π k (Φ)} and A ∈ Ψ, then:
(i) A( x) ⊲ Ψ A( x) ∧ A( x). (ii) A( x) ∨ A( x) ⊲ Ψ A( x).
For the non-deterministic case there is nothing to prove because the left side and the right side are provably equivalent. For the deterministic case of (ii), use induction on the complexity of A. If A ∈ Φ, then there is nothing to
prove. If A = ∀ z ≤ t( x) B( x, z), then since B ∈ Σ k (Φ), by IH we will have B( x, z) ∨ B( x, z) ⊲ Σ k B( x, z). Therefore, it is clear that ∀ u ≤ t( x) B( x, u) ∨ ∀ v ≤ t( x)B( x, v) ⊲ Π k+1 ∀ z ≤ t( x) B( x, z),
because it is enough to add ∀ z ≤ t( x) in front of all formulas in the flow and then witness them by themselves. Hence,
∀ z ≤ t( x) [B( x, z) ∨ B( x, z)] ⊲ Π k ∀ z ≤ t( x) B( x, z).
And then we have to add
∀ u ≤ t( x) B( x, u) ∨ ∀ v ≤ t( x)B( x, v)
as the first formula in the flow to have
∀ u ≤ t( x) B( x, u) ∨ ∀ v ≤ t( x)B( x, v) ⊲ Π k+1 ∀ z ≤ t( x) B( x, z).
Notice that we have to use the variable z as the witness for both of u and v. Therefore, using the Lemma 2.24, we know that there is a flow from
If A = ∃ z ≤ t( x) B( x, z), then note that we have B( u) ∧ ¬B( u) ⊲ d ⊥ and B( v) ∧ ¬B( v) ⊲ d ⊥ and hence by propositional rules (B( u) ∨ B( v)) ∧ ¬B( u) ∧ ¬B( v) ⊲ d ⊥ ( * ) Therefore, there is a flow from [B( u) ∨ B( v)] ∧ ∃i, j ≤ 1 (χ B ( u) = i) ∧ (χ B ( v) = j) to ∃i, j ≤ 1 [(χ B ( u) = i) ∧ (χ B ( v) = j)] ∧ (i = 1 ∨ j = 1) where χ B ( u) = i means (i = 1 → B( u)) ∧ (i = 0 → ¬B( u)). It∃ u, v ≤ t B( u) ∨ B( v) to ∃ u, v ≤ t ∃i, j ≤ 1 [(χ B (u) = i) ∧ (χ B (v) = j)] ∧ (i = 1 ∨ j = 1). Now, it is enough to show that ∃ u, v ≤ t( x) ∃i, j ≤ 1 (i = 1 ∨ j = 1) ∧ (χ B (u) = i) ∧ (χ B (v) = j)
is reducible to ∃ y ≤ t( x) B( x, y). It is enough to read i and j and decide between the cases that i = 1 or j = 1. Then based on that decision we can de-
cide to witness ∃ y ≤ t( x) B( x, y) as ∃ u ≤ t( x) B( x, u) or ∃ v ≤ t( x) B( x, v).
The case (i) is the dual of (ii) and provable by just taking negations.
Lemma 2.27. (Cut and Induction)
(i) If Γ 0 ( x) ⊲ A( x), ∆ 0 ( x) and Γ 1 ( x), A( x) ⊲ ∆ 1 ( x), then Γ 0 ( x), Γ 1 ( x) ⊲ ∆ 0 ( x), ∆ 1 ( x). (ii) If s ∈ T and Γ( x), A(y, x) ⊲ ∆( x), A(y + 1, x), then Γ( x), A(0, x) ⊲ ∆( x), A(s( z, x), x). Proof. For (i), Since Γ 0 ⊲ ∆ 0 , A and Γ 1 , A ⊲ ∆ 1 then Γ 0 ⊲ ∆ 0 ∨ A and Γ 1 ∧ A ⊲ ∆ 1 .
Apply conjunction with Γ 1 on the first one and disjunction with ∆ 0 on the second one to prove
Γ 1 ∧ Γ 0 ⊲( ∆ 0 ∨A)∧ Γ 1 and ( Γ 1 ∧A)∨ ∆ 0 ⊲ ∆ 1 ∨ ∆ 0 . Since ( ∆ 0 ∨A)∧ Γ 1 ≤ ( Γ 1 ∧A)∨ ∆ 0 , by using gluing we will have Γ 1 ∧ Γ 0 ⊲ ∆ 0 ∨ ∆ 1 .
For (ii) we reduce the induction case to the strong gluing case. Since Γ, A(y, x) ⊲ ∆, A(y + 1, x) by definition, Γ ∧ A(y, x) ⊲ ∆ ∨ A(y + 1, x). Therefore, by the Lemma 2.21 we have
( Γ ∧ A(y, x)) ∨ ∆ ⊲ ∆ ∨ A(y + 1, x) ∨ ∆
and by contraction for ∆ we know
∆ ∨ A(y + 1, x) ∨ ∆ ⊲ ∆ ∨ A(y + 1, x). Hence, ( Γ ∧ A(y, x)) ∨ ∆ ⊲ ∆ ∨ A(y + 1, x).
Then by conjunction introduction and the fact that ( Γ ∧ A(y, x)) ∨ ∆) ⊲ Γ ∨ ∆, (( Γ∧A(y, x))∨ ∆), ( Γ∧A(y, x))∨ ∆)⊲( ∆∨A(y+1, x))∧( Γ∨ ∆) By using the propositional, structural and the cut rule, it is easy to prove
(φ ∨ ψ) ∧ (σ ∨ ψ) ⊲ (φ ∧ σ) ∨ ψ.
Hence, by using the contraction we have
( Γ ∧ A(y, x)) ∨ ∆ ⊲ ( Γ ∧ A(y + 1, x)) ∨ ∆.
Now by strong gluing we have
( Γ ∧ A(0, x)) ∨ ∆ ⊲ ( Γ ∧ A(s( z, x), x)) ∨ ∆. But since Γ ∧ A(0, x) ⊲ ( Γ ∧ A(0, x)) ∨ ∆ and ( Γ ∧ A(s( x), x)) ∨ ∆ ≤ ∆ ∨ A(s( z, x), x), we have Γ( x), A(0, x) ⊲ ∆( x), A(s( z, x), x).
Lemma 2.28. (Implication Rules)
(i) If Γ 0 ⊲ ∆ 0 , A and Γ 1 , B ⊲ ∆ 1 then Γ 0 , Γ 1 , A → B ⊲ ∆ 0 , ∆ 1 . (ii) If Γ, A ⊲ ∆, B then Γ ⊲ ∆, A → B.
In the deterministic case, we assume A → B ∈ Π k (Φ).
Proof. For (i), in the deterministic case notice that if A → B ∈ Π k (Φ) then k = 0 and since Φ is closed under subformulas, A, B ∈ Φ and ¬A ∈ Φ. Therefore, by definition, it is easy to see that A → B ≡ ¬A ∨ B. Therefore: For the non-deterministic case note that when A → B ∈ Π then since Π is closed under subformulas, we have A, B ∈ Π. For (i), since Γ 0 ⊲ ∆ 0 , A by conjunction application we have
For (i) since Γ 0 ⊲ ∆ 0 , AΓ 0 , Γ 1 , ¬A ∨ B ⊲ ∆ 0 , ∆ 1 . Since A → B ⊲ ¬A ∨ B, by using cut we haveΓ 0 , Γ 1 , A → B ⊲ ∆ 0 , ∆ 1 .Γ 0 ∧ A → B ⊲ ( ∆ 0 ∨ A) ∧ A → B. Since ( ∆ 0 ∨ A) ∧ (A → B) ⊲ ∆ 0 ∨ (A ∧ (A → B)), and A ∧ A → B ≤ n B, we have ∆ 0 ∨ (A ∧ (A → B)) ⊲ ∆ 0 ∨ B.
And then since Γ 1 ⊲ B, ∆ 1 , by cut on B we have
Γ 0 , Γ 1 , A → B ⊲ ∆ 0 , ∆ 1 .
For (ii), if Γ, A ⊲ B, ∆, then by disjunction application
( Γ ∧ A) ∨ (A → B) ⊲ ∆ ∨ B ∨ (A → B).
And since The following theorem is the main theorem of the theory of flows in bounded theories of arithmetic:
(( Γ ∨ (A → B)) ∧ (A ∨ (A → B)) ⊲ ( Γ ∧ A) ∨ (A → B), we have (( Γ ∨ (A → B)) ∧ (A ∨ (A → B)) ⊲ ∆ ∨ B ∨ (A → B).Theorem 2.29. (Soundness) (i) If Γ( x) ∪ ∆( x) ⊆ Π k (Φ), B(T all , Π k (Φ), A) ⊢ Γ( x) ⇒ ∆( x) and A ⊆ B then Γ ⊲ (Π k (Φ),B) d ∆. (ii) If Π is a π-class, Γ( x) ∪ ∆( x) ⊆ Π, B(T, Π, A) ⊢ Γ( x) ⇒ ∆( x) and A ⊆ B then Γ ⊲ (T,Π,B) n ∆.
Proof. We prove the lemma by induction on the length of the free-cut free proof of Γ( x) ⇒ ∆( x).
(Axioms). If Γ( x) ⇒ ∆( x)
is a logical axiom then the claim is trivial. If it is a non-logical axiom then the claim will be also trivial because all non-logical axioms are quantifier-free and provable in B. Therefore there is nothing to prove. Lemma 2.27. 4. (Propositional). The conjunction and disjunction cases are proved in the Lemma 2.23. The implication and negation cases are proved in the Lemma 2.28.
(Structural Rules). It is proved in the Lemma 2.26.
(Cut). It is proved by
(Bounded Universal Quantifier, Right
). If Γ( x) ⇒ ∆( x), ∀z ≤ p( x)B( x, z)
is proved by the ∀ ≤ R rule by Γ( x), z ≤ p( x) ⇒ ∆( x), B( x, z), then by IH Γ( x), z ≤ p( x)⊲ d ∆( x), B( x, z). Therefore, there exists a term t( x), a formula H(u, x, z) ∈ Π k (Φ) and sequences of terms E 0 E 1 , G 0 , G 1 and F (u) such that the conditions of the Definition 2.12 are provable in B. First of all, extend the sequence by repeating the last formula to reach a majorization t ′ ( x) of t( x, p( x)). This is possible because z ≤ p( x) and t is monotone. Then, define t ′ ( x) ≥ t( x, p( x)) and H ′ (u, x) = ∀z ≤ p( x)H(u, x, z) and finally define E ′ 0 E ′ 1 , G ′ 0 , G ′ 1 and F ′ (u) as functions that read the outmost quantifier ∀z and sends it to itself and then apply the corresponding operations. Since H(u, x, z) ∈ Π k (Φ), then ∀z ≤ p( x)H(u, x, z) ∈ Π k (Φ). The other conditions to check that the new sequence is a (Π k (Φ), B)-flow is straightforward . 5 ′ . For the non-deterministic case, by IH we have Γ( x), z ≤ p( x) ⊲ n ∆( x), B( x, z). Therefore, there exists a term t( x) ∈ T, a formula H(u, x, z) ∈ Π such that the conditions of the Definition 2.13 are provable in B. First of all, extend the sequence by repeating the last formula to reach a majorization t ′ ( x) of t( x, p( x)). This is possible since z ≤ p( x) and t is monotone. Then, define t ′ ( x) ≥ t( x, p( x)) and H ′ (u, x) = ∀z ≤ p( x)H(u, x, z). Since H(u, x, z) ∈ Π then ∀z ≤ p( x)H(u, x, z) ∈ Π. The other conditions to check that the new sequence is a (T, Π, B)-flow is a straightforward consequence of the fact that if B ⊢ ∀u ≤ t ′ ( x)H(u, z, x) → H(u + 1, z, x), then B ⊢ ∀u ≤ t ′ ( x)∀z ≤ p( x)H(u, z, x) → ∀z ≤ p( x)H(u + 1, z, x).
(Bounded Universal Quantifier, Left). Suppose
Γ( x), s( x) ≤ p( x), ∀z ≤ p( x)B( x, z) ⇒ ∆( x) is proved by the ∀ ≤ L rule by Γ( x), B( x, s( x)) ⇒ ∆( x).
Then by IH, Γ( x), B( x, s( x)) ⊲ d ∆( x). Therefore, there exist a term t( x), a formula H(u, x) ∈ Π k (Φ) and sequences of terms E 0 E 1 , G 0 , G 1 and F (u) such that the conditions of the Definition 2.12 are provable in B. Similar to the case 5, w.l.o.g. extend the length to t ′ ( x) ≥ t( x, p( x)). Now define t ′′ ( x) = t ′ ( x) + 1,
H ′ (u, x) = Γ( x) ∧ s( x) ≤ p( x) ∧ ∀z ≤ p( x)B( x, z) u = 0 H(u, y, x) 0 < u ≤ t ′ ( x) + 1
And finally, define E ′ 0 E ′ 1 , G ′ 0 , G ′ 1 and F ′ (u) as sequences of terms that compute the universal quantifier ∀z as t( x). Since t( x) is a terms, it is easy to check that this new sequence is the (Π k (Φ), B)-flow that we wanted.
6 ′ . For the non-deterministic case, since B ⊢ s( x) ≤ p( x) ∧ ∀z ≤ p( x)B( x, z) → B( x, s( x)), we have s( x) ≤ p( x), ∀z ≤ p( x)B( x, z) ⊲ n B( x, s( x)). Since Γ( x), B( x, s( x)) ⊲ n ∆( x), by cut we have Γ( x), s( x) ≤ p( x), ∀z ≤ p( x)B( x, s( x)) ⊲ d ∆( x). 7. (Bounded Existential Quantifier, Right). If Γ( x), s( x) ≤ p( x) ⇒ ∆( x), ∃z ≤ p( x)B( x, z) is proved by the ∃ ≤ R rule by Γ( x) ⇒ ∆( x), B( x, s( x)) then ¬B( x, z) ∈ Π. Therefore, by Lemma 2.25 Γ( x), ¬B( x, s( x)) ⇒ ∆( x). By 6, Γ( x), s( x) ≤ p( x), ∀z ≤ p( x)¬B( x, z) ⇒ ∆( x). Again by Lemma 2.25, Γ( x), s( x) ≤ p( x) ⇒ ∆( x), ∃z ≤ p( x)¬¬B( x, z) which means Γ( x), s( x) ≤ p( x) ⇒ ∆( x), ∃z ≤ p( x)B( x, z).
8. (Bounded Existential Quantifier, Left). If Γ, y ≤ p( x), B( x, y)⊲∆ then since ∃y ≤ p( x)B( x, y) ∈ Π, then B has a negation in Π. By disjunction application (Γ ∧ B( x, y)) ∨ ¬B( x, y) ⊲ ∆ ∨ ¬B( x, y). Since ⊲B( x, y) ∨ ¬B( x, y), then Γ ⊲ (Γ ∧ B( x, y)) ∨ ¬B( x, y). Therefore, Γ, y ≤ p( x) ⊲ ∆ ∨ ¬B( x, y). Now by 5, we have Γ ⊲ ∆, ∀y ≤ p( x)¬B( x, y).
By conjunction application Γ, ∃y ≤ p( x)B( x, y) ⊲ ( ∆ ∨ ∀y ≤ p( x)¬B( x, y)) ∧ ∃y ≤ p( x)B( x, y).
But,
∀y ≤ p( x)¬B( x, y) ∧ ∃y ≤ p( x)B( x, y) ⊲ ⊥.
Hence, ( ∆ ∨ ∀y ≤ p( x)¬B( x, y)) ∧ ∃y ≤ p( x)B( x, y) ⊲ ∆.
Therefore, Γ, ∃y ≤ p( x)B( x, y) ⊲ ∆.
(Induction). It is proved in Lemma 2.27.
Using the soundness theorem we can show that any non-deterministic reduction and hence all non-deterministic flows can be simulated by termlength deterministic flows. Note that even when the length of a non-deterministic flow belongs to some term ideal of the language, then the length of the simulated deterministic flow exceeds all the terms in the term ideal and needs the whole power of terms.
, if A( x), B( x) ∈ Π k (Φ) and A( x) ≤ (T,Π k (Φ),B) n B( x), then A( x) ⊲ (T all ,Π k (Φ),B) d B( x). Proof. If A( x) ≤ Π n B( x) then B ⊢ A( x) ⇒ B( x). By deterministic soundness we have A( x) ⊲ (Π k (Φ),B) d B( x). Corollary 2.31. if A( x), B( x) ∈ Π k (Φ) and A( x) ⊲ (T,Π k (Φ),B) n B( x), then A( x)⊲ (Π k (Φ),B) d B( x). Therefore, the existence of a non-deterministic (T all , Π k (Φ), B)- flow is equivalent to the existence of a deterministic (Π k (T all ), B)-flow. Proof. If A( x)⊲ (T,Π k (Φ),B) n B( x), then by definition, there exist a term t( x) and a formula H(u, x) ∈ Π k (Φ) such that A( x) ≡ n H(0, x), B( x) ≡ n H(t( x), x) and H(u, x) ≤ n H(u + 1, x). By the Theorem 2.30, A( x) ⊲ d H(0, x), B( x) ⊲ d H(t( x), x) and H(u, x)⊲ d H(u+1, x). By strong gluing, H(0, x)⊲ d H(t( x), x) and therefore by gluing A( x) ⊲ d B( x).
We also have the following completeness theorem:
Theorem 2.32. (Completeness) (i) If Γ( x)⊲ (Π k (Φ),B) d ∆( x) and B ⊆ B(T all , Π k (Φ), A), then B(T all , Π k (Φ), A) ⊢ Γ( x) ⇒ ∆( x). (ii) If Γ( x) ⊲ (T,Π,B) n ∆( x) and B ⊆ B(T, Π, A), then B(T, Π, A) ⊢ Γ( x) ⇒ ∆( x).
Proof. For (ii), if Γ( x) ⊲ (T,Π,B) n ∆( x), then by Definition 2.13, there exist a term t( x) ∈ T, and a formula H(u, x) ∈ Π such that we have the following:
(i) B ⊢ H(0, x) ↔ Γ( x), (ii) B ⊢ H(t(x), x) ↔ ∆( x), and (iii) B ⊢ H(u, x) → H(u + 1, x). Since B ⊆ B(T, Π, A), we have B(T, Π, A) ⊢ ∀u ≤ t( x) H(u, x) → H(u + 1, x).
Since H(u, x) ∈ Π and t ∈ T, by induction we have ,
B(T, Π, A) ⊢ H(0, x) → H(t( x), x).
On the other hand, we have
B ⊢ H(0, x) ↔ Γ( x) and B ⊢ H(t( x), x) ↔ ∆( x). Therefore, B(T, Π, A) ⊢ Γ( x) ⇒ ∆( x)
. The proof of the deterministic case, i.e., the case (i), is very similar.
Applications on Bounded Theories
In this section we will use the soundness theorems that we proved in the previous section to extract the computational content of the low complexity statements of some concrete weak bounded theories such as Buss's hierarchy of bounded theories of arithmetic and some strong theories such as I∆ 0 +EXP and PRA. For the beginning, let us focus on the deterministic soundness theorem. The first application is on the fragments of the theory I∆ 0 which are related to the linear time hierarchy:
Corollary 3.1. Let Γ( x) ∪ ∆( x) ⊆Û k . Then, IÛ k ⊢ Γ( x) ⇒ ∆( x) iff Γ ⊲ (Û k ,R) d ∆.
The second application, and maybe the more important one, is the case of Buss's hierarchy of bounded arithmetic.
Corollary 3.2. Let Γ( x) ∪ ∆( x) ⊆Π b k (# n ). Then, T k n ⊢ Γ( x) ⇒ ∆( x) iff Γ ⊲ (Π b k (#n),PV(#n)) d ∆. Specifically, for n = 2, T k 2 ⊢ Γ( x) ⇒ ∆( x) iff Γ ⊲ (Π b k ,PV) d ∆.
Proof. Note that it is enough to know that T k n is axiomatizable byΠ b k (# n )induction.
And also we can apply the soundness theorem on stronger theories with full exponentiation like I∆ 0 + EXP. Consider the theory R augmented with a function symbol for exponentiation with the usual recursive definition and denote it by R(exp). Then:
Corollary 3.3. Let Γ( x) ∪ ∆( x) ⊆ Π b k (open). Then, I∆ 0 + EXP ⊢ Γ( x) ⇒ ∆( x) iff Γ ⊲ (Π b k (open),R(exp)) d ∆.
We can also use the theory of flows to extract the computational content of low complexity sentences of the very strong theories of arithmetic like IΣ n and PA + TI(α). But this is not what we can implement in a very direct way. The reason is that our method is tailored for bounded theories while these theories are unbounded. Hence, to use our theory, we have to find a way to transfer low complexity statements from these theories to some corresponding bounded theories. This is what the continuous cut elimination method makes possible in a very elegant way. It transfers all Π 0 2 consequences of a strong theory T to some quantifier-free extensions of PRA and then makes it possible to apply the flow decomposition technique. To explain how it works, we need some definitions: Definition 3.4. (i) An ordered structure (X, ≺ X ) is called B-representable when there exists a relation ≺ ∈ L B defined by a quantifier-free formula such that:
(i) The order type of ≺ equals ≺ X .
(ii) B proves the axioms of discrete ordered structures for the language.
(ii) An ordered structure (X, ≺ X , + X , · X , − X , ⌊ · · ⌋ X , 0 X , 1 X ) is called B-representable when there exists a quantifier-free relation ≺ ∈ L B and L B -terms +, ·, −, ⌊ · · ⌋ : N × N → N and constants 0, 1 ∈ N such that: (i) The order type of ≺ equals ≺ X .
(ii) B proves the axioms of discrete ordered semi-rings for the language without the commutativity of addition and the axioms which state that ≺ preserves under left addition and left multiplication by a non-zero element.
In this paper we are mainly interested in the cases that B = PV or B = PRA, i.e., the case of polytime representability and the case of primitive recursive representability.
Definition 3.5. Let ≺ be a quantifier-free formula in the language of PRA. By theory PRA+PRWO(≺) we mean PRA plus the axiom schema PRWO(≺) which states ∀ x∃y f ( x, y + 1) ⊀ f ( x, y) for any function symbol f .
The following theory is the skolemization of PRA + PRWO(≺):
Definition 3.6. The language of the theory PRA ≺ consists of the language of PRA plus the scheme which says that for any PRA-function symbol f ( x, y), there exists a function symbol [µy.f ]( x). Then BASIC ≺ is the theory axiomatized by the axioms of PRA and R and the following definitional equations:
f ( x, 1+[µy.f ]( x)) ⊀ f ( x, µy.f ]( x)) and z < [µy.f ]( x) → f ( x, z+1) ≺ f ( x, z).
Finally, PRA ≺ is BASIC ≺ plus the usual induction rule.
We are ready to define Π 0 2 -proof theoretical ordinal of a theory.
Definition 3.7. Let T be a theory of arithmetic. We say that α is a Π 0 2proof theoretical ordinal of T when (α, ≺ α ) is PRA-representable by ≺ and T ≡ Π 0 2 PRA + PRWO(≺). As we have mentioned before, using the continuous cut elimination technique, we can compute the Π 0 2 -ordinal of some specific theories. (See [4] for the sketch of the proof for PA + TI(α). The rest is similar.)
Theorem 3.8. (Continuous Cut Elimination) (i) The Π 0 2 -ordinal of IΣ 1 is ω 2 . (ii) For n > 1, the Π 0 2 -ordinal of IΣ n is ω n . (iii) The Π 0 2 -ordinal of PA is ǫ 0 .
(iv) For any PRA-representable ordinal ǫ 0 ≺ α, the Π 0 2 -ordinal of PA + TI(α) is α. Now we are ready to have the following corollary:
Corollary 3.9. Let Γ( x) ∪ ∆( x) ⊆ Π b k (open), and α T is the Π 0 2 -ordinal of T with a PRA-representation ≺ α T , then T ⊢ Γ( x) ⇒ ∆( x) iff Γ ⊲ (Π b k (open),BASIC≺ α T ) d ∆.
Proof. Note that the existence of the flow is equivalent to the provability of Γ ⇒ ∆ in PRA ≺α T because PRA ≺α T is a bounded theory axiomatizable by the usual induction on formulas in Π b k (open). On the other hand,
Γ( x) ∪ ∆( x) ⊆ Π b k (open)
, which means that the sequent is bounded and hence is in Π 0 2 . Therefore, by the definition of Π 0 2 -ordinal we know that PRA ≺α T ⊢ Γ ⇒ ∆ iff T ⊢ Γ ⇒ ∆ and it completes the proof.
So far, we have used the theory of deterministic flows to decompose first order proofs of bounded theories. In the following we will introduce two different kinds of characterizations and we will use them to reprove some recent results for some specific classes of formulas. The types that we want to use are generalizations of some recent characterizations of some low complexity statements in Buss's hierarchy of bounded arithmetic by Game induction principles [6], [5] and some kind of PLS problems [3]. Definition 3.10. Fix a language L. An instance of the (j, k)-game induction principle, GI j k (L), is given by size parameters a and b, a uniform sequence G 0 , . . . , G a−1 of open (quantifier-free) relations, a term V and a uniform sequence W 0 , . . . , W a−2 of terms. The instance GI(G, V, W, a, b) states that, interpreting G 0 , . . . , G a−1 as k-turn games in which all moves are bounded by b, the following cannot all be true:
(i) Deciding the winner of game G 0 depends only on the first j moves, (ii) Player B can always win G 0 (expressed as a Π j (open) property.) (iii) For i = 0, . . . , a − 2, W i gives a deterministic reduction of G i+1 to G i , (iv) V is an explicit winning strategy for Player A in G a−1 .
In the following theorem, denote Π i (Φ) where Φ is the class of all quantifierfree formulas by Π i and do the similar thing for Σ i (Φ).
(i) H(0, x) ≡ (E 0 ,E 1 ) d ¬A(x). (ii) H(t(x), x) ≡ (I 0 ,I 1 ) d ⊥. (iii) ∀u < t(x)H(u, x) ≤ Fu d H(u + 1, x)
. First of all, note that we can change the definition of H in the following way:
H ′ (u, x) = (u = 0 → ¬A(x)) ∧ (u = 0 → H(u − 1, x)).
And, it is possible to shift also the reductions to have (i) to (iii) for H ′ . But note that the truth of H ′ (0, x) depends only on first j blocks of quantifiers when we write it in the strict Π j (open) form.
W.l.o.g., we assume that all bounds in H ′ (u, x) are the same, say s(x). Since H ′ is strict, we have H ′ (u, x) = ∀ z 1 ≤ s∃ y 1 ≤ s∀ z 2 ≤ s . . . G(u, z 1 , y 1 , z 2 , . . .). Define a = t(x), b = s(x), G i as the game G(i, z 1 , y 1 , z 2 , . . .), W i = F ′ i and V = I ′ 0 . Therefore, we have an instance of the game induction. Now we want to show that A(x) is reducible to this game induction provably in B. Since
B ⊢ ∀u < t(x)H ′ (u, x) ≤ F ′ u d H ′ (u + 1, x) and H(t(x), x) ≡ (I 0 ,I 1 ) d ⊥,
the false part is "player B can always win the game G 0 " which means that H ′ (0, x) is false. Since H ′ (0, x) is equivalent with A provable in B, the reduction of the sentence A to the game induction principle is proved.
Using this generalization it is trivial to reprove the case for Buss's hierarchy of bounded arithmetic: Corollary 3.12. ([6], [5]) For all j ≤ k, ∀Σ b j (T k 2 ) ≡ GI j k .
Now, let us explain the second type of problems, i.e., the generalized local search problems: Definition 3.13. A formalized (Ψ, Λ, B, ≺, t)-GLS problem consists of the following data:
(i) A term N(x, s) ∈ L B as local improvement.
(ii) A term c(x, s) ∈ L B as cost function.
(iii) A predicate F (x, s) ∈ Ψ which intuitively means that s is a feasible solution for the input x.
(iv) An initial term i(x) ∈ L B .
(v) A goal predicate G(x, s) ∈ Λ.
(vi) A quantifier-free predicate ≺∈ L B as a well-ordering.
(vii) A bounding term t(x).
such that B proves that ≺ is a total order and
B ⊢ ∀x F (x, i(x)) B ⊢ ∀xs (F (x, s) → F (x, N(x, s))) B ⊢ ∀xs (N(x, s) = s ∨ c(x, N(x, s)) ≺ c(x, s)) B ⊢ ∀xs (G(x, s) ↔ (N(x, s) = s ∧ F (x, s))) B ⊢ ∀xs (G(x, s) → s ≤ t(x))
for some term t. Moreover, if L PV ⊆ L B and t(x) = 2 p(|x|) for some polynomial p we show the GLS problem by PLS(Ψ, Λ, ≺, B) and if F is quantifier-free in the language of B, G is quantifier-free in the language of PV we show the GLS problem by PLS(≺, B). Finally if B = PV, then we write PLS(≺).
Theorem 3.14. If A ∈ Π k (Φ), then B(T all , Π k+1 (Φ), B) ⊢ ∀x∃y ≤ t(x)A(x, y)
iff the search problem of finding y by x is reducible by a projection to an instance of a GLS(Π k (Φ), {A}, B, ≤, t) provably in B.
Proof.
Assume
B(T all , Π k+1 (Φ), B) ⊢ ∀x∃y ≤ t(x)A(x, y).
Then, we know that ∀y ≤ t(x)¬A(x, y) ⇒ ⊥ is provable in the theory. By soundness theorem 2.29, there exist a term s(x), a formula H(u, x) ∈ Π k+1 (Φ) and sequences of terms E 0 , E 1 , G 0 , G 1 and F (u) such that the following statements are provable in B:
(i) H(0, x) ≡ (E 0 ,E 1 ) d ∀y ≤ t( x)¬A( x, y). (ii) H(t(x), x) ≡ (G 0 ,G 1 ) d ⊥. (iii) ∀u < t(x) H(u, x) ≤ Fu d H(u + 1, x). Since H ∈ Π k+1 (Φ), we have H(u, x) = ∀ v ≤ r( x, u)G(u, v, x) where G( v, u, x) ∈ Σ k (Φ).
Use the deterministic reductions to show the existence of terms U, V and Z such that
(i) B ⊢ A(Z( v), x) → G(0, v, x). (ii) B ⊢ G(t(x), U , x) → ⊥. (iii) B ⊢ ∀u < t(x)G(u, V (u, v, x), x) → G(u + 1, v, x). Now define F (x, u, v, z) = ¬G(u − 1, v) u > 0 z ≤ t ∧ A(z, x) u = 0 and N(x, u, v, z) = (u − 1, V (u, v, x), z) u > 1 (0, v, Z( v)) u = 1 (u, v, z) u = 0 and Goal(x, u, v, z) = [z ≤ t ∧ A(x, z)], i(x) = (t(x), U , 0), and c(u, v) = u.
It is clear to see that this data is a (Π k (Φ), {A}, B, ≤, t)-GLS problem. The answer to this problem is (0, u, v, z) where A(z, x) holds. Note that by a projection we can extract z from it which is the witness for ∃y in A.
Again we have the special case for Buss's hierarchy:
Corollary 3.15. ([3]) For all l ≤ k, ∀Σ b l+1 (T k+1 2 ) ≡ PLS(Π b k , Π b l , PV, ≤).
Remark 3.16. Note that the power of characterizations via these kinds of problems are more limited than the theory of flows'. The reason is that the game induction method relaxes the condition of provability of reductions and GLS problems unwind just one universal quantifier and put the rest into the feasibility predicate.
Using this characterization by GLS problems, we can capture the class of total NP search problems in strong theories: (iv) For any PRA-representable ordinal ǫ 0 ≺ α, TFNP(PA + TI(α)) ≡ PLS(BASIC ≺α , ≤).
Remark 3.21. These characterizations of search problems of strong theories of arithmetic may seem a bit counter-intuitive. The reason is as follows: Assume that we are working with IΣ 1 . Then by Corollary 3.20, we have access to all primitive recursive functions and predicates for our formulas and reductions and what we want to solve is just an NP search problem. Hence, having this huge power, it seems that just one reduction should be enough and it means that our characterization is weak or trivial in some sense. This is not the case and the explanation is as follows: It is correct that we have access to all primitive recursive functions but they act just like oracles in a black box. We can ask our questions but we can not understand their behavior and hence we can not be sure about the truth of their answers. Therefore, we need to use a long sequence of reductions and in each reduction we can be sure of the very limited part of the argument. But if you still think that this incomplete access to complex functions is unbearable even with the mentioned explanation, we will refer you to the next section in which we eliminate the presence of complex function symbols via proof theoretic ordinals.
In the rest of this section we will explain some applications of nondeterministic flows. But first of all let us explain why we need this kind of non-determinism. Assume that we are working in the theory S k 2 which has the polynomial induction and not the usual one. If we want to decompose proofs of this theory to a sequence of reductions, we have to kill the effect of the contraction rule. But simulating contraction needs an exponential sequence of reductions which we can not afford by our polynomial induction.
Hence in this situation and in all the situations that the induction is extremely weaker than the bounds of the formulas, it is natural to work with reductions that handle the contraction rule automatically, and this power is exactly what the non-deterministic reductions provide.
To apply the non-deterministic soundness, let us first define a hierarchy of theories of bounded arithmetic to have a variety of theories with gaps between term bounds and induction lengths:
Definition 3.22.
Define T m as the term ideal consisting of all terms less than terms of the form |t| m . Define the theory R k m,n as B(T m , Π b k (# n ), BASIC(# n )). In the following theorem, we show that it is possible to decompose proofs of R k m,n :
Theorem 3.23. Let Γ, ∆ ⊆ Π b k (# n ), then R k m,n ⊢ Γ ⇒ ∆ iff Γ ⊲ (Tm,Π b k (#n),BASIC(#n)) n ∆.
The previous theorem is useful for some specific cases that we are interested in. For the first application, we can reprove some strong version of Buss's witnessing theorem for the {S k 2 } ∞ k=0 hierarchy: Corollary 3.24. (Strong Witnessing Theorem) The provably Σ b k -definable functions of S k 2 are in p k , provably in PV, i.e. if S k 2 ⊢ ∀ x∃yA( x, y) where A( x, y) ∈ Σ b k , then there exist a machine M computing a function f ∈ p k and polytime function symbol g such that PV ⊢ comp M ( x, w) → A( x, g( x, w)).
Proof. Assume S k 2 ⊢ ∀ x∃yA( x, y). By Parikh theorem we know that there exists a bound for the existential quantifier. Hence S k 2 ⊢ ∀y ≤ t(x) ¬A( x, y) ⇒ ⊥. By Theorem 3.23 we know that there exist a polynomial p( |x|) and a formula H(u, x) ∈ Π b k such that the following statements are provable in R ∪ BASIC:
(i) H(0, x) ↔ ∀y ≤ t(x) ¬A( x, y). (ii) H(p(| x|), x) ↔ ⊥. (iii) ∀u < p( |x|) H(u, x) → H(u + 1, x). Note that H(u, x) = ∀ z ≤ s( x) G(u, x, z) where G(u, x, z) ∈ Σ b k−1 .
Since R∪BASIC is a universal theory, by the generalization of Herbrand's theorem we know that there exists a ∨-expansion of formulas ∀y ≤ t(x) ¬A(x, y) → H(0, x), H(t(x), x) → ⊥ and ∀u < p( |x|) H(u, x) → H(u + 1, x) such that we can witness existential quantifiers by terms. Note that since we have the power to decide all formulas in Σ b k−1 , we can kill the effect of the expansion to find the polytime functions to witness the existential quantifiers such that:
(i) (U( x, z) ≤ t( x) → ¬A( x, U( x, z)) → G(0, x, z). (ii) G(t(x), x, V ) → ⊥. (iii) ∀u < p( |x|) G(u, x, Z( x, z)) → G(u + 1, x, z).
Now, define the algorithm M as the following: Begin with V and do the following for p(|x|) many steps: In each step apply Z, write it somewhere and ask the oracle about G(u, x, Z( x, z)) and save it also somewhere else.
We claim that this M works. If we have the whole computation of M, i.e. w, it is easy to compute the witness in that step, a u and value of G(u, x, a u ) by poly-time functions v(w, u) and j(w, u, v) provably in PV. Hence, the statement j(w, u, v(w, u)) = 0 is provable by length induction on u and therefore provably in PV we know that if w is the computation of M then j(w, p(|x|), v(w, p(|x|))) = 0 and thus ¬G(0, x, v(w, p(|x|))) and hence v(w, p(|x|)) ≤ t(x) and A(x, v(w, p(|x|)). Pick g(w) = v(w, p(|x|)) and we have the claim.
As the second application, note that if we put m = n − 2, Γ = ∅ and ∆ = {∀x∃y ≤ |t(x)| n−2 A(x, y)} where A is (n − 1)-bounded, the previous theorem in the presence of RSUV isomorphism, finds a way to extract the information about NP search problems of higher-order bounded arithmetic expressed in the first order language by using faster growing smash functions.
Ordinal Flows
In the previous sections we investigated bounded theories of arithmetic and we proved that they are sound and complete with respect to their appropriate flow-based interpretations. Now, it is natural to seek for a similar theory for unbounded theories of arithmetic. First of all, note that since we are interested in low-complexity statements and since it is possible to reduce the whole quantifier complexity of unbounded strong enough theories to universal statements via their proof theoretic ordinals, it is natural to restrict our investigations to these universal theories with ordinal induction for universal formulas. Note that here we are not working with bounded theories and hence assuming that the length of a flow is a term seems inappropriate. But clearly, there is also a natural candidate in this case, which is the proof theoretic ordinal. Therefore, in the case of strong enough unbounded theories we will work with flows of universal formulas with ordinal length and we will use them to extract the computational information of the theories.
Definition 4.1. Let L PV be the language of PV. Define the system TI(∀ 1 , ≺) as the usual first order sequent calculus of first order language plus the axioms of PV and the following induction rule:
Γ, ∀γ ≺ β A(γ) ⇒ ∆, A(β) (Indα ) Γ ⇒ ∆, A(δ)
For every A ∈ ∀ 1 where ∀ 1 means the class of all universal formulas.
Using Π 0 2 -ordinal we can transfer Π 0 2 sentences form a theory T to the theory PRA + PRWO(≺) where ≺ is a PRA-representation of α T . The following theorem makes it possible to continue this process of transferring to TI(∀ 1 , ≺) which is a more convenient theory for our technical purpose. Proof. First of all, notice that it is possible to represent any primitive recursive function f by a polynomial time computable predicate F . We will use this definition to interpret all quantifier-free statements in PRA as formulas in ∀ 1 statements in the language of PV. By our way of interpretation the defining axioms in PRA are provable in TI(∀ 1 , ≺). For the induction, it is enough to use induction on ω ≺ α in TI(∀ 1 , ≺). What remains is the axiom PRWO(≺). Note that the interpretation of this axiom is ∀y∀uv (F ( z, y + 1, u) ∧ F ( z, y, v)) → u ≺ v ⇒ ⊥. We know that f (0) exists, i.e. ∀a¬F (0, a) ⇒ ⊥. To prove, use induction on
A(x) = ⊥ F (r(x), q(x)) ∧ x ≺ aω ⊤ o.w.
where q(x) = ⌊ x ω ⌋ and r(x) = x − q(x).
A is inductive because if ∀z ≺ x A(x) is true and A(x) is false, then by definition x ≺ ωa and F (q(x), r(x)). Pick c as f (r(x) + 1) which we know exists. Therefore by the assumption we know ∀y∀uv F ( z, y + 1, u) ∧ F ( z, y, v) → u ≺ v and hence c ≺ f (r(x)). If a = 0 we have a(f (r(x) + 1)) + r(x) + 1 ≺ af (r(x)) + r(x) = x. Hence A(a(f (r(x) + 1)) + r(x) + 1) is ⊥ which contradicts ∀z ≺ x A(x). If a = 0 then f (0) = 0 which contradicts f (1) ≺ f (0) = 0.
It is easy to check that (β ′′ , H ′′ ) is an α-flow from A( x) to C( x).
For (iii) if we have ∀γ ≺ δA(γ, x) ⊲ ∀γ ≺ δ + 1A(γ, x) then there exists β and H(γ, δ, x) such that we have the conditions of the Definition 4.3. Define β ′ = β × θ and I(γ, x) = H(⌊ γ θ ⌋, ⌊ γ θ ⌋, x). It is easy to see that (I, β ′ ) is an α-flow from ⊤ to A(θ, x).
(i) If Γ, A ⊲ ∆ or Γ, B ⊲ ∆, then Γ, A ∧ B ⊲ ∆. (ii) If Γ 0 ⊲ ∆ 0 , A and Γ 1 ⊲ ∆ 1 , B, then Γ 0 , Γ 1 ⊲ ∆ 0 , ∆ 1 , A ∧ B. (iii) If Γ ⊲ ∆, A or Γ ⊲ ∆, B, then Γ ⊲ ∆, A ∨ B.
(iv) If Γ 0 , A ⊲ ∆ 0 and Γ 1 , B ⊲ ∆ 1 , then Γ 0 , Γ 1 , A ∨ B ⊲ ∆ 0 , ∆ 1 .
Proof. The proof is similar to the proof of the theorem 2.23. Note that the proof of the theorem 2.23 is fully based on the weak gluing and conjunction and disjunction applications, hence we can apply the same proof wherever we have those properties. Proof. We prove the lemma by induction on the length of the free-cut free proof of Γ( x) ⇒ ∆( x).
(Universal Quantifier, Right
). If Γ( x) ⇒ ∆( x), ∀zB( x, z) is proved by the ∀R rule by Γ( x) ⇒ ∆( x), B( x, z), then by IH, Γ( x) ⊲ ∆( x), B( x, z). Therefore, there exist an ordinal β and a formula H(γ, x, z) ∈ ∀ 1 such that the conditions of the Definition 4.3 are provable in PV. Define β ′ = β and H ′ (γ, x) = ∀zH(γ, x, z). Since H(γ, x, z) ∈ ∀ 1 then ∀zH(γ, x, z) ∈ ∀ 1 . The other conditions to check that the new sequence is an α-flow is a straightforward consequence of the fact that if PV ⊢ ∀γ ≺ δH(γ, z, x) → γ ≺ δ + 1 H(γ, z, x), then PV ⊢ ∀γ ≺ δ∀zH(γ, z, x) → ∀γ ≺ δ + 1∀zH(γ, z, x).
6. (Universal Quantifier, Left). If Γ( x), ∀zB( x, z) ⇒ ∆( x) is proved by the ∀L rule by Γ( x), B( x, s( x)) ⇒ ∆( x), then since PV ⊢ ∀zB( x, z) → B( x, s( x)), we have ∀zB( x, z) ≤ B( x, s( x)).
And since Γ( x), B( x, s( x)) ⊲ ∆( x), by using cut we have Γ( x), ∀zB( x, z) ⊲ ∆( x).
(Induction)
. The proof is similar to the proof of Lemma 2.27.
And also like in the bounded case we have the completeness theorem: Therefore, using induction on H(δ, x) we have TI(∀ 1 , ≺) ⊢ H(0, x) → H(γ, x).
And hence TI(∀ 1 , ≺) ⊢ H(0, x) → ∀γ ≺ β H(γ, x), and thus TI(∀ 1 , ≺) ⊢ A( x) ⇒ B( x).
In the following we will use the PLS(≺ α ) problems to characterize the NP search problems of any theory with Π 0 2 -ordinal α.
Theorem 4.10. Let T be a theory of arithmetic and α T be its Π 0 2 -ordinal with a PV-representation ≺ α T of the order and a PV-representation of its ordinal arithmetic, then TFNP(T ) ≡ PV PLS(≺ α T ).
Proof. First of all, it is easy to see that ∃s ¬c(N(x, s)) ≺ c(x, s) ∧ F (x, s) is provable in PRA + PRWO(≺). Define f (0) = (c(x, i(x)), i(x)) and f (n+1) = (c(x, N(x, f 0 (n))), N(x, f 0 (n))) c(x, N(x, f 0 (n)) ≺ c(x, f 0 (n)) ∧ F (x, f 0 (n)) f (n) o.w.
where the order on the range of ≺ ′ is the order of the ordinal ot(≺ c(x,i(x)) )×ω.
Since ≺ ′ is a sub-order of ≺, by PRWO(≺) there exists some n such that f (n + 1) ⊀ f (n). By definition of f , this n should impose the property that c(x, N(x, f 0 (n)) ⊀ c(x, f 0 (n)) ∨ ¬F (x, f 0 (n)). It is easy to show by induction on m that F (x, f 0 (m)) for any m, hence c(x, N(x, f 0 (n)) ⊀ c(x, f 0 (n)) ∧ F (x, f 0 (n)). Now it is enough to pick s = f 0 (n). Therefore, PRA+ PRWO(≺ ) ⊢ ∃s N(x, s) = s ∧ F (x, s) and therefore, PRA + PRWO(≺) ⊢ ∃s G(x, s). And finally, PRA + PRWO(≺) ⊢ ∃s |s| ≤ p(|x|) ∧ G(x, s) which by definition means T ⊢ ∃s |s| ≤ p(|x|) ∧ G(x, s).
For the converse, assume that T ⊢ ∀x∃y|y| ≤ p(|x|)A(x, y) where A(x, y) is quantifier-free in the language of PV. Then by definition PRA + PRWO(≺ ) ⊢ ∀x∃y|y| ≤ p(|x|)A(x, y) because ∀x∃y |y| ≤ p(|x|) ∧ A(x, y) ∈ Π 0 2 . Then by Lemma 4.2 we have TI(∀ 1 , ≺ α T ) ⊢ ∀y(|y| ≤ p(|x|) → ¬A(x, y)) ⇒ ⊥.
By Theorem 4.8 we have ∀y(|y| ≤ p(|x|) → ¬A(x, y))⊲⊥. Hence there exists (H, β) such that (i) PV ⊢ ∀y(|y| ≤ p(|x|) → ¬A(x, y)) → ∀γ ≺ 1 H(1, x).
hence by disjunction application we have ( Γ ∧ A) ∨ ¬A ⊲ ∆ ∨ ¬A. By the claim we have ⊲A ∨ ¬A, therefore by conjunction application Γ⊲ Γ ∧ (A ∨ ¬A). But, it is easy to see that Γ ∧ (A ∨ ¬A) ⊲ ( Γ ∧ A) ∨ ¬A. Hence by gluing we have Γ ⊲ ∆ ∨ ¬A. For (ii), we have Γ ⊲ ∆ ∨ A. By conjunction application Γ ∧ ¬A ⊲ ( ∆ ∨ A) ∧ ¬A. By the claim we have A ∧ ¬A ⊲ ⊥ therefore by disjunction application ∆ ∨ (A ∧ ¬A) ⊲ ∆. But, it is clear that ( ∆ ∨ A) ∧ ¬A ⊲ ∆ ∨ (A ∧ ¬A). Hence by gluing, Γ ∧ ¬A ⊲ ∆.Now, we will prove the claim. For the non-deterministic case, the claim is trivial because we have B ⊢ ⊤ → A( x)∨¬A( x) and B ⊢ A( x)∧¬A( x) → ⊥.
is enough to define the sequence of statements in between by the following scheme: If i = j = 1, then use B(u) ∧ B(v). If i = 1 and j = 0 use B(u) ∧ ¬B(v). If i = 0 and j = 1 use ¬B(u) ∧ B(v). And finally if i = j = 0, use the flow from ( * ).
For
(ii), if we have Γ, A⊲∆, B then by the Lemma 2.25 we have Γ, ⊲∆, ¬A, B. Hence by the Lemma 2.23 we have Γ, ⊲∆, (¬A ∨ B), (¬A ∨ B). By contraction, Γ, ⊲∆, (¬A ∨ B). Since ¬A ∨ B ⊲ A → B, by cut Γ, ⊲∆, A → B.
Since B ≤ n (A → B), by contraction and cut we haveB ∨(A → B)⊲A → B. On the other hand, ≤ A ∨ (A → B). Hence Γ ⊲ (( Γ ∨ (A → B)) ∧ (A ∨ (A → B)),and therefore by gluing Γ ⊲ ∆, A → B.
Theorem 2 .
230. (Simulation) Let B be a bounded theory of arithmetic. Then all non-deterministic reductions can be simulated by a term-length sequence of deterministic reductions. In other words
Theorem 3 . 11 .
311Let j ≤ k. Then,∀Σ j (open)(B(T all , Π k (open), B)) ≡ B GI j k (L). Proof. It is clear that B(T all , Π k (open), B) ⊢ GI j k (L). For the converse, assume B(T all , Π k (open), B) ⊢ ∀xA(x) where A ∈ Σ j (open) and j ≤ k.Then, we know that B(T all , Π k (open), B) ⊢ ¬A(x) ⇒ ⊥ and ¬A ∈ Π j (open). By Corollary 3.2, there exist a term t(x), a formula H(u, x) ∈ Π k (open) and sequences of terms E 0 , E 1 , I 0 , I 1 and F (u) such that the following statements are provable in B:
Corollary 3 . 17 .
317TFNP(I∆ 0 + EXP) ≡ PLS(R(exp), ≤).
Lemma 3 . 18 .
318TFNP(PRA ≺ ) ≡ PLS(BASIC ≺ , ≤). Therefore by definition of Π 0 2 -ordinal and the fact that PRA ≺ is a conservative extension of PRA + PRWO(≺), we have: Theorem 3.19. Let T be a theory of arithmetic with Π 0 2 -ordinal α T with a PRA-representation ≺ α T , then TFNP(T ) ≡ PLS(BASIC ≺α T , ≤). And finally by Theorem 3.8 we have: Corollary 3.20. (i) TFNP(IΣ 1 ) ≡ PLS(BASIC ≺ ω 2 , ≤).
( ii )
iiFor all n > 1, TFNP(IΣ n ) ≡ PLS(BASIC ≺ω n , ≤).
(
iii) TFNP(PA) ≡ PLS(BASIC ≺ǫ 0 , ≤).
Lemma 4 . 2 .
42PRA + PRWO(≺) ⊆ TI(∀ 1 , ≺).
Theorem 4.8. (Soundness) If Γ ∪ ∆ ⊆ ∀ 1 and TI(∀ 1 , ≺) ⊢ Γ ⇒ ∆, thenthere exists an α-flow from Γ to ∆.
Theorem 4. 9 .
9(Completeness) If Γ ∪ ∆ ⊆ ∀ 1 and Γ ⊲ ∆, then TI(∀ 1 , ≺) ⊢ Γ ⇒ ∆.Proof. If there exists an α-flow from Γ to ∆ then it means that there exists (H, β) such that(i) PV ⊢ A( x) → ∀γ ≺ H(1, x).
(
ii) PV ⊢ ∀γ ≺ β [∀δ ≺ γ H(δ, x) → ∀δ ≺ γ + 1 H(δ, x)]. (iii) PV ⊢ ∀γ ≺ β H(γ, x) → B( x).
by the Lemma 2.25 we have Γ 0 , ¬A ⊲ ∆ 0 . On the other hand, we have Γ 1 , B ⊲ ∆ 1 . Therefore, by the Lemma 2.23 we have
. (Axioms). If Γ( x) ⇒ ∆( x) is a logical axiom then the claim is trivial. If it is a non-logical axiom then the claim will be also trivial because all non-logical axioms are provable in PV. Therefore there is nothing to prove.2. (Structural Rules). The case for weakening and exchange are trivial. For the contraction, note that all formulas are ∀ 1 which means that having all quantifiers, it is possible to decide in polynomial-time which formula is true and hence we can handle the contraction case.3. (Cut). It is similar to theLemma 2.27. 4. (Propositional). The conjunction and disjunction cases are proved in the Lemma 4.7. The implication and negation cases are trivial because they should be quantifier-free and hence we can manipulate them as in the Lemma 2.25 and 2.28.
Acknowledgment. We wish to thank Pavel Pudlak for his support, his suggestions and the invaluable discussions that we have had since the beginning of this project. We are also genuinely grateful to Sam Buss and Raheleh Jalai for their constructive suggestions and the helpful discussions on the crucial and primitive stages of developing the theory. (x, ∆(γ), y, Z(γ)) γ = 0, ¬G(x, γ, z) (x, 0, y, 0) γ = 0, G(x, γ, z) (x, γ, y, z) γ = 0 and i(x) = (x, Γ, 0, Z) and c(x, γ, y, z) = γ, Goal(x, γ, y, z) = G(x, 0, z). It is easy to see that this new data is a PLS(≺ α T ) problem. Now it is not hard to shift everything for γ ≺ ω one point to the right to add |y| ≤ p(|x|) ∧ A(x, y) to the first point and use Y for its neighborhood. Now we have a PLS(≺ α T ) problem and finally by the answer of the problem namely (x, γ, y, z) we can compute y which is the witness for A and computable just by a projection. Note that this reduction is provable in PV.And as a corollary we have: (iv) For any representable ǫ 0 ≺ α, TFNP(PA + TI(α)) ≡ PLS(≺ α ).Proof. It is enough to have a PV-representation of these ordinals and the basic arithmetic on them which was carried out in[2].
B( x) and H(δ, x) be some formulas in ∀ 1 . A tuple (H, β) is called an α-flow if (i) PV ⊢ A( x). Definition 4.3. Let A( x. → ∀γ ≺ H(1, xDefinition 4.3. Let A( x), B( x) and H(δ, x) be some formulas in ∀ 1 . A tuple (H, β) is called an α-flow if (i) PV ⊢ A( x) → ∀γ ≺ H(1, x).
. Pv ⊢ ∀γ ≺ Β, ∀δ ≺ γ H(δ, x) → ∀δ ≺ γ + 1 H(δ, x)PV ⊢ ∀γ ≺ β [∀δ ≺ γ H(δ, x) → ∀δ ≺ γ + 1 H(δ, x)].
PV ⊢ ∀γ ≺ β H(γ, x) → B( x). (iii) PV ⊢ ∀γ ≺ β H(γ, x) → B( x).
Like the bounded case we need to prove some basic theorems for this new notion. They will help us to prove the soundness theorem for this kind of flowLike the bounded case we need to prove some basic theorems for this new notion. They will help us to prove the soundness theorem for this kind of flow.
Conjunction Application) Let C( x) ∈ ∀ 1 be a formula. If A( x) ⊲ B( x) then A( x) ∧ C( x) ⊲ B( x) ∧ C( x). Lemma 4.4.Lemma 4.4. (Conjunction Application) Let C( x) ∈ ∀ 1 be a formula. If A( x) ⊲ B( x) then A( x) ∧ C( x) ⊲ B( x) ∧ C( x).
B( x), then by Definition 4.3 there exist a term β and a formula H(γ, x) ∈ ∀ 1 such that we have the conditions in the Definition 4.3. Define β ′ = β and H ′ (γ, x) = H(γ, x) ∧ C( x). It is clear that the (H ′ , β ′ ) is an α-flow from A( x) ∧ C( x) to B( x) ∧ C( x)Proof. Since A( x) ⊲ B( x), then by Definition 4.3 there exist a term β and a formula H(γ, x) ∈ ∀ 1 such that we have the conditions in the Definition 4.3. Define β ′ = β and H ′ (γ, x) = H(γ, x) ∧ C( x). It is clear that the (H ′ , β ′ ) is an α-flow from A( x) ∧ C( x) to B( x) ∧ C( x).
Disjunction Application) Let C( x) ∈ ∀ 1 be a formula. If A( x) ⊲ B( x) then A( x) ∨ C( x) ⊲ B( x) ∨ C( x). Lemma 4.5.Lemma 4.5. (Disjunction Application) Let C( x) ∈ ∀ 1 be a formula. If A( x) ⊲ B( x) then A( x) ∨ C( x) ⊲ B( x) ∨ C( x).
then by Definition 4.3, there exist an ordinal β and a formula H(γ, x) ∈ ∀ 1 such that the conditions in the Definition 4.3 is provable in PV. Now define β ′ = β and H ′ (γ, x) = H(γ, x) ∨ C( x). Since A( x) ⊲ B( x). It is easy to see that (H ′ , β ′ ) is an α-flow from A( x) ∨ C( x) to B( x) ∨ C( x)Proof. Since A( x) ⊲ B( x), then by Definition 4.3, there exist an ordinal β and a formula H(γ, x) ∈ ∀ 1 such that the conditions in the Definition 4.3 is provable in PV. Now define β ′ = β and H ′ (γ, x) = H(γ, x) ∨ C( x). It is easy to see that (H ′ , β ′ ) is an α-flow from A( x) ∨ C( x) to B( x) ∨ C( x).
Weak Gluing) If A( x) ⊲ B( x) and B( x) ⊲ C( x), then A( x) ⊲ C( x). Lemma 4.6. (iLemma 4.6. (i) (Weak Gluing) If A( x) ⊲ B( x) and B( x) ⊲ C( x), then A( x) ⊲ C( x).
Strong Gluing) If ∀γ ≺ βA(γ, x) ⊲ γ ≺ β + 1A(γ, x), then ⊤ ⊲ A(θ, x). (Strong Gluing) If ∀γ ≺ βA(γ, x) ⊲ γ ≺ β + 1A(γ, x), then ⊤ ⊲ A(θ, x).
For (i), since A( x) ⊲ B( x) there exist an ordinal β and a formula H(γ, x) ∈ ∀ 1 such that PV proves the conditions in the Definition 4.3. On the other hand since B( x) ⊲ C( x) we have the corresponding data for B( x) to C( x) which we show by β ′ and H ′ (γ, x). Define β ′′ = β + β ′ and H ′′ (γ, x) = H(γ, xProof. For (i), since A( x) ⊲ B( x) there exist an ordinal β and a formula H(γ, x) ∈ ∀ 1 such that PV proves the conditions in the Definition 4.3. On the other hand since B( x) ⊲ C( x) we have the corresponding data for B( x) to C( x) which we show by β ′ and H ′ (γ, x). Define β ′′ = β + β ′ and H ′′ (γ, x) = H(γ, x)
. Pv ⊢ ∀γ ≺ Β, ∀δ ≺ γ H(δ, x) → ∀δ ≺ γ + 1 H(δ, x)PV ⊢ ∀γ ≺ β [∀δ ≺ γ H(δ, x) → ∀δ ≺ γ + 1 H(δ, x)].
. Pv ⊢ ∀γ ≺ Β H(γ, X) → ⊥, PV ⊢ ∀γ ≺ β H(γ, x) → ⊥.
On the other hand, all the conditions are provable in PV which means that we can witness the existential quantifiers by polytime functions. Hence, (i ′ ) PV ⊢ (|Y (x, z)| ≤ p(|x|) → ¬A. Since H ∈ ∀ 1 we have H(γ, x) = ∀zG(γ, x, z). x, Y (x→ G(0, x, z)Since H ∈ ∀ 1 we have H(γ, x) = ∀zG(γ, x, z). On the other hand, all the conditions are provable in PV which means that we can witness the existential quantifiers by polytime functions. Hence, (i ′ ) PV ⊢ (|Y (x, z)| ≤ p(|x|) → ¬A(x, Y (x, z))) → G(0, x, z).
. ( , ′ Pv ⊢ ∀γ ≺ Β, ∆(δ) ≺ γ → G(∆(δ), x, Z(δ)) → δ ≺ γ + 1 → G(δ, x, z)(ii ′ ) PV ⊢ ∀γ ≺ β [∆(δ) ≺ γ → G(∆(δ), x, Z(δ)) → δ ≺ γ + 1 → G(δ, x, z)].
. → , iii ′ ) PV ⊢ (Γ ≺ β → G(Γ, x(iii ′ ) PV ⊢ (Γ ≺ β → G(Γ, x, Z)) → ⊥.
Put δ = γ in (ii ′ ), then we have PV ⊢ ∀γ ≺ β. ∆(γ) ≺ γ → G(∆(γ), x, Z(γ)) → G(γ, x, z)Put δ = γ in (ii ′ ), then we have PV ⊢ ∀γ ≺ β [(∆(γ) ≺ γ → G(∆(γ), x, Z(γ)) → G(γ, x, z)].
. ) = Define, ¬g, x, γ, z) and N(x, γ, yDefine F (x, γ, y, z) = ¬G(x, γ, z) and N(x, γ, y, z) =
A Characterisation of Definable NP Search Problems in Peano Arithmetic, Logic, Language, Information and Computation, 16th International Workshop. A Beckmann, Tokyo, JapanA. Beckmann, A Characterisation of Definable NP Search Problems in Peano Arithmetic, Logic, Language, Information and Computation, 16th International Workshop, WoLLIC 2009, Tokyo, Japan, June 21-24, 2009.
Ordinal Notations and Well-Orderings in Bounded Arithmetic. A Beckmann, S R Buss, C Pollett, Annals of Pure and Applied Logic. 120A. Beckmann, S. R. Buss, C. Pollett, Ordinal Notations and Well- Orderings in Bounded Arithmetic, Annals of Pure and Applied Logic 120( 2002), 197-223.
Polynomial Local Search in the Polynomial Hierarchy and Witnessing in Fragments of Bounded Arithmetic. A Beckmann, S R Buss, Journal of Mathematical Logic. 9A. Beckmann, S. R. Buss, Polynomial Local Search in the Polynomial Hierarchy and Witnessing in Fragments of Bounded Arithmetic, Journal of Mathematical Logic 9, 1 (2009) 103-138.
A short course in ordinal analysis. W Pohlers, Aczel, Simmons and WainerW. Pohlers, A short course in ordinal analysis, in: Aczel, Simmons and Wainer (1992), pp. 27-78.
Higher complexity search problems for bounded arithmetic and a formalized no-gap theorem. N Thapen, Archive for Mathematical Logic. 50N. Thapen, Higher complexity search problems for bounded arithmetic and a formalized no-gap theorem, Archive for Mathematical Logic, Vol 50:7-8, pages 665-680, 2011.
The provably total search problems of bounded arithmetic. A Skelley, N Thapen, Proceedings of the London Mathematical Society. 103A. Skelley, N. Thapen, The provably total search problems of bounded arithmetic, Proceedings of the London Mathematical Society, Vol 103:1, pages 106-138, 2011.
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[
"Phases and collective modes of hardcore Bose-Fermi mixture in an optical lattice",
"Phases and collective modes of hardcore Bose-Fermi mixture in an optical lattice"
]
| [
"S Sinha \nIndian Institute of Science Education Research\nHC Block\nSector III, Salt LakeKolkata-700106India. (\n",
"K Sengupta \nTheoretical Physics Department\nIndian Association for the Cultivation of Sciences\nJadavpurKolkata-700032India\n\nTCMP division\nSaha Institute of Nuclear Physics\n1/AF BidhannagarKolkata-700064India\n"
]
| [
"Indian Institute of Science Education Research\nHC Block\nSector III, Salt LakeKolkata-700106India. (",
"Theoretical Physics Department\nIndian Association for the Cultivation of Sciences\nJadavpurKolkata-700032India",
"TCMP division\nSaha Institute of Nuclear Physics\n1/AF BidhannagarKolkata-700064India"
]
| []
| We obtain the phase diagram of a Bose-Fermi mixture of hardcore spinless Bosons and spinpolarized Fermions with nearest neighbor intra-species interaction and on-site inter-species repulsion in an optical lattice at half-filling using a slave-boson mean-field theory. We show that such a system can have four possible phases which are a) supersolid Bosons coexisting with Fermions in the Mott state, b) Mott state of Bosons coexisting with Fermions in a metallic or charge-density wave state, c) a metallic Fermionic state coexisting with superfluid phase of Bosons, and d) Mott insulating state of Fermions and Bosons. We chart out the phase diagram of the system and provide analytical expressions for the phase boundaries within mean-field theory. We demonstrate that the transition between these phases are generically first order with the exception of that between the supersolid and the Mott states which is a continuous quantum phase transition. We also obtain the low-energy collective excitations of the system in these phases. Finally, we study the particle-hole excitations in the Mott insulating phase and use it to determine the dynamical critical exponent z for the supersolid-Mott insulator transition. We discuss experiments which can test our theory. | 10.1103/physrevb.79.115124 | [
"https://arxiv.org/pdf/0811.4515v2.pdf"
]
| 118,861,746 | 0811.4515 | a1aaf2324229c27a39254545ebcd1d9a820237df |
Phases and collective modes of hardcore Bose-Fermi mixture in an optical lattice
8 Dec 2008
S Sinha
Indian Institute of Science Education Research
HC Block
Sector III, Salt LakeKolkata-700106India. (
K Sengupta
Theoretical Physics Department
Indian Association for the Cultivation of Sciences
JadavpurKolkata-700032India
TCMP division
Saha Institute of Nuclear Physics
1/AF BidhannagarKolkata-700064India
Phases and collective modes of hardcore Bose-Fermi mixture in an optical lattice
8 Dec 2008numbers: 6760Fp6470Tg7322Gk7322Lp
We obtain the phase diagram of a Bose-Fermi mixture of hardcore spinless Bosons and spinpolarized Fermions with nearest neighbor intra-species interaction and on-site inter-species repulsion in an optical lattice at half-filling using a slave-boson mean-field theory. We show that such a system can have four possible phases which are a) supersolid Bosons coexisting with Fermions in the Mott state, b) Mott state of Bosons coexisting with Fermions in a metallic or charge-density wave state, c) a metallic Fermionic state coexisting with superfluid phase of Bosons, and d) Mott insulating state of Fermions and Bosons. We chart out the phase diagram of the system and provide analytical expressions for the phase boundaries within mean-field theory. We demonstrate that the transition between these phases are generically first order with the exception of that between the supersolid and the Mott states which is a continuous quantum phase transition. We also obtain the low-energy collective excitations of the system in these phases. Finally, we study the particle-hole excitations in the Mott insulating phase and use it to determine the dynamical critical exponent z for the supersolid-Mott insulator transition. We discuss experiments which can test our theory.
I. INTRODUCTION
Recent experiments on ultracold trapped atomic gases have opened a new window onto the phases of quantum matter 1 . A gas of Bosonic atoms in an optical or magnetic trap has been reversibly tuned between superfluid (SF) and insulating ground states by varying the strength of a periodic potential produced by standing optical waves 1 . This transition has been explained on the basis of the Bose-Hubbard model with on-site repulsive interactions and hopping between nearest neighboring sites of the lattice. 2,3 . Further, theoretical studies of Bosonic atoms with spin and/or pseudospin have also been undertaken 4,5 . These studies have revealed a variety of interesting Mott 1 and supersolid 6 phases and superfluid-insulator transitions 3 in these systems. On the Fermionic side, the experimental studies have mainly concentrated on the observation of paired superfluid states 7 and the BCS-BEC crossover in such systems near a Feshbach resonance 8 . More recently, it has been possible to generate mixtures of Fermionic and Bosonic atoms in a trap 9 . Several theoretical studies followed soon, which established such Bose-Fermi mixtures to be interesting physical systems in their own right 10,11 , exhibiting exciting Mott phases in the presence of an optical lattice.
Many of the earlier studies of Bose-Fermi mixtures has been restricted to one-dimensional (1D) systems 12 or have concentrated on regimes where the coupling between Bosons and Fermions are weak 13 . The existence of a supersolid (SS) phase in these system in such a weak coupling regime has also been predicted 14 . Other works 10,11 which have looked at the strong coupling regime have restricted themselves to integer filling factors of Bosons and Fermions (per spin) and have therefore not addressed the phenomenon of translational symmetry breaking and possible associated SS phases in the strongly interacting regimes of these systems. More recently, however, the authors of Ref. 15 have studied a mixture of spinless softcore Bosons with an on-site interaction U and spin-polarized non-interacting Fermions at half-filling in a 3D optical lattice using dynamical mean-field theory (DMFT). Several interesting phases, including a SS phase of Bosons and charge-density wave(CDW) states of Fermions, have been found in Ref. 15. In this work, we study a mixture of hardcore spinless Bosons and spin-polarized interacting Fermions in an optical lattice at half-filling using a slave-boson meanfield technique. We concentrate on the case where both the Bosons and the Fermions have a nearest-neighbor density-density repulsive interaction in addition to the usual on-site interaction term between them. We provide an analytical, albeit mean-field, phase diagram for the system and demonstrate that the ground state of such a system consists of four distinct phases, namely, a) a Mott insulating (MI) phase where both Fermions and Bosons are localized at the lattice sites, b) a metal+SF phase where the Fermions are in a metallic phase with a gapless Fermi surface and the Bosons are in a superfluid state, c) a SS phase where the Bosons are in the SS state while the Fermions are localized at the lattice site, and d) a CDW+MI phase of coexisting Fermions with weak density wave order along with Mott insulating Bosons. We show, within mean-field theory, that the transition between these phases are generically first order with the exception of that between the SS and the MI phases which is a continuous quantum phase transition. We also obtain the low-energy collective modes in the metal+SF, CDW+MI and the SS phases and demonstrate that they have linear dispersions with defi-nite group velocities. Further, in the MI phase which has no gapless modes, we find the dispersion of the gapped particle-hole excitations and use it to determine the dynamical critical exponent z for the continuous MI-SS transition. We also discuss realistic experiments which can test our theory.
The plan of rest of the paper is as follows. In Sec. II A, we develop a slave-Boson mean-field theory for the system. This is followed by Sec. II B, where the meanfield phase diagram is charted out. In Sec. III, we obtain the low-energy collective modes of the system in the metal+SF, CDW+MI and the SS phases. This is followed by Sec. IV, where we discuss the gapped particle-hole excitations of the MI phase and use it to determine z for the SS-MI transition. We discuss relevant experiments which can test our theory and conclude in Sec. V
II. SLAVE-BOSON MEAN-FIELD THEORY
A. Mean-field equations
The Hamiltonian of the Bose-Fermi mixture in a ddimensional hypercubic lattice is described by the Hamiltonian
H = H F + H B + H F B (1) H F = −t F ij c † i c j + V F ij n F i n F j (2) H B = −t B ij b † i b j + V B ij n B i n B j(3)H F B = U i n F i n B i (4) where c i (b i ) and n F i = c † i c i (n B i = b † i b i )
denote the annihilation and number operators for Fermions(Bosons) at site i, V F (V B ) and t F (t B ) denote the nearest-neighbor interaction strengths and hopping amplitudes for the Fermions(Bosons) respectively, U represents the amplitude for on-site interaction between the Fermions and the Bosons, and ij represents sum over nearest neighbor ij pairs on the lattice. In what follows, we shall study the Hamiltonian at half-filling. We note at the outset that this constraint of half-filling implies n F i , n B i ≤ 1 at each site.
To obtain an analytical understanding of the phases of these model, we first introduce a slave-boson representation for the Fermions: c i = a i d i , where a i denotes annihilation operator for the slave-bosons and d i represents the annihilation operator for pseudo-fermions. We note that in this representation, the anticommutation relation for the Fermionic operator [c † i c j ] + = δ ij , where δ ij denotes the Kronecker delta, enforces the constraint n d i = n a i on each site. In terms of these slave-bosons and pseudofermions, the Hamiltonians H F and H F B in Eqs. 2 and 4 can be written as
H ′ F = −t F ij d † i a † i a j d j + V F ij n a i n a j + i λ i (n a i − n d i ) (5) H ′ F B = U i n a i n B i(6)
where we have implemented the constraint n a i = n d i using Lagrange multipliers λ i at each site, and have used the fact that n a i = n d i = n F i ≤ 1 at each site i. We note that the Hamiltonian H ′ = H B + H ′ F + H ′ F is exact and is completely equivalent to H (Eq. 1).
To make further progress, we proceed with mean-field approximation of H ′ . To this end, we first decompose the quartic hopping term in Eq. 5 using
t F ij d † i a † i a j d j = t 1 ij a † i a j + t 2 ij d † i d j − N t 2 t 1 t F(7)
where N is the number of sites in the lattice and the hopping amplitudes t 1 and t 2 are given by
t 1 = t F N ij d † i d j 0 , t 2 = t F N ij a † i a j 0 .(8)
Here .. 0 denotes average with respect to the ground state of the system. Next, we use a mean-field approximation for the constraint term and approximate the Lagrange multiplier field λ i = λ 0 + ∆(−1) i , where for sake of definiteness, we take i = i 1 + i 2 + ... + i d to be even for A sublattice sites. Such an ansatz for λ i is motivated by the fact that it is the simplest mean-field ansatz that preserves the basic symmetries of the problem and, at the same time, allows for translational symmetry breaking in the pseudo-fermion sector. With these approximations, the mean-field Hamiltonian for the system can be written as
H MF = −t 2 ij d † i d j + ∆ i (−1) i (n a i − n d i ) + ij V F n a i n a j − t 1 a † i a j + i U n a i n b i + ij V B n B i n B j − t B b † i b j + N t 2 t 1 t F(9)
In writing Eq. 9, we have ignored the term λ 0 i (n d i −n a i ) since it merely renormalizes the chemical potential of the fermions and thus behave like a constant as long as we restrict ourselves to half-filling.
To obtain the ground state energy corresponding to this mean-field Hamiltonian, we now use a variational ansatz for the ground-state wavefunction
|Ψ 0 = i∈A |ψ A ⊗ i∈B |ψ B ⊗ |FS |ψ A = cos(θ)|n B = 0 + sin(θ)|n B = 1
⊗ (cos(γ)|n a = 0 + sin(γ)|n a = 1 ) |ψ B = cos(θ)|n B = 1 + sin(θ)|n B = 0 ⊗ (cos(γ)|n a = 1 + sin(γ)|n a = 0 )
|FS = k θ(|k| − k F )d † k |0(10)
where k F denotes the Fermi wavevector for the pseudofermions and θ and γ are variational parameters which has to be determined by minimizing the ground state energy. Note that the variational wavefunction given by Eq. 10 has a two sublattice structure which allows for the possibility of translational symmetry broken phases. For the current system, where the Fermions and the Bosons both interact via nearest-neighbor density-density interaction terms, Eq. 10 is the simplest possible mean-field variational wavefunction which respects all the symmetries of the Hamiltonian. We would like to point out here that capturing the phases of a Bose-Fermi mixture with such interaction terms is beyond the scope of any single-site mean-field theory including single site DMFT. The variational mean-field energy E v = Ψ|H MF |Ψ of the system can now be easily obtained and is given by
E v N t F = − d 2 Z ′ sin 2 (2γ) + Z sin 2 (2θ) + U ′ 2 cos 2 (γ) cos 2 (θ) + sin 2 (γ) sin 2 (θ) − ∆ ′ 2 cos(2γ) + 2 N FS| i (−1) i d † i d i |FS Z ′ = (t 1 − V F )/t F , Z = (t B − V B )/t F(11)
where we have used t 2 /t F = sin 2 (2γ), ∆ ′ = ∆/t F , U ′ = U/t F , and t 1 /t F has to be determined from Eq. 8. The corresponding mean-field equations which determine the ground-state values of the variational parameters are given by
∂E v /∂θ = ∂E v /∂γ = ∂E v /∂∆ ′ = 0 and yields sin(2θ) U ′ 2 cos(2γ) + 2Zd cos(2θ) = 0 sin(2γ) U ′ 2 cos(2θ) + 2Z ′ d cos(2γ) − ∆ ′ = 0 cos(2γ) + 2 N FS| i (−1) i d † i d i |FS = 0 (12)
Next, we evaluate the effective hopping amplitude t 1 and 2/N FS| i (−1) i d † i d i |FS . To this end, first, let us consider the pseudo-fermion HamiltonianH
= −t 2 ij d † i d j − ∆ i (−1) i n d i . The energy spectrum ofH is given by ±E(k), where E(k) = ǫ(k) 2 + ∆ 2 , ǫ(k) = −2t 2 i=1,d cos(k i a)
is the energy dispersion of free fermions in a hypercubic lattice in d dimensions, and a is the lattice spacing which we shall, from now on, set to unity. The density of states (DOS) corresponding to these fermions are therefore given by
ρ(E) = ρ 0 ( E 2 − ∆ 2 ) √ E 2 − ∆ 2 E ρ 0 (ǫ) = 1 2t 2 d d k (2π) d δ[(ǫ) − d i=1 cos(k i )](13)
where ρ 0 denotes the DOS of free fermions with tightbinding dispersion on a hypercubic lattice,ǫ = ǫ/2t 2 , and we have used the relation ρ(E)dE = ρ 0 (ǫ)dǫ. It is convenient to use Eq. 13 to express the expectation values over pseudo-fermion ground states in Eqs. 8 and 12 and one obtains (15) Eqs. 12, 14 and 15 denotes the complete set of meanfield equations which can be now solved to determine the mean-field phase diagram.
cos(2γ) = −∆I 1 (∆), t 1 = t 2 I 2(14)I 1 (∆) = 0 −1 1 √ǫ 2 +∆ 2 ρ 0 (ǫ)dǫ 0 −1 ρ 0 (ǫ)dǫ I 2 = 0 −1ǫ ρ 0 (ǫ)dǫ 0 −1 ρ 0 (ǫ)dǫ
B. Phase diagram
Eqs. 12 and 14 can be easily solved numerically to obtain the mean-field phase diagram for the system. However, before resorting to numerics, we provide a qualitative discussion of the nature of the phases.
We find that Eqs. 12 and 14 yields four distinct solutions which corresponds to four possible phases of the system. First, we find a MI phase with broken translational symmetry where both the fermions and the Bosons are localized. Such a phase corresponds to the solution 16
θ = 0, γ = π/2,∆ → ∞(16)
Note that the divergence of∆ corresponds to t 2 → 0 which in turn ensures that cos(2γ) = −1. Such a MI state corresponds to a intertwined checkerboard density-wave pattern where the Fermions are localized in sublattice A (n a i = sin 2 (γ) = n d i = 1 for i ∈ A) and the Bosons are localized in sublattice B (n B i = cos 2 (θ) = 1 for i ∈ B). The mean-field energy of this state is E 1 = 0.
Second, we find a SS phase, where the Bosons are in a supersolid phase with coexisting density-wave and superfluid order and the Fermions are localized in a Mott phase. Such a state corresponds to the solution
cos(2θ) = U ′ 4Zd , γ = π/2,∆ → ∞(17)
Such a state has b = sin(2θ)/2 = 0 and (−1) i b † i b i = − cos(2θ) = 0 and thus corresponds to a SS phase for the Bosons. Note that the realization of this state necessarily requires U ′ /4Zd < 1. For U ′ /4Zd = 1, θ = 0 and we recover the MI state where b = 0. The energy of the SS state is per site given by
E 2 = − Zdt F 2 U ′ 4Zd − 1 2(18)
Third, we find the MI+CDW state where the fermions show weak density-wave oscillations whereas the Bosons are localized in the MI state. This corresponds to the solution
θ = 0, γ = γ 0 = 0, π/2(19)
where γ 0 and ∆ are to be determined from a numerical solution of the mean-field equations (20) The energy of this state per site is given by
cos(2γ 0 ) = − U ′ 4Z ′ d (1 − 2∆/U ) = −∆I 1 (∆).E 3 = U cos 2 (γ 0 ) 2 1 − 4Z ′ d U ′ sin 2 (γ 0 )(21)
Finally, we find the state in which the superfluid Bosons coexist with metallic Fermions. This corresponds to the solution
θ = γ = π/4 ∆ = 0(22)
Note that such a phase has b = sin(2θ)/2 = 1 and (−1) i b † i b i = − cos(2θ) = 0 so that the Bosons are in an uniform superfluid state. Also, ∆ = 0 and (−1) i a † i a i = − cos(2γ) = 0 in this state indicating that the Fermions are in a gapless uniform metallic state. The energy of this state per site is given by
E 4 = U 4 1 − 2d U ′ (Z + Z ′ )(23)
The phase boundaries corresponding to these phases can be analytically computed using Eqs. 18, 21 and 23, provided γ 0 and t 1 (which determines Z ′ ) are obtained from numerical solutions of Eqs. 20 and Eq. 8. For the MI phase to occur, we must have E 2 , E 3 , E 4 ≥ E 1 = 0 which yields the conditions (24) Note that the condition U ′ /(4|Z|d) ≤ 1 which is necessary for the realization of the SS phase has to be simultaneously satisfied with the condition Z ≤ 0 to make sure that the SS phase is actually a competing candidate to the MI state. The MI phase can indeed be realized in the parameter regime Z ≥ 0 provided U ′ > 4|Z|d. The first condition (Z + Z ′ ) ≤ U ′ /2d shows that the MI phase is favored over the metal+SF phase for large U/d and predicts a linear phase boundary in the U ′ − Z plane U ′ = 2d(Z + Z ′ ) with a slope of 2d and intercept of 2dZ ′ between these two phases. Note that the MI phase always wins over the metal+SF phase if the nearest-neighbor interactions between the Bosons and Fermions are large compared to their hopping amplitudes making Z + Z ′ negative. The final condition 4Z ′ d U ′ sin 2 (γ 0 ) ≤ 1 indicates that the phase boundary between the MI and MI+CDW phases is independent of Z. The former phase is favored over the latter for larger U and smaller Z ′ .
1 − 2d U ′ (Z + Z ′ ) ≥ 0, Z ≤ 0 and U ′ 4|Z|d ≤ 1, 4Z ′ d U ′ sin 2 (γ 0 ) ≤ 1
Similarly for the SS phase to occur one needs
U ′ /(4|Z|d) < 1 and E 2 ≤ E 1 , E 3 , E 4 , which yields U ′ Zd cos 2 (γ 0 ) 4Z ′ d U ′ sin 2 (γ 0 ) − 1 ≤ U ′ 4Zd − 1 2 Z ≥ 0, 4Z ′ d U ′ sin 2 (γ 0 ) ≤ 1, U ′ 4 √ ZZ ′ d ≥ 1(25)
We note that the SS phase is favored when the nearestneighbor interaction between the Bosons are weak compared to their hopping amplitudes making Z positive and when U ′ is small enough so that U ′ /(4|Z|d) < 1. Also, from the conditions in Eq. 25 (obtained using E 2 ≤ E 4 , E 1 ), we note that for a given Z ′ and d, the boundary between the metal+SF and the SS phases is a parabola in the U ′ − Z plane given by U ′ 2 = 16ZZ ′ d 2 while that between the SS and the MI state is a line given by U ′ = 4Zd.
Finally, the condition for occurrence of the metal+SF phase is given by E 4 ≤ E 1 , E 2 , E 3 and is given by
1 − 2d U ′ (Z + Z ′ ) ≤ 0, U ′ 4 √ ZZ ′ d ≥ 1 and U ′ 4|Z|d ≤ 1, 1 − 2d U ′ (Z + Z ′ ) ≤ 2 cos 2 (γ 0 ) 1 − 4Z ′ d U ′ sin 2 (γ 0 )(26)
The last condition in Eq. 26 determines the phase boundary between the metal+SF and the MI+CDW phases which depends on value of γ 0 . However, numerically, we find that for U ≃ 0 γ 0 ≃ π/4, and in this regime, the phase boundary between these phases occurs at Z ≃ 0 for all Z ′ and d. Note that strictly at U = 0, the Fermionic state is metallic; however a CDW gap opens up in the Fermionic spectrum for an infinitesimal finite U ′ . To verify the above-mentioned qualitative arguments and to find a precise phase diagram for the system, we numerically solve Eqs. 12 ,14, and 15, for d = 2 and for representative values V F /t F = 0, 0.5. We plot the ground state phase diagram as a function of Z and U ′ in Figs. 1 and 2. We find that the numerical results agree well with the qualitative arguments. Figs. 1 and 2 indicate that the phase boundary between the CDW+MI and MI phases is independent of Z as noted earlier. The linear and the parabolic nature of the phase boundaries between the MI and SS phases and the SS and metal+SF phases respectively can also be easily verified from the Figs. and are in accordance with the qualitative discussion. We note that one of the effects of nearest-neighbor repulsion between the Fermions is to enhance the SS phase which occupies a larger region of phase space in Fig. 2 (V F /t F = 0.5) than in Fig. 1 (V F = 0). Such an interaction, for Z ≤ 0, also favors the Mott phase over the CDW+MI phase as can also be seen from Figs. 1 and 2.
To determine the nature of transition between the different phases, we plot the ground state values θ as a function of Z for U ′ = 10 and V F /t F = 0.5 in Fig. 3. Such a plot clearly shows that the transition between the metal+SF and the SS phases is, within the mean-field theory considered here, first order and is accompanied by a jump in the value of θ. In contrast, the SS-MI transition turns out to be continuous. A similar plot of ground state values γ as a function of U ′ for Z/t F = −2 and V F = 0 , shown in Fig. 4, indicates that the transition between the CDW+MI and the MI phase is also discontinuous and is accompanied by a jump in the ground state value of γ.
Next, we compare our phase diagram with that obtained from DMFT in Ref. 15. This can be done in the regime of large positive Z (which correspond to V B /t B → 0) which was the case treated in Ref. 15. We find that the two phase diagrams qualitatively agree in the sense that both yield SS and MI phases in these limit. The difference lies in the fact that our mean-field predicts a second-order transition between the two phases whereas FIG. 2: Same as in Fig. 1 but for VF /tF = 0.5. The interaction between the Fermions favors the SS phase as can be seen by comparing Fig. 1 and 2. DMFT yields a narrow region of coexistence. This is presumably an effect of quantum fluctuation which is not captured within the mean-field theory. In addition, we also find a region of metal+SF phase at low U which was not seen in Ref. 15.
Finally, we would like to point out that the slave-boson mean-field phase diagram obtained above yields qualitatively correct phase diagram, but not a quantitatively correct one. This can be most clearly seen by noting that our Hamiltonian reduces to an effective Falicov-Kimball (FK) model 17 in the limit Z = t B = V B = V F = 0. This is most easily seen by writing our starting Hamiltonian
H (Eq. 1) for t B = V B == V F = 0 H F K = −t F ij c † i c j + U i c † i c i n B i(27)
At half-filling the Bosons are localized in the B sublattice so that n B i = (1 − (−1) i )/2. Thus the Fermions have a CDW instability even for an infinitesimal U due to nesting for half-filling on a square lattice. Consequently, the FK model at half-filling is insulating for any infinitesimal U , as known from several earlier studies 18 . Such a CDW instability, which can be easily captured by weakcoupling mean-field theory, is not straightforward to obtain in our strong coupling slave-boson mean-field approach which predicts a finite critical U for the transition from metal+SF to the MI phase. Note however that the slave-boson mean-field theory does predict a CDW+MI state for weak U , but for small negative Z, as can be seen from Fig. 1. This indicates that the phase diagram obtained has qualitatively, but not quantitatively, correct features.
III. COLLECTIVE MODES
The phases of the Bose-Fermi mixture discussed in the previous section allows for two types of excitations. The first type is the low-energy gapless collective modes that are present in the metal+SF, CDW+MI, and the SS phases of the system. These are the gapless Goldstone modes corresponding to the Boson and the slave-Boson fields. For all of these modes, the pseudo-Fermion sector remain gapped and do not contribute to their dispersion.
The gapped modes corresponds to particle-hole excitations in the CDW+MI, SS and the Mott states. In this section, we concentrate on the gapless collective modes in the CDW+MI, SS, and the metal+SF phases.
To obtain the dispersion, we first consider a timedependent variational wave-function
|Ψ d (t) 0 = cos(θ i )|n B i = 0 + sin(θ i )e −iχi |n B i = 1 ⊗ cos(γ i )|n F i = 0 + sin(γ i )e −iφi |n F i = 1(28)
where θ i , χ i , γ i , and φ i are space-time dependent fields. Note that in the static limit, the ground-state of the system corresponds to
θ i = θ i0 = θ 0 (π/2 − θ 0 ), γ i = γ i0 = γ 0 (π/2 − γ 0 ) for i ∈ A(B) sites, and χ i = φ i = 0, so that |Ψ d (t) reduces to |Ψ . The Lagrangian L = i Ψ d (t)|i∂ t − H ′ + µ B i n B i + µ F i n F
I |Ψ d can now be computed using the variational wave-function and one obtains
L = i ∂ t χ i sin 2 (θ i ) + ∂ t φ i sin 2 (γ i ) −U sin 2 (θ i ) sin 2 (γ i ) + ij 1 4 (t B sin(2θ i ) sin(2θ j ) cos(χ i − χ j ) +t 1 sin(2γ i ) sin(2γ j ) cos(φ i − φ j ) −{V F sin 2 (γ i ) sin 2 (γ j ) + V B sin 2 (θ i ) sin 2 (θ j )} + i ∆ − sin 2 (γ i )(−1) i + n F i +µ B sin 2 (θ i ) + µ F sin 2 (γ i )(29)
where n F i = F S|(−1) i d † i d i |F S /N is the Fermion number density different on A and B sublattices and all time dependence of the fields are kept implicit for the sake of clarity. To obtain the collective modes, we now write θ i (t) = θ i0 + δθ i (t), γ i (t) = γ i0 + δγ i (t), and expand the Lagrangian to quadratic order in δθ i (t), δγ i (t), φ i (t) and χ i (t). Then a variation of this Lagrangian with respect to δθ i (t), δγ i (t), φ i (t) and χ i (t) and consequent adjustment of values of the parameters µ B and µ F following Ref. 19, yields the equations for the low-energy collective modes (we set = 1 from now on)
∂ t δγ k + t 1 d sin(2γ 0 ) (1 − c(k)) φ k = 0 (30) ∂ t φ k − α 1 (k)δγ k − U sin(2θ 0 )δθ k = 0 (31) ∂ t δθ k + t B d sin(2θ 0 ) (1 − c(k)) χ k = 0 (32) ∂ t χ k − α 2 (k)δθ k − U sin(2γ 0 )δγ k = 0(33)
where we have taken the Fourier transform of all the fields, c(k) = j=1,d cos(k j )/d, and α 1 and α 2 are given by
α 1 (k) = 4t 1 d sin(2γ 0 ) 1 + c(k) V F t 1 sin 2 (2γ 0 ) + cos 2 (2γ 0 ) α 2 (k) = 4t B d sin(2θ 0 ) 1 + c(k) V B t B sin 2 (2θ 0 ) + cos 2 (2θ 0 )(34)
It is important to note that Eqs. 30 and 31 holds when γ 0 = 0, π/2 while Eqs. 32 and 33 holds when θ 0 = 0, π/2. Thus none of these equations are valid in the MI phase which do not support any low-energy collective modes. The gapped modes of the MI phases will be obtained in the next section.
In the SS phase where cos(2θ 0 ) = U ′ /(4Zd) = 0, 1 and γ 0 = π/2, the collective mode corresponds to the low-energy excitations of the Bosons and are given by Eqs. 32 and 33. A simple set of standard manipulations of these equations yield the dispersion of the collective modes ω 2 = 2v 2
1 (k)(1 − c(k)), where v 2 1 (k) = t B d sin(2θ 0 )α 2 (k)/2(35)
Note that for low momentum, we get a gapless linearly dispersing collective mode with velocity v ss = v 1 (k = 0). Similarly for the MI+CDW phase, where θ 0 = 0 and γ = γ 0 , the collective mode corresponds to the low energy excitations of the pseudo-bosons and can be obtained by solving Eqs. 30 and 31. Since the pseudo-Fermion sector is always gapped in this phase (∆ = 0), the collective mode here corresponds to the density-wave mode of the real Fermions. These modes have linear dispersion
ω 2 = 2v 2 2 (k)(1 − c(k)) where v 2 2 (k) = t 1 d sin(2γ 0 )α 1 (k)/2(36)
Thus for low momenta, we again get a gapless linearly dispersing collective mode with velocity v CDW = v 2 (k = 0). Finally for the metal+SS phase, all the Eqs. 30..33 hold and they need to be solved simultaneously. In this phase since γ 0 = θ 0 = π/4, we find that α 1 (k) =
4t 1 d[1 + c(k)V F /t 1 ] and α 2 (k) = 4t B d[1 + c(k)V B /t B ].
Solving these equations, one finds two collective modes with linear dispersions ω 2
± = 2v 2 ± (k(1−c(k)) where v ± (k) are given by v 2 ± (k) = 1 4 (α 1 (k)t 1 d + α 2 (k)t B d) ± (α 1 (k)t 1 d − α 2 (k)t B d) 2 + 16(U t B t 1 d) 2(37)
These collective modes result from the hybridization of the Bogoliubov modes of the Bosons and the densitywave modes for the metallic Fermions. This fact can be easily checked by putting U = 0 in Eq. 37 by which one can retrieve these modes with velocities v 2 B (k) = α 2 (k)t B d/2 for the Bosons and v 2 F (k) = α 1 (k)t 1 d/2 for the Fermions. As U ′ increases, the hybridization between these modes become stronger until the velocity v − (k = π) touches zero at U ′ = 4 |Z ′ Z|d which is precisely the condition for the metal+SF phase to become unstable to the SS phase.
IV. GAPPED MODES IN THE MI PHASE
The MI phase, in contrast to the other three phases of the system, do not support a gapless mode. The lowestlying excitations of such a state with conserved number density are particle-hole excitations. Such excitations, are of two types. The first type, shown in second panel of Fig. 5, involves particle and hole excitations that spatially well-separated while the second type, shown in second panel of Fig. 6, involves particle and hole excitations in nearest-neighbor sites which forms a dipole. In what follows, we first compute the energies of both these excitations using perturbation theory up to second order in t B/F /V B/F which are supposed to small in the MI phase.
Such an energy estimate can be easily carried out by strong-coupling perturbation theory developed in Ref. 20 in context single species Bose-Hubbard model. The generalization is largely trivial, except for one important detail. In the standard Bose-Hubbard model studied in Ref. 20, any particle/hole excitation could have lowering of energy via nearest-neighbor hopping which is O(t/U ) process. In contrast, as can be seen from Figs. 5 and 6, it is not possible for the particle-hole or dipole excitations to directly hop to the next site since such a direct hop always take us out the low energy manifold of states in the Mott limit. In particular, we note that any kinetic energy gain of the particle-hole or dipole excitation must occur via hopping of the partice/hole to the second-neighbor sites and hence necessarily leads to O(t 2 /V 2 ) energy gain.
We first compute the excitation energy of the Bosonic(Fermionic) particle-hole pair when they are far apart. The on-site energy of creating such a pair is E B/F on−site = 4dV B/F + U while the energy-lowering due to hopping of each of the particle and the hole is given by E
B/F hopping = −2d(2d−1)t 2 B/F /[2(2d−2)V B/F +U ].
The energy of the Mott state to second order in perturbation
theory is E MI = −d(t 2 B /[2(2d − 1)V B + U ] + t 2 F /[2(2d − 1)V F + U ])
so that the excitation energy of the particlehole pair is
E B/F p−h = 4dV B/F + U − 4d(2d − 1)t 2 B/F 2(2d − 2)V B/F + U + dt 2 B 2(2d − 1)V B + U + dt 2 F 2(2d − 1)V F + U(38)
We note that in the limit of large d, where the mean-field results are expected to be accurate, we have
E B/F p−h ≃ 4dV B/F + U − 8d 2 t 2 B/F 4dV B/F + U(39)
The Mott state is destabilized in favor of the SS phase when E B/F p−h = 0. Next, we compute the excitation energy of the Bosonic/Fermionic dipole state. We are going to do this in the limit of V F = V B = V . We note at the outset that once such a dipolar excitation is created, it remains stable, i.e., the hole can not hop away arbitrarily far away from the particle. It can be easily verified from Fig. 6 that such hoppings generate higher-energy end states and takes one out of the low energy manifold of states.
The on-site energy cost for creating such an excitation is E on−site = 2(2d − 1)V + U while the hopping process, shown schematically in Fig. 6, necessarily involves hopping of both Fermions and Bosons and leads to an energy gain of E d hopping = t B t F /2(2d− 1)V . Thus the net energy of such a dipole excitation is given by
E dipole = 2(2d − 1)V + U − t B t F 2(2d − 1)V + dt 2 B 2(2d − 1)V + U + dt 2 F 2(2d − 1)V + U(40)
We note that for d ≫ 1, where our mean-field theory holds, it is always energetically favorable to create particles and holes well-separated since E B/F hopping ≪ E d hopping . However, the dipolar excitations may becomes favorable in low dimension and for large U/V . In this case, hopping ≫ E d hopping for U ≫ V and in this limit, the dipolar excitations would be preferred in destabilizing the MI phase. We shall not discuss this issue here any further since this is clearly beyond the scope of our mean-field theory.
Finally, we compute the dispersion of the gapped particle-hole excitations within mean-field theory where the particle and the hole are spatially well-separated and do not interact. To this end, we temporarily relax the constraint of conservation of particle-number and consider the energy of excitations of adding a particle E p and a hole E h to the Mott state. The physical particlehole excitation energy can then be computed from E ph = E p + E h . To compute the energy of these particel/hole excitations, we adapt a time-dependent variational approach as done in Ref. 19 for single species Bosons in an optical lattice. We begin with the variational wavefunction of Bosons and slave Bosons(Fermions) given by
|ψ(t) = |ψ B (t) × |ψ F (t) |ψ B (t) = f α 0 (t)|n B = 0 + f α 1 (t)|n B = 1 |ψ F (t) = g α 0 (t)|n a = 0 + g α 1 (t)|n a = 1(41)
where α = A, B denotes sublattice indices. The coefficients f and g satisfy the normalization condition |f 0 | 2 + |f 1 | 2 = 1 and |g 0 | 2 + |g 1 | 2 = 1. We note that at equilibrium f A 0 = sin θ = 0, f A 1 = cos θ = 1, g A 0 = cos γ = 0, g A 1 = sin γ = 1, f B 0(1) = f A 1(0) , and g B 0(1) = g A 1(0) for the MI phase. The Lagrangian of the Bose-Fermi mixture can then be written as
L ′ = j i f * 0jḟ 0j + f * 1jḟ 1j + g * 0jġ 0j + g * 1jġ 1j − ij − t B f * 1i f 0i f * 0j f 1j − t 1 g * 1i g 0i g * 0j g 1j +V B |f 1i | 2 |f 1j | 2 + V F |g 1i | 2 |g 1j | 2 −U i |f 1i | 2 |g 1i | 2 + µ b i |f 1i | 2 +µ f i |g 1i | 2 − i λ i (|f 0i | 2 + |f 1i | 2 − 1) − i ν i (|g i0 | 2 + |g 1i | 2 − 1)(42)
where λ i and ν i are variational parameters used for implementing the constraint whose values are to be determined from proper choice of the saddle point which in the MI phase yields λ A = µ b and λ B = 0 for the Bosons and ν A = 0 and ν B = µ f for the slave bosons. The saddle-point equations for the variational coefficients f i (t) and g i (t) then reads
iḟ 0i + t B f 1i j f * 1j f 0j − λ i f 0i = 0 iġ 0i + t 1 g 1i j g * 1j g 0j − ν i g 0i = 0 iḟ 1i − − t B f 0i j f * 0j f 1j + 2V B f 1i j |f 1j | 2 +U f 1i |g 1i | 2 − µ b f 1i − λ i f 1i = 0 iġ 1i − − t 1 f 0i j g * 0j g 1j + 2V F g 1i j |g 1j | 2 +U g 1i |f 1i | 2 − µ f g 1i − ν i g 1i = 0(43)
Next we implement the two sublattice structure, shift to momentum and frequency space, and expand f
A/B ak = δf A/B ak + f A/B a and g A/B ak = δg A/B ak + g A/B a
where a = 0, 1. Note that since δf A/B and δg A/B corresponds to deviation of particle number of the MI state, these dispersion corresponding to their eigenmodes must represent the particle and hole excitations over the MI phase. In the MI phase, we find that the equation of motions for the Bosons and the pseudobosons decouple at linear order δf A/B ak and δg A/B ak . For the Bosons, we obtain, to linear order in δf
A/B ak − ωδf A 0k = −2dt B c(k)δf * B 1k + λ A δf A 0k −ωδf B 1k = −2t B dc(k)δf A 0k + 2V B zδf B 1k +U δf B 1k − (µ b − λ B )δf B 1k(44)
which yields two physical excitation dispersion corre-sponding to particle and hole excitations
E p(h) = +(−)(2dV B + U 2 − µ b ) + (2dV B + U 2 ) 2 − (2t B dc(k)) 2(45)
The energy of a particle-hole excitation which conserves particle number is therefore obtained by adding E p and E h and is given by
E p−h = 2 (2dV B + U 2 ) 2 − (2t B dc(k)) 2(46)
Note that E p−h vanishes along the line Z = U ′ /4d which agrees with the mean-field result for the SS-MI phase boundary. Also, expanding Eq. 46 to O(t 2 B ) for k = 0 leads exactly to Eq. 39 which shows that the secondorder perturbation theory discussed earlier agrees to the present calculation in the high d limit. Further, at small wave-vector, we find E p−h ∼ |k| which shows that the SS-MI quantum phase transition has a dynamical critical exponent z = 1. Similar dispersion can be obtained for the pseudo-bosons by considering collective modes corresponding to δg ak . These modes have the same dispersion as Eqs. 45 and 46 with V B and t B replaced by V F and t 1 respectively.
V. DISCUSSION
Experimental realization of Bose-Fermi mixtures have long been achieved in ultracold atomic systems. These mixtures can be easily tuned to a regime where the onsite intra-species interaction between both the Fermions and the Bosons are large so that they effectively behave as hard-core particles. Thus experimental realization of a Bose-Fermi mixture with V B = V F = 0 is relatively straightforward. However, most such mixtures do not have sufficiently large nearest-neighbor repulsion and thus it might be difficult to realize mixtures which has large V F or V B . Some progress in this direction has recently been made in Ref. 21. Also, use of spin-polarized 52 Cr atoms for the Fermionic part of the mixture may help since these atoms have significant dipole moment which may provide the requisite interaction.
Once such a Bose-Fermi system is realized, several predictions of the present work can be verified by realizable experiments that are commonly used for ultracold systems. First, we note that since the Bosons are spinless and the Fermions are spin-polarized, the Bosonic and the Fermionic part of the mixture can be separated by applying a standard Stern-Gerlach field during a standard time-of-flight experiment as done earlier in Ref. 22 in the context of spinor Bosons in optical traps. Such a procedure allows us to separately study the momentum distribution functions of the Bosons and the Fermions using time-of-flight experiments 1 . For the Bosonic cloud, the distinction between the SF and the MI phases can be easily done by measuring the presence or absence of coherence peaks in its momentum distribution as measured in a standard time-of-flight experiment. The precise nature of the broken translational symmetry in the MI and the SS phases for the Bosons can also be determined by studying noise-correlations of the expanding clouds as already proposed in Ref. 23. Thus, the MI, SS and the SF phases for the Bosons can be qualitatively distinguished by these experiments. As for the Fermions, the presence/absence of a Fermi surface for the Fermions in a trap can be easily distinguished in time-of-flight measurements as performed for ultracold Fermions in Ref. 24. Thus, these experiments should allow one to qualitatively distinguish between all four predicted phases. One of the central predictions of our theory is that for any finite U , the metallic state of the Fermions shall always be accompanied by a SF phase of the Bosons. In terms of time-of-flight experiments this means that any measurement on Fermions which sees a gapless Fermi surface shall always be accompanied by corresponding coherence peak (and no density wave ordering) for the Bosons. The collective modes computed in this work can also be verified experimentally using standard inelastic light scattering experiments 25 . Such experiments can detect lowenergy collective modes and should thus detect either two ( metal+SF phase) or one (SS or MI+CDW phases) linearly dispersing collective mode(s). The MI phase will be characterized by absence of any low energy collective modes of the system.
There are several possible extension of our analysis.
The first and the simplest extension would be to study the phases of the Bose-Fermi mixture away from halffilling. This would require a more careful handling of the chemical potential µ B and µ F of the Bosons and the Fermions. In particular one would need to determine t 1 in a self-consistent manner as a function of µ F . Second, it would be interesting to look at the phase diagram by relaxing the hard-core constraint on the Bosons by putting a finite on-site repulsion between them. Of particular interest in this respect is to check if the slave-boson meanfield can provide any indication of the phase separation found in such a system in Refs. 14,15. Finally, it would be useful to study the phase diagram of mixture of spinpolarized Fermions with spin-one and spin-two Bosons with nearest-neighbor interactions. Such system are expected to have a much richer phase diagram and have not been theoretically studied so far.
To conclude, in this work, we have carried out a slaveboson mean-field analysis of a mixture of hardcore spinless Bosons and spin-polarized Fermions in an optical lattice. Our analysis provides the mean-field phase diagram of the system and shows the presence of four distinct phases. We have also computed the low-energy collective modes of three of these phases (metal+SF, CDW+MI and SS) and studied the gapped particle-hole excitation of the fourth (MI). We have discussed experiments which can be used to test our theory and possible extension of our theory to other systems.
We thank Jim Freericks for drawing our attention to the Falicov-Kimball limit of the present model and for several useful discussions.
FIG. 1 :
1Ground state phase diagram as a function of Z and U ′ for noninteracting Fermions (VF = 0). The phase boundaries coincides with the analytical mean-field phase boundaries (see text for details).
FIG. 3 :
3(Color online) Variation of ground state value of θ with Z for U ′ = 10 and VF /tF = 0.5. The discontinuity in θ at Z = 3.38 indicates a first order transition between the metal+SF and the SS phases. The transition between the SS and the MI phases is continuous.FIG. 4: (Color online) Variation of the ground state value of γ with U ′ for Z = −2. and VF = 0. The discontinuity in γ at U ′ = 2 indicates a first order transition between the CDW+MI and the MI phases.
FIG. 5 :
5(Color online) Cartoon representation of the MI state. A) The Mott state at half-filling. The red filled circles represent Bosons and the empty blue circles indicate Fermions. B) A particle and a hole excitation which are not nearest neighbors. C) An intermediate virtual high energy state which assists hopping of holes. D) A state where the hole has hopped to the next-neighbor site. This state is identical in energy to the state B.
FIG. 6 :
6(Color online) Cartoon representation of the MI state and the associated dipole excitations. All symbols are same as in Fig. 5. A) The Mott state at half-filling. B) A dipole excitation over the Mott state C) An intermediate virtual high energy state which assists hopping of dipoles. D) A state where the dipole has hopped to an adjacent link. This state is identical in energy to the state B when VF = VB.
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| []
|
[
"Greenberger-Horne-Zeilinger theorem for N qudits",
"Greenberger-Horne-Zeilinger theorem for N qudits",
"Greenberger-Horne-Zeilinger theorem for N qudits",
"Greenberger-Horne-Zeilinger theorem for N qudits"
]
| [
"Junghee Ryu \nInstitute of Theoretical Physics and Astrophysics\nUniversity of Gdańsk\n80-952GdańskPoland\n\nDepartment of Physics\nHanyang University\n133-791SeoulKorea\n",
"Changhyoup Lee \nDepartment of Physics\nHanyang University\n133-791SeoulKorea\n\nCentre for Quantum Technologies\nNational University of Singapore\n3 Science Drive 2117543Singapore\n",
"Marekżukowski \nInstitute of Theoretical Physics and Astrophysics\nUniversity of Gdańsk\n80-952GdańskPoland\n",
"Jinhyoung Lee \nDepartment of Physics\nHanyang University\n133-791SeoulKorea\n\nCenter for Macroscopic Quantum Control\nSeoul National University\n151-742SeoulKorea\n",
"Junghee Ryu \nInstitute of Theoretical Physics and Astrophysics\nUniversity of Gdańsk\n80-952GdańskPoland\n\nDepartment of Physics\nHanyang University\n133-791SeoulKorea\n",
"Changhyoup Lee \nDepartment of Physics\nHanyang University\n133-791SeoulKorea\n\nCentre for Quantum Technologies\nNational University of Singapore\n3 Science Drive 2117543Singapore\n",
"Marekżukowski \nInstitute of Theoretical Physics and Astrophysics\nUniversity of Gdańsk\n80-952GdańskPoland\n",
"Jinhyoung Lee \nDepartment of Physics\nHanyang University\n133-791SeoulKorea\n\nCenter for Macroscopic Quantum Control\nSeoul National University\n151-742SeoulKorea\n"
]
| [
"Institute of Theoretical Physics and Astrophysics\nUniversity of Gdańsk\n80-952GdańskPoland",
"Department of Physics\nHanyang University\n133-791SeoulKorea",
"Department of Physics\nHanyang University\n133-791SeoulKorea",
"Centre for Quantum Technologies\nNational University of Singapore\n3 Science Drive 2117543Singapore",
"Institute of Theoretical Physics and Astrophysics\nUniversity of Gdańsk\n80-952GdańskPoland",
"Department of Physics\nHanyang University\n133-791SeoulKorea",
"Center for Macroscopic Quantum Control\nSeoul National University\n151-742SeoulKorea",
"Institute of Theoretical Physics and Astrophysics\nUniversity of Gdańsk\n80-952GdańskPoland",
"Department of Physics\nHanyang University\n133-791SeoulKorea",
"Department of Physics\nHanyang University\n133-791SeoulKorea",
"Centre for Quantum Technologies\nNational University of Singapore\n3 Science Drive 2117543Singapore",
"Institute of Theoretical Physics and Astrophysics\nUniversity of Gdańsk\n80-952GdańskPoland",
"Department of Physics\nHanyang University\n133-791SeoulKorea",
"Center for Macroscopic Quantum Control\nSeoul National University\n151-742SeoulKorea"
]
| []
| We generalize Greenberger-Horne-Zeilinger (GHZ) theorem to an arbitrary number of Ddimensional systems. Contrary to conventional approaches using compatible composite observables, we employ incompatible and concurrent observables, whose common eigenstate is still a generalized GHZ state. It is these concurrent observables which enable to prove a genuinely N -partite and D-dimensional GHZ theorem. Our principal idea is illustrated for a four-partite system with D which is an arbitrary multiple of 3. By extending to N qudits, we show that GHZ theorem holds as long as N is not divisible by all nonunit divisors of D, smaller than N . | 10.1103/physreva.88.042101 | [
"https://arxiv.org/pdf/1303.5326v3.pdf"
]
| 118,501,635 | 1303.5326 | c268168cdc51babe2aec5bf2d642badbc5ed8f0a |
Greenberger-Horne-Zeilinger theorem for N qudits
7 Oct 2013
Junghee Ryu
Institute of Theoretical Physics and Astrophysics
University of Gdańsk
80-952GdańskPoland
Department of Physics
Hanyang University
133-791SeoulKorea
Changhyoup Lee
Department of Physics
Hanyang University
133-791SeoulKorea
Centre for Quantum Technologies
National University of Singapore
3 Science Drive 2117543Singapore
Marekżukowski
Institute of Theoretical Physics and Astrophysics
University of Gdańsk
80-952GdańskPoland
Jinhyoung Lee
Department of Physics
Hanyang University
133-791SeoulKorea
Center for Macroscopic Quantum Control
Seoul National University
151-742SeoulKorea
Greenberger-Horne-Zeilinger theorem for N qudits
7 Oct 2013
We generalize Greenberger-Horne-Zeilinger (GHZ) theorem to an arbitrary number of Ddimensional systems. Contrary to conventional approaches using compatible composite observables, we employ incompatible and concurrent observables, whose common eigenstate is still a generalized GHZ state. It is these concurrent observables which enable to prove a genuinely N -partite and D-dimensional GHZ theorem. Our principal idea is illustrated for a four-partite system with D which is an arbitrary multiple of 3. By extending to N qudits, we show that GHZ theorem holds as long as N is not divisible by all nonunit divisors of D, smaller than N .
I. INTRODUCTION
The inconsistency of (local) hidden variable theories with quantum mechanics fascinates many researchers. It has been discussed in many theoretical [1] and experimental works [2]. Bell's theorem, one of the most profound discoveries concerning the foundations of quantum mechanics, states that any local realistic theory is incompatible with quantitative predictions of quantum mechanics. Even though Bell's theorem was studied mostly in terms of statistical inequalities, a more striking conflict, without inequalities, was also shown for a multiqubit system by Greenberger, Horne, and Zeilinger (GHZ) [3]. They derived an all-versus-nothing contradiction based on perfect correlations for so-called GHZ states. This leads to a direct refutation of Einstein-Podolsky-Rosen (EPR) ideas on the relation between locality and elements of reality with quantum mechanics [4]. This is a striking blow right into the very basic ideas linked with local hidden variables. After all, EPR used the concept of (local) elements of reality to support their claim that quantum mechanics is incomplete. All this can be best explained using the three particle GHZ paradox. Take a state |GHZ = 1 √ 2 (|+++ −|−−− ), where |± denotes states associated with the eigenvalues ±1 of the local Pauli σ z operator. The operators σ x ⊗ σ x ⊗ σ x , σ x ⊗ σ y ⊗ σ y , σ y ⊗ σ x ⊗ σ y , and σ y ⊗ σ y ⊗ σ x all commute, and their eigenstate is |GHZ . The eigenvalues are −1, 1, 1, and 1, respectively, which signify perfect (GHZ)-EPR correlations. This would please any local realist. Assume that the particles are far away from each other, and three distant independent observers can perform experiments on them, choosing at will the observables. For example, the first and the second one may choose σ x and their measurement results are 1 and −1, respectively. In such a case, they can together predict with certainty what would have been the result of the third observer had he or she chosen to measure also σ x . Simply the local results must multiply to the eigenvalue of the joint observable σ x ⊗ σ x ⊗ σ x , and this is −1. Thus, the third observer, if the hypothetical case of him or her choosing to measure σ x really happens, must for sure get 1. Thus, as EPR would say, this value is an element of reality, because in no way the distant choices, and obtained results can influence anything that happens at the location of the third observer (especially if measurements actions, are spatially separated events in the relativistic sense of this term, and the measurement choices are made some time after the particles are emitted from a common, say central, source). For such counterfactual reasonings one can use any of the four perfect correlation cases for the joint measurements given above, and apply to each observer. Thus, it seems that one can ascribe elements of reality of to all local situations, no matter whether the local observable is σ x or σ y . Note that these are incommensurable. Let us denote such elements of reality, related with a single emission act of three particles, by r k w , where w = x, y denotes the observable, and k = 1, 2, 3 denotes the observer. Obviously r k w = ±1. For the four cases of GHZ-EPR perfect correlations one therefore must have r 1 x r 2
x r 3 x = −1, and r 1 x r 2 y r 3 y = 1, r 1 y r 2 x r 3 y = 1, r 1 y r 2 y r 3 x = 1. If one multiplies these four relations side by side, one gets 1 = −1. Thus an attempt of introducing EPR elements of reality leads to a nonsense. Ergo, elements of reality are a nonsense. No other argument against local realism could be more striking.
Extending Bell's theorem to more complex systems such as multipartite and/or high-dimensional systems [5] is important not only for a deeper understanding of foundations of quantum mechanics. It is associated with developing new applications in quantum information processing, such as quantum cryptography, secret sharing, quantum teleportation, reduction of communication complexity, quantum key distribution, and random numbers generation [6][7][8][9][10]. Similarly to Bell's theorem, also all-versus-nothing tests, which we call GHZ theorem, have been generalized to higher dimensional systems. For the sake of convenience, we shall use the tuple (N, M, D) to denote N parties, M measurements for each party, and D distinct outcomes for each measurement. In Ref. [11], the GHZ theorem was derived for a (D + 1, 2, D) problem. A probabilistic but conclusive GHZ-like test was shown for (D, 2, D) in Ref. [12]. The (N, 2, D) problem for odd N > D and even D was studied by Cerf et al. [13]. Lee et al. showed the GHZ theorem for more general cases, (odd N, 2, even D), by an unconventional approach using incompatible observables [14]. Recently, Tang et al. generalized GHZ theorem to the N (≥ 4)-partite case and even-D dimensional systems with the help of GHZ graphs [15]. Despite such an intensive progress in extending GHZ theorem, many cases of N -partite and Ddimensional systems remain still as open problems.
We generalize the GHZ theorem to three or higher Ddimensional systems. To this end, we employ concurrent composite observables which, in contrast with the standard approach, are mutually incompatible but still have a common eigenstate, here a generalized GHZ state. They can be realized by multiport beam splitters and phase shifters, as it is shown in Refs. [11,14]. We first illustrate our principal idea with four 3d-dimensional systems and then provide a systematic method, so as to extend it to three or higher D-dimensional systems. Finally, we show a GHZ-type contradiction, as long as N is not divided by all nonunit divisors of D, smaller than N . Our generalization is genuinely N -partite and D-dimensional and can reproduce the previous results [3,11,13,14]. This approach can lead to a general GHZ theorem for N qudits.
II. CONCURRENT OBSERVABLES
Some sets of observables have a common eigenstate. If a system is prepared in the eigenstate, the measurement results for such observables are concurrently appearing with certainty. Such observables are called "concurrent" [14]. For a quantum system of dimension D(> 2), consider two Hermitian operators andB such that = a|ψ ψ| + ⊥ ψ andB = b|ψ ψ| +B ⊥ ψ witĥ A ⊥ ψ (B ⊥ ψ )|ψ = 0. The state |ψ is then a common eigenstate of both observables asÂ( [16]. Such concurrent observables can be constructed by the method introduced in Ref. [14]. Consider a unitary oper-atorÛ , which is of the form ofÛ = e iφ |ψ ψ| +Û ⊥ ψ witĥ
B)|ψ = a(b)|ψ , even if [Â,B] = [Â ⊥ ψ ,B ⊥ ψ ] = 0U ⊥ ψ |ψ = 0. HereÛ ⊥ ψ is a unitary operator on a space H ⊥ ψ which is defined by the requirement H = H ψ ⊕H ⊥ ψ ,
where H ψ is the one-dimensional space containing |ψ . Every such unitary operator leaves the state |ψ unchanged, up to a global phase: If the state |ψ satisfiesÂ|ψ = λ|ψ , then all transformed operatorsB U =ÛÂÛ † are concurrent withÂ.
Consider N qudits prepared in a generalized GHZ state |ψ = 1 √ D D−1 n=0 N k=1 |n k , where |n denotes a basis state for a qudit. This GHZ state is a common eigenstate with the unity eigenvalue of any composite ob-servableX ⊗N ≡X ⊗X ⊗ · · · ⊗X asX ⊗N |ψ = |ψ , where the local observableX is defined by applying quantum Fourier transformationF on a reference unitary observableẐ = D−1 n=0 ω n |n n| with ω = exp(2πi/D), that isX =FẐF † [17]. An eigenvector ofX associated with the eigenvalue ω n is given by |n
x =F |n = 1 √ D D−1 m=0 ω nm |m . With the standard basis set {|n }, the observableX is written asX = D−1 n=0 |n n + 1|, where |n ≡ |n mod D .
To construct a set of concurrent composite observables, we employ a unitary operation in the form
ofÛ = N k=1P k (f k ) with a phase shifterP k = D−1 n=0 ω f k (n) |n n|. If "phases" f k (n) satisfy a condition, N k=1 f k (n) ≡ 0 mod D,(1)
for each n, then the unitary operatorÛ leaves the GHZ state |ψ invariant. This simple invariance condition enables one to construct a large number of concurrent observables which have a common eigenstate of the generalized GHZ state. Let us apply the unitary operationÛ with the phases f k (n) = α k n with rational numbers α k toX ⊗N . If the phases f k (n) satisfy the invariance condition (1)
, the transformed observableÛX ⊗NÛ † =X(α 1 ) ⊗X(α 2 ) ⊗ · · ·⊗X(α N ) is concurrent withX ⊗N , i.e.,ÛX ⊗NÛ † |ψ = |ψ .
For each eigenvalue ω n , the eigenvector of the local observableX(α) is given by applying the phase shifterP on |n x as |n α =P |n
x = 1 √ D D−1 m=0 ω (n+α)m |m .
The observableX(α) can be written in the standard basis set {|n } aŝ
X(α) = ω −α D−2 n=0 |n n + 1| + ω αD |D − 1 0| . (2)
Note that if α is an integer, the measurement basis set {|n α } ofX(α) will be the same as {|n x } ofX except the ordering, i.e., |n α = |n + α x . Thus,X(α) = ω −αX . That is, the observableX(α) is equivalent toX, up to a phase factor ω −α . Let |n α be the eigenstate ofX(α) associated with eigenvalue ω n , and |m β the eigenstate ofX(β) associated with ω m . If and only if α differs from β by an integer, then two measurement bases sat-
isfy | α n|m β | 2 = δ D (γ), where γ = m − n + β − α.
Here δ D (γ) = 1 if γ is congruent to zero modulo D and otherwise δ D (γ) = 0. That is, if β − α is not an integer, two local observablesX(α) andX(β) are inequivalent.
III. GENERALIZED GHZ THEOREM
A. Four-qudit system
We first illustrate our idea by considering a four-qudit system. Already this case goes significantly beyond the previous studies [11,[13][14][15]. Take a four-qudit GHZ state |ψ = 1 √ D D−1 n=0 |n, n, n, n (here D is assumed to be an integral multiple of 3). The qudits are distributed to four sufficiently separated parties. Each party performs one of two nondegenerate local measurements on his or her qudit, each of which produces distinguishable D outcomes. We represent the measurement for the k-th party byM k , and the eigenvalues of the observables are of the form ω m k , where m k is an integer. One can denote a joint probability that each party obtains the result ω m k by P(m 1 , m 2 , m 3 , m 4 ), and define a correlation function E QM (M 1 , M 2 , M 3 , M 4 ) = ψ|M 1 ⊗M 2 ⊗M 3 ⊗M 4 |ψ , equivalently the quantum average of products of the measurement results:
D−1 m1=0 · · · D−1 m4=0 ω 4 i=1 mi P(m 1 , m 2 , m 3 , m 4 ).
When all measurements areX, that is, eachM i =X, sincê X ⊗X ⊗X ⊗X|ψ = |ψ , one has E QM (X, X, X, X) = 1. This implies that we have a perfect GHZ correlation. If arbitrary three parties know their own outcomes ω xi of measurementsX, then they can predict with certainty the remaining party's outcome. We will denote such a perfect correlation by
C QM (x 1 + x 2 + x 3 + x 4 ≡ 0).(3)
The sum is taken modulo D; such a convention is used below in all formulas. Let us construct concurrent composite observables from the observablev 0 =X ⊗4 , by applying a unitary operatorÛ 1 =P 1 ⊗P ⊗3 2 , with the phase shiftersP k of phases f 1 (n) = (D − 1)n and f 2 (n) = n/3. One of the new observables isv 1 ≡Û 1v0Û † 1 =X(D − 1) ⊗X(1/3) ⊗3 . The phases f k (n) satisfy the invariance condition (1), f 1 (n) + 3f 2 (n) ≡ 0 mod D. Thus, the observablev 1 has |ψ as its eigenstate with eigenvalue 1. By Eq. (2) one hasX(D − 1) = ωX(0), andŶ ≡X(1/3) is given bŷ
Y = ω −1/3 D−2 n=0 |n n + 1| + ω D/3 |D − 1 0| . (4)
The observableŶ =X(1/3) is not equivalent toX = X(0) [see the explanation below Eq. (2]. The other three concurrent observables are obtained byv l ≡Û lv0Û † l , l ∈ {2, 3, 4}, where the unitary operatorsÛ l are composed of the cyclic permutations of the local phase shifters:Û 2 = P 2 ⊗P 1 ⊗P 2 ⊗P 2 ,Û 3 =P 2 ⊗P 2 ⊗P 1 ⊗P 2 , andÛ 4 =P 2 ⊗ P 2 ⊗P 2 ⊗P 1 . The perfect correlations forv i (i = 1, . . . , 4) are, respectively, given by
C QM (x 1 + y 2 + y 3 + y 4 ≡ −1), C QM (y 1 + x 2 + y 3 + y 4 ≡ −1), C QM (y 1 + y 2 + x 3 + y 4 ≡ −1), C QM (y 1 + y 2 + y 3 + x 4 ≡ −1),(5)
where ω yi is an outcome of measurementŶ for the i-th party.
Local realistic theories assume that the outcomes of the measurements are predetermined, before the actual measurements. This implies that the values of local realistic predictions for the correlations (5), for each experimental run, must satisfy:
ω x1 ω y2 ω y3 ω y4 = ω −1 , ω y1 ω x2 ω y3 ω y4 = ω −1 , ω y1 ω y2 ω x3 ω y4 = ω −1 , ω y1 ω y2 ω y3 ω x4 = ω −1 .(6)
If local realism is also to reproduce quantum perfect correlations for the case ofv 0 =X ⊗4 , one must have for each run,
ω x1 ω x2 ω x3 ω x4 = 1.(7)
However, one sees that this is possible only provided ω −3 4 i=1 yi−4 = 1 if one multiplies side-by-side all equations (6). As ω = exp(2πi/D), if D = 3d where d is an integer, one can use the fact that an elementary algebra shows that there is no integer solution of the equation 3y + 4 ≡ 0 mod D. Thus local realistic correlation (7) is impossible.
It is worth noting that the approach with concurrent observables relaxes the restrictions of early studies requiring compatible observables. This enables one to generalize GHZ contradictions beyond the case N > D studied in Refs. [11,13].
Note, that to prove the four-partite GHZ contradiction, we chose the local dimension D and the number of the observablesŶ 's in the considered correlation functions, N 2 , such that the greatest common divisor (gcd) of D and N 2 , here gcd(D, N 2 ) = 3, does not divide the number of parties N = 4, or equivalently the number N 1 of the observablesX (here equal to 1). This mathematical property plays a central role in the generalization of the GHZ contradiction to an arbitrary number of parties.
B. Extending to N qudits system
We extend our approach into a general case of N qudits, N ≥ 3, such that N is nondivisible by any nonunit divisor of D, smaller than N . To this end, we use a set of (N + 1) concurrent observables given byv 0 = X ⊗N and N observables of the following forms:
v 1 = ωX ⊗N1 ⊗Ŷ ⊗N2 andv k =Ŷ ⊗k−1 ⊗ ωX ⊗N1 ⊗Ŷ ⊗N2−k+1 for k = 2, . . . , N 2 +1 and finallyv k =X ⊗k−N2−1 ⊗Ŷ ⊗N2 ⊗ ωX ⊗N −k+1 for k = N 2 + 2, . . . , N .
The composite observablev 1 is obtained by a unitary transformation,Û 1 =P 1 ⊗ 1 1 ⊗N1−1 ⊗P ⊗N2 2 , of the observablev 0 , i.e.,v 1 =Û 1v0Û † 1 , with the phase shiftersP 1 andP 2 of phases f 1 (n) = (D − 1)n and f 2 (n) = n/N 2 , respectively. The local observablesX =X(0) and Y =X(1/N 2 ) are given by Eq. (2). Likewise, we obtain the other concurrent observablesv l (2 ≤ l ≤ N ) by cyclic permutations in the unitary operatorÛ 1 , as it was done for the four-partite case. The phases satisfy the invariance condition (1) as f 1 (n) + N 2 f 2 (n) ≡ 0 mod D. Thus, the N -partite generalized GHZ state |ψ = 1 √ D D−1 n=0 N k=1 |n k is a common eigenstate of all the (N + 1) concurrent observablesv l (l = 0, . . . , N ), with the same eigenvalue 1. This leads to the following values of correlation functions (for later convenience we use party indices i = 1, . . . , N , which will later on allow us to get a more concise notation in formulas). If all local observables areX, that is, for globalv 0 , one has E QM (X, X, . . . , X) = 1. Thus we have a perfect correlation which can be denoted, in the way introduced earlier, as C QM ( N i=1 x i ≡ 0). Forv k , where k = 1, 2, . . . , N , one has perfect correlations of the following forms: for k = 1,
C QM ( N1 i=1 x i + N i=N1+1 y i ≡ −1), and for k = 2, . . . , N 2 + 1, C QM ( k−1 i=1 y i + N1+k−1 i=k x i + N i=N1+k y i ≡ −1)
, and finally for k = N 2 + 2, . . . , N ,
C QM ( k−N2−1 i=1 x i + k−1 i=k−N2 y i + N i=k x i ≡ −1)
. Following similar arguments as in the case of the fourpartite GHZ contradiction, we obtain the following condition for the local realistic correlation function for the composite observablev 0 =X ⊗N , to have value equal to the quantum prediction, that is, 1. It reads (modulo D)
N 1 N i=1 x i ≡ −N 2 N i=1 y i − N ≡ 0.(8)
However, if N 2 is an integral multiple of g but N cannot be divided by g, then there are no solutions of y = i y i to the equation N 2 y+N ≡ 0 mod D. The greatest common divisor of N 2 and D is an integral multiple of g, i.e., gcd(N 2 , D)=kg for some positive integer k but kg cannot divide N as N is not an integer multiple of g. Thus we have a contradiction with the quantum prediction.
In order to show a GHZ contradiction for N -partite and D-dimensional system, we choose that (a) D = dg, (b) N 2 = ηg, where d and η are positive integers, and (c) N cannot be divided by g. Choosing the integer g, a nonunit divisor D, plays a crucial role. For example, consider four six-dimensional systems. The nonunit divisors g, smaller than N = 4, are 2 and 3. If we choose g = 2, then we are unable to see any four-partite GHZ contradiction as the greatest common divisor (gcd) of N 2 and D, gcd(N 2 = 2, D = 6) = 2, divides N = 4. On the other hand, if we choose g = 3, the four-partite GHZ contradiction can be proved as gcd(N 2 = 3, D = 6) = 3 and g = 3 does not divide N = 4. This is a specific example of a GHZ contradiction for a (4, 2, 3d) problem. As a consequence, we conclude that one is always able to prove the GHZ contradiction for the N (≥ 3)-partite and D(≥ 2)-dimensional systems as long as N cannot be divided by all nonunit divisors of D.
For appropriate values of N and D, our approach reproduces the previous works [11,13,14]. A GHZ contradiction for (D + 1) qudits shown in Ref. [11] is reproduced by choosing N 1 = 1 and N 2 = D in our method, and noticing the fact that N 1 = 1 is indivisible by D = gcd(N 2 = D, D). The case of (odd N, 2, even D) studied in Refs. [13,14] can be also proved by choosing a nonunit divisor g = 2 of D and an arbitrary odd integer N 1 . One can also easily check that if N 2 = D = 2 and N 1 = 1, then our contradiction is reduced to the original GHZ theorem [3].
C. Genuinely N -partite D-dimensional case
The GHZ theorem for the two-dimensional systems seems to be fully understood. However, for more complex systems this is not so. Cerf et al. suggested a criterion for a genuinely N -partite (D-dimensional) GHZ contradiction [13]. It arises only for the given full N -partite (D-dimensional) system, but not for any n(< N )-partite subset, or for an effectively lower dimensionality of the involved observables. For example, the three-qubit classic GHZ theorem can be put as the theorem for three qutrits, and specific entangled GHZ states involving only two-dimensional subspaces for each qutrit.
The GHZ contradiction we show here is a genuinely N -partite one, as it is constructed using a set of composite observables composed of cyclic permutations. Let us explain this with the four-partite GHZ contradiction, where we used the five concurrent composite observables: X ⊗X ⊗X ⊗X,X ⊗Ŷ ⊗Ŷ ⊗Ŷ ,Ŷ ⊗X ⊗Ŷ ⊗Ŷ , Y ⊗Ŷ ⊗X ⊗Ŷ , andŶ ⊗Ŷ ⊗Ŷ ⊗X. In such circumstances, if we eliminate one of the parties, we are unable to show a GHZ contradiction with the remaining observables. The four-partite GHZ state is no longer their common eigenstate. Similar argument can be put forward in the case of our N -partite theorem.
The genuine D dimensionality is reflected by the fact that the operators are undecomposable to a direct sum of any subdimensional observables [14]. In other words, if two local observablesX andŶ can be simultaneously block diagonalized by some similarity transforma-tionŜ such thatŜXŜ † =X 1 ⊕ · · · ⊕X K andŜŶŜ † = Y 1 ⊕ · · · ⊕Ŷ K , then there exist some eigenstates |n α of X and |m β ofŶ such that α n|Ŝ †Ŝ |m β = 0 and one can find a sub-dimensional GHZ contradiction. However, there are no such eigenstates in our method because for every n and m, | α n|m β | 2 = sin 2 (πξ) D 2 sin 2 [(π/D)ξ] > 0, where ξ = m − n + β − α. As, the local observablesX =X(α) andŶ =X(β) are such that β − α is not an integer, ξ is a nonintegral rational number. Thus, our GHZ contradiction is genuinely D dimensional.
IV. SUMMARY
We construct a generalized GHZ contradiction for multipartite and high-dimensional systems. The GHZ theorem holds as long as N is not divisible by all nonunit divisors of D, smaller than N . We also demonstrate that our formulation of a generalized GHZ contradiction is genuinely N partite and D dimensional. For this purpose, we employ concurrent composite observables, which have a generalized GHZ state as a common eigenstate (even though these observables are incompatible). Our approach, by using concurrent observables, enables us to find a broader class of GHZ contradictions. There remain still more possibilities for constructing concurrent observables, which may help in further extension of the GHZ theorem. We hope that our approach of concurrent observables would be useful in the search of other kinds of quantum correlations, which are impossible classically.
ACKNOWLEDGMENTSWe thankČ. Brukner for discussions. The work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (Grants No. 2010-0015059 and No. 2010-0018295). J.R. and M.Ż. are supported by the Foundation for Polish Science TEAM project cofinanced by the EU European Regional Development Fund and a NCBiR-CHIST-ERA Project QUASAR. C.L. is supported by the National Research Foundation and Ministry of Education, Singapore.
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Note that compatible observables are clearly concurrent. Note that compatible observables are clearly concurrent.
Here, we shall use local unitary observablesV = D−1. Here, we shall use local unitary observablesV = D−1
Therefore the complex eigenvalues ofV can be associated with the measurement results, denoted by real eigenvalues ofĤ. Such a unitary representation leads to a simplification of mathematics without changing any. in Ddimensional Hilbert space. The observableV is unitary, however, one can uniquely relate it with a Hermitian ob-servableĤ by requiringV = exp(iĤ). physical results [13, 14n=0 ω n |n v n|, where ω = exp(2πi/D), in D- dimensional Hilbert space. The observableV is unitary, however, one can uniquely relate it with a Hermitian ob- servableĤ by requiringV = exp(iĤ). Therefore the com- plex eigenvalues ofV can be associated with the measure- ment results, denoted by real eigenvalues ofĤ. Such a unitary representation leads to a simplification of math- ematics without changing any physical results [13, 14].
| []
|
[]
| [
"María Laura Barbagallo \nDepartamento de Matemática\nFCEN\nUniversidad de Buenos Aires\nArgentina\n\nIMAS\nCONICET-UBA\nArgentina\n",
"Gabriela Jeronimo \nDepartamento de Matemática\nFCEN\nUniversidad de Buenos Aires\nArgentina\n\nIMAS\nCONICET-UBA\nArgentina\n",
"Juan Sabia \nDepartamento de Ciencias Exactas\nCBC\nUniversidad de Buenos Aires\nArgentina\n\nIMAS\nCONICET-UBA\nArgentina\n"
]
| [
"Departamento de Matemática\nFCEN\nUniversidad de Buenos Aires\nArgentina",
"IMAS\nCONICET-UBA\nArgentina",
"Departamento de Matemática\nFCEN\nUniversidad de Buenos Aires\nArgentina",
"IMAS\nCONICET-UBA\nArgentina",
"Departamento de Ciencias Exactas\nCBC\nUniversidad de Buenos Aires\nArgentina",
"IMAS\nCONICET-UBA\nArgentina"
]
| []
| We present a new procedure to count the number of real zeros of a class of univariate Pfaffian functions of order 1. The procedure is based on the construction of Sturm sequences for these functions and relies on an oracle for sign determination. In the particular case of Epolynomials, we design an oracle-free effective algorithm solving this task within exponential complexity. In addition, we give an explicit upper bound for the absolute value of the real zeros of an E-polynomial. | 10.1016/j.jalgebra.2015.11.050 | [
"https://arxiv.org/pdf/1506.07406v2.pdf"
]
| 119,150,047 | 1506.07406 | 2314a773f0a00cee8bfb03c253d72018fcbf9d74 |
12 Jan 2016 January 13, 2016 (2013/2016).
María Laura Barbagallo
Departamento de Matemática
FCEN
Universidad de Buenos Aires
Argentina
IMAS
CONICET-UBA
Argentina
Gabriela Jeronimo
Departamento de Matemática
FCEN
Universidad de Buenos Aires
Argentina
IMAS
CONICET-UBA
Argentina
Juan Sabia
Departamento de Ciencias Exactas
CBC
Universidad de Buenos Aires
Argentina
IMAS
CONICET-UBA
Argentina
12 Jan 2016 January 13, 2016 (2013/2016).Zero counting for a class of univariate Pfaffian functionsPfaffian functionszero countingSturm sequencescomplexity * Partially supported by the following grants: PIP 099/11 CONICET and UBACYT 20020120100133
We present a new procedure to count the number of real zeros of a class of univariate Pfaffian functions of order 1. The procedure is based on the construction of Sturm sequences for these functions and relies on an oracle for sign determination. In the particular case of Epolynomials, we design an oracle-free effective algorithm solving this task within exponential complexity. In addition, we give an explicit upper bound for the absolute value of the real zeros of an E-polynomial.
Introduction
Pfaffian functions, introduced by Khovanskii in the late '70 (see [6]), are analytic functions that satisfy first order partial differential equation systems with polynomial coefficients. A fundamental result proved by Khovanskii ([7]) states that a system of n equations given by Pfaffian functions in n variables defined on a domain Ω has finitely many non-degenerate solutions in Ω, and this number can be bounded in terms of syntactic parameters associated to the system.
From the algorithmic viewpoint, [5] presents a summary of quantitative and complexity results for Pfaffian equation systems essentially based on Khovanskii's bound. The known elimination procedures in the Pfaffian structure rely on the use of an oracle (namely, a blackbox subroutine which always gives the right answer) to determine consistency for systems of equations and inequalities given by Pfaffian functions. However, for some classes of Pfaffian functions the consistency problem is algorithmically decidable: for instance, an algorithm for the consistency problem of systems of the type f 1 (x) ≥ 0, . . . , f k (x) ≥ 0, f k+1 (x) > 0, . . . , f l (x) > 0, where x = (x 1 , . . . , x n ), f i (x) = F i (x, e h(x) ) and F i (1 ≤ i ≤ l) and h are polynomials with integer coefficients, is given in [16]. This result allows the design of algorithms to solve classical related geometric problems (see, for example, [14]). More generally, the decidability of the theory of the real exponential field (i.e. the theory of the structure R exp = R; +, ·, −, 0, 1, exp, < ) was proved in [8] provided Shanuel's conjecture is true.
In this paper, we design a symbolic procedure to count the exact number of zeros in a real interval of a univariate Pfaffian function of the type f (x) = F (x, ϕ(x)), where F is a polynomial in Z[X, Y ] and ϕ is a univariate Pfaffian function of order 1 (see [5,Definition 2.1]). The procedure is based on the construction of a family of Sturm sequences associated to the given function f (x), which is done by means of polynomial subresultant techniques (see, for instance, [1]). As it is usual in the literature on the subject, we assume the existence of an oracle to determine the sign a Pfaffian function takes at a real algebraic number. Sturm sequences in the context of transcendental functions were first used in [13] to extend the cylindrical decomposition technique to non-algebraic situations. In [19], this approach was followed to count the number of real roots of exponential terms of the form p(x)+q(x)e r(x) , where p, q and r are real polynomials. Later in [9], the same technique is applied to treat the case of functions of the type F (x, e x ), where F is an integer polynomial.
A function of the form f (x) = F (x, e h(x) ),
where F and h are polynomials with real coefficients, is called an E-polynomial ( [16]). For these particular functions, we give an effective symbolic algorithm solving the zero-counting problem with no calls to oracles. To this end, we construct a subroutine to determine the sign of univariate E-polynomials at real algebraic numbers. Our algorithms only perform arithmetic operations and comparisons between rational numbers. In order to deal with real algebraic numbers, we represent them by means of their Thom encodings (see Section 2.2). The main result of the paper is the following: Finally, we prove an explicit upper bound for the absolute value of the real zeros of an E-polynomial in terms of the degrees and absolute values of the coefficients of the polynomials involved. This bound could be used to separate and approximate the real zeros of an E-polynomial. It provides an answer to the 'problem of the last root' for this type of functions. Previously, in [18], the existence of such a bound was established for general exponential terms, but even though it is given by an inductive argument with a computable number of iterations, the bound is not explicit. Algorithms for the computation of upper bounds for the real roots of functions of the type P (x, e x ) or, more generally, P (x, trans(x)), with P an integer polynomial and trans(x) = e x , ln(x) or arctan(x) are given in [9] and [10] respectively. The paper is organized as follows: in Section 2, we fix the notation and recall some basic theoretical and algorithmic results on univariate polynomials. Section 3 is devoted to the construction of Sturm sequences for the Pfaffian functions we deal with. In Section 4, we present our general procedure for zero counting. Finally, in Section 5, we describe the algorithms and prove our main results on E-polynomials.
Preliminaries
Basic notation and results
Throughout the paper, we will deal with univariate and bivariate polynomials. For a polynomial F ∈ Z[X, Y ], we write deg X (F ) and deg Y (F ) for the degrees of F in the variables X and Y respectively, H(F ) for its height, that is, the maximum of the absolute values of its coefficients in Z, and cont(F ) ∈ Z[X] for the gcd of the coefficients of F as a polynomial in Z[X] [Y ].
Note that, if p 1 , p 2 ∈ Z[X] are polynomials with degrees bounded by d 1 and d 2 , and heights bounded by H 1 and H 2 , then H(
p 1 p 2 ) ≤ (min{d 1 , d 2 } + 1)H 1 H 2 .
If f is a real univariate analytic function, we denote its derivative by f ′ and, for k > 1, its kth successive derivative by f (k) .
For γ = (γ 0 , . . . , γ N ) ∈ R N +1 with γ i = 0 for every 0 ≤ i ≤ N , the number of variations in sign of γ is the cardinality of the set {1 ≤ i ≤ N : γ i−1 γ i < 0}. For a tuple γ of arbitrary real numbers, the number of variations in sign of γ is defined as the number of variations in sign of the tuple which is obtained from γ by removing its zero coordinates. Given x ∈ R and a sequence of univariate real functions f = (f 0 , . . . , f N ) defined at x, we write v(f , x) for the number of variations in sign of the (N + 1)−tuple (f 0 (x), . . . , f N (x)).
We recall some well-known bounds on the size of roots of univariate polynomials (see [11,Proposition 2.5.9 and Theorem 2.5.11]).
Lemma 2 Let p = d j=0 a j X j ∈ C[X], a d = 0. Let r(p) := max{|z| : z ∈ C, p(z) = 0}. Then: i) r(p) < 1 + max a j a d : 0 ≤ j ≤ d − 1 ii) r(p) < 1 + 0≤j≤d−1 a j a d 2 1/2
We will also use the following lower bound for the separation of the roots of a univariate polynomial with integer coefficients (see [11,Theorem 2.7
.2]):
Lemma 3 Let p ∈ Z[X] be a polynomial of degree d ≥ 2, and α 1 , . . . , α d be all the roots of p.
Then min{|α i − α j | : α i = α j } > d − d+2 2 (d + 1) 1−d 2 H(p) 1−d .
A basic tool for our results is the well-known theory of subresultants for univariate polynomials with coefficients in a ring and its relation with polynomial remainder sequences (see [1,Chapter 8]
SRes e−1 = −Remainder((−1) (d−e−1)(d−e)/2 lc(G) d−e+1 F, G),
where lc(G) is the leading coefficient of G and, for an index i with 1 ≤ i ≤ d such that SRes i−1 is non-zero of degree j:
• If SRes j−1 = 0, then SRes i−1 = gcd(F, G) up to a factor in Z[X].
• If SRes j−1 = 0 has degree k,
s j t i−1 SRes k−1 = −Remainder(s k t j−1 SRes i−1 , SRes j−1 ) and the quotient lies in Z[X][Y ].
Here, s l denotes the lth subresultant coefficient of F and G as defined in [1,Notation 4.22] and t l is the leading coefficient of SRes l .
We define a sequence of integers as follows:
• n 0 = d + 1, n 1 = d. • For i ≥ 1, if SRes n i −1 = 0, then n i+1 = deg(SRes n i −1 ).
The polynomials
R i := SRes n i −1 are proportional to the polynomials in the Euclidean remainder sequence associated to F and G. Moreover, the following relations hold:
(−1) (d−e)(d−e+1)/2 lc(G) d−e+1 R 0 = R 1 C 1 − R 2 (1) s n i+2 t n i+1 −1 R i = R i+1 C i+1 − s n i+1 t n i −1 R i+2 for i ≥ 1 (2) where C i ∈ Z[X][Y ]
for every i.
Algorithms and complexity
The algorithms we consider in this paper are described by arithmetic networks over Q (see [2]). The notion of complexity of an algorithm we consider is the number of operations and comparisons in Q. The objects we deal with are polynomials with coefficients in Q, which are represented by the array of all their coefficients in a pre-fixed order of their monomials.
To estimate complexities we will use the following results (see [3]). The product of two polynomials in Q[X] of degrees bounded by d can be done within complexity O(M (d)), where M (d) = d log(d) log log(d). Interpolation of a degree d polynomial in Q[X] requires O(M (d) log(d)) arithmetic operations. We will use the Extended Euclidean Algorithm to compute the gcd of two polynomials in Q[X] of degrees bounded by d within complexity O(M (d) log(d)). We will compute subresultants by means of matrix determinants, which enables us to control both the complexity and output size (an alternative method for the computation of subresultants, based on the Euclidean algorithm, can be found in [1,Algorithm 8.21]). For a matrix in Q n×n , its determinant can be obtained within complexity O(n ω ), where ω < 2.376 (see [3,Chapter 12]).
For a polynomial in Z[X], we will need to approximate its real roots by rational numbers and to isolate them in disjoint intervals of pre-fixed length with rational endpoints. There are several known algorithms achieving these tasks (see, for instance, [15] and the references therein). Here we use a classical approach via Sturm sequences. The complexity of the algorithm based on this approach is suboptimal. However, the complexity order of the procedures in which we use it as a subroutine would not change even if we replaced it with the one with the best known complexity bound.
Lemma 4 Let p ∈ Z[X] be a polynomial of degree bounded by d and ǫ ∈ Q, ǫ > 0. There is an algorithm which computes finitely many pairwise disjoint intervals I j = (a j , b j ] with a j , b j ∈ Q and b j − a j ≤ ǫ such that each I j contains at least one real root of p and every real root of p lies in some I j . The complexity of the algorithm is of order O(d 3 log(H(p)/ǫ)). • Determine, for each of the intervals J r and J l , whether p has a real root in that interval or not. Keep the intervals that contain real roots of p.
The recursion finishes when the length of all the intervals is at most ǫ. The output consists of all the intervals of length at most ǫ containing roots of p, including the intervals I appearing at intermediate steps.
In order to determine whether p has a real root in a given interval, we use the Sturm sequence of p and p ′ (see [1,Theorem 2.50]), which is computed within complexity O(M (d) log(d)) by means of the Euclidean Algorithm.
At each step of the recursion, we keep at most d intervals together with the number of variations in sign of the Sturm sequence evaluated at each of their endpoints. For each of these intervals, the procedure above requires at most 2d + 1 additional evaluations of polynomials of degrees at most d. Then, the complexity of each recursive step is of order O(d 3 ).
Since the length of the intervals at the kth step is at most 1+H(p) 2 k−1 , the number of steps is at most 1 + ⌈log( 1+H(p) ǫ )⌉. Therefore, the overall complexity is O(d 3 log(H(p)/ǫ)).
In order to deal with real algebraic numbers in a symbolic way, we will use Thom encodings. We recall here their definition and main properties (see [1,Chapter 2]). Given p ∈ R[X] and a real root α of p, the Thom encoding of α as a root of p is the sequence (sign(p ′ (α)), . . . , sign(p (deg p) (α))), where we represent the sign with an element of the set {0, 1, −1}. Two different real roots of p have different Thom encodings. In addition, given the Thom encodings of two different real roots α 1 and α 2 of p, it is possible to decide which is the smallest between α 1 and α 2 (see [1, Proposition 2.28]).
For a polynomial p ∈ R[X], we will denote
Der(p) := (p, p ′ , . . . , p (deg p) )
A useful tool to compute Thom encodings and manipulate real algebraic numbers is an effective procedure for the determination of feasible sign conditions on real univariate polynomials. For p 1 , . . . , p s ∈ R[X], a feasible sign condition for p 1 , . . . , p s on a finite set Z ⊂ R is an s-tuple (σ 1 , . . . , σ s ) ∈ {=, >, <} s such that {x ∈ Z : p 1 (x)σ 1 0, . . . , p s (x)σ s 0} = ∅.
Lemma 5 (see [12,Corollary 2]) Given p 0 , p 1 , . . . , p s ∈ R[X], p 0 ≡ 0, deg p i ≤ d for i = 0, . . . , s, the feasible sign conditions for p 1 , . . . , p s on {p 0 = 0} can be computed algorithmically within O(sd 2 log 3 (d)) operations. Moreover, if p 0 has m roots in R, this can be done within O(smd log(m) log 2 (d)) operations. The output of the algorithm is a list of s-tuples in {0, 1, −1} s , where 0 stands for =, 1 for > and −1 for <.
Sturm sequences and zero counting for Pfaffian functions
Following [4], we introduce the notion of a Sturm sequence for a continuous function in a real interval: ) is said to be a Sturm sequence for f 0 in the interval (a, b) if the following conditions hold:
Definition 6 Let f 0 : (a, b) → R be a continuous function of a single variable. A sequence of continuous functions f = (f 0 , . . . , f N ) on (a, b1. If f 0 (y) = 0, there exists ǫ > 0 such that f 1 (x) = 0 for every x ∈ (y − ǫ, y + ǫ) ⊆ (a, b), x = y, f 0 (x)f 1 (x) < 0 for y − ǫ < x < y and f 0 (x)f 1 (x) > 0 if y < x < y + ǫ.
For every
i = 1, . . . , N − 1, if f i (x) = 0 for x ∈ (a, b), then f i−1 (x)f i+1 (x) < 0. 3. f N (x) = 0 for every x ∈ (a, b).
Recalling that, for a given x ∈ R, v(f , x) denotes the number of variations in sign of the (N + 1)-tuple (f 0 (x), . . . , f N (x)), we have the following analog of the classical Sturm theorem:
Theorem 7 ([4, Theorem 2.1]) Let f 0 : (a, b) → R be a continuous function of a single variable. Let f = (f 0 , . . . , f N ) be a Sturm sequence for f 0 in the interval (a, b) and let a < c < d < b. Then, the number of distinct real zeros of f 0 in the interval (c, d] is v(f , c) − v(f , d).
The aim of this section is to build Sturm sequences for a particular class of Pfaffian functions we introduce below. For the definition of Pfaffian functions in full generality and the basic properties of these functions see, for instance, [5].
Given a polynomial Φ ∈ Z[X, Y ] with deg Y (Φ) > 0, let ϕ be a function satisfying the differential equation
ϕ ′ (x) = Φ(x, ϕ(x)).(3)
Note that ϕ is analytic on its domain, which may be a proper subset of R.
We are going to work with Pfaffian functions of the type
f (x) = F (x, ϕ(x)), where F ∈ Z[X, Y ].
Taking into account that the first derivative of such a function is
∂F ∂X (x, ϕ(x)) + ∂F ∂Y (x, ϕ(x)).Φ(x, ϕ(x)),
we define, for any F ∈ Z[X, Y ], the polynomial F ∈ Z[X, Y ] (associated with Φ) as follows:
F (X, Y ) = ∂F ∂X (X, Y ) + ∂F ∂Y (X, Y )Φ(X, Y ).(4)
Thus, we have that
f ′ (x) = F (x, ϕ(x)).
Due to the following result, in order to count the number of real zeros of a function f as above, we will assume from now on, without loss of generality, that Res Y (F, F ) = 0.
Lemma 8 Let Φ, ϕ be as in equation (3) and let F ∈ Z[X, Y ] with deg Y (F ) > 0.
There exists a polynomial P ∈ Z[X, Y ] such that Res Y (P, P ) = 0 and P (x, ϕ(x)) has the same real zeros as F (x, ϕ(x)). Moreover, the polynomial P can be effectively computed from F and Φ.
Proof. Without loss of generality, we may assume that F is square-free. Suppose that Res
Y (F, F ) = 0. Write F = cont(F ) F 0 . Then, Res Y (F 0 , F 0 ) = 0 and so, the greatest common divisor of F 0 and F 0 is a polynomial S ∈ Z[X, Y ] of positive degree in Y . If F 0 = S U and F 0 = S V for U, V ∈ Z[X, Y ], we have that f 0 (x) = F 0 (x, ϕ(x)) = S(x, ϕ(x)) U (x, ϕ(x)) and f ′ 0 (x) = F 0 (x, ϕ(x)) = S(x, ϕ(x)) V (x, ϕ(x)), which implies that a zero ξ of f 0 which is not a zero of U (x, ϕ(x)) satisfies that mult(ξ, f 0 ) = mult(ξ, S(x, ϕ(x))) ≤ mult(ξ, f ′ 0 ), leading to a contradiction. Then, f 0 and U (x, ϕ(x)) have the same zero set in R. As F 0 = (S U ) = S U + S U , it follows that, if T ∈ Z[X, Y ] is a common factor of U and U with positive degree in Y , then T divides F 0 = S V .
Since U and V are relatively prime polynomials, then T divides S and, therefore T 2 divides F 0 , contradicting the fact that F 0 is square-free. The lemma follows considering the polynomial P = cont(F ) U .
We will apply the theory of subresultants introduced in Section 2 in order to get Sturm sequences for f .
Let
F 1 = Remainder(lc(F ) D F , F ) ∈ Z[X][Y ],
where D is the smallest even integer greater than or equal to 1
+ deg Y ( F ) − deg Y (F ).
Notation 9 Following Section 2.1, for i = 0, . . . , N , let
R i := SRes n i −1 ∈ Z[X][Y ] be the (n i − 1)th subresultant polynomial associated to F and F 1 , τ i := t n i −1 ∈ Z[X] be the leading coefficient of R i and, for i = 2, . . . , N +1, let ρ i := s n i ∈ Z[X] be the n i th subresultant coefficient of F and F 1 .
Definition 10 For an interval I = (a, b) containing no root of the polynomials τ i for i = 0, . . . , N or ρ i for i = 2, . . . , N + 1, we define inductively a sequence (σ I,i ) 0≤i≤N ∈ {1, −1} N +1 as follows:
• σ I,0 = σ I,1 = 1,
• σ I,2 = (−1) 1 2 (deg Y (F )−deg Y (F 1 ))(deg Y (F )−deg Y (F 1 )+1) sg I (lc(F 1 )) deg Y (F )−deg Y (F 1 )+1 , • σ I,i+2 = sg I (ρ i+2 τ i+1 ρ i+1 τ i )σ I,i ,
where, for a continuous function g of a single variable with no zeros in I, sg I (g) denotes the (constant) sign of g in I. For i = 0, . . . , N , we define
F I,i = σ I,i R i ∈ Z[X, Y ].
Finally, if I is contained in the domain of ϕ, we introduce the sequence of Pfaffian functions
f I = (f I,i ) 0≤i≤N defined by f I,i (x) = F I,i (x, ϕ(x))
.
Proposition 11 Let F ∈ Z[X, Y ], deg Y (F ) > 0, and let ϕ be a Pfaffian function satisfying ϕ ′ (x) = Φ(x, ϕ(x)), where Φ ∈ Z[X, Y ] with deg Y (Φ) > 0. Consider the function f (x) = F (x, ϕ(x)). Let F ∈ Z[X, Y ] be defined as in (4). Assume that the resultant Res Y (F, F ) ∈ Z[X]
is not zero. With the notation and assumptions of Definition 10, the sequence of Pfaffian
functions f I = (f I,i ) 0≤i≤N is a Sturm sequence for f in I = (a, b).
Proof. For simplicity, as the interval I is fixed, the subindex I will be omitted throughout the proof.
First we prove that f 0 and f 1 do not have common zeros in I. Suppose α ∈ I is a common zero of f 0 and f 1 . Then F (α, ϕ(α)) = 0 and F 1 (α, ϕ(α)) = 0; therefore, ρ N +1 (α) = Res Y (F, F 1 )(α) = 0, contradicting the assumptions on I.
From this fact, taking into account that f 0 = f , and f 1 has the same sign as f ′ at any zero of f lying in I, condition 1 of Definition 6 follows.
To prove that condition 2 holds, first note that if f j (α) = 0 and f j+1 (α) = 0 for some α ∈ I, since ρ i and τ i do not have zeros in I, by identities (1) and (2), α is a common zero of all f i s, contradicting the fact that f 0 and f 1 do not have common zeros in I. Then, condition 2 in Definition 6 follows from the definition of the signs σ i and identities (1) and (2). Condition 3 follows from the assumption that τ N , which equals f N up to a sign, does not have zeros in I.
In order to count the number of zeros of a Pfaffian function in an open interval, provided that the function is defined in its endpoints, we introduce the following:
Notation 12 Let f : J → R be a non-zero analytic function defined in an open interval J ⊂ R and let c ∈ J. We denote Note that sg(f, c + ) is the sign that f takes in (c, c + ε) and sg(f, c − ) is the sign that f takes in (c − ε, c) for a sufficiently small ε > 0. Then, by Theorem 7, we have:
sg(f, c + ) = sign(f (c)) if f (c) = 0 sign(f (r) (c)) if mult(c, f ) = r and sg(f, c − ) = sign(f (c)) if f (c) = 0 sign((−1) r f (r) (c)) if mult(c, f ) = r where mult(c, f ) isPropositionI = (a, b) equals v(f I , a + ) − v(f I , b − ).
As a consequence, we get a formula for the number of zeros of the Pfaffian function f in any bounded interval:
Theorem 14 Let f (x) = F (x, ϕ(x)), where F ∈ Z[X, Y ], deg Y (F ) > 0, and ϕ is a Pfaffian function satisfying ϕ ′ (x) = Φ(x, ϕ(x)) for a polynomial Φ ∈ Z[X, Y ] with deg Y (Φ) > 0. Assume Res Y (F, F ) = 0. Consider a bounded open interval (α, β) ⊂ R such that [α, β] is contained in the domain of ϕ.
Let ρ i and τ i be the polynomials in Z[X] introduced in Notation 9. If α 1 < α 2 < · · · < α k are all the roots in (α, β) of ρ i and τ i , the number of zeros of f in [α, β] equals
#{0 ≤ j ≤ k + 1 : f (α j ) = 0} + k j=0 v(f I j , α + j ) − v(f I j , α − j+1 ),
where α 0 = α, α k+1 = β and, for every 0 ≤ j ≤ k, I j = (α j , α j+1 ) and f I j is the sequence of functions introduced in Definition 10.
Algorithmic approach
Let ϕ be a Pfaffian function satisfying
ϕ ′ (x) = Φ(x, ϕ(x)) for a polynomial Φ ∈ Z[X, Y ]. Let δ Y := deg Y (Φ) > 0 and δ X := deg X (Φ).
In this section, we describe an algorithm for counting the number of zeros in a bounded interval contained in the domain of ϕ of a function of the type
f (x) = F (x, ϕ(x)), where F ∈ Z[X, Y ] with deg Y (F ) > 0.
To estimate the complexity of the algorithm, we need an upper bound for the multiplicity of a zero of a function of this type. Here, we present a bound in our particular setting which takes into account the degrees in each of the variables X and Y of the polynomials involved in the definition of the functions. A general upper bound on the multiplicity of Pfaffian intersections depending on the total degrees of the polynomials can be found in [5,Theorem 4.3]. Even though both bounds are of the same order, our bound may be smaller when the total degrees are greater than the degrees with respect to each variable.
Lemma 15 With the previous notation, let g(x) = G(x, ϕ(x)) with G ∈ Z[X, Y ] be a nonzero Pfaffian function. For every α ∈ R such that g(α) = 0, we have
mult(α, g) ≤ 2 deg X (G) deg Y (G) + deg X (G)(δ Y − 1) + (δ X + 1) deg Y (G).
Proof. Assume first that G is irreducible in Z[X, Y ]. If g(α) = 0, then mult(α, g) > mult(α, g ′ ).
As g ′ (x) = G(x, ϕ(x)), then G does not divide G and, therefore, R := Res Y (G, G) = 0. Let S, T ∈ Z[X, Y ] be such that R = SG + T G. We have that
R(x) = S(x, ϕ(x)). g(x) + T (x, ϕ(x)). g ′ (x).
If α is a multiple root of g, the previous identity implies that mult(α, g) ≤ mult(α, R)
+ 1 ≤ deg(R) + 1. Taking into account that deg(R) ≤ deg X (G) deg Y ( G) + deg X ( G) deg Y (G), deg X ( G) ≤ deg X (G) + δ X and deg Y ( G) ≤ deg Y (G) − 1 + δ Y , we conclude that mult(α, g) ≤ 2 deg X (G) deg Y (G) + deg X (G)(δ Y − 1) + δ X deg Y (G) + 1. In the general case, write G = c(X) 1≤i≤t G i (X, Y ) m i , where c(X) = cont(G) and G 1 , . . . , G t ∈ Z[X, Y ] are irreducible polynomials. For every i, let g i (x) = G i (x, ϕ(x)). From the previous bound, we deduce mult(α, g) = mult(α, c) + 1≤i≤t m i mult(α, g i ) ≤ ≤ deg X (c) + 1≤i≤t m i (2 deg X (G i ) deg Y (G i ) + deg X (G i )(δ Y − 1) + δ X deg Y (G i ) + 1) ≤ 2 deg X (G) deg Y (G) + deg X (G)(δ Y − 1) + (δ X + 1) deg Y (G).
The theoretical results in the previous section enable us to construct the following algorithm for zero counting for a function f (x) = F (x, ϕ(x)), where F ∈ Z[X, Y ]. By Lemma 8, we will assume that Res Y (F, F ) = 0.
Algorithm ZeroCounting
INPUT: A function ϕ satisfying a differential equation ϕ ′ (x) = Φ(x, ϕ(x)), a polynomial F ∈ Z[X, Y ] such that Res Y (F, F ) = 0, and a closed interval [α, β] ⊂ Dom(ϕ) with α, β ∈ Q.
OUTPUT: The number of zeros of f (x) = F (x, ϕ(x)) in [α, β].
1. Let F 1 (X, Y ) := F (X, Y ) if deg Y ( F ) < deg Y (F ) Remainder(lc(F ) D F , F ) otherwise
, where D is the smallest even integer greater than or equal to 1
+ deg Y ( F ) − deg Y (F ).
2. Compute the polynomials R i and τ i , for 0 ≤ i ≤ N , and ρ i , for 2 ≤ i ≤ N + 1, associated to F and F 1 as in Notation 9.
3. Determine and order all the real roots α 1 < α 2 < · · · < α k lying in the interval (a, b) of the polynomials τ i , for 0 ≤ i ≤ N , and ρ i , for 2 ≤ i ≤ N + 1.
4. For every 0 ≤ j ≤ k, compute the Sturm sequence f I j = (f I j ,i ) 0≤i≤N for f in I j = (α j , α j+1 ) as in Definition 10, where α 0 = α and α k+1 = β.
5. Decide whether f (α j ) = 0 for every 0 ≤ j ≤ k + 1 and count the number of zeros.
For every
0 ≤ j ≤ k, compute v j := v(f I j , α + j ) − v(f I j , α − j+1 ). 7. Compute #{0 ≤ j ≤ k + 1 : f (α j ) = 0} + k j=1 v j .
Complexity analysis:
Let d X := deg X (F ), d Y := deg Y (F ) and, as before, δ X := deg X (Φ), δ Y := deg Y (Φ).
Step
1. Note that deg Y (F 1 ) < d Y . In the case when deg Y ( F ) ≥ d Y , in order to bound deg X (F 1 ), notice that deg X (lc(F ) D F ) ≤ D deg(lc(F )) + d X + δ X .
Then, the polynomial F 1 can be obtained by means of at most D successive steps, each consisting of subtracting a multiple of F with degree in X bounded by (D − i) deg X (lc(F )) + (i + 1)d X + δ X from a polynomial whose degree in X is bounded by
(D − i + 1) deg X (lc(F )) + i d X + δ X . Then, deg X (F 1 ) ≤ (D + 1)d X + δ X ≤ (δ Y + 2)d X + δ X .
In order to perform the computations (as polynomials in the variable Y ) avoiding division of coefficients (which are polynomials in X), we do not expand the product of the coefficients of F times lc(F ) D at the beginning, and at the ith step, we write each coefficient of the remainder as a multiple of lc(F ) D−i . Thus, at each step, we compute at most d Y + δ Y polynomials in X: for the first d Y of them, we compute the difference of two products of a coefficient of F (whose degree is at most d X ) by a polynomial of degree bounded by (i + 1)d X + δ X , and for the other ones, the product of the leading coefficient of F by a polynomial of degree bounded by (i + 1)d X + δ X . Then, the overall complexity of this step
is O((d Y + δ Y )d X δ Y (δ Y d X + δ X )).
Step 2. Each subresultant of F and F 1 is a polynomial in the variable Y whose coefficients are polynomials of degree bounded by (d Y − 1)d X + d Y ((δ Y + 2)d X + δ X ) in the variable X. We compute it by means of interpolation: for sufficiently many interpolation points, we evaluate the coefficients of F and F 1 , we compute the corresponding determinant (which is a polynomial in Y with constant coefficients) and, finally we interpolate to obtain each coefficient.
For each interpolation point, the evaluation of the coefficients of F and F 1 can be performed
within complexity O(d Y d X + (d Y − 1)((δ Y + 2)d X + δ X )) = O(d Y (δ Y d X + δ X )
). Then, we compute at most 2d Y − 1 determinants of matrices of size bounded by 2d Y − 2 within complexity O(d ω+1 Y ), we multiply them by the polynomials Y j F or Y j F 1 evaluated at the point and we add the results in order to obtain the specialization of the subresultant at the point, which does not modify the complexity order. This is repeated for d Y ((δ Y + 3)d X + δ X ) points. Finally, each of the at most d Y coefficients of the subresultant polynomial is computed by interpolation from the results obtained. Each polynomial interpolation can be done within complexity
O(M (d Y (δ Y d X + δ X )) log(d Y (δ Y d X + δ X ))
). Then, the computation of the at most d Y coefficients of each subresultant can be achieved within complexity
O((d Y (δ Y d X + δ X ) + d ω+1 Y )d Y (δ Y d X + δ X ) + d Y M (d Y (δ Y d X + δ X )) log(d Y (δ Y d X + δ X ))) = O(d ω+2 Y (δ Y d X + δ X ) 2 ).
As we have to compute at most d Y subresultants, the overall complexity of the computation of all the required subresultants is of order O(d ω+3
Y (δ Y d X + δ X ) 2 ).
Note that we may compute successively only the polynomials R i = SRes n i −1 . The index n i+1 indicating the next subresultant to be computed is the degree of R i , and the polynomial τ i is its leading coefficient. Finally, the polynomials ρ i ∈ Z[X] are subresultant coefficients of F and F 1 , which are also computed by interpolation. The complexity of these computations does not modify the order of the overall complexity of this step.
Step 3. Consider the polynomial
L(X) = 0≤i≤N τ i 3≤i≤N +1 ρ i .(5)
Note that ρ 2 = (−1)
1 2 (deg Y (F )−deg Y (F 1 ))(deg Y (F )−deg Y (F 1 )+1) lc(F 1 ) deg Y (F )−deg Y (F 1 )
; so, it has the same zeros as τ 1 = lc(F 1 ).
We determine the Thom encodings of the roots of L in the interval (a, b) by computing the realizable sign conditions on Der(L), X − α, β − X, where Der(L) = (L, L ′ , . . . , L deg(L) ).
The degree of L is bounded by
(2d 2 Y − d Y )((δ Y + 3)d X + δ X )
. We compute its coefficients by interpolation: the specialization of L at a point can be computed within
O(d 2 Y (δ Y d X + δ X )
) operations by specializing its factors and multiplying, and this is done for deg(L) + 1 points; then, the total complexity of evaluation and interpolation is of order
O(d 4 Y (δ Y d X + δ X ) 2 )
. The complexity of computing the realizable sign conditions is of Lemma 5). Finally, we can order the roots of L in (α, β) by comparing their Thom encodings (see [1,Proposition 2.28
order O(d 6 Y (δ Y d X + δ X ) 3 log 3 (d 2 Y (δ Y d X + δ X ))) (see]) within complexity O(d 4 Y (δ Y d X + δ X ) 2 log(d 2 Y (δ Y d X + δ X ))
) using a sorting algorithm. The overall complexity of this step is of order
O(d 6 Y (δ Y d X + δ X ) 3 log 3 (d 2 Y (δ Y d X + δ X ))).
Step 4. The Sturm sequences (f I j ) 0≤j≤k are obtained by multiplying the polynomials (R i ) 0≤i≤N by the corresponding signs (σ I j ,i ) 0≤i≤N as stated in Definition 10. Note that if p is a univariate polynomial having a constant sign in I j = (α j , α j+1 ), to determine this sign it suffices to determine sg(p, α + j ) or sg(p, α − j+1 ), which can be obtained from the signs of p and its successive derivatives at α j or α j+1 respectively.
Then, in order to compute the required signs, we compute the realizable sign conditions on the family
Der(L), X − α, β − X, Der(ρ i ) 3≤i≤N , Der(τ i ) 1≤i≤N −1 which consists of O(d 2 Y (δ Y d X + δ X )) polynomials of degrees bounded by (2d 2 Y − d Y )((δ Y + 3)d X +δ X ). The complexity of this computation is of order O(d 6 Y (δ Y d X +δ X ) 3 log 3 (d 2 Y (δ Y d X + δ X ))
). Going through the list of realizable sign conditions, we determine the signs σ I j ,i and, from them, the Sturm sequences f I j within the same complexity order.
The overall complexity of Steps 1 -4 is of order
O(d 6 Y (δ Y d X + δ X ) 3 log 3 (d 2 Y (δ Y d X + δ X ))).
Steps 5 and 6. These steps require the determination of the sign of Pfaffian functions of the type G(x, ϕ(x)), with G ∈ Z[X, Y ], at real algebraic numbers given by their Thom encodings (more precisely, at the real roots α j of L lying on (α, β) and at the endpoints α and β of the given interval). We assume an oracle is given to achieve this task.
At
Step 5, we need k + 2 ≤ deg(L)
+ 2 = O(d 2 Y (δ Y d X + δ X )
) calls to the oracle for the Pfaffian function defined by the polynomial F , having degrees deg X (F ) = d X and deg Y (F ) = d Y .
At Step 6, we use the oracle for Pfaffian functions defined by polynomials with degrees in X bounded by d Y ((δ Y +3)d X +δ X ) and degrees in Y bounded by d Y . Taking into account the bound for the multiplicity of a zero of such a function given by Lemma 15, it follows that the determination of sg(f I j ,i , α + ℓ ) and sg(f
I j ,i , α − ℓ ) requires at most O(d Y (d Y +δ Y )(δ Y d X +δ X )) calls to the oracle. Then, the oracle is used at most O(d 4 Y (d Y + δ Y )(δ Y d X + δ X ) 2 )
times. Therefore, we have the following:
Proposition 16 Let f (x) = F (x, ϕ(x)) be defined from a polynomial F ∈ Z[X, Y ] and a Pfaf- fian function ϕ satisfying ϕ ′ (x) = Φ(x, ϕ(x)), where Φ ∈ Z[X, Y ] with deg Y (Φ) > 0. Let d X := deg X (F ), d Y := deg Y (F ), δ X := deg X (Φ) and δ Y := deg Y (Φ). Then, Algorithm ZeroCounting computes the number of zeros of f in a closed interval [α, β] ⊂ Dom(ϕ) (α, β ∈ Q) within O(d 6 Y (δ Y d X + δ X ) 3 log 3 (d 2 Y (δ Y d X + δ X ))
) arithmetic operations and comparisons, and us-
ing at most O(d 4 Y (d Y + δ Y )(δ Y d X + δ X ) 2 )
calls to an oracle for determining the signs of Pfaffian functions of the type G(x, ϕ(x)), with G ∈ Z[X, Y ], at real algebraic numbers.
As a consequence of the previous algorithm we deduce an upper bound for the number of zeros of the Pfaffian functions under consideration in a bounded interval:
Corollary 17 Let f (x) = F (x, ϕ(x)) be defined from a polynomial F ∈ Z[X, Y ] and a Pfaffian function ϕ satisfying ϕ ′ (x) = Φ(x, ϕ(x)), where Φ ∈ Z[X, Y ] with deg Y (Φ) > 0. Let d X := deg X (F ), d Y := deg Y (F ), δ X := deg X (Φ) and δ Y := deg Y (Φ). Then, for any open interval I ⊂ Dom(ϕ), the number of zeros of f in I is at most (d Y + 1)(2d 2 Y − d Y )((δ Y + 3)d X + δ X )
. An alternative bound can be obtained from Khovanskii's upper bounds for the number of non-degenerate zeros of univariate Pfaffian functions and for the multiplicity of an arbitrary zero of these functions (see [5]). Keeping our previous notation, for a polynomial F ∈ Z[X, Y ] with deg(F ) = d, if deg(Φ) = δ, using Khovanskii's bounds, it follows that both the number of non-degenerate zeros and the multiplicity of an arbitrary zero of f (x) = F (x, ϕ(x)) are at most d(δ + d). We can get an upper bound for the total number of zeros of f by bounding the number of non-degenerate zeros of f and of its successive derivatives of order at most d(δ + d) − 1.
Following (4), we have that f ′ is defined by a polynomial of degree at most d + δ − 1 and so, for every k ∈ N, f (k) is given by a polynomial of degree at most d + k(δ − 1). Then, the total number of zeros of f is at most
d(δ+d)−1 k=0 (d + k(δ − 1))(δ + d + k(δ − 1)) ≤ 1 2 d 3 δ 2 (δ + d) 3 .
Note that the bound from Corollary 17 is of lower order than this one.
E-polynomials
In this section, we will deal with the particular case of E-polynomials, namely when ϕ(x) = e h(x) for a polynomial h ∈ Z[X] of positive degree. We will first show how to perform steps 5 and 6 of Algorithm ZeroCounting (that is, we will give an algorithmic procedure to replace the calls to an oracle). Finally, we will prove a bound for the absolute value of the zeros of an E-polynomial.
Sign determination
The main goal of this section is to design a symbolic algorithm which determines the sign that an E-polynomial takes at a real algebraic number given by its Thom encoding. To do this, we will use two subroutines. The first one, which follows [16,Lemma 15], determines the sign of an expression of the form e β − α for real algebraic numbers α and β. The second one allows us to locate a real number of the form e h(α) , for a real algebraic number α, between two consecutive real roots of a given polynomial.
Algorithm SignExpAlg
INPUT: Real algebraic numbers α and β given by their Thom encodings σ P 1 (α) and σ P 2 (β) with respect to polynomials P 1 , P 2 ∈ Z[X] such that deg(P 1 ), deg(P 2 ) ≤ d (d ≥ 2) and H(P 1 ), H(P 2 ) ≤ H.
OUTPUT: The sign s := sign(e β − α).
1. Let c := (2 d+1 (d + 1)H) −2 41 d 6 (5d+4⌈log(H)⌉) .
2. Compute w ∈ Q such that |e β − w| < c as follows:
(a) Compute w 1 ∈ Q such that |β − w 1 | < c 2. 3 H+2 (b) Compute w ∈ Q such that |e w 1 − w| < c 2 3. Compute s = sign(w − α).
Proof of correctness and complexity analysis:
Step 1. We will show that, for the chosen value of c, the inequality |e β − α| > c holds.
As shown in [17], if α and β are algebraic numbers of degrees bounded by θ and heights bounded by ν, then |e β − α| > e −2 42 θ 6 ln(ν+e e )(ln(ν)+ln ln(ν))
Note that e 2 42 θ 6 ln(ν+e e )(ln(ν)+ln ln(ν)) ≤ (ν + 16) 2 42 θ 6 (ln(ν)+ln ln(ν)) ≤ (ν + 16) 2 43 θ 6 ln(ν)
It is clear that the degree of an algebraic number is bounded by the degree of any polynomial which vanishes at that number. With respect to the height, by [1, Propositions 10.8 and 10.9], we have
H(α) ≤ 2 d ||P 1 || ≤ 2 d (d + 1) 1/2 H,
and, similarly, it follows that the same bound holds for H(β). Here, ||P 1 || stands for the norm 2 of the vector of the coefficients of P 1 .
The required inequality is deduced by taking θ = d, ν = 2 d (d + 1) 1/2 H, and using the bounds
2 d (d + 1) 1/2 H + 16 ≤ 2 d+1 (d + 1)H and ln(2 d (d + 1) 1/2 H) ≤ 5 4 d + ⌈log(H)⌉.
Step 2.(a) Applying the algorithm from Lemma 4 to the polynomial P 2 with ǫ = c 3 H+3 , we get intervals I j = (a j , b j ] with a j , b j ∈ Q and b j − a j < ǫ (1 ≤ j ≤ κ) such that β ∈ I j 0 for some j 0 . We determine the index j 0 by computing the feasible sign conditions for Der(P 2 ), X − a 1 , X − b 1 , . . . , X − a κ , X − b κ . Finally, we take w 1 = b j 0 . The complexity of this step is of order O(d 3 (log(H.3 H+3 .c −1 ) + log 3 (d))) = O(d 3 H + d 9 (d + log(H)) 2 ).
By the mean value theorem, the inequality |β − w 1 | < c 2. 3 H+2 implies that |e β − e w 1 | < c 2 .
Step 2.(b) Following [16,Lemma 14], in order to obtain w, we compute the Taylor polynomial centered at 0 of the function e x of order t := 8(⌈log(2/c)⌉ + 1 + H) specialized in w 1 . The complexity of this step is bounded by O(d 7 (d + log(H)) 2 + H).
Step 3. The fact that sign(w −α) = sign(e β −α) is a consequence of the inequalities |e β −α| > c and |e β − w| < c. In order to determine this sign, we compute the feasible sign conditions on Der(P 1 ), X − w and look for the one which corresponds to the Thom encoding of α.
The complexity of this step is of order O(d 3 log 3 (d)).
The overall complexity of this subroutine is O(d 3 H + d 9 (d + log(H)) 2 ).
The second subroutine is the following:
The resultant computation in Step 1 can be done within complexity O(ℓ(ℓ+δ) ω ) by interpolation, noticing that deg(S) ≤ ℓ. Applying Lemma 5, the complexity of Step 2 is O(ℓ 3 δ log(ℓ) log 2 (ℓδ)). Finally, taking into account that H(S) ≤ (ℓ + δ)! H(L) δ (2H(h)) ℓ , defining H := max{H(M ), (ℓ + δ)! H(L) δ (2H(h)) ℓ }, the complexity of Step 3 is O m max{η, ℓ} 3 H + max{η, ℓ} 6 (max{η, ℓ} + log(H)) 2 .
The overall complexity of the algorithm is of the same order as the complexity of Step 3.
Now we are ready to introduce the main algorithm of this section.
Algorithm E-SignDetermination
INPUT: Polynomials G ∈ Z[X, Y ], h ∈ Z[X], deg(h) > 0, L ∈ Z[X]
and Thom encodings σ L (α 1 ), . . . , σ L (α t ) of real roots α 1 , . . . , α t of L.
OUTPUT: The signs of G(α j , e h(α j ) ) for 1 ≤ j ≤ t.
Step 2. The complexity of the computation of R is of order O(M (max{ℓ, δ}) log(max{ℓ, δ})) and the realizable sign conditions on Der(L), R, G(X, 1) can be found within complexity O(ℓ 2 max{ℓ, d X } log(ℓ) log 2 (max{ℓ, d X })).
Step 3. In order to compute M (Y ), evaluate G(X, y) at sufficiently many values y, compute the corresponding determinants and interpolate. Taking into account that deg(M ) ≤ ℓd Y , the total cost of this step is of order O(ℓd Y (d X + ℓ) ω + M (ℓd Y ) log(ℓd Y )).
Step 4. The computation of the Thom encodings of the real roots of M can be done within O((ℓd Y ) 3 log 3 (ℓd Y )) operations. Then, we order the real roots of M by means of their Thom encodings within complexity of order O((ℓd Y ) 2 log(ℓd Y )).
Step 5. Following the proof of [1,Proposition 8.15], it follows that
H(M ) ≤ (ℓ + d X )!((d Y + 1)H(G)) ℓ H(L) d X . Recall that deg(M ) ≤ ℓd Y . (a) The complexity of this step is O((ℓd Y ) 4 (H + (ℓd Y ) 6 (ℓd Y + log(H)) 2 )), where H = max{(ℓ + δ)!H(L) δ (2H(h)) ℓ , (ℓ + d X )!H(L) d X ((d Y + 1)H(G)) ℓ }.
(b) By applying Lemma 4 to the polynomial M and a lower bound ǫ for the minimum distance between two different roots of M , we obtain pairwise disjoint intervals (a i , b i ] with rational endpoints such that λ i ∈ (a i , b i ] for i = 1, . . . , m. Following Lemma 3,
we can take ǫ = (ℓd Y ) − ℓd Y +2 2 (ℓd Y + 1) 1−ℓd Y 2 ((ℓ + d X )! H(L) d X ((d Y + 1)H(G)) ℓ ) 1−ℓd Y . Let w j := b i j .
The complexity of this step is O((ℓd Y ) 4 ((ℓ+d X ) log(ℓ+d X )+ℓ(log(H(G))+log(d Y ))+ d X log(H(L)))).
(c) We compute the coefficients of G(X, w j ) within complexity O(d X d Y ). Then, we compute the feasible sign conditions of Der(L), G(X, w j ), which enable us to determine the sign of G(α j , w j ), within O(ℓ 2 max{ℓ, d X } log(ℓ) log 2 (max{ℓ, d X }))) additional operations.
The overall complexity of the algorithm is O(t(ℓd Y ) 4 (H + (ℓd Y ) 6 (ℓd Y + log(H)) 2 )).
The previous complexity analysis leads to:
Proposition 18 Given polynomials G ∈ Z[X, Y ], h ∈ Z[X], deg(h) > 0, L ∈ Z[X]
with degrees bounded by d and height bounded by H, and Thom encodings σ L (α 1 ), . . . , σ L (α t ) of real roots α 1 , . . . , α t of L, we can determine #{1 ≤ j ≤ t : G(α j , e h(α j ) ) = 0} within complexity O(d 3 log 3 (d)). Moreover, the signs of G(α j , e h(α j ) ), for 1 ≤ j ≤ t, can be computed within complexity O(t 8 d d 3d+8 H 2d ).
Zero counting for E-polynomials
Here, we will apply Algorithm E-SignDetermination from the previous section as a subroutine in Algorithm ZeroCounting described in Section 4 to obtain a zero counting algorithm for E-polynomials with no calls to oracles. In order to estimate complexities we will need upper bounds for the degrees and heights of polynomials defining the successive derivatives of an E-polynomial.
Remark 19
For a Pfaffian function g(x) = G(x, e h(x) ), given by a polynomial G ∈ Z[X, Y ], we have that g ′ (x) = G(x, e h(x) ) is given by the polynomial G :
= ∂G ∂X + h ′ (X)Y ∂G ∂Y . If deg X (G) = d X , deg Y (G) = d Y and deg(h) = δ, we have that deg X ( G) ≤ δ − 1 + d X , deg Y ( G) = d Y H( G) ≤ H(G)(d X + d Y δ 2 H(h))
Applying these bounds recursively, we get that the successive derivatives of g can be obtained as Proof. In order to prove the theorem, we adapt Algorithm ZeroCounting introduced in Section 4 to count the number of zeros of an E-polynomial with no call to oracles. It suffices to show how to perform Steps 5 and 6 of the algorithm and estimate the complexity of the procedure.
g (ν) (x) = ν G(x, e h(x) ) for polynomials ν G ∈ Z[X, Y ] such that deg X ( ν G) ≤ ν(δ − 1) + d X , deg Y ( ν G) = d Y H( ν G) ≤ H(G) ν−1 j=0 (j(δ − 1) + d X + d Y δ 2 H(h)
Step 5 can be achieved by means of Steps 1 and 2 of Algorithm E-SignDetermination. As in this case deg(L) ≤ 10d 3 , the complexity is of order O(d 9 log 3 (d)).
To achieve Step 6 of the algorithm, we apply the algorithm E-SignDetermination to the polynomials defining the functions f I j ,i and their successive derivatives, for 0 ≤ i ≤ N . These functions are defined, up to signs, by the polynomials R i introduced in Notation 9, and ν R i , 0 ≤ i ≤ N, ν ∈ N.
Since deg Y ( F ) = deg Y (F ), then F 1 = lc(F ) 2 . F −lc( F )lc(F )F and so, deg X (F 1 ) ≤ 4d−1 and
H(F 1 ) ≤ 4d(d + 1)H 3 (d + d 3 H) ≤ 8(d + 1)d 4 H 4 .
Taking into account the determinantal formula for the subresultants, it follows that for every k, deg X (SRes k ) ≤ 5d 2 − 2d and H(SRes k ) ≤ (2d − 1)!2 5d−2 (d + 1) 2d−2 d 5d−1 H 5d−1 ≤ 3 2d−1 2 5d−2 d 9d−3 H 5d−1 , which are therefore, upper bounds for deg X (R i ) and H(R i ) for all i. Finally, recalling that L is the product of at most 2d polynomials of degrees at most 5d 2 − 2d that are coefficients of the subresultants SRes k , we have that
H(L) ≤ (5d 2 ) 2d−1 (3 2d−1 2 5d−2 d 9d−3 H 5d−1 ) 2d ≤ 3 4d 2 2 10d 2 −2d d 18d 2 −2d−2 H 10d 2 −2d .
Taking into account the bound for the multiplicity of a zero of a Pfaffian function from Lemma 15, we will apply the algorithm E-SignDetermination to the polynomials R i (0 ≤ i ≤ N ) and ν R i for ν ≤ 10d 3 −3d 2 , to determine the signs of the corresponding Pfaffian functions at the zeros of L. The bounds from Remark 19 applied to the polynomials R i imply that, for ν ≤ 10d 3 − 3d 2 ,
deg X ( ν R i ) ≤ (10d 3 − 3d 2 )(d − 1) + 5d 2 − 2d ≤ 10d 4 − 5d 3 H( ν R i ) ≤ H(R i )(10d 4 + (H − 5)d 3 ) 10d 3 −3d 2
Then, the complexity of applying the algorithm to each of these polynomials is of order
O(d 19 (H + d 24 (d 4 + log H) 2 )) where H ≤ (10d 4 + 5d 3 )!H(L) 10d 4 −5d 3 ((d + 1)3 2d−1 2 5d−2 d 9d−3 H 5d−1 (10d 4 + (H − 5)d 3 ) 10d 3 −3d 2 ) 10d 3 = (2dH) O(d 6 ) .
This sign computation is done for at most d(10d 3 − 3d 2 ) polynomials. Finally, for each interval I j , the signs sg(f I j ,i , α + j ) and sg(f I j ,i , α − j+1 ) are obtained easily following Definition 10. Therefore, the overall complexity of the algorithm is of order
(2dH) O(d 6 ) .
The previous procedure can be slightly modified to count algorithmically the total number of real zeros of an E-polynomial. To do this, we consider the signs of E-polynomials at +∞ and −∞.
Let By applying this remark, we conclude that the total number of zeros of an E-polynomial in R can be determined within the same complexity order as in Theorem 20.
g(x) = G(x, e h(x) ) be an E-polynomial. Assume G(X, Y ) = d Y j=0 a j (X)Y j with a d
Remark 22
The assumption Res Y (F, F ) = 0 in Theorem 20 can be removed by using the construction in the proof of Lemma 8. Taking into account the increase of height and degree, it follows that the overall complexity of the root counting algorithm is of order (2dH) d O(1) as stated in Theorem 1.
Bound for the size of roots
The following proposition provides an interval which contains all the zeros of an E-polynomial and whose endpoints are determined by the degrees and heights of the polynomials involved in its definition. Using this bound, applying successively our algorithm for zero counting, it is possible to separate and approximate the roots of an E-polynomial.
Proposition 23 Let f (x) = F (x, e h(x) ) be an E-polynomial defined by F ∈ Z[X, Y ] and h ∈ Z[X] such that deg(F ) ≤ d, deg(h) = δ > 0 and H(F ), H(h) ≤ H. Then, for every zero α ∈ R of f , we have that |α| ≤ M (d, δ, H) := 1 + (d + 1)H 2 max{(d + 1)(1 + 2H 2 ), 2⌊ 2d δ + 1⌋!}.
Proof. Let F (X, Y ) = d Y j=0 a j (X)Y j ∈ Z[X, Y ] with deg(a j ) ≤ d X for every 0 ≤ j ≤ d Y and a d Y = 0.
Let α ∈ R be a zero of f . If a d Y (α) = 0, then |α| ≤ r(a d Y ) < 1 + H (see Lemma 2) and so, the bound in the statement holds. Similarly, if a 0 (α) = 0, the bound holds.
Assume now that a d Y (α) = 0 and a 0 (α) = 0. Then e h(α) is a root of F (α, Y ) ∈ R[Y ] and e −h(α) is a root of Y d Y F (α, Y −1 ) ∈ R[Y ]. By Lemma 2, it follows that e 2h(α) < 1 +
0≤j≤d Y −1 a j (α) a d Y (α) 2 and e −2h(α) < 1 + 1≤j≤d Y a j (α) a 0 (α) 2 .
We are going to prove that, for α > M (d, δ, H), one of the previous inequalities fails to hold. Note that in both cases, the right hand side of the inequality is given by a rational function, 0≤j≤d Y a j (X) 2 a d Y (X) 2 and 0≤j≤d Y a j (X) 2 a 0 (X) 2 respectively, where the numerator and the denominator are integer polynomials of degrees at most 2d X and coefficients of size bounded by (d Y + 1)(d X + 1)H(F ) 2 and (d X + 1)H(F ) 2 respectively. Moreover, the degree of the denominator is less than or equal to the degree of the numerator. First, assume that the leading coefficient of h is positive.
Let p(X) = 0≤j≤d Y a 2 j (X) and q(X) = a 2 d Y (X) so that p(X) q(X) = 1 + 0≤j≤d Y −1 a j (X) a d Y (X)
2
. and let C > 0 be the quotient of the leading coefficients of p and q. Note that |C| ≤ (d Y + 1)H(F ) 2 .
If deg(p) = deg(q), for every x > max{r(q), r(p − (C + 1)q)}, we have that p(x) q(x) < C + 1.
On the other hand, for x > r(2h − ln(C + 1)), we have that e 2h(x) > C + 1. We conclude that, for x > max{r(q), r(p − (C + 1)q), r(2h − ln(C + 1)}, the inequality e 2h(x) > p(x) q(x) holds.
If deg(p) > deg(q), let d 0 := deg(p)−deg(q). For x > max{r(q), r(p−2Cx d 0 q)}, we have that p(x) q(x) < 2Cx d 0 . Note that e 2h(x) > e x δ for x > r(2h − X δ ). As e x δ >
⌊ d 0 δ +1⌋ k=0 1 k! x δk > 2Cx d 0 for x > r( ⌊ d 0 δ +1⌋ k=0 1 k! X δk − 2CX d 0 )
, it follows that p(x) q(x) < e 2h(x) for x > max{r(q), r(p − 2Cx d 0 q), r(
⌊ d 0 δ +1⌋ k=0 1 k! X δk − 2CX d 0 )}.
Using again Lemma 2, we obtain:
• r(q) < 1 + (d X + 1)H(F ) 2
• r(p − (C + 1)q)) < 1 + (d X + 1)H(F ) 2 (d Y + (d Y + 1)H(F ) 2 )
• r(2h − ln(C + 1)) < 1 + H(h) + 1 2 ln((d Y + 1)H(F ) 2 + 1)
• r(p − 2CX d 0 q) < 1 + (d X + 1)(d Y + 1)H(F ) 2 (1 + 2H(F ) 2 )
• r(2h − X δ ) < 1 + 2H(h)
• r ⌊ d 0 δ +1⌋ k=0 1 k! X δk − 2CX d 0 < 1 + 2⌊ 2d X δ + 1⌋!(d Y + 1)H(F ) 2
and, therefore, we conclude that, for α > M (d, δ, H), the following inequality holds
e 2h(α) > 1 + 0≤j≤d Y −1 a j (α) a d Y (α) 2 .
If the leading coefficient of h is negative, applying the previous argument to −h, we have that, for α > M (d, δ, H), the following inequality holds
e −2h(α) > 1 + 1≤j≤d Y a j (α) a 0 (α) 2 .
Finally, noticing that α is a zero of F (x, e h(x) ) if and only if −α is a zero of F (−x, e h(−x) ) we conclude that every zero α of f satisfies α ≥ −M (d, δ, H).
Theorem 1
1Let f (x) = F (x, e h(x) ) be an E-polynomial defined by polynomials F ∈ Z[X, Y ] and h ∈ Z[X] with degrees bounded by d and coefficients of absolute value at most H, and let I = [a, b] be a closed interval or I = R. There is an algorithm that computes the number of zeros of f in I within complexity (2dH) d O(1) .
Proof. The algorithm works recursively. Starting with the interval J = (−(1 + H(p)), 1 + H(p)], which contains all the real roots of p (see Lemma 2), at each intermediate step, finitely many intervals are considered. Given an interval J = (a, b] with {p = 0} ∩ J = ∅ and |J| > ǫ, the procedure runs as follows: • Let c = a+b 2 and J r = (c, b]. • If p(c) = 0, let J l = (a, c]. • If p(c) = 0 and c − ǫ > a, let I = (c − ǫ, c] and J l = (a, c − ǫ]. If p(c) = 0 and c − ǫ ≤ a, take I = (a, c]. (Note that, in any case, I contains a real root of p and has length at most ǫ.)
the multiplicity of c as a zero of f . For a sequence of non-zero analytic functions f = (f 0 , . . . , f N ) defined in J, we write v(f , c + ) for the number of variations in sign in (sg(f 0 , c + ), . . . , sg(f N , c + )) and v(f , c − ) for the number of variations in sign in (sg(f 0 , c − ), . . . , sg(f N , c − )).
13 With the assumptions and notation of Proposition 11, if, in addition, the closed interval [a, b] is contained in the domain of ϕ, the number of zeros of the function f in the open interval
Y = 0 and let j 0 = min{j : a j = 0}. We definesg(g, +∞) = sign(lc(a j 0 )) if lc(h) < 0 sign(lc(a d Y )) if lc(h) > 0 and sg(g, −∞) = sign((−1) deg(a j 0 ) lc(a j 0 )) if (−1) deg(h) lc(h) < 0 sign((−1) deg(a d Y ) lc(a d Y )) if (−1) deg(h) lc(h) > 0For a sequence of E-polynomials f = (f 0 , . . . , f N ), we write v(f , +∞) for the number of variations in sign in (sg(f 0 , +∞), . . . , sg(f N , +∞)) and v(f , −∞) for the number of variations in sign in (sg(f 0 , −∞), . . . , sg(f N , −∞)). Remark 21 Following Notation 9 and Definition 10, let f I +∞ and f I −∞ be Sturm sequences for f (x) = F (x, e h(x) ) in the intervals I +∞ = (M, +∞) and I −∞ = (−∞, −M ) where M is an upper bound for the absolute values of the roots of τ i for i = 0, . . . , N and ρ i for i = 2, . . . , N + 1. Then, the number of zeros of f in I +∞ equals v(f , M + ) − v(f , +∞) and the number of zeros of f in I −∞ equals v(f , −∞) − v(f , −M − ).
). Let F (X, Y ) and G(X, Y ) be polynomials in Z[X, Y ] of degrees d and e in the variable Y respectively. Assume e < d. Following [1, Notation 8.33], for every −1 ≤ j ≤ d, let SRes j be the jth signed subresultant of F and G considered as polynomials in Z[X][Y ]. By the structure theorem for subresultants (see [1, Theorem 8.34 and Proposition 8.40]), we have that
) .
)Now, we can state the main result of this section.Theorem 20 Let f (x) = F (x, e h(x) ) be an E-polynomial defined by F ∈ Z[X, Y ] and h ∈ Z[X] with deg(F ), deg(h) ≤ dand H(F ), H(h) ≤ H, and let [a, b] be a closed interval. Assume that Res Y (F, F ) = 0. There is an algorithm that computes the number of zeros of f in [a, b] within complexity (2dH) O(d 6 ) .
Acknowledgements. The authors wish to thank the referees for their detailed reading and helpful comments.Algorithm RootBoxINPUT: A polynomial h ∈ Z[X], an algebraic number α ∈ R such that h(α) = 0, given by its Thom encoding as a root of a polynomial L ∈ Z[X], and a polynomial M ∈ Z[X] together with the ordered list of Thom encodings of all its real roots λ 1 < λ 2 < · · · < λ m .Proof of correctness and complexity analysis:Note that h(α) is a root of the polynomial S ∈ Z[T ] computed in Step 1. Therefore, in Step 2, the sign condition on Der(L), S(h), S ′ (h), . . . , S (deg(S)) (h) having the Thom encoding of α as a root of L in the first coordinates has the Thom encoding of h(α) as a root of S in the last ones.Assume that deg(L) ≤ ℓ, deg(h) ≤ δ and deg(M ) ≤ η.
For every 1 ≤ j ≤ t, determine whether G(α j , Y ) ≡ 0. If this is the case, the sign of G(α j , e h(α j ) ) is 0For every 1 ≤ j ≤ t, determine whether G(α j , Y ) ≡ 0. If this is the case, the sign of G(α j , e h(α j ) ) is 0.
Going through the list, determine the sign of G(α j , e h(α j ) ) = G(α j , 1) for every j such that G(α j , Y ) ≡ 0 and R. Compute R = gcd(L, h) and the list of realizable sign conditions on Der(L). R, G(X, 1). α j ) = 0Compute R = gcd(L, h) and the list of realizable sign conditions on Der(L), R, G(X, 1). Going through the list, determine the sign of G(α j , e h(α j ) ) = G(α j , 1) for every j such that G(α j , Y ) ≡ 0 and R(α j ) = 0.
. Compute M (Y ) := Res X. Compute M (Y ) := Res X (L(X), G(X, Y )).
Compute the Thom encodings of the real roots of M and order them: λ 1 < · · · < λ m. Compute the Thom encodings of the real roots of M and order them: λ 1 < · · · < λ m .
For every 1 ≤ j ≤ t such that G(α j , Y ) ≡ 0 and R(α j ) = 0: (a) Determine the index 0 ≤ i j ≤ m such that λ i j < e h(α j ) < λ i j +1 by applying subroutine RootBox. where λ 0 := −∞ and λ m+1 := +∞For every 1 ≤ j ≤ t such that G(α j , Y ) ≡ 0 and R(α j ) = 0: (a) Determine the index 0 ≤ i j ≤ m such that λ i j < e h(α j ) < λ i j +1 by applying subroutine RootBox, where λ 0 := −∞ and λ m+1 := +∞.
Compute the sign of the polynomial G(X, w j ) at X = α j . This is the sign of G(α j , e h(α j ) ). Compute the sign of the polynomial G(X, w j ) at X = α j . This is the sign of G(α j , e h(α j ) ).
deg(L) ≤ ℓ and deg(h) ≤ δ. Due to Lindemann's theorem, if α ∈ R is an algebraic number and h(α) = 0, then e h(α) is transcendental over Q. Therefore, for an algebraic number α ∈ R, G(α, e h(α) ) = 0 if and only if either G(α, Y ) ≡ 0 or h(α) = 0 and G(α, 1) = 0. Then, Steps 1 and 2 enable us to determine all the indices j. Assume that deg X (G) ≤ d X , deg Y (G) ≤ d Y ,such that G(α j , e h(α j ) ) = 0Proof of correctness and complexity analysis: Assume that deg X (G) ≤ d X , deg Y (G) ≤ d Y , deg(L) ≤ ℓ and deg(h) ≤ δ. Due to Lindemann's theorem, if α ∈ R is an algebraic number and h(α) = 0, then e h(α) is transcendental over Q. Therefore, for an algebraic number α ∈ R, G(α, e h(α) ) = 0 if and only if either G(α, Y ) ≡ 0 or h(α) = 0 and G(α, 1) = 0. Then, Steps 1 and 2 enable us to determine all the indices j such that G(α j , e h(α j ) ) = 0.
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| []
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[
"BIG BANG NUCLEOSYNTHESIS AND THE HELIUM ISOTOPE RATIO",
"BIG BANG NUCLEOSYNTHESIS AND THE HELIUM ISOTOPE RATIO"
]
| [
"Ryan J Cooke "
]
| []
| []
| The conventional approach to search for departures from the standard model of physics during Big Bang Nucleosynthesis involves a careful, and subtle measurement of the mass fraction of baryons consisting of helium. Recent measurements of this quantity tentatively support new physics beyond the standard model but, historically, this method has suffered from hidden systematic uncertainties. In this letter, I show that a combined measurement of the primordial deuterium abundance and the primordial helium isotope ratio has the potential to provide a complementary and reliable probe of new physics beyond the standard model. Using the recent determination of the primordial deuterium abundance and assuming that the measured pre-solar 3 He/ 4 He meteoritic abundance reflects the primordial value, a bound can be placed on the effective number of neutrino species, N eff (BBN) = 3.01 +0.95 −0.76 (with 95 per cent confidence). Although this value of N eff supports the standard model, it is presently unclear if the pre-solar 3 He/ 4 He ratio reflects the primordial value. New astrophysical measurements of the helium isotope ratio in near-pristine environments, together with updated calculations and experimental values of several important nuclear reactions (some of which are already being attempted), will lead to much improved limits on possible departures from the standard model. To this end, I describe an analysis strategy to measure the 3 He i flux emitted from nearby low metallicity H ii regions. The proposed technique can be attempted with the next generation of large telescopes, and will be easier to realize in metal-poor H ii regions with quiescent kinematics. | 10.1088/2041-8205/812/1/l12 | [
"https://arxiv.org/pdf/1510.02801v1.pdf"
]
| 118,670,689 | 1510.02801 | 9e2563bff89615cabf34e0fd4de2638caeb10115 |
BIG BANG NUCLEOSYNTHESIS AND THE HELIUM ISOTOPE RATIO
October 13, 2015 October 13, 2015
Ryan J Cooke
BIG BANG NUCLEOSYNTHESIS AND THE HELIUM ISOTOPE RATIO
October 13, 2015 October 13, 2015Draft version Preprint typeset using L A T E X style emulateapj v. 5/2/11 Draft versionSubject headings: cosmological parameters -galaxies: abundances -methods: numerical -primordial nucleosynthesis
The conventional approach to search for departures from the standard model of physics during Big Bang Nucleosynthesis involves a careful, and subtle measurement of the mass fraction of baryons consisting of helium. Recent measurements of this quantity tentatively support new physics beyond the standard model but, historically, this method has suffered from hidden systematic uncertainties. In this letter, I show that a combined measurement of the primordial deuterium abundance and the primordial helium isotope ratio has the potential to provide a complementary and reliable probe of new physics beyond the standard model. Using the recent determination of the primordial deuterium abundance and assuming that the measured pre-solar 3 He/ 4 He meteoritic abundance reflects the primordial value, a bound can be placed on the effective number of neutrino species, N eff (BBN) = 3.01 +0.95 −0.76 (with 95 per cent confidence). Although this value of N eff supports the standard model, it is presently unclear if the pre-solar 3 He/ 4 He ratio reflects the primordial value. New astrophysical measurements of the helium isotope ratio in near-pristine environments, together with updated calculations and experimental values of several important nuclear reactions (some of which are already being attempted), will lead to much improved limits on possible departures from the standard model. To this end, I describe an analysis strategy to measure the 3 He i flux emitted from nearby low metallicity H ii regions. The proposed technique can be attempted with the next generation of large telescopes, and will be easier to realize in metal-poor H ii regions with quiescent kinematics.
INTRODUCTION
The standard model of cosmology and particle physics provides our current best physical description of the Universe. Two of the most remarkable predictions of this model, now experimentally confirmed, include the production of the lightest chemical elements during the first minutes after the Big Bang (known as Big Bang Nucleosynthesis, or BBN; Alpher, Bethe, & Gamow 1948), and the existence of a relic cosmic microwave background (CMB) radiation (Penzias & Wilson 1965).
Seconds after the Big Bang, ordinary matter consisted mostly of free neutrons and protons. As the Universe expanded and cooled, some neutrons and protons combined to form deuterium (D); soon after, BBN entered full production. Nearly all of the deuterium nuclei were fused to form the more tightly bound helium-4 ( 4 He) nuclide. Other light nuclides were also produced, but in much lower abundance, including the lighter isotope of helium ( 3 He) and a very small amount of lithium-7 ( 7 Li). After ∼20 minutes, the nuclear reactions were quenched by the declining temperature and density of the Universe, and BBN was complete (for a review, see Steigman 2007;Cyburt et al. 2015).
The relative abundances of the primordial nuclides produced during BBN are sensitive to the universal expansion rate, and the cosmic density of ordinary matter (i.e. the cosmic baryon density, Ω B,0 ). The expansion rate is set by the total energy density of CMB photons and relativistic particles, which for the standard model includes electrons, positrons and three flavors of neutrino. Typically, the various contributions to the expansion rate are collected and parameterized by an "effective number of neutrino species", N eff , where the standard model value corresponds to N eff = 3.046 (Mangano et al. 2005). Deviations from this value could indicate new physics not presently captured by the standard model.
The dependence of each primordial element ratio on Ω B,0 and N eff can be determined from detailed numerical calculations of BBN. Historically, each primordial nuclide is compared relative to hydrogen. For example, the primordial deuterium abundance (D/H) has commonly been used to estimate the cosmic baryon density (Wagoner, Fowler, & Hoyle 1967), while the mass fraction of all baryons consisting of 4 He (Y P ) depends strongly on N eff and is relatively insensitive to Ω B,0 (Steigman, Schramm, & Gunn 1977).
Similar measurements of Ω B,0 and N eff are also available through a careful analysis of the CMB temperature fluctuations, recently recorded in exquisite detail by the Planck Collaboration et al. (2015). Their analysis provides Ω B,0 h 2 (CMB) = 0.02229 +0.00039 −0.00040 and N eff (CMB) = 3.04±0.36 (95 per cent confidence limits), where h is the Hubble constant in units of 100 km s −1 Mpc −1 . Recent measurements of D/H (Pettini & Cooke 2012;Cooke et al. 2014) provide an independent bound on Ω B,0 h 2 (BBN) that is in good agreement with the values determined from the CMB. However, the latest measurements of Y P suggest tentative, but inconclusive evidence favoring N eff (BBN) > N eff (CMB). Specifically, the standard model value of the primordial 4 He mass fraction, inferred from the CMB, is Y P = 0.24668 ± 0.00007 (68 per cent confidence limits; Planck Collaboration et al. 2015). A recent survey conducted by Izotov, Thuan, & Guseva (2014) found Y P = 0.2551 ± 0.0022, which constitutes a 3.8σ deviation from the standard model expectation. Using a subset of the Izotov, Thuan, & Guseva (2014) survey data, Aver, Olive, & Skillman (2015) derived a value Y P = 0.2449 ± 0.0040, which is more consistent with the standard model. Thus, the different values of Y P derived by these authors might be due to systematic differences in the analysis strategies adopted. At present, it is unclear if additional, unaccounted for systematic uncertainties are biasing the best measurements of Y P , and therefore masquerading as non-standard physics (see e.g. Figure 8 from Steigman 2012). To alleviate this concern, a new, sensitive and reliable probe is required to complement the measurement of Y P in the search for possible departures from standard BBN. Until now, the power of combining measurements of the primordial D/H and 3 He/ 4 He ratios has not been fully appreciated.
In this letter, I investigate the sensitivity and observational prospects for uncovering new physics beyond the standard model using the D/H and 3 He/ 4 He abundance ratios set by BBN. In Section 2, I derive the dependence of these ratios on Ω B,0 h 2 and N eff , and summarize the current best observational determinations. In Section 3, I discuss the importance of obtaining new experimental values of several important reaction cross sections. I also discuss the future potential of measuring the He isotope ratio in near-primordial environments, before highlighting the main conclusions of this work in Section 4.
THE BBN ISOTOPES
The relationships between Ω B,0 h 2 , N eff , and the primordial abundances of D/H and 3 He/ 4 He are derived from calculations (Iocco et al. 2009) that use the PArthENoPE code (Pisanti et al. 2008). These results are in good agreement with recent calculations (Cyburt et al. 2015) using the latest determinations of the neutron lifetime and the reaction rate cross-sections (Xu et al. 2013). The primordial hydrogen and helium isotope number abundance ratios are given by:
y H,He = n m a nm ω n B ∆N m eff(1)
where y H = 10 5 × D/H, y He = 10 4 × 3 He/ 4 He, ω B = Ω B,0 h 2 , and ∆N eff = N eff − 3.046. The coefficients a nm are provided in Table 1 and Table 2. I adopt a conservative 5% standard error in y H (Cyburt et al. 2015) and a 3% standard error in y He (Steigman 2007) due to uncertainties in the nuclear reaction rates relevant for the BBN calculations. As discussed further below, this uncertainty dominates the current error budget on Ω B,0 h 2 (BBN) and N eff (BBN). The D/H number abundance ratio can be determined in near-primordial environments to high precision. The employed technique (Adams 1976) requires a rare, chance alignment between a near-pristine "cloud" of neutral gas and a bright, usually unrelated, background source (typically a quasar). The foreground gas cloud imprints the Lyman series absorption lines of neutral D and H on the light from the unrelated background quasar, allowing the relative column density of D i and H i to be measured. In this work, I adopt the latest determination D/H = (2.53 ± 0.04) × 10 −5 by Cooke et al. (2014), derived from the highest precision measurements currently available. The helium isotope ratio ( 3 He/ 4 He), on the other hand, has received relatively little attention from both the theoretical and observational BBN community. The current best estimate of the He isotope ratio is measured from the socalled 'quintessence' phase (He-Q) from solar system meteorite samples (Lewis, Srinivasan & Anders 1975). The isotope ratios of the noble gases that are incorporated into 'phase Q' are believed to represent the values that preceded the formation of the solar system, when the Universe was less than 67 per cent of its current age (i.e. less than ∼ 9 billion years after the Big Bang). Although He-Q may not represent the primordial 3 He/ 4 He (see Section 3.2), this estimate provides an illustrative value that can be used in the present work.
Extracting the noble gases contained in phase Q requires a two stage process (Busemann, Baur, & Wieler 2000). In the first step, a meteorite is exposed to hydrochloric and hydrofluoric acid (i.e. demineralization), which leaves a resistant carbonaceous residue. This residue is subsequently exposed to nitric acid, which oxidizes the (presently unknown) carrier 'Q', thereby releasing the noble gases, including both He isotopes. The ideal meteorites for determining the phase Q isotopic abundances are those that have the lowest cosmic ray exposure age, since the isotope ratio will be less affected by cosmogenic He production. The best current determination of the He-Q isotope ratio comes from the Isna meteorite (Busemann, Baur, & Wieler 2000, 2001, which has a remarkably short cosmic ray exposure age, T 21 = 150 000 yrs (Scherer & Schultz 2000). The Isna He-Q isotope ratio is 3 He/ 4 He = (1.23 ± 0.02) × 10 −4 .
The BBN contours for the above observations and calculations are presented in the left panel of Fig. 1. This figure highlights some of the benefits of using the He isotope ratio to test the standard model; the D/H and 3 He/ 4 He abundance ratios offer almost orthogonal bounds on Ω B,0 h 2 and N eff . The determination of N eff therefore depends almost equally on D/H and 3 He/ 4 He, unlike the combination of D/H+Y P , where Y P drives the determination of N eff . Moreover, D/H, Y P , and 3 He/ 4 He all exhibiting a very different dependence on Ω B,0 h 2 and N eff , and their combined measurement will provide a highly complementary approach to identify physics beyond the standard model.
Assuming that the pre-solar 3 He/ 4 He ratio reflects the primordial value, and adopting a conservative uncertainty in the BBN calculations (5% for y H , 3% for y He ), the following limits are placed on the baryon density and the effective number of neutrino species: Ω B,0 h 2 (BBN) = 0.0227 +0.0016 −0.0013 and N eff (BBN) = 3.01 +0.95 −0.76 (95 per cent confidence limits). These results agree remarkably well with the standard model of cosmology and particle physics, and the Planck CMB analysis He/ 4 He (green) contours are displayed, where dark and light shades represent the 68 and 95 per cent confidence contours respectively. Note that the measured 3 He/ 4 He isotope ratio adopted here is derived from solar system meteorite samples, and is intended to be illustrative; this determination of 3 He/ 4 He may not reflect the primordial value. The gray contours illustrate the results from the Planck satellite observations of the CMB temperature fluctuations (Planck Collaboration et al. 2015). The red contours show the combined D/H and 3 He/ 4 He (BBN only) confidence regions. The BBN contours in the left panel use a 5% and 3% standard error respectively for the BBN calculations, due to uncertainties in the nuclear reaction rates. The right panel illustrates the same contours, now assuming a 1% uncertainty on the nuclear reaction rates, and the same observational measures. Note that the BBN contours in the right panel are comparable in size to the latest CMB results.
(gray contours in Fig. 1).
FUTURE PROSPECTS
Improving the BBN Reaction Rates
By improving a few key BBN nuclear reaction rates (discussed further below), significantly tighter bounds on Ω B,0 h 2 (BBN) and N eff (BBN) will be possible using observational measures of comparable precision. For example, assuming the same central values for the BBN reaction rates, a 1 per cent standard error in these rates would reduce the uncertainty on N eff (BBN) by a factor of 2. The BBN contours for this level of uncertainty in the reaction rates are illustrated in the right panel of Fig. 1, and are competitive with that of the Planck CMB experiment.
In order for this level of precision to be realized, several BBN reaction rates must be revisited and measured using modern facilities. The three most uncertain reaction rates that determine the D/H abundance are d(p,γ) 3 He, d(d,n) 3 He, and d(d,p) 3 H (Fiorentini et al. 1998;Nollett & Burles 2000;Cyburt 2004;Serpico et al. 2004). For the He isotope ratio, the most important reactions are d(p,γ) 3 He and 3 He(d,p) 4 He. New experimental data for the crucial d(p,γ) 3 He reaction, which contributes the dominant uncertainty for both the D/H and 3 He/ 4 He abundance ratios, are currently being acquired by the Laboratory for Underground Nuclear Astrophysics (LUNA; Broggini et al. 2010;Di Valentino et al. 2014). New data for the remaining reaction rates are now needed.
Measuring 3 He/ 4 He in near-pristine environments
The Isna He-Q measurement currently provides our best indication of the primordial He isotope ratio. However, it remains unclear if He-Q is a true reflection of the primordial BBN value. A measurement of the He isotope ratio in a nearpristine environment, where the post-BBN chemical evolution of 3 He is expected to be negligible, will provide a more reliable determination of the truly primordial 3 He / 4 He ratio. In principle, such a measurement should be possible in known metal-poor H ii regions by comparing the profiles of two He i emission lines, since the isotope shift is different for all He i transitions (a relative shift of up to 40 km s −1 for the optical and near-infrared emission lines; Morton, Wu, & Drake 2006). For example, the 3 He 3D → 2P singlet transition (He i λ6679) has an isotope shift of +22.5 km s −1 , and the 3 He 2P → 2S triplet transition (He i λλ10833) has an isotope shift of +35.2 km s −1 .
The above combination of emission lines is perhaps the best suited to obtain a direct measurement of the 3 He i flux from a metal-poor H ii region. Assuming a flux ratio of F ( 3 He/ 4 He) = 10 −4 , the centroid of the 3 He i line would need to be ideally 5σ from the centroid of the 4 He i line to ensure that the 3 He i emission dominates the signal. For the 1.0833 µm He i line, this 5σ shift corresponds to a 4 He i profile with a full width at half maximum, FWHM 18 km s −1 . For comparison, thermal broadening of the 4 He line at a temperature of 15,000 K, which is typical of H ii regions (Izotov, Thuan, & Guseva 2014; Aver, Olive, & Skillman 2015), would produce a profile with FWHM 13 km s −1 .
To detect the 3 He i feature at 5σ confidence, the signalto-noise (S/N) ratio of the 4 He i feature must exceed 5 × F ( 4 He/ 3 He), or S/N 500. This S/N requirement would need to be increased if the stellar continuum of the target galaxy is bright, or if the red wing of the 4 He i emission line extends over the 3 He i emission line. In any case, the primary requirement is that the 4 He i profile is smooth and narrow. To confirm that an observed feature is due to 3 He i emission, and not a coincident weak 4 He i emission feature, another He i emission line with a different isotope shift (e.g. He i λ6679) can be inspected. An example comparison of two He i emission lines is presented in Fig. 2 for a Gaussian profile with FWHM = 13 km s −1 .
Measuring the 3 He i flux from a nearby metal-poor H ii region might be possible with the next generation of 30+ m telescopes. The current instrument concepts of the HIRES and HROS echelle spectrographs, to be mounted on the European Extremely Large Telescope and the Thirty Meter Telescope respectively, are well-designed for this experiment. The wavelength range covered by these instrument concepts includes the best available He i emission lines with a single instrument setup. Moreover, the large telescope aperture greatly facilitates the final S/N that can be achieved. However, to realize this measurement, emphasis should now be placed on Both profiles are normalized to the peak flux of the profile. The vertical red dashed lines at a velocity of +35.2 km s −1 in both panels indicate the He isotope shift of the λλ10833 line. Note that the He isotope shift of the λ6679 line is +22.5 km s −1 . Therefore, the detection of an emission feature at +35.2 km s −1 in the λλ10833 profile, together with the absence of a feature at the same velocity in the λ6679 profile, can be used to confirm the identification as 3 He i emission. discovering suitable low metallicity H ii regions with quiescent kinematics.
SUMMARY AND CONCLUSIONS
The primordial abundance of helium-3 ( 3 He/H), is generally considered to provide very little constraining power on the physics of BBN, since it is less sensitive to the baryon density than the primordial deuterium abundance, is less affected by the expansion rate than the 4 He mass fraction, and a reliable estimate of the primordial value has not been measured. This paper highlights the possibility of combining the deuterium abundance and the helium isotope ratio ( 3 He/ 4 He) to simultaneously estimate Ω B,0 h 2 and N eff at the time of BBN. The following conclusions are drawn:
(i) The D/H and 3 He/ 4 He isotope ratios set by BBN offer almost orthogonal bounds on Ω B,0 h 2 and N eff , unlike Y P which is essentially independent of Ω B,0 h 2 . Therefore, by combining D/H and 3 He/ 4 He, evidence for physics beyond the standard model is less influenced by a single measurement.
(ii) If the pre-solar meteoritic value of 3 He/ 4 He reflects the primordial value, the following bounds can be placed on the baryon density, Ω B,0 h 2 (BBN) = 0.0227 +0.0016 −0.0013 , and the effective number of neutrino species, N eff (BBN) = 3.01 +0.95 −0.76 with 95 per cent confidence, assuming a conservative uncertainty for the measured BBN reaction rates (5% for y H and 3% for y He ).
(iii) In order to achieve bounds on Ω B,0 h 2 and N eff that are competitive with the latest results from the Planck satellite, several important BBN reaction rates must be redetermined, including d(p,γ) 3 He, d(d,n) 3 He, d(d,p) 3 H, and 3 He(d,p) 4 He.
(iv) It is presently unclear to what degree the He-Q isotope ratio is affected by post-BBN nucleosynthesis. It is therefore necessary to obtain a measurement of the He isotope ratio in a near-pristine environment. Nearby metal-poor H ii regions could be the most promising systems to estimate the primordial 3 He/ 4 He ratio. The He i optical and near-infrared emission lines typically seen in H ii regions exhibit a variety of isotope shifts (up to ∼40 km s −1 ), allowing the 3 He emission to be unambiguously identified. With a view to this goal, I present a possible strategy to measure the 3 He i flux from nearby metal-poor H ii regions.
Although such a delicate measurement will have to wait for the next generation of 30+ m telescope facilities, it is critical that the relevant BBN reaction rates are now measured with high precision. Once this goal is achieved, the combined information provided by the primordial D/H abundance and the 3 He/ 4 He isotope ratio has the potential to deliver a reliable probe of possible departures from the standard model of physics during the early Universe.
Fig. 1 .
1-Combined confidence contours of the baryon density and the effective number of neutrino species. The D/H (blue) and 3
Fig. 2 .
2-Synthetic line profiles for the He i triplet (λλ10833; left panel) and singlet (λ6679; right panel) emission.
TABLE 1 H
1isotope ratio coefficientsm
n
0
1
2
3
0
29.428
3.7763
−0.18882
0.045346
1
−3490.0
−437.76
38.547
−8.7506
2
1.8127 × 10 5
21897.0
−2849.5
624.51
3 −4.4923 × 10 6 −5.1455 × 10 5
90285.0
−19402.0
4
4.3247 × 10 7
4.6528 × 10 6
−1.0411 × 10 6 2.2120 × 10 5
TABLE 2
2He isotope ratio coefficientsm
n
0
1
2
3
0
50.278
−2.6670
0.11296
0.11005
1
−4705.2
519.14
−0.45387
−30.061
2
2.3843 × 10 5
−41405.0
−221.73
2564.1
3 −5.9383 × 10 6
1.4212 × 10 6
5602.0
−90887.0
4
5.7966 × 10 7
−1.7735 × 10 7
7849.5
1.1535 × 10 6
I thank M. Pettini and J. X. Prochaska for useful discussions about the work described in this paper, and for suggesting comments on an earlier draft. I am grateful to the anonymous referee who provided constructive comments that improved the presentation and clarity of this work. R. J. C. is currently supported by NASA through Hubble Fellowship grant HST-HF-51338.001-A, awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. The majority of the analysis and figures presented in this paper were prepared using publicly available python packages, including Astropy, NumPy, SciPy, Cython, and Matplotlib (Astropy Collaboration et al. 2013; van der Walt, Colbert, & Varoquaux 2011;Behnel et al. 2011;Hunter 2007).
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| []
|
[
"Theory of Tunneling Effect in 1D AIII-class Topological Insulator (Nanowire) Proximity Coupled with a Superconductor",
"Theory of Tunneling Effect in 1D AIII-class Topological Insulator (Nanowire) Proximity Coupled with a Superconductor"
]
| [
"Ryoi Ohashi \nDepartment of Applied Physics\nNagoya University\n464-8603NagoyaJapan\n",
"Yukio Tanaka \nDepartment of Applied Physics\nNagoya University\n464-8603NagoyaJapan\n",
"Keiji Yada \nDepartment of Applied Physics\nNagoya University\n464-8603NagoyaJapan\n"
]
| [
"Department of Applied Physics\nNagoya University\n464-8603NagoyaJapan",
"Department of Applied Physics\nNagoya University\n464-8603NagoyaJapan",
"Department of Applied Physics\nNagoya University\n464-8603NagoyaJapan"
]
| []
| We study the tunneling effect in an AIII-class insulator proximity coupled with a spin-singlet s-wave superconductor, in which three phases are characterized by the integer topological invariant N . By solving the Bogoliubov-de Gennes equation explicitly, we analytically obtain a normal reflection coefficient R σσ ′ and an Andreev reflection coefficient A σσ ′, and derive a charge conductance formula, where σ(σ ′ ) is the spin index of a reflected (injected) wave. The resulting conductance indicates a wide variety of line shapes: (i)gap structure without coherence peaks for N = 0, (ii)quantized zero-bias conductance peak (ZBCP) with height 2e 2 /h for N = 1, and (iii)ZBCP spitting for N = 2. At zero bias voltage eV = 0, σσ ′ R σσ ′ = σσ ′ A σσ ′ is satisfied and the spin direction of an injected electron is rotated at approximately 90 • for the N = 1 state. Meanwhile, A σσ ′ = 0 is satisfied for the N = 2 state, and the spin rotation angle can become 180 • . | 10.7566/jpsj.89.054701 | [
"https://arxiv.org/pdf/1904.01586v2.pdf"
]
| 102,480,815 | 1904.01586 | f19319252cdaf27ba4af1d798c55e835b4a5f80a |
Theory of Tunneling Effect in 1D AIII-class Topological Insulator (Nanowire) Proximity Coupled with a Superconductor
2 Apr 2019
Ryoi Ohashi
Department of Applied Physics
Nagoya University
464-8603NagoyaJapan
Yukio Tanaka
Department of Applied Physics
Nagoya University
464-8603NagoyaJapan
Keiji Yada
Department of Applied Physics
Nagoya University
464-8603NagoyaJapan
Theory of Tunneling Effect in 1D AIII-class Topological Insulator (Nanowire) Proximity Coupled with a Superconductor
2 Apr 2019
We study the tunneling effect in an AIII-class insulator proximity coupled with a spin-singlet s-wave superconductor, in which three phases are characterized by the integer topological invariant N . By solving the Bogoliubov-de Gennes equation explicitly, we analytically obtain a normal reflection coefficient R σσ ′ and an Andreev reflection coefficient A σσ ′, and derive a charge conductance formula, where σ(σ ′ ) is the spin index of a reflected (injected) wave. The resulting conductance indicates a wide variety of line shapes: (i)gap structure without coherence peaks for N = 0, (ii)quantized zero-bias conductance peak (ZBCP) with height 2e 2 /h for N = 1, and (iii)ZBCP spitting for N = 2. At zero bias voltage eV = 0, σσ ′ R σσ ′ = σσ ′ A σσ ′ is satisfied and the spin direction of an injected electron is rotated at approximately 90 • for the N = 1 state. Meanwhile, A σσ ′ = 0 is satisfied for the N = 2 state, and the spin rotation angle can become 180 • .
I. INTRODUCTION
The tunneling effect in a normal metal/superconductor (N/S) junction has been considered to be a basic quantum phenomenon since the discovery of superconductivity. Blonder, Tinkham, and Klapwijk (BTK) established that tunneling conductance can be expressed by the coefficients of the Andreev reflection and normal reflection in ballistic junctions [1]. By extending the BTK theory, a conductance formula has been developed for unconventional superconductors [2,3], where a pair potential changes sign on the Fermi surface and possesses the so-called surface Andreev-bound states (SABSs). This formula has clarified that the sharp zero-bias conductance peak (ZBCP) observed in many experiments of high T C cuprate [4][5][6][7][8][9][10] stems from the zero energy surface Andreev-bound states (ZESABSs) [11][12][13][14][15] in a d-wave superconductor. Applying this formula for a spin-triplet chiral p-wave superconductor, a broad ZBCP has been obtained reflecting on the linear dispersion of the SABS [16][17][18]. This result is consistent with the tunneling experiments of Sr 2 RuO 4 [19,20] and supports the realization of spin-triplet superconductivity in Sr 2 RuO 4 [21,22].
It is known that the physical origin of these SABSs stems from the chiral edge state protected by the topological invariant defined in the bulk Hamiltonian [23][24][25], and the high T C cuprate and Sr 2 RuO 4 are regarded as topological superconductors [26]. In the last decade, it has been established that topologically protected SABS can be generated based on a lowdimensional electron system with strong spin-orbit coupling without using unconventional pairings. For example, for the s-wave superconductor/ferromagnet junction on the surface of a topological insulator (TI), a chiral edge mode is generated similar to Sr 2 RuO 4 . Previously, one of the authors of this study, YT, derived a conductance formula for this hybrid system and clarified that the slope of the dispersion of the chiral edge mode is tunable by the gate voltage applied on the TI [27]. The derivation of the conductance formula is useful to capture the low energy charge transport in newly developed designed topological superconductors and superconducting TIs [28,29].
The AIII-class topological insulator has a winding number in one dimension, as shown in the topological periodic table [30]. By inducing the s-wave pair potential on an AIII-class TI, this system becomes a topological superconductor belonging to the BDI-class, which is characterized by the topological number N [31]. The phase diagram of this BDI topological superconductor is shown in Fig. 1 [31,32] superconductor can be regarded as a one-dimensional version of the quantum anomalous Hall/superconductor hybrid system [32]. Since there are many researches about Quantum anomalous Hall / superconductor hybrid systems [33][34][35][36][37], to clarify the BDI superconductor has a sufficient value. It is remarkable that topological phase transition is tunable by changing the so-called mass parameter defined in the AIII topological insulator model, where a topological insulator is realized for m < 0 [32]. [31,38]. Simultaneously, the reflection coefficients of the Andreev reflection disappear. Although several theoretical studies have been done regarding this system [31,38], they are based on a low-energy effective model or numerical calcula-tions in a finite system, and the charge conductance formula has not been derived analytically. It is beneficial to solve the scattering problem of a normal metal/one-dimensional (1D) BDI superconductor junction analytically and derive a conductance formula similar to other topological superconductors [27,28]. The aim of this study was to solve the BogoliubovâĂŞ-de Gennes (BdG) equation of a normal metal/BDI superconductor junction based on the AIII TI in one dimension. We selected a standard normal metal with parabolic dispersion. We analytically obtained both the normal reflection coefficient R σσ ′ and Andreev reflection coefficient A σσ ′ . Here, σ is a spin index of a reflected electron (hole) for a normal (Andreev) reflection, and σ ′ is a spin index of an injected electron. The resulting conductance shows a wide variety of line shapes. For the N = 0 state, the conductance exhibits a gap-like structure without sharp coherence peaks in contrast to the standard U-shaped line shape of differential conductance in tunneling spectroscopy of an s-wave superconductor. For the N = 1 state, σσ ′ R σσ ′ = σσ ′ A σσ ′ is satisfied and the charge conductance has a quantized ZBCP of peak height 2e 2 /h. The width of this peak depends on the magnitude of m 0 and ∆ 0 . For the N = 2 state, the charge conductance has a ZBCP splitting at eV = 0 and becomes zero; this is consistent with previous results. Further, we clarified the spin rotation at zero bias voltage eV = 0, when an injected electron exhibits a spin polarization along the z-axis. For the N = 1 state, the spin direction of the normal-reflected electron and Andreev-reflected hole is directed along the y-axis. Meanwhile, for the N = 2 state, the Andreev reflection is absent and the spin rotation angle can become 180 • for the normal-reflected electron.
The remainder of this paper is organized as follows. In section 2, we present the theory and the method to derive the conductance formula. In section 3, we detail the resulting conductance. In section 4, we demonstrate the spin rotation from the obtained reflection coefficients. Our results are summarized in section 5.
II. FORMULATION
We consider a normal metal/BDI superconductor (N/BDI) junction, as shown in Fig. 2 [31]. The one-dimensional limit SC AⅢ TI FIG. 2. Normal metal/BDI superconductor junction. BDI superconductor is realized in the AIII topological insulator region that is proximity coupled with a spin-singlet s-wave superconductor [31]. of a quantum anomalous Hall/superconductor hybrid system can be regarded as a BDI superconductor. The Hamiltonian is given by
H = H N θ(−x) + Uδ(x) + H SC θ(x),(1)
where θ(x) and δ(x) are the Heaviside step function and delta function, respectively. The second term indicates the barrier potential with barrier parameter U. The Hamiltonian of a normal metal is defined as a standard free electron model with parabolic dispersion.
H N (k) = 2 2m N k 2 − µ τ z ,(2)
where µ, m N , and τ z are the chemical potential, effective mass of the normal metal, and Pauli matrices in the Nambu space. Subsequently, the Fermi wave number is given by
k F = 2m N µ/ 2 . The Hamiltonian of the BDI superconduc- tor is H SC (k) = h AIII (k) iσ y ∆ 0 −iσ y ∆ 0 −h * AIII (−k) ,(3)h AIII (k) =m(k)σ z + A 0 kσ x , m(k) = m 0 + B 0 k 2 ,(4)
where h AIII (k) is the AIII-class TI. A 0 and B 0 are material parameters. ∆ 0 is the pair potential of the spin-singlet s-wave superconductor. Here, m 0 , σ z , and A 0 denote the effective mass, z component of the Pauli matrix, and spin-orbital coupling, respectively. Herein, the chemical potential of the AIII-class insulator is fixed at zero. Therefore, the Fermi level is located in the middle between the conduction and valence band. For m 0 < 0, the AIII-class insulator becomes a topological hosting edge state [31,32]. We normalize each parameter using k F in the remainder of this paper
m 0 = m 0 /μ ∆ 0 = ∆ 0 /µ A = A 0 / 2 k F 2m N B = B 0 / 2 2m N Z = 2U/ 2 k F 2m N .(5)
The eigenenergy E ± and eigenfunction ψ ± (k) of H SC are obtained:
E ± = A 2 0 k 2 + (m(k) ± ∆ 0 ) 2 (6) ψ ± (k) = 1 Q ± (k) ±Q ± (k) ±1 , Q ± (k) = − m(k) ± ∆ 0 − E A 0 k .(7)
We can confirm that the bulk energy gap of the BDI superconductor closes at m 0 = ±∆ 0 , as calculated from the eigenenergy. Thus, the topological phase transition occurs at m 0 = ±∆ 0 . In the following section, we study the scattering problem of the N/BDI junction. We assume that the spin direction of an injected electron is along the z axis. In the 1D BDI superconductor, the wave function is satisfied, as follows:
Ψ N (x) = Ψ in (x) + Ψ ref (x) (x < 0)(8)Ψ SC (x) = Ψ tra (x) (x > 0) (9) Ψ in = δ ↑σ δ ↓σ 0 0 e ik F x , Ψ ref = b ↑σ b ↓σ 0 0 e −ik F x + 0 0 a ↑σ a ↓σ e ik F x ,(10)Ψ tra =t 1+ ψ ± (k 1+ )e ik 1+ x + t 2+ ψ ± (k 2+ )e ik 2+ x + t 1− ψ ± (k 1− )e ik 1− x + t 2− ψ ± (k 2− )e ik 2− x ,(11)
where σ is the spin of an injected electron; b σ ′ σ and a σ ′ σ are the amplitudes of the normal and Andreev reflections with σ ′ =↑ (↓); t 1± and t 2± are the corresponding transmission amplitudes. The wave number in the BDI superconductor is calculated from the eigenenergy:
E ± k 1,± (E) 2 = 1 2B 2 0 −(2B 0 (m 0 ± ∆ 0 ) + A 2 0 ) + A 4 0 + 4B 0 (m 0 ± ∆ 0 )A 2 0 + 4B 2 0 E 2 (12) k 2,± (E) 2 = 1 2B 2 0 −(2B 0 (m 0 ± ∆ 0 ) + A 2 0 ) − A 4 0 + 4B 0 (m 0 ± ∆ 0 )A 2 0 + 4B 2 0 E 2 ,(13)
where E is the energy measured from the Fermi level. The sign of the wave number is determined by the group velocity such that the wave function does not diverge for x → ∞. The boundary condition of the wave function at x = 0 is given as follows:
Ψ SC (x = 0) − Ψ N (x = 0) = 0 (v SC {Ψ SC (x)} | x=+0 −v N {Ψ N (x)} | x=−0 ) = −2iUτ z Ψ N (x = 0) .(14)
Here,v is the velocity operatorv = 1 ∂H ∂(−i∂ x ) .
III. TUNNELING EFFECT
The charge conductance Γ in the N/BDI junction can be expressed using the reflection coefficients
Γ = e 2 h 2 − σ,σ ′ (R σσ ′ − A σσ ′) .(15)
The amplitude of the normal reflection b σ = (b ↑σ , b ↓σ ) and that of the Andreev reflection a σ = (a ↑σ , a ↓σ ) for an injected electron with σ =↑, ↓ are expressed by two component vectors obtained from the boundary condition (14) b σ = I + iZ K + + γ * I −1
+ K − + γ * I −1 −1 I − γ * K + + γ * I −1 + K − + γ * I −1 u σ (16) a σ = − 2σ x I + iZ K + + γI −1 + K − + γI −1 −1 K + + γI −1 − K − + γI −1 u σ ,(17)
with u σ = (δ ↑σ , δ ↓σ ). Here, γ ≡ 2 + iZ with the barrier parameter Z, I is a 2 × 2 unit matrix, andK ± is a 2 × 2 matrix with respect to the wave number (12)(13) using the factor of the wave function (7).
K ± ≡Aσ x + 2Bσ zQ± k 1± /k F 0 0 k 2± /k F Q −1 ±(18)Q ± ≡ 1 1 Q ± (k 1± ) Q ± (k 2± )
.
The matrices of the reflection coefficients of the normal reflection R σσ ′ and that of the Andreev reflection A σσ ′ are given by
R σσ ′ = | b ↑↑ | 2 | b ↑↓ | 2 | b ↓↑ | 2 | b ↓↓ | 2(20)A σσ ′ = | a ↑↑ | 2 | a ↑↓ | 2 | a ↓↑ | 2 | a ↓↓ | 2 ,(21)
We calculate the conductance Γ analytically with bias voltage V where E = eV is satisfied. We selected various mass parameters m 0 for a fixed ∆ 0 , as shown in Fig.3. For N = 0 cases, the obtained conductance never becomes exactly zero for any V (Γ 0) owing to the Andreev reflection. The sharp coherent peaks that appear in the case of conventional tunneling spectroscopy of the s-wave superconductor is absent in the conductance. This is because the spin-singlet s-wave pair potential is induced in the insulating AIII phase. The conductance shows a two-gap behavior at eV = |m 0 ± ∆ 0 |. For the N = 1 state, the ZBCP appears with its peak height of 2e 2 /h. The peak width increases with the decrease in m 0 and is approximately proportional to W − , which is the gap width of the energy band E − defined in eq. (6)
W − = |m 0 − ∆ 0 | m 0 − ∆ 0 > −A 2 0 /2B 0 A 2 0 B 0 |m 0 − ∆ 0 | − A 2 0 4B 0 m 0 − ∆ 0 < −A 2 0 /2B 0 .(22)
At m 0 = ∆ 0 , the width of the peak becomes zero and the energy spectrum of the BDI superconductor becomes gapless corresponding to a topological transition. This ZBCP is due to the ZESABS that manifests as a Majorana fermion at the edge of the BDI superconductor [31,32]. For the N = 2 state, the charge conductance exhibits a ZBCP splitting. The height of the peaks at a nonzero voltage is suppressed with the decrease in the value of m 0 . At eV = 0, the conductance at zero voltage becomes exactly zero, consistent with previous results [31,37,38].
ZERO BIAS VOLTAGE
Next, we focus on the conductance at zero voltage where more compact formula of coefficients of the Andreev and normal reflections and conductance can be available. For convenience, we introduce the following:
D ± ≡ A 2 + 4B(m 0 ±∆ 0 ).(23)
The amplitudes of both the normal and Andreev reflections can be expressed as
b σ = (γ * 2 +D + D − ) (|γ | 2 −D + D − ) 2 +4(D + +D − ) 2 (|γ| 2 − D + D − ) − 2i(D + + D − )σ z u σ a σ = − 2(D + −D − ) (|γ | 2 −D + D − ) 2 +4(D + +D − ) 2 2(D + + D − )σ x + (|γ| 2 − D + D − )σ y u σ (if N = 0) b σ = 1 2 γ * 2 −D 2 + |γ | 2 +D 2 + − sgn(A 0 ) γ * 2 +D 2 + |γ | 2 +D 2 + σ y −i 2γ * D + |γ | 2 +D 2 + σ z u σ a σ = − i 2 sgn(A 0 ) 2Z D + |γ | 2 +D 2 + + i |γ | 2 −D 2 + |γ | 2 +D 2 + σ x −i 4D + |γ | 2 +D 2 + σ y − sgn(A 0 )σ z u σ (if N = 1) b σ = − 1 γ iZ + 2sgn(A 0 )σ y u σ a σ = 0 (if N = 2).(24)
For N = 0, both amplitudes of the normal and Andreev reflections exist. For N = 1, after some straightforward calculations, we obtain
σ,σ ′ R σσ ′ = σ,σ ′ A σσ ′ .(25)
This implies that the contributions of the normal and Andreev reflections are completely balanced. This property is unique although the ZBCP exists in both the Andreev and normal reflections. It is qualitatively different from the previous normal metal/unconventional junctions with the perfect resonant case, where only the Andreev reflection exists at zero voltage [2,14]. Consequently, differences are observed between the heights of these ZBCPs. For the N = 2 state, although the ZESABS exists, the Andreev reflection is completely suppressed. Using the results above, we obtain Γ as follows:
Γ = e 2 h × 16(D + −D − ) 2 (|γ | 2 −D + D − ) 2 +4(D + +D − ) 2 (if N = 0) 2 (if N = 1) 0 (if N = 2) .(26)
For the N = 1 state, we can demonstrate analytically that the charge conductance becomes 2e 2 h .
IV. SPIN ROTATION
We have obtained the normal and Andreev reflection coefficients analytically; thus, we can analyze the detailed property of the reflected particles. In this section, we study the spin rotation through the scattering processes at the interface. Here, we consider the normal and Andreev reflections at zero bias voltage to use the formulae in the previous section. Subsequently, the reflection amplitude b σ , a σ is expressed by the spin rotational operator exp iθ 2 ·σ , whereθ = θn denotes the rotational axisn and rotational angle θ. The results depend highly on the topological phase in the BDI superconductor.
For the N = 1 state, the reflection coefficients are denoted by a linear combination of spinors that are expressed by two types of spin rotations
b σ = 1 2 exp iθ 1b1 2 ·σ − i exp iθ 1b2 2 ·σ u σ a σ = sgn(A) 2 exp i π 2 σ z − i exp iθ 1a 2 ·σ u σ (if N = 1),(27)
where the rotation angles are as follows:
θ 1b1 /2 = arctan 4 Z 2 + D 2 + 4 − Z 2 − D 2 + 0 sgn(A)Z −D + 0 sgn(A)Z −D + (28) θ 1b2 /2 = arctan − (4 − Z 2 + D 2 + ) 2 + 4Z 2 D 2 + 4Z 0 (4 − Z 2 + D 2 + )sgn(A) 2Z D + 0 (4 − Z 2 + D 2 + )sgn(A) 2Z D + (29) θ 1a /2 = arctan sgn(A) |γ| 4 + 2D 2 + (4 − Z 2 ) + D 4 + 2Z D + |γ| 2 − D 2 + −4D + 0 |γ| 2 − D 2 + −4D + 0 .(30)
Fig .5 shows the spin direction of the normal and Andreev reflections with the injection of an up-spin electron, where θ y and φ xz are the polar angle from the y-axis and the azimuth angle in the xz-plane, respectively. In the case of A > 0, the spin direction for both the normal and Andreev reflections are almost along the direction of −ŷ because of θ y ∼ π for any Z, as shown in Fig.5(a). Here,ŷ is a unit vector along the y-direction. Meanwhile, the spin directions are the opposite in the A < 0 case. This implies that the spin direction depends on the chirality of the BDI superconductor. Additionally, we confirm that the spin direction for the normal and Andreev reflections with a down-spin injection is the same as those for an up-spin injection. In other words, the spin directions of the reflected waves are polarized. This spin polarization phenomenon is caused by the chirality of the Majorana fermion at the edge of the BDI superconductor. Obviously, the electrons in an isolated normal metal exhibits time-reversal symmetry; however, by applying the bias voltage in the normal metal/1D BDI superconductor junctions, spin polarization occurs. For the N = 2 state, the reflection amplitudes are denoted as follows:
b σ = R 2b exp iθ 2b 2 · σ u σ a σ = 0 (if N = 2),(31)
where the coefficient and rotation angle are as follows:
R 2b = −i 2 − iZ √ 4 + Z 2 ,θ 2b /2 = arctan −sgn(A) 2 Z 0 1 0 .(32)
According to Eq.(32), normal reflection depends only on the sign of A and does not depend on other parameters of the BDI superconductor. Fig.6 shows the spin direction of the normal reflection in an up-spin injection, where θ z and φ xy are the polar angle from the z-axis and the azimuth angle in the xy-plane, respectively. As shown in Fig.6(a), the spins of the reflections are directed to −ẑ, i.e., π-rotation at Z = 0, but do not change through the scattering for Z = ∞. Here, z is a unit vector. This is because the couplings between the injected electron and the edge state of the BDI becomes weak with increasing barrier strength. From Z = 0 to ∞, the spin direction of the normal reflection rotates in the xz-plane. It is noteworthy that the azimuth angle depends on the sign of A, as shown in Fig.6(b). It is known that a giant spin rotation appears in the normal metal/quantum spin Hall junction depending on the edge states [39]. Similarly, spin rotations appear for a weak barrier strength. Finally, for the N = 0 state, the reflection amplitudes are expressed as follows:
b σ = R 0b exp iθ 0b 2 ·σ u σ a σ = R 0a exp iθ 0a 2 ·σ u σ (if N = 0),(33)
where the coefficients and rotation angles are as follows:
R 0b = γ * 2 +D + D − √ (|γ | 2 −D + D − ) 2 +4(D + +D − ) 2 θ 0b /2 = arctan 2(D + +D − ) |γ | 2 −D + D − 0 0 1 (34) R 0a = 2i(D + −D − ) √ (|γ | 2 −D + D − ) 2 +4(D + +D − ) 2 θ 0a /2 = π 2 2(D + + D − ) |γ| 2 − D + D − 0 2(D + + D − ) |γ| 2 − D + D − 0 .(35)
When the spin direction for the injected electrons is along thê z-axis, any spin flipping or rotation does not appear because θ 0b directsẑ and the rotational axis is along the z-axis. Further, the Andreev reflection is flipped simply,i.e., ↑ to ↓ or vice versa. This spin rotation is similar to that of conventional tunneling of the s-wave superconductor. The results above are summarized in Table I.
Z = 0 Z = ∞ Z = 0 Z = ∞ 0 +ẑ +ẑ −ẑ −ẑ 1 −sgn(A)ŷ −sgn(A)ŷ −sgn(A)ŷ −sgn(A)ŷ 2 −ẑ +ẑ - -
V. CONCLUSION
We have studied the tunneling effect in a topological superconductor based on a 1D AIII-class TI that is proximity coupled with a spin-singlet s-wave superconductor. This topologi-cal superconductor belongs to the BDI-class and topologically different phases are characterized by the topological invariant N for a bulk BDI superconductor. By solving the BdG equation of the normal metal/BDI superconductor junction, we have analytically obtained both the normal reflection coefficient R σσ ′ and Andreev reflection coefficient A σσ ′ , where σ(σ ′ ) is a spin index of the reflected (injected) wave. The resulting conductance indicates a wide variety of line shapes. For the N = 0 state, the obtained conductance exhibits a Gaplike structure without sharp coherence peaks at their maxima, in contrast to the standard U-shaped line shape in s-wave superconductor tunneling spectroscopy. For the N = 1 state, σσ ′ R σσ ′ = σσ ′ A σσ ′ is satisfied. The obtained conductance exhibits a ZBCP of height 2e 2 /h. The width of this peak depends on the magnitudes of m 0 and ∆ 0 . With the decrease in the value of m 0 for a fixed ∆ 0 , the width of the peak increases up to m 0 = −∆ 0 . For the N = 2 state, the charge conductance exhibits a ZBCP spitting; at eV = 0, it became zero, consistent with previous results. Further, we have calculated the spin rotation at zero bias voltage eV = 0, when an injected electron exhibited a spin polarization along the z-axis. For the N = 0 state, the spin direction of the reflected electron is along the z-axis and that of the hole is in the opposite direction. For the N = 1 state, the spin directions of the reflected electron and hole are directed along the y-axis. Meanwhile, for the N = 2 state, A σσ ′ = 0 is always satisfied and the reflected electron exhibited a spin rotation. The spin rotation angle can become 180 • in the extreme case when no barrier exists at the boundary.
FIG. 3 .
3Calculated conductance for various m 0 . We selected ∆ 0 = 0.03, A = B = 0.1, and Z = 1. (a): The values of (m 0 , ∆ 0 ) in the phase diagram of the BDI superconductor. (b): N = 0 with (i)m 0 = 0.06 and (i)m 0 = 0.04. (c): N = 1 with (iii)m 0 = 0.02 and (iv)m 0 = −0.02. (d): N = 2 with (v)m 0 = −0.04 and (vi)m 0 = −0.06.
FIG. 4 .
4Conductance with zero bias voltage for∆ 0 = 0.25, A = B = 0.1, Z = 1.
FIG. 5 .
5Calculated spin rotation for N = 1 at eV = 0. We selected ∆ 0 = 0.03, A = B = 0.1,m 0 = 0 and Z = 1. (a): Plot of θ y that is the polar angle of the spin direction from the y-axis. The normal and Andreev reflections has the same θ y value. (b): Plot of φ xz that is the azimuth angle of the spin direction in the xz-plane.
rotation injected up-spin for N = 2 at eV = 0. We selected∆ 0 = 0.03, A = B = 0.1,m 0 = 0, and Z = 1. (a): Plot of θ z that is the polar angle of the spin direction from the z-axis. In the A > 0 and A < 0 cases, the same value is observed for θ z . (b): Plot of φ xy that is the azimuth angle of the spin direction in the x y-plane.
. The present BDI topological FIG. 1. Phase diagram of BDI topological superconductor
TABLE I .
ISpin direction of the normal and Andreev reflections with injected up-spin electron.ŷ andẑ are unit vectorsN
Normal
Andreev
ACKNOWLEDGMENTSWe would like to thank valuable discussion with A. Yamakage. This work was supported by Grant-in-Aid for Scientific Research on Innovative Areas, Topological Material Science (Grant Nos. JP15H05851, JP15H05853) and JSPS KAKENHI Grant Numbers JP18K03538 and JP18H01176 from the Ministry of Education, Culture, Sports, Science, and Technology, Japan (MEXT).
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| []
|
[
"CRITERIA FOR EXISTENCE OF RIESZ BASES CONSISTING OF ROOT FUNCTIONS OF HILL AND 1D DIRAC OPERATORS",
"CRITERIA FOR EXISTENCE OF RIESZ BASES CONSISTING OF ROOT FUNCTIONS OF HILL AND 1D DIRAC OPERATORS"
]
| [
"Plamen Djakov ",
"Boris Mityagin "
]
| []
| []
| We study the system of root functions (SRF) of Hill operator Ly = −y ′′ + vy with a singular potential v ∈ H −1 per and SRF of 1D Dirac operator Ly = i, subject to periodic or anti-periodic boundary conditions.Series of necessary and sufficient conditions (in terms of Fourier coefficients of the potentials and related spectral gaps and deviations) for SRF to contain a Riesz basis are proven. Equiconvergence theorems are used to explain basis property of SRF in L p -spaces and other rearrangement invariant function spaces. | 10.1016/j.jfa.2012.07.003 | [
"https://arxiv.org/pdf/1106.5774v1.pdf"
]
| 119,593,508 | 1106.5774 | 5591ae4952a843cde840c21d3af4ca2df577caaa |
CRITERIA FOR EXISTENCE OF RIESZ BASES CONSISTING OF ROOT FUNCTIONS OF HILL AND 1D DIRAC OPERATORS
28 Jun 2011
Plamen Djakov
Boris Mityagin
CRITERIA FOR EXISTENCE OF RIESZ BASES CONSISTING OF ROOT FUNCTIONS OF HILL AND 1D DIRAC OPERATORS
28 Jun 2011Hill operatorssingular potentialsDirac operatorsspectral decompositionsRiesz basesequiconvergence 2010 Mathematics Subject Classification: 47E0534L40
We study the system of root functions (SRF) of Hill operator Ly = −y ′′ + vy with a singular potential v ∈ H −1 per and SRF of 1D Dirac operator Ly = i, subject to periodic or anti-periodic boundary conditions.Series of necessary and sufficient conditions (in terms of Fourier coefficients of the potentials and related spectral gaps and deviations) for SRF to contain a Riesz basis are proven. Equiconvergence theorems are used to explain basis property of SRF in L p -spaces and other rearrangement invariant function spaces.
Introduction
1. In the case of ordinary differential operators with strictly regular boundary conditions (bc) on a finite interval the system {u k } of eigen-and B. Mityagin acknowledges the support of the Scientific and Technological Research Council of Turkey and the hospitality of Sabanci University, April-June, 2011. associated functions could contain only finitely many linearly independent associated functions. The well-defined decompositions (1.1) k c k (f )u k = f ∀f ∈ L 2 ([0, π]), do converge; moreover, convergence is unconditional, i. e., {u k }, u k = 1, is a Riesz basis in L 2 ([0, π]). These facts and phenomena have been well understood in the early 1960's after the works of N. Dunford [15,16], V. P. Mikhailov [41] and G. M. Keselman [27].
Maybe the simplest case of regular but not strictly regular bc comes if we consider a Hill operator L bc .
(1.2) Ly = −y ′′ + v(x)y, 0 ≤ x ≤ π,
where v(x) = v(x+π) is a complex-valued smooth function, and bc is periodic (bc = P er+) or anti-periodic (bc = P er−), i. e., (a) periodic P er + : y(0) = y(π), y ′ (0) = y ′ (π); (b) anti-periodic P er − : y(0) = −y(π), y ′ (0) = −y ′ (π); (Later we will consider non-smooth v as well, say v ∈ L 2 or L 1 , and v ∈ H −1/2 or v ∈ H −1 per , -see in particular Section 4.1.) Recently, i.e., in the 2000's, many authors [18,26,29,34,33,35,37,38,39,40,52] focused on the problem of convergence of eigenfunction (or more generally root function) decompositions in the case of regular but not strictly regular bc.
The free operators L 0 bc = d 2 /dx 2 , with bc = P er ± have infinitely many double eigenvalues λ 0 n = n 2 , (with n even for bc = P er + and n odd for if bc = P er − ), the corresponding two-dimensional eigenspaces E 0 n are mutually orthogonal and we have the spectral decomposition of the space
L 2 ([0, π]) = ⊕E 0 n or f = n P 0 n f ∀f ∈ L 2 ([0, π]),
where P 0 n is the orthogonal projection on E 0 n . The operator L bc (v) = L 0 bc + v is a "perturbation" of the free operator; its spectrum is discrete and for large enough n, say n > N, close to λ 0 n = n 2 there are exactly two eigenvalues λ − n , λ + n (counted with multiplicity). Moreover, if E n is the corresponding two-dimensional invariant subspace and P n = 1 2πi Cn (z − L bc ) −1 dz is the corresponding Cauchy projection, then we have the spectral decomposition (1.3)
S N f + k>N P k f = f ∀f ∈ L 2 ([0, π]),
where S N is the (finite-dimensional) projection on the invariant subspace corresponding to "small" eigenvalues of L bc (v), and the series in (1.3) converges unconditionally. However, even if all eigenvalues λ − n , λ + n , n > N are simple, there is a question whether we could use the corresponding eigenfunctions to give an expansion like (1.1). The same questions for P er ± in the case of 1D periodic Dirac operators could be asked. Interesting conditions on potentials v (or on its Fourier coefficients), which guarantee basisness of {u k }, -with or without additional assumptions about the structure or smoothness of a potential v -have been given by A. Makin [37,38,39,40], A. Shkalikov [52], O. Veliev [59,60,61,1], P. Djakov and B. Mityagin [4,8,12,13,14].
2. In our papers [24,3,4,9,14] we analyzed the relationship between smoothness of a potential v in (1.2) and the rate of "decay" of sequences of (1.4) spectral gaps γ n = λ + n − λ − n and (1.5) deviations δ n = µ n − 1 2
(λ + n + λ − n ).
This analysis is based on the Lyapunov-Schmidt projection method: by projecting on the n-th eigenvalue space E 0 n of the free operator L 0 the eigenvalue equation Ly = λy is reduced locally, for λ = n 2 + z with |z| < n/2 to an eigenvalue equation for a 2 × 2 matrix α n (v, z) β − n (v; z) β + n (v; z) α n (v, z)
. The entries of this matrix are functionals (depending analytically on v and z), which are given by explicit formulas in terms of the Fourier coefficients of the potential v (see (2.37) and (2.38) below). They played a crucial role in proving estimates for and inequalities between γ n , δ n , β ± n and (1.6) t n (z) := |β − n (v, z)|/|β + n (v, z)| -see [4], Lemma 49 and Proposition 66.
Moreover, it turns out that there is an essential relation between the Riesz basis property of the system of root functions and the ratio functionals t n (v, z) which made possible to give criteria for existence of (Riesz) bases consisting of root functions not only for Hill operators but for Dirac operators as well (see, for example, [13,Theorem 1] or [12,Theorem 2] for Hill, or [14,Theorem 12] for Dirac operators). These criteria are quite general and applicable to wide classes of potentials. For example, we proved that if
(1.7) v(x) = 5e −4ix + 2e 2ix − 3e 2ix + 4e 4ix ,
then neither for bc = P er + nor for bc = P er − the root function system of L bc contains a basis in L 2 ([0, π]). To apply our criterion we had to overcome a few analytic difficulties. This was done on the basis of our results and techniques from [5].
In this paper we extend and slightly generalize these criteria. We claim, both for Hill operators with singular H −1 per -potentials and Dirac operators with L 2 -potentials the following.
Criterion. The root system of functions of the operator L P er ± (v) has the Riesz basis property (i.e., contains a Riesz basis) if and only
(1.8) ∃C > 0 : 1/C ≤ t n (z * n ) ≤ C if λ − n = λ − n , n ∈ Γ bc , |n| > N * .
(See the definition of Γ bc in Section 2, Formulas (2.10) and (2.26).) 3. Recently F. Gesztesy and V. Tkachenko [19,Theorem 1.2] gavein the case of Hill operators with L 2 potentials -a criterion of basisness in the following form:
The system of root vectors for bc = P er + or bc = P er − , contains a Riesz basis if and only if
(1.9) R bc = sup |µ n − λ + n | |λ + n − λ − n | : n ∈ Γ bc , λ + n = λ − n < ∞.
One can prove, by using the estimates of |λ + n − λ − n | and |µ n − λ + n | in terms of |β − n (v, z)| and |β + n (v, z)| (see [4,Theorem 66, Lemma 49] and [9, Theorem 37, Lemma 21]) that the conditions (1.8) and (1.9) are equivalent.
However, we directly show (see Theorem 24 in Section 7), using the fundamental inequalities proven in [24,3,4,9], that (1.9) gives necessary and sufficient conditions of Riesz basisness of root system with bc = P er + or bc = P er − both (A) in the case of 1D periodic Dirac operators with L 2 potential, and (B) in the case of Hill operators with potential in H −1 per .
4. Criterion for L p -spaces, 1 < p < ∞, given in [19,Theorem 1.4] can be essentially improved and extended as well. We take any separable rearrangement invariant function space E on [0, π] (see [28,32]) squeezed between L a and L b , 1 < a ≤ b < ∞. If (1.9) holds. In the case of Hill operators with v ∈ H −1/2 the hypothesis (1.10) could be weakened to
(1.11) 1/a − 1/b < 1.
Of course for L p , 1 < p < ∞, we can put a = b = p, so (1.10) and (1.11) hold.
The structure of this paper and the topics discussed in different sections are shown in Content, see p. 1.
Localization of spectra and Riesz projections for Hill and Dirac operators
For basic facts of Spectral Theory of ordinary differential operators we refer to the books [30,45,36]. But let us introduce some notations and remind a few properties of Hill and Dirac operators on a finite interval.
1. We consider the Hill operator
(2.1) Ly = −y ′′ + v(x)y, x ∈ I = [0, π],
with a (complex-valued) potential v ∈ L 2 (I), or more generally with a singular potential v ∈ H −1 per of the form
(2.2) v = w ′ , w ∈ L 2 loc (R), w(x + π) = w(x).
For v ∈ L 2 , we consider the following bc (boundary conditions):
(a) periodic P er + : y(0) = y(π), y ′ (0) = y ′ (π); (b) anti-periodic P er − : y(0) = −y(π), y ′ (0) = −y ′ (π); (c) Dirichlet Dir : y(0) = 0, y(π) = 0. For each bc = P er ± , Dir the operator L generates a closed operator L bc with
(2.3)
Dom(L bc ) = {f ∈ W 2 2 (I) : f satisfies bc}. In the case of singular potentials (2.2) A. Savchuk and A. Shkalikov [47,48] suggested to use the quasi-derivative y [1] = y ′ − w y in order to define properly the boundary conditions and corresponding operators. In particular, the periodic and anti-periodic boundary conditions P er ± have the form (a * ) P er + : y(π) = y(0), y [1] (π) = y [1] (0), (b * ) P er − : y(π) = −y(0), y [1] (π) = −y [1] (0). The Dirichlet boundary condition has the same form (c) as in the classical case. Of course, in the case where w is a continuous function, P er + and P er − coincide, respectively, with the classical periodic boundary condition (a) and (b).
We refer the reader to our papers [6,7,9] for definitions of the operators L bc and their domains in the case of H −1 per -potentials. (We followed [47,48] and further development of A. Savchuk -A. Shkalikov's approach by R. Hryniv and Ya. Mykytyuk [21,22,23] to justify Fourier method in analysis of Hill-Schrödinger operators with singular potentials.
If v = 0 we denote by L 0 bc the corresponding free operator. Of course, it is easy to describe the spectra and eigenfunctions for L 0 bc . Namely, we have (i) Sp(L 0 P er + ) = {n 2 , n = 0, 2, 4, . . .}; its eigenspaces are E 0 n = Span{e ±inx } for n > 0 and E 0 0 = {const}, dim E 0 n = 2 for n > 0, and dim E 0 0 = 1. 2. Localization of spectra in the case of Hill operators. Proposition 1. (localization of spectra) Consider L bc (v) with bc = P er ± , Dir and with potential v ∈ L 2 or v ∈ (2.2). Then, for large enough N * = N * (v) ∈ 2N, we have
(2.6) Sp (L bc ) ⊂ Π N * ∪ n>N * , n∈Γ bc D(n 2 , r n ), where (2.7) Π N = {z = x + iy ∈ C : |x|, |y| < N 2 + 1 2 N, (2.8) D(a, r) = {z ∈ C : |z − a| < r}, with (2.9) r n = N * /2 if v ∈ L 2 , r n = n/4 if v ∈ H −1 per , and (2.10) Γ bc = {0} ∪ 2N bc = P er + , 2N − 1 bc = P er − , N bc = Dir.
With the resolvent R(z) = (z − L bc ) −1 well defined in the complement of Sp (L bc ), we set
(2.11) S N * = 1 2πi ∂Π N * (z − L bc ) −1 dz,
(2.12) P n = 1 2πi |z−n 2 |=rn (z − L bc ) −1 dz, n > N * , n ∈ Γ bc , and (2.13) S N = S N * + N n = N * + 1 n ∈ Γ bc P n .
Then
(2.14) dim P n = 2 n even, bc = P er + , 2 n odd, bc = P er − , 1 n ∈ N, bc = Dir, and (2.15) dim S N * = N * + 1 bc = P er + , N * bc = P er − or Dir.
In each case the series
(2.16) S N * f + n>N * , n∈Γ bc P n f = f ∀f ∈ L 2 (I)
converges unconditionally, so the system of projections is a Riesz system.
The latter is true not only for potentials v ∈ L 2 but in the case v ∈ H −1 per as well. It has been proven by A. Savchuk and A. Shkalikov [48,Theorem 2.8]. An alternative proof is given by the authors in [7], see Theorem 1 and Proposition 8.
3. Next we remind the basic fact about spectra decompositions and spectral decompositions for Dirac operators
(2.17) Ly = i 1 0 0 −1 dy dx + vy, (2.18) v(x) = 0 P (x) Q(x) 0 , y = y 1 y 2 ,
with L 2 -potential v, i.e., P, Q ∈ L 2 (I).
We consider three types of boundary conditions: (a) periodic P er + : y(0) = y(π), i.e., y 1 (0) = y 1 (π) and y 2 (0) = y 2 (π); (b) anti-periodic P er − : y(0) = −y(π), i.e., y 1 (0) = −y 1 (π) and y 2 (0) = −y 2 (π);
(c) Dirichlet Dir : y 1 (0) = y 2 (0), y 1 (π) = y 2 (π). The corresponding closed operator with a domain
(2.19) ∆ bc = f ∈ (W 2 1 (I)) 2 : F = f 1 f 2 ∈ (bc)
will be denoted by L bc . If v = 0, i.e., P ≡ 0, Q ≡ 0, we write L 0 bc . Of course, it is easy to describe the spectra and eigenfunctions for L 0 bc : (a) Sp(L 0 P er + ) = {n even} = 2Z; each number n ∈ 2Z is a double eigenvalue, and the corresponding eigenspace is
(2.20) E 0 n = Span{e 1 n , e 2 n }, where (2.21) e 1 n (x) = e −inx 0 , e 2 n (x) = 0 e inx ; (b) Sp(L 0 P er − ) = {n odd} = 2Z + 1
; the corresponding eigenspaces E 0 n are given by (2.20) and (2.21) but n ∈ 2Z + 1;
(c) Sp(L 0 Dir ) = {n ∈ Z}; each eigenvalue n is simple. The corresponding normalized eigenfunction is
(2.22) g n (x) = 1 √ 2 e 1 n + e 2 n , n ∈ Z,
so the corresponding (one-dimensional) eigenspace is
(2.23) G 0 n = Span{g n }.
4. Localization of spectra in the case of Dirac operators.
Proposition 2. (localization of spectra) For Dirac operators
L bc (v) with bc = P er ± , Dir, there is N * = N * (v), such that (2.24) Sp (L bc ) ⊂ Π N * ∪ n>N * , n∈Γ bc D(n 2 , 1/4), where (2.25) Π N = {z = x + iy ∈ C : |x|, |y| < N 2 + 1 4 , and (2.26) Γ bc = 2Z bc = P er + , 1 + 2Z bc = P er − , Z bc = Dir. With the resolvent R(z) = (z − L bc ) −1 well defined in the complement of Sp (L bc ), we set (2.27) S N * = 1 2πi ∂Π N * (z − L bc ) −1 dz, (2.28) P n = 1 2πi |z−n|=1/4 (z − L bc ) −1 dz, |n| > N * , n ∈ Γ bc , and (2.29) S N = S N * + N * + 1 ≤ |n| ≤ N n ∈ Γ bc P n .
Then
(2.30) dim P n = 2 n even, bc = P er + , 2 n odd, bc = P er − , 1 n ∈ Z, bc = Dir, and (2.31) dim S N * = 2N * + 2 bc = P er + , 2N * bc = P er − 2N * + 1 bc = Dir.
In each case the series
(2.32) S N * f + |n|>N * , n∈Γ bc P n f = f ∀f ∈ L 2 (I)
converges unconditionally, so
(2.33) {S N * , P n , |n| > N * , n ∈ Γ bc } is a Riesz system of projections.
The latter is proven in [8,Theorem 5.1]. (Under more restrictive assumption on the potential v ∈ H α , α > 1/2, the fact that (2.33) is a Riesz system of projections has been proven in [43,Theorem 8.8].) Propositions 1 and 2 guarantee the existence of the level N * = N * (v) when all formulas for P n , S N , etc. become valid if n > N * , n ∈ N (or |n| > N * , n ∈ Z in the Dirac case). In the next sections, there are other formulas which are valid for large enough n and require different levels N * = N * (v). But throughout the paper we use one and the same letter N * to indicate by the inequalities n > N * or |n| > N * that formulas hold for sufficiently large indices.
5. Propositions 1 and 2 allows us to apply the Lyapunov-Schmidt projection method (see [4,Lemma 21]) and reduce the eigenvalue equation Ly = λy to a series of eigenvalue equations in two-dimensional eigenspaces E 0 n of the free operator.
This
Lemma 3. (a) Let L be a Hill operator with a potential
v ∈ L 2 or v ∈ H −1
per . Then, for large enough n ∈ N, there are functions α n (v, z) and β ± n (v; z), |z| < n such that a number λ = n 2 + z, |z| < n/4, is a periodic (for even n) or anti-periodic (for odd n) eigenvalue of L if and only if z is an eigenvalue of the matrix
(2.34) α n (v, z) β − n (v; z) β + n (v; z) α n (v, z)
.
(b) Let L be a Dirac operator with a potential v ∈ L 2 . Then, for large enough |n|, n ∈ Z, there are functions α n (v, z) and β ± n (v; z), |z| < 1 such that a number λ = n+z, |z| < 1/4, is a periodic (for even n) or anti-periodic (for odd n) eigenvalue of L if and only if z is an eigenvalue of the matrix (2.34).
(c) A number λ = n 2 + z * , |z| < n/4, (respectively, λ = n + z, |z| < 1/4 in the Dirac case) is a periodic (for even n) or anti-periodic (for odd n) eigenvalue of L of geometric multiplicity 2 if and only if z * is an eigenvalue of the matrix (2.34) of geometric multiplicity 2.
The functionals α n (z; v) and β ± n (z; v) are well defined for large enough |n| by explicit expressions in terms of the Fourier coefficients of the potential (see for Hill operators with Here we provide formulas only for β ± n (v; z) in the case of Hill operators with H −1 per -potentials. Let v be a singular potential as in (2.2), and
(2.35) v = w ′ , w = m∈2Z W (m)e imx .
Then the Fourier coefficients of v are given by
(2.36) V (m) = im W (m), m ∈ 2Z,
and by [9, Formulas (3.21)-(3.33)] we have
(2.37) β ± n (v; z) = V (±2n) + ∞ k=1 S ± k (n, z), with (2.38) S ± k (n, z) = j 1 ,...,j k =±n V (±n − j 1 )V (j 1 − j 2 ) · · · V (j k−1 − j k )V (j k ± n) (n 2 − j 2 1 + z) · · · (n 2 − j 2 k + z)
.
Next we summarize some basic properties of α n (z; v) and β ± n (z; v).
Proposition 4. Let v be a H −1
per -potential of the form (2.2), and let L P er ± be the corresponding Hill operator.
(a) The functionals α n (z; v) and β ± n (z; v) depend analytically on z for |z| < n. There exists a sequence of positive numbers ε n → 0 such that for large enough n
(2.39) |α n (v; z)| + |β ± n (v; z)| ≤ n · ε n , |z| ≤ n/2, and (2.40) ∂α n ∂z (v; z) + ∂β ± n ∂z (v; z) ≤ ε n , |z| ≤ n/4.
(b) For large enough n (even, if bc = P er + or odd, if bc = P er − ), a number λ = n 2 + z, |z| < n/4, is an eigenvalue of L P er ± if and only if z satisfies the basic equation
(2.41) (z − α n (z; v)) 2 = β + n (z; v)β − n (z; v). (c)
For large enough n, the equation (2.41) has exactly two roots in the disc |z| < n/4 counted with multiplicity.
Proof. Part (a) is proved in [9,Proposition 15]. Lemma 3 implies Part (b). By (2.39), sup{| 1 z α n (z)|, |z| = n/4} → 0 and sup{| 1 z β ± n (z)|, |z| = n/4} → 0. Therefore, Part (c) follows from the Rouché theorem.
Proposition 5. Let L P er ± be a Dirac operator with L 2 -potential.
(a) The functionals α n (z; v) and β ± n (z; v) depend analytically on z for |z| < 1. There exists a sequence of positive numbers ε n → 0 such that for large enough |n|
(2.42) |α n (v; z)| + |β ± n (v; z)| ≤ ε n , |z| ≤ 1/2, and (2.43) ∂α n ∂z (v; z) + ∂β ± n ∂z (v; z) ≤ ε n , |z| ≤ 1/4. (b)
For large enough |n|, (n even, if bc = P er + or odd, if bc = P er − ), the number λ = n + z, z ∈ D = {ζ : |ζ| ≤ 1/4}, is an eigenvalue of L P er ± if and only if z ∈ D satisfies the basic equation Proof. Part (a) is proved in [4,Proposition 35]. Lemma 3 implies Part (b). By (2.42), sup D |α n (z)| → 0 and sup D |β ± n (z)| → 0 as n → ∞. Therefore, Part (c) follows from the Rouché theorem.
(2.44) (z − α n (z; v)) 2 = β + n (z; v)β − n (z, v),(c)
Elementary geometry of bases in a Banach space
In this section we give a few well-known facts about geometry and bases in Banach and Hilbert spaces -see [25,31,32,2,28].
1. Let {u k ∈ X, ψ k ∈ X ′ } k∈N be a biorthogonal system in a Banach space X, i. e.,
(3.1) ψ k (u j ) = 1, k = j, 0, k = j j, k ∈ N.
The system {u k } is called a basis, or a Shauder basis in Y , its closed linear span if
(3.2) lim N →∞ N k=1 ψ k (y)u k = y, ∀y ∈ Y. Put (3.3) Q m = q 2m−1 + q 2m , where q j (x) = ψ j (x)u j , j ∈ N are one-dimensional projections so (3.4) q j = u j · ψ j .
Let us assume that
(3.5) lim M →∞ M m=1 Q m y = y ∀y ∈ Y.
In this case, certainly
(3.6) sup m Q m = C < ∞.
Notice that partial sums in (3.5) are equal to partial sums in (3.2) with even indices. But
(3.7) 2t+1 k=1 ψ k (y)u k = t m=1 Q m y + ψ 2t+1 (y)u 2t+1 .
These elementary identities together with (3.1) explain the following.
Lemma 6. If {u k } ∞ 1 is a basis in Y, i.e., (3.2) holds then (3.8) T ≡ sup j q j < ∞.
Under the assumption (3.5) if (3.8) holds then {u k } ∞ 1 is a basis in Y.
What does happen inside of 2D subspaces
E m = Ran Q m , m ∈ N?
Let {u 1 , u 2 }, u j = 1, be a basis in E m and let ψ 1 , ψ 2 be the corresponding biorthogonal functionals, so
(3.9) h = ψ 1 (h)u 1 + ψ 2 (h)u 2 ∀h ∈ E m .
To avoid any confusion let us notice that for j = 2m − 1, 2m
(3.10) ψ j (y) = ψ j (Q m y) ∀y ∈ Y,
and if (3.5) holds then with (3.6)
(3.11) Q m y ≤ C y .
Therefore,
(3.12) ψ j ≥ sup{|ψ j (w)| : w = 1, w ∈ E m } ≥ sup{| 1 C Q m y, ψ j | : y = 1, y ∈ Y } = 1 C ψ j , so (3.13) ψ j ≤ Cκ j , κ j ≤ ψ j ,
i.e.,
(3.14) κ j ≡ ψ j |E m ≤ ψ j |Y ≤ Cκ j .
In a Hilbert space case, elementary straightforward estimates show that for j = 1, 2
(3.15) κ j = sup{|ψ j (w)| : w = 1, w ∈ C 2 } = (1 − | u 1 , u 2 |) −1/2 .
We use this fact when analyzing subspaces E m and their bases {u 2m−1 , u 2m }, m ∈ N.
3. Now we consider separable Hilbert spaces H. We say that the system {Q m } ∈ (3.3) is a Riesz system, or an unconditional 2D-block basis in Y if for some C > 0
(3.18) q j Q m = q j if j = 2m − 1, 2m 0 otherwise so for j = 2m − 1, 2m (3.19) q j y = q j Q m y ≤ M Q m y ≤ ( q 2m−1 y + q 2m y ) ≤ 2M Q m y .
Therefore (3.20) 1 4M 2 q 2m−1 y 2 + q 2m y 2 ≤ Q m y 2 ≤ 2M 2 q 2m−1 y 2 + q 2m y 2 and with C 1 = 2M the condition (3.16) holds for the system of 1D projections {q j }. It guarantees that {q j } is a Riesz system and {u k } is an unconditional basis in Y .
4. Now we are ready to claim the following. When we analyze systems of projections {P n , |n| ≥ N * } coming from Hill or Dirac operators, then it is a fundamental fact that they are Riesz systems.
If v ∈ L 2 this has been understood since 1980's ( [49,50,51] per and bc = P er ± . An alternative proof has been given by the authors -see Theorem 1 and Proposition 8 in [10].
Finally, in the case of one dimensional Dirac operators we proved (4.2) -(4.3) if v ∈ L 2 and bc = P er ± or Dir (see [8], Theorems 3.1 and 5.1). Later we proved (4.2) -(4.3) for arbitrary regular boundary condition -see Theorems 15 and 20 in [11]; however, we do not use these results from [11] in the present paper. Certainly in all these cases (4.4) P n − P 0 n 2 → 0 and P n 2 ≤ 3/2 for |n| > N * . These bibliography references justify applicability of Criterion 9 when we are trying to give different analytic criteria for Riesz basis property of the root function system of specific differential operators.
Of course, Corollary 10 indicates that in a Hilbert space there is no separate question about Schauder basis property. If {Q m }, or {S N ; P n , |n| ≥ N } is a Riesz system such that dim Q m = 2, dim P n = 2, then the properties of the system {u 2m−1 , u 2m } to be a Riesz basis or to be a Schauder basis are identical. Therefore, to talk about two properties is semantically artificial.
2. Let us define the root function system {u j } which will play a special role in our analysis in Sections 5 and 6 and in Main Theorem (Theorem 24). Section 3 and Criterion 9 use an indexation by natural numbers, i. e., m ∈ N. But in the case of Riesz bases (or unconditional convergence of series) it means that we can ignore the ordering in N, consider any countable set of indices and use all related statements from Section 3. Of course, in the case of bases which are not Riesz bases we should be accurate when we use statements from Section 3 -this is important in Section 6.
Remark 11. In the case of Hill operators, Γ bc ∈ (2.10) as a subset of N has a natural ordering and we have no confusion in defining the sum in (
(4.6) M = {m ∈ Γ bc : |m| > N * , λ + m − λ − m = 0}, (4.7) M 1 = {m ∈ Γ bc : |m| > N * , λ + m − λ − m = 0, P m L bc P m = λ · 1 Em }, i. e., λ +
m is a double eigenvalue of algebraic and geometric multiplicities 2; (4.8) M 2 = {m ∈ Γ bc : |m| > N * , λ + m −λ − m = 0, P m L bc P m is a Jordan matrix}, i. e., λ + m is a double eigenvalue of algebraic multiplicity 2 and geometric multiplicity 1.
If m ∈ M, we choose (u 2m−1 , u 2m ) in such a way that (4.9) Such a pair (u 2m−1 , u 2m ), m ∈ M 2 -as for m ∈ M 1 -is a nice basis in E m , so it will not be an obstacle for Riesz basisness of the larger system (see Lemmas 7 and 8) which contains {u 2m−1 , u 2m }.
Lu 2m = λ + m u 2m , Lu 2m−1 = λ − m u 2m−1 ,
Option 2. We choose u 2m as in Option 1, and we choose u 2m−1 ∈ (4.11) to be an associated function, i.e., (4.14)
L bc u 2m = λ + m u 2m , L bc u 2m−1 = λ + m u 2m−1 + u 2m . Since we choose u 2m−1 to satisfy (4.14) and (4.11), it is uniquely defined but its norm u 2m−1 is out of our control.
For Hill operators with potentials in L 1 A. Shkalikov
∃C > 0 0 < 1 C ≤ u 2m−1 ≤ C < ∞, ∀m ∈ M 2 .
Example 12. Take Gasymov type [17] singular potential This example is in a quite curious contrast with the case v ∈ L 2 or v ∈ L 1 -see (4.15) above. We prove the claims (i) and (ii) But still we need to define u j for small j, |j| ≤ N * . This system will be a basis in E * = RanS N * . Of course dim E * < ∞, so this choice has no bearing on whether the entire system will or will not be a Riesz basis (or a basis) in L 2 or another function space. We want it to be a system of root functions, so we choose the system of eigen-and associated functions of a finite-dimensional operator S * L bc S * , S * = S N * (We omit elementary linear algebra details.) 5. L p -spaces and other rearrangement invariant function spaces 1. In Sections 3 and 4 we discussed (criteria of) convergence of decompositions
(5.1) S N * f + n>N * ,n∈Γ bc P n f = f ∀f ∈ L 2
in L 2 . Convergence of such series or of eigenfunction decompositions in L p , p = 2, or other rearrangement invariant function spaces (see [28,44]) is not an independent from convergence in L 2 question because of the following two reasons of very general nature: (A) In the case of free operator L 0 its decompositions (5.1) are standard (or slight variations of) Fourier series. These decompositions
(5.2) S 0 N * f + n>N * ,n∈Γ bc P 0 n f = f ∀f ∈ E converge in E if E is a separable rearrangement invariant function space
where Hilbert transform is bounded. This is certainly the case if
(5.3) L a ⊃ E ⊃ L b for some a, b with 1 < a ≤ b < ∞.
See [28,Theorem 2.7.2], [44,62], and more about Boyd indices in [32], Theorem 2.c. 16 There are different versions of equiconvergence -see the survey paper of A. Minkin [42]. For example, J. Tamarkin [54,55] and M. Stone [53] proved the following.
Lemma 14.
If v ∈ L 1 then for any f ∈ L 1
(5.5) (S N − S 0 N )f ∞ → 0.
This lemma helps to cover the case of Hill operator with v ∈ L 1 . For v ∈ H −1 per see Proposition 16 below. Equiconvergence in the case of Dirac operator with potentials v ∈ L c , c > 4/3, is proven in [43, Theorem 6.2 (a)]. As a corollary it is noticed there [43,Theorem 6.4,(6.105)] that the series (5.6) converges in L p (I, C 2 ) 1 < p < ∞.
2. Now we can combine (A) and (B) to conclude the following.
Proposition 15.
If v ∈ L 2 and (5.3) holds then (5.6) S N * f + n>N * ,n∈Γ bc
P n f = f ∀f ∈ E Proof. Indeed (5.7) S N f = S 0 N f + (S N − S 0 N )f but with (5.3) g E ≤ g ∞ so for f ∈ L 1 (5.8) (S N − S 0 N )f E ≤ (S N − S 0 N )f ∞ → 0. Now (5.
2) and (5.8) together imply (5.6).
Of course in the case of Hill operators we want to cover potentials
v ∈ H −1
per as well. This is possible because the following equiconvergence statement is true.
Proposition 16. Let v ∈ H −1
per , W be coming from (2.35) and (2.36), and
(5.9) 1 < a ≤ b < ∞ with δ = 1/2 − (1/a − 1/b) > 0.
Then for any N > N * (v)
(5.10) S N − S 0 N : L a → L b ≤ C(δ) N −τ + E N (W ) , where (5.11) τ = δ if 1 < a ≤ 2 ≤ b < ∞; 1 − 1/a if 1 < a ≤ b ≤ 2; 1/b if 2 ≤ a ≤ b < ∞. and (5.12) E N (W ) = |m|≥N |W (m)| 2 1/2 .
Proof with all details is to be given in another paper we will submit shortly.
Proposition 17. If v ∈ H −1
per and E is a s.r.i.f.s. such that (5.3) and (5.9) hold then (5.6) hold.
Proof. Now with g a ≤ g E ≤ g b (5.10) and (5.7) imply (5.13)
(S N −S 0 N )f E ≤ (S N −S 0 N )f L b ≤ S N −S 0 N : L a → L b · f L b ≤ ε(N ) f E , where ε(N ) = C(δ) N −δ + E N (w) → 0,
so (5.6) holds.
4.
Terms P m f in (5.6) are vectors in two-dimensional subspaces
(5.14) E m = Lin Span{u 2m−1 , u 2m },
with {u j } defined in Section 4.2, (4.19). Fact (C). In these 2D subspaces L 1 norms and L ∞ norms are uniformly equivalent, i.e., with B = B(v) < ∞
(5.15) F ∞ ≤ B F 1 if F ∈ E m , m ≥ N (v)
This is proven in [43,Theorem 8.4,p.185] for Dirac operators with V ∈ L p , 1 < p, and in [9, Theorem 51, p.159] for Hill operators with v ∈ H −1 per . Section 4.2 explains that with conditions (3.5) and (3.6)
ψ j |E m ≤ ψ j |E ≤ C ψ j |E m .
-see (3.11) -(3.14). By Lemma 6, the system {u j } is a basis in Y ⊂ E if and only if
(5.16) sup j u j E · ψ j E < ∞.
But Fact (C) shows that (5.16) holds -or does not hold -for all s.r.i.f.s. E such that
(5.17) L 1 ([0, π]) ⊃ E ⊃ L ∞ ([0, π])
simultaneously. Any condition which is good to guarantee basisness in one E is automatically good for all E ′ s. Therefore, we can immediately to claim the following.
Theorem 18. Let E be a separable r.i.f.s. and
(5.18) L a ([0, π]) ⊃ E ⊃ L b ([0, π]), 1 < a ≤ b < ∞.
The system {u j } defined in (4.19) is a basis in E (or E 2 ) if and only if {u j } is a basis in L 2 ([0, π]) ( or (L 2 ([0, π])) 2 .
Criteria in terms of Fourier coefficients of potentials
1. Let L = L P er ± (v) be a Hill operator with H −1 per -potential, or Dirac operator with L 2 -potential, subject to periodic P er + or anti-periodic P er + boundary conditions.
Recall that the eigenvalues λ ± n , µ n and the related functions β ± n (v, z) are well defined for large enough |n|. Let
(6.1) t n (z) = |β − n (z)/β + n (z)| if β + n (z) = 0, ∞ if β + n (z) = 0, β − n (z) = 0, 1 if β + n (z) = 0, β − n (z) = 0. |n| > N * .
Then the following criterion for existence of a Riesz basis consisting of root functions of L holds. This theorem implies that Condition (7) in Theorem 24 is equivalent to Conditions (1) -(6) there.
(6.2) 0 < lim inf n∈M t n (z * n ), lim sup n∈M t n (z * n ) < ∞, where z * n = 1 2 (λ − n + λ + n ) − λ 0
Proof. In view of Remark 13 we need to prove only (a).
For Dirac operators, [14,Theorem 12] proves, in the case N \ M is finite, that Condition (6.2) implies the existence of a Riesz basis in L 2 ([0, π], C 2 ) which consists of eigenfunctions and at most finitely many associated functions of the operator L P er ± (v). The same proof explains that (6.2) implies (a) for arbitrary infinite set of indices M not only for Dirac operators but also for Hill operators with H −1 per -potentials. If (6.2) fails, then one may follow, with a slight modification, the proof of [4,Theorem 71] in order to show that (a) fails. We provide all details of such a modification below. Suppose (6.2) fails. Then there is a subsequence of indices (n k ) in M such that either (6.3) t n k (z * n k ) → 0 as k → 0, or t n k (z * n k ) → ∞. Next we consider only the case (6.3) because the other one is symmetric -if 1/t n k (z * n ) → 0, then one may exchange the roles of β + n and β − n and use the same argument.
Lemma 20. In the above notations, if (6.3) holds then there is a sequence (η k ) of positive numbers such that
(6.4) t n k (z) ≤ η k → 0 ∀ z ∈ [z − n k , z + n k ], where [z − n , z + n ]
denotes the segment with end points z − n and z + n . Proof. By [9,Lemma 20] (in the case of Hill operators with H −1 per -potentials) or by [4,Lemma 40] (in the case of Dirac operators), for large enough |n| we have |γ n | ≤ 2(|β − n (z * n )| + |β + n (z * n )|). Therefore, (6.3) implies that for large enough k
(6.5) |γ n k | ≤ 2(|β − n k (z * n k )| + |β + n k (z * n k )|) ≤ 4|β + n k (z * n k )|.
In view of (2.40) in Proposition 4 or (2.43) in Proposition 5, for each z ∈ [z − n , z + n ] and all n ∈ M with large enough |n| we have
(6.6) |β ± n (z) − β ± n (z * n )| ≤ sup [z − n ,z + n ] ∂β ± n ∂z (z) · |z − z * n | ≤ ε n |γ n |,
with ε n → 0 as |n| → ∞. Therefore, by (6.5) and (6.6) it follows that
(6.7) |β + n k (z)| ≥ |β + n k (z * n k )| − 4ε n k |β + n k (z * n k )| = (1 − 4ε n k )|β + n k (z * n k )|.
On the other hand, (6.5) and (6.6) imply that
|β − n k (z)| ≤ |β − n k (z) − β − n k (z * n k )| + |β − n k (z * n k )| ≤ 4ε n k |β + n k (z * n k )| + |β − n k (z * n k )|. Thus, since ε n k → 0, we obtain |β − n k (z)| |β + n k (z)| ≤ 4ε n k |β + n k (z * n k )| + |β − n k (z * n k )| (1 − 4ε n k )|β + n k (z * n k )| = 4ε n k + t n k (z * n k ) 1 − 4ε n k → 0,
i. e., (6.4) holds with η k = 4εn k +tn k (z * n k ) 1−4εn k . Now one may follow p. 754 in [4] (in Russian original p. 170) in order to complete the proof.
2. Theorem 19 provides a general criterion for Riesz basis property of the system of root functions of Hill operator or Dirac operator subject to periodic or anti-periodic boundary conditions. It extends and slightly generalizes [13, Theorem 1] (or [12,Theorem 2]) in the case of Hill operators, and [14,Theorem 12] in the case of Dirac operators.
Theorem 19 is an effective criterion for analyzing the existence or nonexistence of Riesz bases consisting of root functions of Hill or Dirac operators. We refer to our papers [12,13,14] for concrete applications (see also [4,Theorem 71]). Now we give examples of classes of Hill operators with singular potentials which system of root functions has (or has not) the Riesz basis property.
Example 21. Let A ⊂ (0, π) be countable, and let
(6.8) v(x) = k∈Z α∈A g(α)δ(x − α − kπ) − 1 π α∈A g(α)
with (6.9) ∃α * : |g(α * )| > α∈A\{α * } |g(α)|.
Then the system of root functions of L P er ± (v) has the Riesz basis property. Proof. Indeed, (6.8) implies that the Fourier coefficients of v (6.10) V (k) = 1 π α∈A g(α)e ikα , k ∈ 2Z, satisfy (6.11)
∃A > 0 : 1 A ≤ |V (k)| ≤ A, ∀k ∈ 2Z.
Recall that by (2.37) β ± n (v, z) = V (±2n) + ∞ k=1 S ± k , with S k defined by (2.38). In view of (2.38) and (6.11),
|S ± k | ≤ j 1 ,...,j k =±n A k+1 |n 2 − j 2 1 + z| · · · |n 2 − j 2 k + z| .
For |z| < n/2, we have
|n 2 − j 2 + z| ≥ |n 2 − j 2 | − n/2 ≥ 1 2 |n 2 − j 2 | for j = ±n, j − n ∈ 2Z.
Therefore,
|S ± k | ≤ j 1 ,...,j k =±n (2A) k+1 |n 2 − j 2 1 | · · · |n 2 − j 2 k | ≤ (2A) k+1 j =±n 1 |n 2 − j 2 | k .
Now, by the elementary inequality
j =±n 1 |n 2 − j 2 | ≤ 2 log n n , n ≥ 3, it follows that |S ± k | ≤ (4A) k+1 log n n k .
Thus, ∞ 1 |S ± k | = O((log n)/n), so we obtain (6.12) β ± n (v, z) = V (±2n) + O((log n)/n). In view of (6.11) the latter formula implies (6.2), thus the system of root functions of L P er ± (v) has the Riesz basis property.
3. Next we use (6.12) to explain the claims in Example 12.
Proof of Claims (i) and (ii) in Example 12.
Proof of (i). In view of (4.17), the Fourier coefficients V (m), m ∈ 2Z, of the potential v in Example 12 are given by
V (m) = 0 m ≤ 0, c(m/2) m > 0.
Since V (m) = 0 for m ≤ 0, one can easily see from Formulas (2.37) and (2.38) that β − n (v; z) ≡ 0 ∀n > N * , |z| ≤ n. On the other hand, by (4.18),
∃A > 0 : 1/A ≤ |V (m)| ≤ A ∀m ∈ 2N,
so the same argument as above proves that (6.12) holds. Since, by (4.18), we have |V (2n)| > 1/A, it follows that (6.13) β + n (v; z) = V (±2n) + O((log n)/n) = 0 if n > N * . Fix an n > N * . By Proposition 4, the equation (2.41), that is
(z − α n (z; v)) 2 = β + n (z; v)β − n (z; v)
has exactly two (counted with multiplicity) roots in the disc |z| < n/4. Since β − n (v; z) ≡ 0, now this equation has one double root, say z * n , and the matrix Proof of (ii). By the proof of (i) we have, for large enough n, (6.14) γ n = 0, β − n (v; z * n ) = 0,
α n (v, z * n ) − z * n β − n (v; z * n ) β + n (v; z * n ) α n (v, z)α n (v, z * n ) − z * n = 0 0 β + n (v; z * n ) 01 2A ≤ |β + n (v; z * n )| ≤ 2A.
Therefore, by [9, Theorem 37, (7.30)] it follows for n > N * * that
(6.15) 1 144A ≤ 1 72 |β + n (v; z * n )| ≤ |µ n − λ + n | ≤ 58|β + n (v; z * n )| ≤ 116A. We set f n = u 2n , ξ n = u 2n−1 −1 , ϕ n = ξ n · u 2n−1 .
Then (4.14) takes the form L f n = λ + n f n , L ϕ n = λ + n ϕ n + ξ n · f n , so now we are using the notations of [9, Lemma 30] (or [4,Lemma 59]) and can apply the related Fundamental Inequalities.
By the inequalities |µ n − λ + n | ≤ 4ξ n + 4|γ n | ξ n ≤ 4|γ n | + 2(|β − n (v; z * n )| + |β + n (v; z * n )|) (see [4], p. 741; p. 156 in Russian original) it follows, in view of (6.14) and (6.15), that ξ n ∼ |µ n − λ + n | ∼ |β + n (v; z * n )|). Therefore, 0 < inf{ξ n }, sup{ξ n } < ∞, so the system {u 2n , u 2n−1 , n > N * } is a Riesz basis in its closed linear span. This completes the proof of Claim (ii) in Example 12.
Fundamental inequalities and criteria for Riesz basis property
1. Now we have to analyze carefully 2D-blocks, P m , E m = RanP m and pairs of root-functions {u 2m−1 , u 2m }.
As a matter of fact it has been done -just in the form which perfectly fits to our needs coming from Criterion 9 -in our papers [24,3,4,9]. T. Kappeler and B. Mityagin [24,Theorem 4.5], in the case of Hill operator with L 2 -potential proved the inequality (Notice that the constants may change because in [24] and [3] the interval I = [0, 1], not [0, π] as in the present paper.) All these results are presented in [4] and the proofs are written in the way which covers the case of 1D Dirac operator as well -see Section 4.2 and 4.3 there. Moreover, these proofs could be extended to the case of Hill operators with H −1 per potentials as soon as we prove (4.4) for the deviations P n − P 0 n . This is done in [9, Section 9.2, Proposition 44 and Theorem 45] even in a stronger form (7.3) P n − P 0 n L 1 →L ∞ → 0 as n → ∞ -see [9, (9.7), (9.8), (9.84)]. Analogues of the inequalities (7.1) and (7.2) are inside of the proof of Lemma 30 there.
2. We fix m to consider E = E m = RanP m , dimE = 2, with m large enough. For a while we suppress an index m and write
(7.4) f = u 2m , h = u 2m−1 , γ = λ + m − λ − m = 0 with (7.5) L bc f = λ + f, L bc h = λ − h, f = h = 1
and such a normalization that (7.6) h = af + bϕ, ϕ, f = 0, a ≥ 0, b > 0, a 2 + b 2 = 1.
Notice that
(7.7) u 2m , u 2m−1 = f, h = a, κ := (1 − a 2 ) −1/2 = 1/b.
Moreover,
(7.8) L bc ϕ = (λ + − γ)ϕ + ξf, ξ = − a b γ.
For µ = µ m put (7.9) L Dir g = µg, g = 1.
Then -see [4, formula (4.32)] and the lines which follow -for some τ, 1/2 ≤ |τ |, by [4, (4.28)]
(7.10) τ (µ − λ + )g = b(ξP Dir f − γP Dir ϕ). Put (7.11) r = |µ − λ + | |λ + − λ − | , i.e., |γ| = 1 r |µ − λ + |; then (7.12) µ − λ + = 1 τ b (ξ P Dir f, g − γ P Dir ϕ, g )
and with P Dir ≤ 3/2 by (4.4) we have
(7.13) |µ − λ + | ≤ 2 3 2 |ξ| + 3 2 · 1 r |µ − λ + | .
If r ≥ 6 it follows that (7.14) |µ − λ + | ≤ 6|ξ| = 6a |γ|/b ≤ 6 b · |γ|, and (7.15) r ≤ 6κ, κ ∈ (7.7).
If r ≤ 6 of course (7.15) holds because κ ≥ 1. These relations (7.14)-(7.15) hold for any m ∈ M, Proof. With proper adjustments of indexation (see the remark in the first paragraph of Section 4.2) Criterion 9, Formula (3.22), imply that if U ∆ is a basis then (7.19) holds. By (7.14)-(7.15) for each individual m ∈ ∆ (7.21)
r m = |µ m − λ + m | |λ + m − λ − m | ≤ 6κ m .
Taking supremum over m ∈ ∆ we get (7.20).
3. Now we want to complement the inequality (7.14)-(7.15) with estimates of κ = 1/b from above in terms of r ∈ (7.11) (m is suppressed). It immediately follows from the inequality (7.22) |ξ| ≤ 8|γ| + 36|µ − λ + | -see lines after formula (4.59) on p. 745 in [4] (p. 161 in Russian original). Indeed with γ = 0 (7.22) together with (7.8) and (7.11) imply 4. Fundamental inequalities (7.14) and (7.22) for individual m and Propositions 22 and 23 where a subset ∆ could be chosen as we wish emphasize that neither Dirichlet eigenvalues µ m , m ∈ ∆, nor P er + or P er − eigenvalues λ ± for m ∈ ∆ could have any effect on R ∆ or κ(∆). In particular, Dirichlet eigenvalues with even (or odd) indices have no effect whatsoever when convergence of spectral decompositions related to P er − (or P er + correspondingly) is considered.
(7.23) |ξ| ≤ 1 b | γ| ≤ (8 + 36r)|γ| so b ≥ √ 3 2 , 1 b ≤ 2 √ 3 < 2, or b ≤ √ 3 2 ,
We can combine Propositions 22 and 23 and claim (for all four cases listed in Section 4.2 in the line prior to (4.5)) the following.
Theorem 24. Let L P er ± (v) be either the Hill operator with L 2 or H −1 perpotential v or the Dirac operator with L 2 -potential v, subject to periodic P er + or anti-periodic P er − boundary conditions. Then the following conditions are equivalent:
(1) The system of root functions of L P er ± (v) contains a Riesz basis in L 2 ([0, π]) (respectively in L 2 ([0, π]) 2 .
(2) The system {u j } defined in (4.19) is a Riesz basis in L 2 ([0, π]) (respectively in (L 2 ([0, π])) 2 .
(3) The system {u j } is a basis in L 2 ([0, π]) (respectively in (L 2 ([0, π])) 2 . (6) The system {u j } is a basis in a separable r.i.f.s. E such that for some
1 < a ≤ b < ∞ L a ⊃ E ⊃ L b , g L a ≤ g E ≤ g L b ∀g ∈ L ∞ .(7)
With β ± n (v, z) defined in (2.34), and t n (z) = |β − n (v, z)/β + n (v, z)| (7.28) 0 < lim inf n∈M t n (z * n ), lim sup n∈M t n (z * n ) < ∞, where z * n = 1 2 (λ + n + λ − n ) − n 2 in the Hill case and z * n = 1 2 (λ + n + λ − n ) − n in the case of Dirac operators.
(Recall that β ± n (v; z) are introduced in Section 2.5, Lemma 3; see their basic properties in Propositions 4 and 5).
Proof. The equivalence of Conditions (1) -(5) follows from Propositions 22 and 23 and Corollary 10. Conditions (6) and (7), and their equivalence to (1) -(5) are explained in Sections 5, Theorem 18 and Section 6, Theorem 19.
the above cases (A) and (B) the root function system contains a basis in E if and only
=
(ii) Sp(L 0 P er − ) = {n 2 , n = 1, 3, 5, . . .}; its eigenspaces are E 0 n = Span{e ±inx }, and dim E 0 n = 2. (iii) Sp(L 0 Dir ) = {n 2 , n ∈ N}; each eigenvalue n 2 is simple; the corre-Span{s n }.
For large enough |n|, the equation(2.44) has exactly two (counted with multiplicity) roots in D.
≤ C for any finite subset F ⊂ N.
Lemma 7 .
7Assume the system of 2D projections Q m ∈ (3.3) in a Hilbert space H is a Riesz system, i. e.,(3.16) holds. If {u k } ∞ 1 is a basis in Y ⊂ H then it is an unconditional basis in Y.Proof. Proof is based on the Orlicz[46] lemma:Lemma 8. (3.16) holds for the system Q m ∈ (3.3) in a Hilbert space if and only if for some constant C the norms of 1D projections q j are uniformly bounded. By (3.1)
Criterion 9 .
9With notations(3.1),(3.3) let us assume that the system of 2D projections {Q m } is a Riesz system in a Hilbert space. If a normalized system(3.21) {u k }, u k = 1,is a basis in Y then(3.22) κ := sup {(1 − | u 2m−1 , u 2m | 2 ) −1/2 : m ∈ N} < ∞.If the condition(3.22) holds then {u k } is a normalized unconditional basis, that is a Riesz basis in Y .
Corollary 10 .
10If(3.16) holds in a Hilbert spaceH the system {u k } ∞ 1 ∈ (3.21), (3.1) is a Riesz basis if and only if it is a basis.4. Moving from geometric criterion to Hill and Dirac operators1. The basic assumption in the geometric Criterion 9 is the property of a system of projections {Q m } in a Hilbert space to be a Riesz system.
( 4 .
410) u j = 1, j ∈ N. If m ∈ M 1 choose any pair of orthogonal normalized vectors in E m(4.11) u 2m−1 , u 2m = 0.3. For m ∈ M 2 we consider two different options to choose root functions for a basis.
Option 1 .
1If m ∈ M 2 , then there is only one (up to constant factor) normalized eigenvector f ∈ E m ,
.13) u 2m = f, u 2m−1 ⊥ u 2m , u 2m−1 = 1.
∃A > 0 : 1/A ≤ |c(k)| ≤ A ∀k ∈ N.Then we have:(i) M 2 = Γ bc ∩ {n : n > N * } for bc = P er + and P er − , i. e., all E m with m > N * are Jordan;(ii) with choices by Option 2 the condition (4.16) holds, and the system of eigen-and associated functions {u 2m−1 , u 2m } is a Riesz basis in L 2 .
Theorem 19 .
19Let M = {n : |n| ≥ N * , λ − n = λ + n }, and let {u 2n−1 , u 2n } be a pair of normalized eigenfunctions corresponding to the eigenvalues λ − n , λ + n . (a) The system {u 2n−1 , u 2n , n ∈ M} is a Riesz basis in its closed linear span if and only if
n with λ 0 n = n 2 for Hill operators and λ 0 n = n for Dirac operators. (b) The system of root functions of L contains a Riesz basis if and only if (6.2) holds.
(
The function v in (6.8) lies in H −1 per as it follows from [21, Theorem 3.1 and Remark 2.3] or [7, Proposition 1].)
is Jordan. In view of Lemma 3(c), this implies that all E m with m > N * are Jordan, i.e., (i) in Example 12 holds.
− λ + | ≤ 2K 10 (|ξ| + 2|γ|) (see notations in (7.4) -(7.9) below). P.Djakov and B. Mityagin [3, Lemma 10, Inc. (4.32)] succeeded to go to the opposite direction and proved the inequality (7.2)|ξ| ≤ 6|γ| + 8|µ − λ + |
( 7 .
716) M = {n : γ n = λ + n − λ − n = 0, n ≥ N * }. For ∆ ⊂ M set (7.17) U ∆ = {u 2m−1 , u 2m : m ∈ ∆} and (7.18) H ∆ = the closure of Lin Span U ∆ .Proposition 22. If the system U ∆ is a basis in H ∆ then(7.19) κ(∆) = sup{(1 − | u 2m−1 , u 2m | 2 ) −1/2 : m ∈ ∆} < ∞is finite, and(7.20) R ∆ = sup m∈∆ |µ − λ + | |λ + − λ − | ≤ 6κ(∆) < ∞.
( 4 )
4κ(M) := sup {(1 − | u 2m−1 , u 2m | 2 ) −1/2 : m ∈ M} < ∞.
( 5 )
5R(M) := sup |µm−λ + m | |λ + m −λ − m | : m ∈ M < ∞.
But in all four cases -P er + and P er − for both Hill and Dirac operators the systems of projections2.16)
-this is
lim
N →∞
N * < n ≤ N
n ∈ Γ bc
if this limit does exists. However for Dirac operators Γ bc ∈ (2.26) are subsets
in Z; we have to accept convention to define the sum in (2.32) as
lim
N →∞
N * < |n| ≤ N
n ∈ Γ bc
and
lim
N →∞
−N < n ≤ N + 1
n ∈ Γ bc , |n| > N *
if both these limits exist and are equal. Such understanding is in accordance
with the choice of contours in (2.25) and (2.27).
(4.5)
{S N * , P n , |n| > N * , n ∈ Γ bc }
given in (2.11) -(2.15) or (2.27) -(2.31) are Riesz systems of projections
as (2.16) and (2.32) tell us.
Now we define three sets of indices:
With these inequalities Criterion 9, its second part, implies with notations (7.19), (7.20) the following.Proposition 23. If R ∆ < ∞ then (7.26) κ(∆) ≤ 16 + 72R ∆ and the system U ∆ is a Riesz basis in H ∆ .Proof. Again, individual inequalities (7.27) κ m ≤ 16 + 72r m , m ∈ ∆ hold by(7.25). With R ∆ being finite if we take supremum over m ∈ ∆ in (7.25) we get (7.26). Then Criterion 9 claims that U ∆ is a Riesz basis in H ∆ .and
(7.24)
1
2b
≤
√
1 − b 2
b
≤ 4(2 + 9r);
Therefore, in either case
(7.25)
κ =
1
b
≤ 16 + 72r.
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| []
|
[
"Probing two-level systems with electron spin inversion recovery of defects at the Si/SiO 2 interface",
"Probing two-level systems with electron spin inversion recovery of defects at the Si/SiO 2 interface"
]
| [
"M Belli \nLaboratorio MDM\nIMM-CNR\nVia C. Olivetti, 220864Agrate Brianza (MB)Italy\n",
"M Fanciulli \nLaboratorio MDM\nIMM-CNR\nVia C. Olivetti, 220864Agrate Brianza (MB)Italy\n\nDipartmento di Scienza dei Materiali\nUniversità degli Studi di Milano-Bicocca\nVia R. Cozzi 5320126MilanoItaly\n",
"R De Sousa \nDepartment of Physics and Astronomy\nUniversity of Victoria\nV8W 2Y2VictoriaBritish ColumbiaCanada\n"
]
| [
"Laboratorio MDM\nIMM-CNR\nVia C. Olivetti, 220864Agrate Brianza (MB)Italy",
"Laboratorio MDM\nIMM-CNR\nVia C. Olivetti, 220864Agrate Brianza (MB)Italy",
"Dipartmento di Scienza dei Materiali\nUniversità degli Studi di Milano-Bicocca\nVia R. Cozzi 5320126MilanoItaly",
"Department of Physics and Astronomy\nUniversity of Victoria\nV8W 2Y2VictoriaBritish ColumbiaCanada"
]
| []
| The main feature of amorphous materials is the presence of excess vibrational modes at low energies, giving rise to the so called "boson peak" in neutron and optical spectroscopy. These same modes manifest themselves as two level systems (TLSs) causing noise and decoherence in qubits and other sensitive devices. Here we present an experiment that uses the spin relaxation of dangling bonds at the Si/(amorphous)SiO2 interface as a probe of TLSs. We introduce a model that is able to explain the observed non-exponential electron spin inversion recovery and provides a measure of the degree of spatial localization and concentration of the TLSs close to the interface, their maximum energy and its temperature dependence. | 10.1103/physrevresearch.2.033507 | [
"https://arxiv.org/pdf/1904.01984v1.pdf"
]
| 102,487,960 | 1904.01984 | 879427556bf2d35a6ee2c659118bb193b16c50e4 |
Probing two-level systems with electron spin inversion recovery of defects at the Si/SiO 2 interface
M Belli
Laboratorio MDM
IMM-CNR
Via C. Olivetti, 220864Agrate Brianza (MB)Italy
M Fanciulli
Laboratorio MDM
IMM-CNR
Via C. Olivetti, 220864Agrate Brianza (MB)Italy
Dipartmento di Scienza dei Materiali
Università degli Studi di Milano-Bicocca
Via R. Cozzi 5320126MilanoItaly
R De Sousa
Department of Physics and Astronomy
University of Victoria
V8W 2Y2VictoriaBritish ColumbiaCanada
Probing two-level systems with electron spin inversion recovery of defects at the Si/SiO 2 interface
(Dated: April 4, 2019)arXiv:1904.01984v1 [cond-mat.mes-hall] 3 Apr 2019
The main feature of amorphous materials is the presence of excess vibrational modes at low energies, giving rise to the so called "boson peak" in neutron and optical spectroscopy. These same modes manifest themselves as two level systems (TLSs) causing noise and decoherence in qubits and other sensitive devices. Here we present an experiment that uses the spin relaxation of dangling bonds at the Si/(amorphous)SiO2 interface as a probe of TLSs. We introduce a model that is able to explain the observed non-exponential electron spin inversion recovery and provides a measure of the degree of spatial localization and concentration of the TLSs close to the interface, their maximum energy and its temperature dependence.
Several properties of amorphous materials can be explained by assuming the presence of additional low energy vibrational modes on top of the usual phonon density of states. In neutron scattering and Raman spectroscopy these modes appear as a universal boson peak with average energy increasing with temperature [1,2]. At low temperatures, these modes give rise to an anomalous contribution to the specific heat. A convenient assumption is to model the excess modes at the low energy tail of the boson peak as an ensemble of tunnelling twolevel systems (TLSs), each with energy splitting E. Assuming their energy density scales as a power law with exponent α (ρ ∝ E α ) leads to specific heat scaling as T 1+α [3,4]. The coefficient α gives a measure of the degree of amorphousness of the material.
The TLSs are often responsible for the origin of noise, decoherence, and dielectric energy loss in all kinds of devices for solid state quantum computation, including superconducting Josephson devices [5,6] and spin qubits [7]. As these devices are generally made from high quality materials, the TLSs usually appear close to surfaces and interfaces, where the degree of crystallinity is quite hard to control. In addition several architectures are based on semiconductor/amorphous oxide interfaces.
In contrast most experimental measurements of the boson peak and TLSs are "bulk probes". To date the few experiments that are able to probe TLSs at thin films or interfaces still lack the energy resolution to measure properties such as the exponent α and the temperature dependence of the boson peak [8,9]. As devices become smaller, the interface plays an increasing role; this, combined with the fact that the interface is inevitably amorphous, motivates the development of a method that detects TLSs at surfaces and interfaces.
Here we describe an experiment that uses danglingbond spins as a probe of TLSs at the Si/SiO 2 interface. Unsaturated dangling bonds (DBs), generically called P b -centers, appear at the Si/SiO 2 interfaces. [10][11][12] Their structure is quite well understood. [13] We mea-sure spin-lattice relaxation of the DB spin magnetization, S z (t) , using inversion-recovery experiments with echo detection. We show that S z (t) approaches thermal equilibrium in a highly non-exponential fashion, leading to a wealth of information on the spatial distribution and energetics of TLSs nearby the DB spin. The signal intensity is measurable thanks to a nanostructuring of the interface into nanowires, instead of a flat surface, greatly increasing the surface-to-volume ratio. [14,15] It is in fact well known that DBs can act as a probe of TLSs because their spin relaxation rate 1/T 1 is strongly dominated by TLS dynamics, even at higher temperatures [16,17]. However, previous experiments [17] were unable to interpret the long time non-exponential decay. Below we describe our experiment and propose a theoretical model based on a Poisson distribution of TLSs within a radius of each DB. This model is able to capture the long time non-exponential dynamics thus allowing the extraction of much more information on TLS parameters than previous approaches. As a result we are able to obtain a clear picture of TLSs at the interface, including the measurement of their degree of spatial localization, one of the most important unsolved problems in the physics of the boson peak.
Experiment.-Silicon nanowires were prepared by a metal-assisted chemical etching (MACE) process starting from two different types of seed metal deposited on intrinsic [001] silicon. Details of the two samples under investigation are reported in Table I. For sample A, pinholes in a 3.8 nm thick gold layer were used to realize the nanowires. For sample B, the seed metal consisted in Ag nanoparticles, deposited by electroless deposition, as explained in Refs. 15 and 18, where also details of the etching process are reported. At the end of the MACE process and complete metal removal a dense carpet of straight silicon nanowires, fully passivated with H and with structural parameters dependent on the process details, cover the whole sample surface ( Figure 1). The two samples were chosen out of many different batches for the present investigation. Both samples were annealed in vacuum for 15 minutes at 550 • C to induce depassivation of surface defects from the naturally hydrogen-passivated state formed during the MACE process. Details of the depassivation process have been reported elsewhere. [14,15] The samples were then characterized by continuouswave and pulsed electron paramagnetic resonance (Bruker Elexsys E580 system, X-band). After the depassivation treatment the typical electron paramagnetic resonance signal of DB defects at the interface, the so called P b defects, was observed to increase. [14,15] Inversion recovery experiments, with the magnetic field H [001], were performed in the temperature range 4-300 K to obtain information on the relaxation rate. A typical example of an inversion recovery curve is reported in Figure 2, together with a fit attempt with a single exponential recovery.
Such a model evidently fails, especially at low temperature, though the resulting thermal trend of the spinlattice relaxation rate determined assuming a single exponential recovery, may allow comparisons with data re-Inversion Recovery Delay (s) ported in the literature. Generally, the inefficiency of a single exponential recovery fit is neglected and the analysis focuses on the thermal variation of the resulting spin-lattice relaxation rate 1/T 1 , which is reported to follow a power-law trend ∝ T 2+α with α in the range ∼ 0.3 − 1.5. [17,19] The lowest value reported to the authors' knowledge is α = −0.2, in Ref. 16. In our case, the fit would result in even lower α values ranging from −0.45 to −0.55, which are quite unusual, at least for bulk materials. Fitting attempts with a stretched exponential recovery model were attempted and seemed indeed more successful, though they essentially shift the whole information on the dominating relaxation mechanisms into the temperature dependence of other two physical quantities: the stretched relaxation rate and the stretched exponent, which require further interpretation. We think that a deeper understanding of the non-exponential recovery is necessary. The temperature dependence of the spin lattice relaxation rate, obtained either by a single exponential or by a stretched inversion recovery, was attributed to the presence of TLSs. We need therefore a broader theoretical framework modelling the role of the TLSs already at the level of the recovery curves.
Theoretical model.-Previous models for DB spin relaxation [7,17] assumed a low disorder prescription: Each DB spin was assumed to relax through cross-relaxation with only one TLS. For the case of an interface we generalize this model to account for an arbitrary number of TLSs surrounding each P b DB. A key aspect is that different P b spins may be surrounded by different numbers of TLSs, leading to a much greater level of disorder. In this high disorder prescription the fraction of DBs with n TLSs within a "coupling radius" follows a Poisson dis-tribution,
p n =N n n! e −N ,(1)
whereN is the average number of TLSs coupled to each DB. We assume that the TLSs do not interact with each other and are independently distributed with energy splitting in the interval E ∈ [0, E max ], each with density proportional to E α . The average spin magnetization for a DB interacting with n TLSs is given by
S z (t) n = Emax 0 dE 1 E α 1 · · · Emax 0 dE n E α n e − n i=1 Γ(Ei,T )t Emax 0 dE 1 E α 1 · · · Emax 0 dE n E α n = ( S z (t) 1 ) n ,(2)
where Γ(E i , T ) is the spin relaxation rate for a DB interacting with one TLS of energy E i , and
S z (t) 1 = α + 1 E α+1 max Emax 0 dEE α e −Γ(E,T )t(3)
the associated magnetization decay. Taking an average over the number n of TLSs nearby each DB leads to
S z (t) = ∞ n=0 p n S z (t) n = exp −N [1 − S z (t) 1 ] .
(4) This expression shows that the Poisson distribution of TLSs makes DB spin relaxation highly non-exponential in time.
In order to complete the model we need to obtain a suitable expression for the relaxation rate Γ(E, T ), the rate for a DB spin to achieve thermal equilibrium with a single TLS of energy E. The mechanism is based on spinorbit induced cross-relaxation [7]. When a TLS switches, the local environment around the DB spin fluctuates; spin-orbit coupling translates this switch into a fluctuating magnetic field that flips the spin.
The energy eigenstates of each TLS are denoted | ± with energies ±E/2, and the transition rate for a TLS to switch from state | ± to state | ∓ is denoted r ± (the subscript refers to the initial state in the transition).
When the magnetic field is low so that DB Zeeman energy is much less than both k B T and E, the rate for crossrelaxation is well approximated by [7]
Γ ±↑ = Γ ±↓ ≈ A 2 r ± ,(5)
with A ≪ 1 a dimensionless spin-orbit coupling parameter. Here Γ +↑ denotes the rate for a cross switch from the TLS-DB state |+, ↑ into the state |−, ↓ . These cross rates are much stronger than non-cross spin flips because they couple states that are not the time reversal of each other (|+, ↑ is not the time reversal of |−, ↓ ). The thermalized rate Γ(E, T ) is obtained by averaging over TLS and DB spin states with their corresponding Boltzmann occupations. Denote p(i|σ) the probability of finding TLS in state i = +, − given that the DB spin is known to be in state σ =↑, ↓. In the limit of DB Zeeman energy much smaller than k B T, E we get p(i| ↑) ≈ p(i| ↓) hence
Γ(E, T ) = i=+,− σ=↑,↓ p(i|σ)Γ iσ ≈ 2A 2 (p + r + + p − r − ) = 4A 2 r + r − r + + r − .(6)
Note how the DB spin relaxation rate Γ(E, T ) is solely determined by the TLS rates r ± . To describe the physics up to quite high temperatures we generalized the theory for r ± described in [7] to processes involving one and two acoustic phonons. The final result is
Γ(E, T ) = a E/k B sinh (βE) + b (E/k B ) 5 1 + e βE 0.00714 + 2930 (βE) 7 × 1 + βE 2 + (βE) 2 10 + (βE) 3 100 ,(7)
where β = 1/(k B T ), and a and b model the linear and quadratic dependence of TLS parameters on the phonon dilation strain, respectively. Therefore, a models the efficacy of TLS flipping following the emission/absorption of a single acoustic phonon and b models the same effect involving two phonons. Equation (7) was plugged into Eqs. (3) and (4) leading to an explicit analytic expression for the measured inversion recovery curve,
I I 0 = 1 − 2 exp −N 1 − (βE max ) −(α+1) (βEmax) α+1 0 dx e −Γ x 1 α+1 ,T t .(8)
We stress again the highly non-exponential form of the model. This expression has five fitting parameters: a, b, α,N , E max . The fitting was done by assuming the first four parameters independent of temperature, with E max temperature dependent according to The assumed polynomial fit for the T dependence of E max may be seen as an approximation to the complex TLS thermal activation. The fit results are reported in Table II. The common value ofN ≈ 0.7, i.e. less than one, reveals either a relatively diluted distribution of TLSs at or close to the interface and/or strong space localization for the TLS atomic configuration. The differences between the two samples may be tentatively ascribed to their different structural characteristics -mainly diameter and nanowire density on the surface. This applies to a and b values, which have a higher impact on the effective typical times of the inversion recovery. Differences can be observed also in the parameters describing the thermal evolution of E max , though the two leading terms, c and d, are relatively similar. This implies that the difference between the two samples is more relevant in the higher temperature range.
E max (T ) = c + dT + f T 2 + gT 3 .(9)
Conclusions.-We exploited the high interface area of silicon nanowires to detect, with good signal-to-noise ratio, the electron spin inversion recovery of P b centers at the Si/SiO 2 interface. A novel model was developed to describe the non-exponential character of the inversion recovery, attributed to a relatively dilute density of TLSs. The fact that each dangling-bond center interacts, on the average, with less than one TLS (N < 1) indicates the TLSs are highly localized and/or dilutely distributed across the interface. The proposed method provides information on the TLSs and can be extended to other relevant systems. A comparison of the results with theoretical models of specific TLSs, such as amorphous modes associated to hydrogen or other point defects in the oxide, may lead to the still missing identification of the microscopic nature of the TLSs.
M.F. and M.B. acknowledge financial support from the CARIPLO Foundation (Italy), ELIOS project, and the Italian Ministry of Defense, QUDEC project. R.d.S. acknowledges financial support from NSERC (Canada) through its Discovery (RGPIN-2015-03938) and Collaborative Research and Development programs . We thank J. Fabian for useful discussions.
FIG. 1 .
1Scanning Electron Microscope images of the two systems under investigation. Images on the left refer to sample A, while images on the right refer to sample B. The images were taken on twin samples obtained from the same batches of the ones used for magnetic resonance investigations.
FIG. 2 .
2(Color online) Comparison between a single exponential inversion recovery fit at 5 K for sample B and a fit according to the model outlined in Eq.(8).
TABLE I .
IStructural characteristics of the investigated SiNW samples and average distance and concentration of the corresponding P b -like defects. f is an estimate of the surface-tovolume increase factor with respect to the case of the flat surface of the bulk.Sample A
Sample B
Catalyzer
Au layer
Ag NPs
Length
(4.2 ± 0.3) µm
(17 ± 1) µm
Diam. range
8 nm -30 nm
50 nm -200 nm
< r >
3.7(1) nm
3.89(5) nm
[P b ]
8.0(8) × 10 11 cm −2 7.2(2) × 10 11 cm −2
f
∼ 800
∼ 3000
TABLE II .
IIFitted parameters according to model described
in Equations 7, 8, 9.
Sample A
Sample B
a
2337(3) Hz/K
244.2(2) Hz/K
b/k 4
B (58000 ± 8000) J −4 (1800 ± 300) J −4
c
613(1) µeV
409(2) µeV
d
569.5(1) µeV/K
558.9(1) µeV/K
f
−619(2) neV/K 2 −2035(1) neV/K 2
g
0.263(9) neV/K 3 1.530(6) neV/K 3
α
3.074(6)
2.127(5)
N
0.7290(1)
0.68900(9)
χ 2
r
1.3157
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| []
|
[
"Observability of characteristic binary-induced structures in circumbinary disks",
"Observability of characteristic binary-induced structures in circumbinary disks"
]
| [
"R Avramenko \nInstitute of Theoretical Physics and Astrophysics\nUniversity of Kiel\nLeibnizstrasse 1524118KielGermany\n",
"S Wolf \nInstitute of Theoretical Physics and Astrophysics\nUniversity of Kiel\nLeibnizstrasse 1524118KielGermany\n",
"T F Illenseer tillense]@astrophysik.uni-kiel.de \nInstitute of Theoretical Physics and Astrophysics\nUniversity of Kiel\nLeibnizstrasse 1524118KielGermany\n"
]
| [
"Institute of Theoretical Physics and Astrophysics\nUniversity of Kiel\nLeibnizstrasse 1524118KielGermany",
"Institute of Theoretical Physics and Astrophysics\nUniversity of Kiel\nLeibnizstrasse 1524118KielGermany",
"Institute of Theoretical Physics and Astrophysics\nUniversity of Kiel\nLeibnizstrasse 1524118KielGermany"
]
| []
| Context.A substantial fraction of protoplanetary disks form around stellar binaries. The binary system generates a time-dependent non-axisymmetric gravitational potential, inducing strong tidal forces on the circumbinary disk. This leads to a change in basic physical properties of the circumbinary disk, which should in turn result in unique structures that are potentially observable with the current generation of instruments. Aims. The goal of this study is to identify these characteristic structures, constrain the physical conditions that cause them, and evaluate the feasibility of observing them in circumbinary disks. Methods. To achieve this, first we perform 2D hydrodynamic simulations. The resulting density distributions are post-processed with a 3D radiative transfer code to generate re-emission and scattered light maps. Based on these distributions, we study the influence of various parameters, such as the mass of the stellar components, mass of the disk, and binary separation on observable features in circumbinary disks. Results. We find that the Atacama Large (sub-)Millimetre Array (ALMA) as well as the European Extremely Large Telescope (E-ELT) are capable of tracing asymmetries in the inner region of circumbinary disks, which are affected most by the binary-disk interaction. Observations at submillimetre/millimetre wavelengths allow the detection of the density waves at the inner rim of the disk and inner cavity. With the E-ELT one can partially resolve the innermost parts of the disk in the infrared wavelength range, including the disk's rim, accretion arms, and potentially the expected circumstellar disks around each of the binary components. | 10.1051/0004-6361/201630084 | [
"https://arxiv.org/pdf/1702.02862v2.pdf"
]
| 119,372,315 | 1702.02862 | c19767c66861c54b62ce467388cb511d131a0695 |
Observability of characteristic binary-induced structures in circumbinary disks
R Avramenko
Institute of Theoretical Physics and Astrophysics
University of Kiel
Leibnizstrasse 1524118KielGermany
S Wolf
Institute of Theoretical Physics and Astrophysics
University of Kiel
Leibnizstrasse 1524118KielGermany
T F Illenseer tillense]@astrophysik.uni-kiel.de
Institute of Theoretical Physics and Astrophysics
University of Kiel
Leibnizstrasse 1524118KielGermany
Observability of characteristic binary-induced structures in circumbinary disks
Astronomy & Astrophysics manuscript no. 30084_am c ESO 2018 November 9, 2018Accretion, accretion disks -Binaries: general -Hydrodynamics -Radiative transfer
Context.A substantial fraction of protoplanetary disks form around stellar binaries. The binary system generates a time-dependent non-axisymmetric gravitational potential, inducing strong tidal forces on the circumbinary disk. This leads to a change in basic physical properties of the circumbinary disk, which should in turn result in unique structures that are potentially observable with the current generation of instruments. Aims. The goal of this study is to identify these characteristic structures, constrain the physical conditions that cause them, and evaluate the feasibility of observing them in circumbinary disks. Methods. To achieve this, first we perform 2D hydrodynamic simulations. The resulting density distributions are post-processed with a 3D radiative transfer code to generate re-emission and scattered light maps. Based on these distributions, we study the influence of various parameters, such as the mass of the stellar components, mass of the disk, and binary separation on observable features in circumbinary disks. Results. We find that the Atacama Large (sub-)Millimetre Array (ALMA) as well as the European Extremely Large Telescope (E-ELT) are capable of tracing asymmetries in the inner region of circumbinary disks, which are affected most by the binary-disk interaction. Observations at submillimetre/millimetre wavelengths allow the detection of the density waves at the inner rim of the disk and inner cavity. With the E-ELT one can partially resolve the innermost parts of the disk in the infrared wavelength range, including the disk's rim, accretion arms, and potentially the expected circumstellar disks around each of the binary components.
Introduction
The development of circumstellar disks around single stars and their formation during the collapse of molecular clouds were studied extensively in the past three decades (McKee & Ostriker 2007;Andrews et al. 2010). However, at the same time it was found that a substantial fraction of protoplanetary disks form around binaries (Duquennoy & Mayor 1991;Kraus & Hillenbrand 2009).
The assumed mechanism by which a interstellar cloud evolves into a binary is either by multiple fragmentation or stellar capture (Bate 2000;Bate et al. 2002;Wolf et al. 2001). During the collapse both components can develop accretion disks in the same way as a single star system (Regály et al. 2011;Müller & Kley 2012). In addition to that, the dust and gas from outside of the binary orbit form a third disk, the so-called cirumbinary disk (Rodriguez et al. 2010;Romero et al. 2012;Lines et al. 2015), which revolves around the centre of mass of the binary. Furthermore, it is possible for a newly developed single star system to capture a transiting star to form a binary. In this case the primary could retain a part of its original disk and the captured star would accrete a disk of its own. The remaining matter would form the circumbinary disk.
Regardless of the exact formation mechanism the major difference from single star systems is that the orbiting binary generates a time-dependent non-axisymmetric gravitational potential, inducing strong tidal forces on the surrounding gas and dust (Artymowicz & Lubow 1994). This leads to a change in basic physical properties of the circumbinary disk, which should in turn result in unique structures such as a cavity in the disk centre, accretion arms, density waves, and spiral arms (Günther & Kley 2002;Günther et al. 2004). Previously it was shown that some of these structures, such as the cavity and parts of the accretion arms, are observable with the current generation of instruments (Ruge et al. 2015). There are also examples of more complex systems, such as GG Tau, which is a triple star system (Piétu et al. 2011;Di Folco et al. 2014). Some of the disk structures, for example spiral arms potentially associated with embedded planets, have also been observed (Muto et al. 2012;Garufi et al. 2013;Stolker et al. 2016).
The presence of a planet further complicates the case (Pierens & Nelson 2013). A binary can also feature two types of planetary orbits: planets that revolve around a single binary component and planets that orbit around both stars Orosz et al. 2012;Schwamb et al. 2013). Furthermore, the growth of planetesimals from smaller objects might be hindered (Meschiari 2012;Marzari et al. 2013). The migration behaviour of planets in a binary system can also be problematic since the planet can be ejected if it enters a 4:1 resonance with the binary (Kley & Haghighipour 2014;Pierens & Nelson 2008).
As mentioned above, the binary exerts a tidal force on the surrounding disk, which should result in characteristic structures. However, specific questions regarding the exact nature of these structures are still open: How do the characteristic spatial scales and timescales depend on parameters such as the mass and binary separation? How does the overall density and temperature structure compare to single star systems? Can structures induced Article number, page 1 of 16 arXiv:1702.02862v2 [astro-ph.SR] 21 Feb 2017 A&A proofs: manuscript no. 30084_am by the binary be observed with the current or the next generation of instruments/observatories?
To answer the above questions, we simulate the density distribution of a circumbinary disk, using the 2D hydrodynamic code Fosite (Illenseer & Duschl 2009). Subsequently, radiative transfer simulations are performed to obtain the corresponding temperature structure of these disks with the code Mol3D (Ober et al. 2015). Furthermore, scattered light and re-emission maps of these systems are generated. In each step the unique structures and quantities generated by the asymmetrical gravitational potential of the binary are discussed. Finally, a feasibility study to observe these quantities with the Atacama Large Millimeter/submillimeter Array (ALMA) and the future European Extremely Large Telescope (E-ELT) operating at optical/infrared wavelengths is performed.
The paper is divided into three parts. First, we provide a general overview of the underlying 2D hydrodynamic simulations and 3D radiative transfer calculations (Sect. 2). Subsequently, the structures caused by the binary-disk interaction are analysed (Sect. 3.1). In the third part (Sect. 3.2 and 3.3) we discuss the feasibility of observing distinctive structures resulting from the binary-disk interaction.
Hydrodynamic and radiative transfer simulations
In this chapter the applied hydrodynamic and radiative transfer simulation software is introduced. Furthermore, the general simulation set-up is presented and discussed.
Two-dimensional hydrodynamic simulation
We simulate the density distribution of the gas and dust around a binary system for a fixed parameter set. We consider the heating by the star by calculating the temperature for an initial density distribution after the system has reached a quasi-stationary stage. This density distribution is calculated using a time-and spatialconstant temperature profile. With the new time-independent temperature profile a second density distribution, which is used in our study, is calculated. With this iteration a more realistic hydrodynamic simulation can be performed compared to a simulation with a constant temperature distribution. Similar work was performed by Günther & Kley (2002) and Günther et al. (2004). The major difference between these studies and our approach is the calculation of the temperature distribution. In case of Günther & Kley (2002) and Günther et al. (2004), the heating via the viscosity and the stellar irradiation were considered in each time step of the hydrodynamic simulation. By avoiding the calculation of the temperature in each time step of the hydrodynamic simulation we can decrease the required simulation time and achieve a higher spatial resolution. The viscous heating of the disk can be neglected, since the stellar irradiation is the dominant form of heating in circumbinary disks as was shown by Günther et al. (2004).
The hydrodynamic simulations are performed with Fosite (Illenseer & Duschl 2009). The code solves the 2D continuity and Navier-Stokes equations
∂Σ ∂t + ∇ · (Σv) = 0 (1) ∂Σv ∂t + ∇ · (Σv ⊗ v + ΠI) = ∇ · T − Σ∇Φ(2)
for the vertically integrated surface density Σ, pressure Π, and gas velocity v. The quantity T is the viscous stress tensor and Φ the gravitational potential of the binary, since the self-gravitation of the disk is neglected. The hydrostatic scale height H of the disk depends on the midplane temperature T c and the gravitational potential φ(r) = i=1,2 GM i √ |r−r i | and can be written as follows (see Günther & Kley 2002):
H(r) = i=1,2 GM i c 2 s | r − r i | 3 − 1 2 ,(3)
where c s is the speed of sound and M i the masses of the components.
Binary system: The masses of the primary and secondary star are chosen to be equal to reduce the parameter space and inner radius r in of the computational domain (see below). A small inner radius is chosen to trace the disk structures up to the binary orbit. Furthermore, with equally massive components the system obtains a point symmetry, making it easy to spot possible nonphysical behaviour in the results. As was shown by Bate et al. (2002), the mass tends to be accreted by the lower mass component, as it sweeps a larger area of the disk. This mechanism could lead to binary systems with mass ratio q ≈ 1. Long period systems with mass ratios ∼ 1 have also been observed (Raghavan et al. 2010). The mass ratio of the primary and secondary is q = M sec /M prim and we denote from now on M prim = M sec = M B . Three different semi-major axes a are applied to study their influence on the structures generated by the binaries, such as the inner cavity, accretion arms, and radial inhomogeneities in the density distribution (a = 10 AU, 20 AU, and 30 AU). Furthermore, for a separation of 20 AU three different stellar masses M B are applied to examine the influence of the stellar mass on the disk structures. The full set of parameters is summarised in Table 1.
The eccentricity of the orbit is set to zero, which reduces the parameter space and simplifies the system. Simultaneously, it allows one to reduce the inner radius r in (see below), since the grid centre is located at the centre of mass of the binary. At this point it is important to note that although there are some examples of binary systems with eccentricity ε < 0.1, the majority of them have a short orbital period P < 1 yr. Systems with larger values of P tend to have smaller eccentricities (Raghavan et al. 2010).
Temperature: All calculations are performed in a locally isothermal mode based on a temperature profile T c . We derive the temperature profile by performing a low-resolution hydrodynamical simulation for each disk set-up with a constant temperature profile as a starting condition. Subsequently, the dust temperature distribution is calculated with the 3D radiative transfer code Mol3D (Ober et al. 2015, see. Sect. 2.2). The resulting azimutally averaged temperature profile is then used as default temperature T c for the high resolution hydrodynamic calculation.
Initial and boundary conditions: The initial disk mass is concentrated in a Gaussian surface density profile with its maximum at 150 AU. The mass is fixed at M disk = 0.02 M for all hydrodynamic simulations. The gas is set in a near Keplerian motion that accounts for the potential asymmetry with a higher velocity to counter additional gravitational attraction.
In order to reduce the impact of boundary conditions on the density distribution at the relevant region (inner 300 AU), the outer radius is set to 1000 AU for all simulations. The density as well as the gas velocities are sufficiently small at this boundary so as not to influence the relevant areas of the disk. Furthermore, the choice of the inner radius has an impact on the disk structure (Pierens & Nelson 2013). It cannot be chosen to be zero because the sources of gravitational potential have to be outside of the simulated region. An inner radius of r in = a · 0.6 AU is used, where a is the semi-major axis of the binary. This ensures that no singularities occur inside the grid. It also permits us to trace the accretion flow nearly up to the binary orbits. This is important, as the biggest differences to a circumstellar disk are expected to be in the inner disk region. The inner boundary condition allows for a mass flux from inside the computational domain through the boundary, but prevents any flow in the opposite direction. This has a distinct disadvantage of preventing the circumprimary or circumsecondary disks from forming. The lack of matter in the direct vicinity of both stars should result in short wavelength radiation deficiency in the radiative transfer simulations. The accretion rate at the inner boundary amounts to ∼ 10 −8 M /yr and hence does not significantly change the mass of the binary or the disk for the duration of the simulation.
Grid: In order to achieve a sufficient resolution for the subsequent radiative transfer simulation, especially in the inner regions, a polar grid (r, φ) with logarithmic spacing for the r coordinate and a grid resolution of (N r = 508, N φ = 508) is chosen.
Simulation time: The evolution of the system is calculated over a period of t sim = 5 · 10 4 yr for all simulations. For the subsequent temperature calculations, density distributions with time steps between 2.2 · 10 4 yr and 2.7 · 10 4 yr are used. This decision is based on two constraints. First, the gas has to be sufficiently mixed to ensure that the initial state does not influence the resulting distribution. The timescale related to the advective mixing processes can be estimated using the dynamical timescale,
t dyn ∼ r v φ ∼ Ω −1 .(4)
Here, v φ is the φ component of the velocity and Ω is the angular velocity. By choosing a simulation time step corresponding to over 100 binary periods this condition is met. The second constraint is to avoid losing too much of the initial mass. The timescale for this process is the viscous timescale, which can be estimated using the radial drift velocity v r ,
t visc ∼ r v r .(5)
This timescale is only reached in the simulation in the innermost regions of the disk. The viscous timescale at 200 AU, which is the outer radius of the disk in radiative transfer simulations, is of the order of 10 6 years. The averaged accretion rate is of the order of 5 · 10 −10 M /yr, which is consistent with observations (Bary & Petersen 2014). The result of the hydrodynamic simulations are the circumbinary disk surface density Σ, scale height H, and the positions of the binary components. Based on those, radiative transfer simulation are performed.
Radiative transport simulation
In the following we describe the radiative transfer simulation. The density distribution from the previous section is used to derive the corresponding temperature distribution and to calculate synthetic scattered light and thermal re-emission maps. These provide the basis for the assessment of whether the generated structures can be observed.
The radiative transfer simulations are performed with the 3D continuum and line radiative transfer code Mol3D (Ober et al. 2015). First, a temperature is calculated solving the radiative transfer equation using a Monte Carlo algorithm and taking the optical properties of the dust into account. The temperature is then used to simulate emission maps. In addition, scattered light maps are calculated with the Monte Carlo algorithm as well.
Grid: A spherical grid is used with a logarithmic scaling of the r coordinate to ensure that small structures near the binaries, in particular the temperature gradient at the inner disk rim and accretion arms, are sufficiently resolved.
While the inner radius r in is identical to that of the hydrodynamic simulations, the outer radius r out is set to 200 AU because most of the gas and all significant density structures are located inside this radius. All simulations are performed for a face-on inclination of i = 0 • .
Gas and dust distribution: Mol3D employs a perfectly mixed gas and dust with a gas-to-dust mass ratio of 100:1. Based on the isothermal hydrodynamically calculated 2D surface density distribution, the 3D density distribution is constructed from
= Σ H exp − 1 2 z H 2 ,(6)
where z is the z coordinate (Lynden-Bell 1969). The normalised density distribution calculated with Fosite is scaled accordingly, allowing us to consider three different disk masses M disk = 10 −1 M , 10 −2 M , 10 −3 M . This provides the basis for studying the effect of optical depth on the emission maps.
Dust: The dust is considered to be homogeneous spheres of radius a and with the following size distribution (Dohnanyi 1969):
dn(a) ∼ a −3.5 da .(7)
Here, a minimum radius of a min = 5 nm and maximum radius a max = 250 nm (Mathis et al. 1977) are assumed. A composition of 62.5% astronomical silicate and 37.5% graphite is used with optical data from Weingartner & Draine (2001). Applying Mie theory, the optical properties are calculated at 100 logarithmically distributed wavelengths λ sim in the interval between 0.05 and 2000 µm (Wolf & Voshchinnikov 2004).
Binary luminosity: To conduct the radiative transfer simulations of the dust density distribution constructed beforehand, the luminosities of the binary components are required. The stars are considered to be main sequence stars with a same chemical composition as the sun. The stellar luminosities corresponding to the masses are derived from the main sequence tracks from Siess et al. (2000). The full parameter set for the radiative transfer simulation is shown in Table 2.
Results
In this chapter, we present the results of the simulations described in Sect. 2. We discuss the most notable features of binary systems and their influence on the radiation emitted by the surrounding disk, i.e. observable quantities.
Characteristic structures in the density distribution of circumbinary disks
First we present the results of the hydrodynamic simulations and discuss the structures caused by the disk-binary interaction and the mechanism creating them. Fig. 1 shows an image of the surface density distribution for a semi-major axis a = 30 AU. The most notable feature of the distribution is the cavity in the inner region of the disk. The cavity appears if the binary can exert enough torque on the surrounding disk to counter the torque generated by the viscous stresses. As can be gathered from Fig. 2, the radius of the cavity is approximately 2 − 2.5 times the binary separation. This is consistent with the theoretical predictions (Artymowicz & Lubow 1994). As mentioned in Sect. 2.1 the choice of equally massive stars and an eccentricity of ε = 0 leads to an almost circular cavity centred on the barycentre of the binary. Since the accretion arms have to stretch from the inner edge of the disk to the binary orbit, larger cavities result in longer accretion arms. The accretion arms also show a distinct point symmetry that remains undisturbed for at least 100 binary orbits. This finding is in agreement with the results of Günther & Kley (2002).
The hydrostatic scale height H only depends on the temperature and gravitational potential in our simulations. These are either provided as a fixed initial condition or only changes slightly for radii r >> a during the simulation. Consequently, the hydrostatic scale height profile is smooth and monotonically is increasing. This means that in the current model only the shadowing effects that result from the increase of the surface density, in contrast to variation of H, can be studied. Another striking feature of the investigated binary systems are the density waves that extend radially from the inner edge of the disk. In Fig. 2 the surface density Σ, azimuthally averaged and normalised by the total mass is shown for three different values of the binary semi-major axis a. We identify two important quantities: the wavelength of the surface density oscillation λ a , measured as the distance between two maxima, and its magnitude M a , measured as the difference of the maximum and its subsequent minimum. One can clearly see that both quantities depend on the semi-major axis a. To quantify this dependence, the wavelength λ a was calculated by first averaging over the distance of the maxima in each time step of the simulation and averaging the result over 100 time steps in a time interval of 10 4 years. In Table 3 the resulting wavelength λ a and magnitude M a are shown. One can see a trend of increasing wavelength λ a with increasing a. The same applies to the magnitude M a of the density oscillation, i.e. an increase of the semi-major axis leads to an increase of the magnitude of the density oscillation.
In Fig. 3 the impact of binaries with different masses on the circumbinary disk is illustrated. We find that the radius of the cavity does not depend on the binary mass. By applying the same procedure as for the calculation of wavelength λ a and the magnitude Parameter At this point it appears necessary to discuss the theoretical basis of these density waves. Starting with the 2D continuity and Navier-Stokes equations we calculate
λ a [AU] M a M B = 1.0 M , a = 10 AU 5.69 1.13 · 10 −4 M B = 1.0 M , a = 20 AU 8.36 4.61 · 10 −4 M B = 1.0 M , a = 30 AU 11.0 6.52 · 10 −4 λ M B [AU] M M B M B = 0.5 M ,∂ 2 t Σ − 1 r ∂ r [r∂ r (Σc 2 s )] = 0,(8)
which is a 1D wave equation in polar coordinates (full derivation can be found in App. A). The interpretation of this result is the following: while the gas is accreted inward with velocity v, it provides a moving frame for density waves. These density waves are driven by the binary in the centre and propagate outward with sound speed c s . This interpretation also explains the trend of wavelength (λ a , λ M B ) with regard to the semi-major axis a and binary mass M B . The system behaves as a driven harmonic oscillator that swings with the frequency of the instigator f . This frequency is coupled with the binary period P. Neglecting the dispersion, one would get the formula for the wavelength λ = c s / f ≈ c s · P. By increasing the semi-major axis a or decreasing the binary mass M B , we increase the orbital period and with that the wavelength (λ a , λ M B ). The magnitude of the oscillation (M a , M M B ) should be proportional to the amount of torque that the binary can exert on the disk. This seems to be the case with increasing values of the semi-major axis a. However, one would assume that a binary with higher mass M B would be able to generate a stronger torque. Instead, we find that the magnitude M M B first increases with increasing binary mass, but then decreases for value of M B = 1.5 M . The exact nature of this phenomenon should be investigated in a future study.
Observability of characteristic structures: Analysis of ideal observations
The goal of this study is to investigate whether the characteristic structures in a circumbinary disk can be observed with currently operating and future instruments/observatories. For this purpose we simulate various observable quantities, such as scattered light and re-emission images, as well as the spectral energy distribution (SED) in the wavelength range between 0.05 µm and 2000 µm of the circumbinary disks discussed in Sect. 3.1. A distance of 140 parsec to the object is assumed for all simulations.
In the hydrodynamic simulations it was shown that the most striking differences from an undisturbed circumstellar disk are the large gas and dust depleted cavities in the centre, accretion arms in the cavity, and density waves at the inner rim of the disk. Of course, a central cavity is also characteristic for circumstellar transitional disk. However, in the case of a transitional disk this cavity results from disk evolution. With the calculated surface density and scale height it is now possible to derive a 3D density distribution (r, θ, φ) according to Eq. 2.2.1. This density is post-processed with Mol3D to derive observable quantities (see Sect. 2.2 for details). Fig. 4 shows an exemplary density distribution for a = 20 AU and M B = 1 M . Although the hydrostatic scale height H is fairly smooth as a function of the radial distance to the central star, a variation of the disk profile in the plane perpendicular to the midplane can be observed. From Eq. 2.2.1 it is obvious that the variation of the surface density causes a shadow that screens parts of the inner disk edge (Fig. 5). Accretion arms caused by the presence of a binary, although not very dense, have enough mass to reach optical depths τ 1 (see Fig. 5).
In the following we discuss the influence of the binary on the observational properties of the disk.
Spectral energy distribution
The cavity in the centre of the disk is clearly visible in all calculated density distributions and emission maps and its observability is the topic of Sect. 3.3. Because of the cavity, circumbinary disks tend to have less dust, which can be heated to a high temperature. This results in much lower flux at near-and midinfrared spectrum in comparison to circumstellar disks (Fig. 6). Because the cavity radius scales with the semi-major axis a, the flux at short wavelengths is in indirect proportion to a (Fig. 7). The flux of the corresponding circumstellar disk (i.e. same disk mass and net stellar mass) is higher for all wavelengths (Fig. 6). It was simulated using the standard set-up presented in Sect. 2.1 and a star with the mass of M star = 2 M . The luminosity of this star is higher than the combined luminosity of two stars with individual masses M star = 1 M . This leads to an overall higher temperature and thus to higher fluxes. This opens a potential way for distinguishing between disks around binaries and single stars: The outer edges of the circumbinary disk behave nearly Keplerian. Through the line broadening one can determine the mass of the central star. For the same mass of the central star (or binary), the decreased flux due to the cavity and the lower disk temperature (due to the lower net stellar luminosity) result in a lower infrared emission in the case of the circumbinary disk. At longer far-infrared to millimetre wavelengths the differences of the fluxes arising from circumbinary disks with different cavity radii (i.e. different semi-major axes of the central binary) become negligible (Fig. 7). This is expected since, for large distances from the central binary, the disks structures are similar.
The dependence on the disk mass can be seen in Fig. 8. Both the short wavelength and long wavelength radiation increases with disk mass, although the increase is caused by different mechanisms. The presence of more hot dust in the vicinity of the binary leads to higher levels of short wavelength radiation. For long wavelength radiation, for which the disk is optically thin in the vertical direction, higher dust mass leads to a higher column density of radiating particles. Single star Binary re-emitted radiation depend solely on the radial density distribution, an interpretation of the SED to reveal the characteristic disk structures discussed in Sect. 3.1 is rather limited. We now investigate the impact of these characteristic structures on spatially resolved observations in various wavelength regimes.
Disk mass: The effect the disk mass has on observable features is best seen in the (sub)millimetre regime, since the observed column density is much higher. Fig. 9 shows the two surface brightness distribution for two disk masses at λ = 324 µm. The cavity in the centre of the disk is not affected by the mass increase. As the disks are optically thin in this wavelength range, the re-emission flux scales approximately linearly with the disk mass.
Flux maximum: Fig. 10 shows the spatial flux density distribution at different wavelengths. The near-and mid-infrared radiation reaches its maximum along the accretion arms, which are the innermost structures, directly exposed to the stellar radiation. The thermal re-emission from the inner disk edge, where the density waves are strongest reaches its maximum at submillimeter wavelengths.
Density waves: In Sect. 3.1 we presented a quantitative description of the dependence of the density waves generated by the binary on the semi-major axis a and the binary mass M B . We now investigate the wave structure in the resulting surface brightness distributions employing a similar algorithm as in Sect. 3.1. At first, we average the brightness distribution in azimuthal direction. Subsequently, we determine the locations of maxima and minima and calculate the flux difference between the maxima and their surrounding minima. From both differences, the smaller one is chosen, which ensures that in case of sufficient sensitivity at least one flux maximum is detected. After performing this procedure for all maxima, the largest difference for each parameter configuration, defined by the binary mass M B , disk mass M disk , semi-major axis a, and wavelength λ, is chosen. This allows us to derive the lower boundary for the sensitivity required to detect at least one flux maximum. Fig. 11 shows the flux differences as a function of the wavelength of the emitted radiation. Each figure represents a different semi-major axis a and the colours denote different disk masses for each binary configuration. Fig. 12 depicts the same quantities for different binary masses M B .
We start at the long wavelength end of these graphs to find an explanation for the trend observed here. At wavelengths λ > 1 mm the disks are optically thin (τ ≈ 1 in the most dense regions for M disk = 10 −1 M ). The linear increase of the flux towards shorter wavelengths is due to the higher fluxes at these wavelengths (see e.g. Fig. 8). Additionally, the amplitude of the density waves is the highest at the inner disk rim (Fig. 2 and 3). As the flux maximum moves inward for shorter wavelengths, the flux at the location of the largest magnitudes increases, resulting in the further increase of the flux differences. At about 100 µm the flux difference decreases abruptly. This is the wavelength at which the disk becomes optically thick. At even shorter wavelengths, one no longer traces the full column density, but only the wavelengths-dependent photosphere of the disk. This interpretation also explains why this decrease happens at shorter wavelengths for disks with less mass. The remaining feature to be discussed is the bump at wavelengths between the increase of the optical depth and about 10 µm. In this wavelength range the flux difference shows a more complex dependence on the parameters than in the cases discussed previously. It increases with disk mass, but not linearly (in a log-log diagram) as before. Here, the temperature at the inner disk rim influences the maximum differences. These are larger for smaller semi-major axis a values and higher binary masses M B . In both cases the temperature increases owing to the proximity of the binary orbit to the disk inner rim or owing to higher stellar luminosity. At these short wavelengths the regions of the disk contributing most to the flux no longer have azimuthal symmetry that is necessary for the applied algorithm. For this reason these parts of the graphs are not discussed further.
Accretion arms: As discussed earlier, the appearance of the disk at infrared wavelengths is dominated by the thermal reemission and scattered light of the accretion arms (see Fig. 10). In this part of the disk, the gravitational potential differs signifi-cantly from a standard Keplerian potential. This leads to a high temporal variability and to the lack of the azimuthal symmetry. For those reasons, we abstain from attempting to quantify our results as we did with density waves. Instead, we only outline general trends with regard to the various parameter spaces.
Figs. 13 -17 show the surface brightness distribution for semi-major axes a = 10, 20, 30 AU, M B = 1 M , and masses M B = 0.5, 1.5 M , a = 20 AU at two different wavelength, λ = 4.5 µm and λ = 20 µm.
Here we can note a major trend that was partially mentioned before, namely that greater values of the semi-major axis a lead to greater distances of disk edge to the binary orbit and therefore lower fluxes. Similarly, a higher binary mass leads to a higher stellar luminosity and consequently to a higher thermal re-emission flux. A further trend is the increase of flux with wavelength (see . In Table 4 the fluxes originating from inside a circle with radius of 2 × a are compiled. From these we can deduce that the binary mass M B and thus its luminosity has the biggest impact on the fluxes.
Taking the trends discussed above into account, one can derive predictions for the observability of circumbinary disk fea- tures (at this stage we disregard the influence of a realistic observing instruments; this is the topic of the following sections).
-The accretion arms can easiest be observed in the infrared, where we find the highest fluxes for them. Systems with smaller semi-major axis values a and higher binary masses M B are preferable for such observations. -The flux of a disk with a higher dust mass is significantly larger, which leads to the conclusion that more massive disks are more suitable for observations in infrared and in millimetre radiation. -For observations at submillimetre/millimetre wavelengths we find that higher values of the semi-major axis a result in larger flux differences for the density waves. The binary mass has no significant impact on the flux difference, but one should expect the overall flux to be higher for systems with higher binary masses.
Simulated observations: Specific instrument studies
We now study synthetic observations based on the analysis of ideal surface brightness distributions. First, we consider simulated observations at submillimeter/millimetre wavelengths. Subsequently, simulated observations with the future E-ELT are presented, which will operate at optical to infrared wavelengths.
ALMA
In this subsection we evaluate the observability of the characteristic structures discussed in Sect. 3.2 with the (sub)-millimetre interferometer ALMA. In particular these are the inner cavity and density waves. At first, we specify basic technical data assumed in this observational feasibility study. The on-source time for all simulated observations is set to 6 h with the thermal noise option enabled and recommended water vapours values applied. A declination representative for the nearby star-forming region in Taurus is used (δ ≈ 22 • ).
Density waves: The goal of this study is to determine whether the density waves can be observed with ALMA and, if so, which configuration and band are the most suitable to perform these observations with regard to the considered binary system. We begin with an example of a synthetic ALMA observation (see Figs. 18). For this, we choose the shortest possible wavelength ALMA is capable of observing (λ = 320 µm, band 10) and the configuration with the maximum baseline in that band (D = 3697 m), which results in an angular resolution of θ res = 0.024 . A quick look reveals that, at least for the semimajor axis a = 30 AU and disk mass of M disk = 0.1 M , the sensitivity and angular resolution are sufficient for the detection of the density waves (see Fig. 18). As a next step we quantify the flux difference between the wave maxima and minima in units of sensitivity σ, employing an algorithm similar to that applied in Sect We begin the discussion of the results with the highest values of the flux difference, which are measured in units of sensitivity σ. For all configurations the maximum is reached at λ = 849 µm. Similarly, the higher values of the flux difference are concentrated at longer wavelengths and configurations with shorter maximum baselines. However, the resolution at those wavelengths and configurations is not sufficient to resolve the density waves. What we detect is the flux increase with the radius, which corresponds to the inner disk cavity. The combination of wavelength and configuration for which we can detect at least one density maximum are labeled with the appropriate value of flux difference. The detectability of the density waves increases with the semi-major axis a. This is not surprising taking into account the results of Tab. 3. The wavelength and the magnitude of the density wave increase with the semi-major axis a. However, it is surprising, that the detectability is higher for systems with lower binary mass. From Tab. 3 we can infer that the wavelength is increasing for decreasing binary mass, which results in more pronounced maxima. However, the magnitude for a binary mass M B = 1 M is almost a factor of two greater than for M B = 0.5 M . From this we conclude that the wavelength of the density wave has a greater influence on the detectability than the wave magnitude. Finally, we want to compare our results to a similar study performed by Ruge et al. (2015). One of the major differences of our work compared to the results of Ruge et al. (2015) is the lack of density waves. There are three major reasons for this. The first reason is the difference in the simulations. Ruge et al. (2015) conducted a SPH simulation of the inner disk regions. The hydrodynamic simulation performed in this study achieves a better resolution, which allows us to track smaller structures. The second reason is the parameters of the systems. A semi-major axis of a = 2 AU, considered by Ruge et al. (2015), should result in much smaller density wave wavelength than studied here. The last reason are the considered observing wavelengths. The shortest wavelength employed by Ruge et al. (2015) was 750 µm. As can be seen in Fig. 11, the flux difference between the maximum and minimum decreases with wavelength. Thus, a longer wavelength should decrease the feasibility of detecting the density waves. The variability in flux distribution on the inner disk rim and spiral arm seen in Fig. 7 in the study of Ruge et al. (2015) is caused by the eccentric orbit of the binary (ε = 0.3).
European Extremely Large Telescope
In Sect. 3.2.2 it became clear that the accretion arms are best observed at infrared wavelengths. To simulate an observation with the E-ELT instrument METIS, we convolve the ideal maps with the wavelength-dependent point spread function of an ideal 39 m telescope with a circular aperture. The goal here is to determine whether it is possible to observe the highly variable structures, such as the inner disk rim and accretion arms. The brightness ratio between the individual binary component and the brightest structures amounts to ∼ 10 2 − ∼ 10 4 in the considered cases. For this reason, the binaries are removed from the maps before the convolution (see .
All images at λ = 4.5 µm are dominated by the flux from the binary. However, in all systems, with the exception of the case M B = 1 M and a = 10 AU, it is still possible to spatially resolve the accretion arms. Similarly, at λ = 20 µm one would only detect a bright ring.
In all but one system the accretion arms are resolved in both wavelength regimes. In the case of M B = 1 M and a = 10 AU, we only detect a bright ring that corresponds to the inner disk rim. For λ = 4.5 µm we find two small flux maxima that are caused by hot dust in the direct vicinity of the binary at the edge of the computational domain. In Table 5 nating from inside a circle with radius of 2 × a are compiled. We find no major differences from the original maps caused by the convolution process (see Tab. 4).
Summary
The goal of this work was to study the observability of characteristic structures in circumbinary disks, generated by binary-disk interaction. To achieve this goal, a 2D hydrodynamic simulation was performed with the code Fosite to calculate a surface density distribution. The results were applied to a 3D grid-based Monte Carlo code Mol3D and a temperature profile; in addition, resulting scattered light and re-emission maps were simulated. These images were used to simulate imaging observations in the near to mid-infrared wavelength range (E-ELT) and at submillimeter/millimetre wavelengths (ALMA).
We found that the torque exerted on the circumbinary disk by the binary generates unique features that can be used to distinguish them from protoplanetary disks around single stars. In particular these are the inner cavity, accretion flows from disk inner rim to the binary orbit, and the density waves on the disk inner edge. We quantified the dependence of those features on binary mass M B , semi-major axis a, and disk mass M disk . Furthermore, we derived a wave equation governing the behaviour of the density waves, which can be transformed into a standard 1D wave solution in two dimensions.
We have shown that those features alter the thermal and radiation balance of the disk to a sufficiently high degree, which can be seen in the re-emission and scattered light maps. Here we once again studied the dependence of the observable features on the parameters and derived predictions with regard to the successful observation of those features. Subsequently synthetic observations of the simulated circumbinary disks with ALMA and E-ELT were generated. We could show that ALMA configurations with the highest angular resolution are capable of observing the density waves generated by the binary with a = 20 AU and a = 30 AU. Furthermore, we made a parameter study that showed that even a configuration with a far smaller maximum baseline is sufficient to observe the inner cavity of those disks. Simulated observations with E-ELT have shown that it will be possible to resolve the accretion arms at the considered wavelengths of 4.5 µm and 20 µm.
Further studies of this kind would need to include systems with binary mass ratio q = M sec /M prim 1 and an eccentricity value greater than 0, as they comprise the majority of the known systems (Raghavan et al. 2010) and those parameters have a large impact on the disk structure (Ruge et al. 2015). Furthermore, this study was focused only on dust emissions. However, the gas of which leaves us with the 2D continuity and the r component of the Navier-Stokes equations in polar coordinates,
∂ t Σ + 1 r ∂ r (rΣv r ) = 0 , (A.1) Σ∂ t v r + Σv r ∂ r v r = −∂ r (c 2 s Σ). (A.2)
The parameter Σ is the vertically integrated surface density; v r is the radial drift velocity, which is in this case generated by the disks differential rotation combined with the sheer viscosity; and c s is the sound speed of the gas. By applying the time derivative on A.1 we get
∂ 2 t Σ + 1 r ∂ r (rΣ∂ t v r + rv r ∂ t Σ) = 0. (A.3)
The time derivation of the surface density can be substituted using A.1 once again and A.2 to substitute Σ∂ t v r , Assuming the drift velocity v r and its derivation ∂ r v r are much smaller than the sound speed c s and its derivative ∂ r c s , Eq. (A.5) can be reduced to Eq. (3.1.1) A full summary of the solution for the simplified Eq. (3.1.1) can be found in Polyanin (2002), p. 296-297.
Fig. 1 .
1Normalised surface density for a = 30 AU, M B = 1M . The crosses indicate the positions of the binary components.
Fig. 2 .Fig. 3 .
23M a , the values of λ M B and M M B can be determined.Here, λ M B is the oscillation wavelength of the surface density and M M B magnitude with regard to binary mass M B . We find that with increasing binary mass M B the wavelength of the oscillation λ M B decreases. At the same time the trend for the massdependent magnitude M B appears to be more complex, as the Normalised by the total mass and azimuthally averaged surface density for semi-major axes a = 10 AU, 20 AU, and 30 AU; Azimuthally averaged normalised surface density for a = 20 AU, and M B = 0.5 M , 1 M , and 1.5 M .
for M B = 0.5 M and M B = 1.5 M are smaller than for M B = 1 M . This suggests that there is a value of M B for which the magnitude reaches a maximum. Similar density waves were shown in Fig. 2 of the work from Günther et al. (2004).
Fig. 4 .
4Normalised dust density distribution for a = 20 AU, M B = 1 M . The crosses indicate the positions of the binary components.
Fig. 5 .
5Left: Optical depth τ at λ = 526 nm calculated in the midplane. Right: resulting midplane temperature of the same system. Black line denotes where the optical depth reaches values of τ = 1 and τ = 10 4 , respectively.
Fig. 6 .
6Spectral energy distribution for two simulated disks with M disk = 10 −1 M , stellar mass of 2 M . Blue: single star; red: binary with two components of M B = 1 M . Inner cavity radius for the binary ≈ 40 AU.
Fig. 7 .
7Dependence of the SED on the semi-major axes a = 10, 20, 30 AU (M B = 1 M ).
Fig. 8 .
8Dependence of the SED on the binary disk mass. a = 20 AU, M B = 1 M , and M disk = 10 −1 − 10 −3 M
Fig. 9 .Fig. 10 .
910Surface brightness distribution at λ = 324 µm for disk masses M disk = 10 −1 M (left) and M disk = 10 −3 M (right). Surface brightness distribution at λ = 10.5 µm (left) and 324 µm (right) (a = 20 AU, M B = 1 M ).
Fig. 11 .
11Flux difference for M B = 1 M and a = 10, 20, 30 AU (see Sect. 3.1 for details).
Fig. 12 .
12Flux difference for M B = 0.5, 1.5 M and a = 20 AU (see Sect. 3.1 for details).
Table 4 .
4Integrated flux of the scattered and re-emitted radiation in units of mJy originating from inside a circle with the radius of 2×a. The direct stellar radiation is not considered. Parameter λ = 4.5µm λ = 20µm M B = 1.0M , a = 10 AU 0.43 2819.76 M B = 1.0M , a = 20 AU 0.27 2137.45 M B = 1.0M , a = 30 AU 0.1 1363.06 M B = 0.5M , a = 20 AU 0.079 152.56 M B = 1.5M , a = 20 AU 1.71 7129.55
Fig. 13 .
13ALMA set-up: We perform simulations within the frame of capabilities in Observational Cycle 4, which employs 40 of the 12 m antennas in nine different configurations with maximum baselines between 155 m and 12 644 m (seven wavelength bands between 0.32 mm and 3.6 mm). Despite the fixed number of antennas and configurations, the results presented here are of general nature because they only depend on the simulated scattered light and re-emission maps and the chosen angular resolution Surface brightness distribution at λ = 4.5 µm (left) and λ = 20 µm (right) for M B = 1 M and a = 10 AU. and sensitivity. The synthetic observations are performed with the ALMA simulation tool kitCASA (McMullin et al. 2007).
Fig. 14 .
14. 3.2.2. The results are shown in Figs. 19 -23. Surface brightness distribution at λ = 4.5 µm (left) and λ = 20 µm (right) for M B = 1 M and a = 20 AU.
Fig. 15 .
15Surface brightness distribution at λ = 4.5 µm (left) and λ = 20 µm (right) for M B = 1 M and a = 30 AU.
Fig. 16 .
16Surface brightness distribution at λ = 4.5 µm (left) and λ = 20 µm (right) for M B = 0.5 M and a = 20 AU.
Fig. 17 .
17Surface brightness distribution at λ = 4.5 µm (left) and λ = 20 µm (right) for M B = 1.5 M and a = 20 AU.
Fig. 18 .
18Synthetic ALMA observation at λ = 320 µm with configuration # 7; M B = 1 M and a = 30 AU.
Fig. 19 .
19Flux difference between density wave maximum and minimum in units of sensitivity for a = 10 AU, M B = 1 M , and M disk = 10 −1 M .
Fig. 20 .
20Flux difference between density wave maximum and minimum in units of sensitivity for a = 20 AU, M B = 1 M , and M disk = 10 −1 M .
Fig. 21 .
21Flux difference between density wave maximum and minimum in units of sensitivity for a = 30 AU, M B = 1 M , and M disk = 10 −1 M .
Fig. 22 .Fig. 23 .Fig. 24 .Fig. 25 .Fig. 26 .Fig. 27 .Fig. 28 .
22232425262728Flux difference between density wave maximum and minimum in units of sensitivity for a = 20 AU, M B = 0.5 M , and M disk = 10 −1 M . Flux difference between density wave maximum and minimum in units of sensitivity for a = 20 AU, M B = 1.5 M , and M disk = 10 −1 M . the circumbinary disk is influenced in the same manner. If realised, the observation of binary-induced structures in molecular lines would allow one to observe velocity fields in the disk and in the accretion flows in the disk centre. Flux after convolution at λ = 4.5 µm (left) and λ = 20 µm (right) for M B = 1 M and a = 10 AU. Flux after convolution at λ = 4.5 µm (left) and λ = 20 µm (right) for M B = 1 M and a = 20 AU. Flux after convolution at λ = 4.5 µm (left) and λ = 20 µm (right) for M B = 1 M and a = 30 AU. Flux after convolution at λ = 4.5 µm (left and λ = 20 µm (right) for M B = 0.5 M and a = 20 AU. Flux after convolution at λ = 4.5 µm (left) and λ = 20 µm (right) for M B = 1.5 M and a = 20 AU.
∂ r [−rΣv r ∂ r v r − r∂ r (c 2 s Σ) − v r ∂ r (rΣv r )] = 0,
Table 1 .
1Hydrodynamic simulation parameters and ranges.Parameter
Parameter value/range
Mass of individual binary components
M B [M ]
0.5, 1, 1, 5
Binary mass ratio
q
1
Initial disk mass
M disk [M ]
0.02
Semi-major axis
a [AU]
10, 20, 30
Inner boundary radius
r in [AU]
0.6 · a
Outer boundary radius
r out [AU]
1000
Eccentricity
ε
0
Simulation time
t sim [yr]
5 · 10 5
Table 2. Radiative transfer simulation parameters and ranges.
Parameter
Parameter value/range
Binary temperature
T B [K]
4500, 5600, 6600
Binary radius
R B [R ]
0.42, 0.98, 1.4
Binary luminosity
L B [M ]
0.45, 0.87, 8.2
Inner radius
r in [AU]
0.6 · a
Outer radius
r out [AU]
200
Simulation wavelengths λ sim [µm]
0.05 . . . 2000
Table 3 .
3Wavelength and magnitudes for different semi-major axes a and binary masses M B .
Table 5 .
5Integrated flux of the scattered and re-emitted radiation in units of mJy originating from inside a circle with the radius of 2×a. The direct stellar radiation is not considered.Parameter λ = 4.5µm λ = 20µm M B = 1.0M , a = 10 AU0.42
2702.1
M B = 1.0M , a = 20 AU
0.27
2031.33
M B = 1.0M , a = 30 AU
0.16
1252.79
M B = 0.5M , a = 20 AU
0.078
144.97
M B = 1.5M , a = 20 AU
1.66
6610.53
Article number, page 5 of 16 A&A proofs: manuscript no. 30084_am
.2.2. Surface brightness distribution So far, we only considered the synthetic SEDs of the simulated disks. However, as both the contributions of the scattered and Article number, page 6 of 16 Avramenko, Wolf, Illenseer: Observability of structures in circumbinary disks
Article number, page 8 of 16Avramenko, Wolf, Illenseer: Observability of structures in circumbinary disks
Article number, page 12 of 16 Avramenko, Wolf, Illenseer: Observability of structures in circumbinary disks Acknowledgements. We thank all the members of the Astrophysics Department Kiel for helpful discussions and remarks and for their language corrections. This study was funded by the German Science Foundation (DFG), grant: WO 857/12-1.A&A proofs: manuscript no. 30084_amAppendix A: Derivation of the wave equationIn this section we present a detailed derivation of the wave equation mentioned in Sect. 3.1.Since the wave propagates in r direction we neglect the φ component and terms containing the derivative with regard to φ,
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| In this short note we observe that the recent examples of derivedequivalent Calabi-Yau 3-folds with different fundamental groups also have different Brauer groups, using a little topological K-theory. Some years ago Gross and Popescu [12] studied a simply-connected Calabi-Yau 3-fold X fibered in non-principally polarized abelian surfaces. They expected that its derived category would be equivalent to that of the dual abelian fibration Y , which is again a Calabi-Yau 3-fold but with π 1 (Y ) = (Z 8 ) 2 , the largest known fundamental group of any Calabi-Yau 3fold. This derived equivalence was later proved by Bak [2] and Schnell[23]. Ignoring the singular fibers it is just a family version of Mukai's classic derived equivalence between an abelian variety and its dual[19], but of course the singular fibers require much more work. As Schnell pointed out, it is a bit surprising to have derived-equivalent Calabi-Yau 3-folds with different fundamental groups, since for example the Hodge numbers of a 3-fold are derived invariants [22, Cor. C].Gross and Pavanelli[11]showed that Br(X) = (Z 8 ) 2 , the largest known Brauer group of any Calabi-Yau 3-fold. In this note we will show that the finite abelian group H 1 (X, Z) ⊕ Br(X) is a derived invariant of Calabi-Yau 3-folds; thus in this example we must have Br(Y ) = 0, and in particular the Brauer group alone is not a derived invariant. This too is a bit surprising, since the Brauer group is a derived invariant of K3 surfaces: if X is a K3 surface then Br(X) ∼ = Hom(T (X), Q/Z) [7, Lem. 5.4.1], where T (X) = N S(X) ⊥ ⊂ H 2 (X, Z) is the transcendental lattice, which is a derived invariant by work of Orlov[20].Since an earlier version of this note first circulated, Hosono and Takagi [14] have found a second example of derived-equivalent Calabi-Yau 3-folds with different fundamental groups. Their X and Y are constructed from spaces of 5×5 symmetric matrices in what is likely an instance of homological projective duality[15], and one has π 1 (X) = Z 2 and π 1 (Y ) = 0. While Br(X) and Br(Y ) are not known, from our result we see that Br(Y ) ∼ = | 10.1007/978-3-319-46852-5_1 | [
"https://arxiv.org/pdf/1306.6538v4.pdf"
]
| 119,126,118 | 1306.6538 | 02c5b724466bd96399c4bfc785aed4ca21bc2dce |
28 Dec 2013
Nicolas Addington 28 Dec 2013arXiv:1306.6538v3 [math.AG] The Brauer group is not a derived invariant
In this short note we observe that the recent examples of derivedequivalent Calabi-Yau 3-folds with different fundamental groups also have different Brauer groups, using a little topological K-theory. Some years ago Gross and Popescu [12] studied a simply-connected Calabi-Yau 3-fold X fibered in non-principally polarized abelian surfaces. They expected that its derived category would be equivalent to that of the dual abelian fibration Y , which is again a Calabi-Yau 3-fold but with π 1 (Y ) = (Z 8 ) 2 , the largest known fundamental group of any Calabi-Yau 3fold. This derived equivalence was later proved by Bak [2] and Schnell[23]. Ignoring the singular fibers it is just a family version of Mukai's classic derived equivalence between an abelian variety and its dual[19], but of course the singular fibers require much more work. As Schnell pointed out, it is a bit surprising to have derived-equivalent Calabi-Yau 3-folds with different fundamental groups, since for example the Hodge numbers of a 3-fold are derived invariants [22, Cor. C].Gross and Pavanelli[11]showed that Br(X) = (Z 8 ) 2 , the largest known Brauer group of any Calabi-Yau 3-fold. In this note we will show that the finite abelian group H 1 (X, Z) ⊕ Br(X) is a derived invariant of Calabi-Yau 3-folds; thus in this example we must have Br(Y ) = 0, and in particular the Brauer group alone is not a derived invariant. This too is a bit surprising, since the Brauer group is a derived invariant of K3 surfaces: if X is a K3 surface then Br(X) ∼ = Hom(T (X), Q/Z) [7, Lem. 5.4.1], where T (X) = N S(X) ⊥ ⊂ H 2 (X, Z) is the transcendental lattice, which is a derived invariant by work of Orlov[20].Since an earlier version of this note first circulated, Hosono and Takagi [14] have found a second example of derived-equivalent Calabi-Yau 3-folds with different fundamental groups. Their X and Y are constructed from spaces of 5×5 symmetric matrices in what is likely an instance of homological projective duality[15], and one has π 1 (X) = Z 2 and π 1 (Y ) = 0. While Br(X) and Br(Y ) are not known, from our result we see that Br(Y ) ∼ =
Z 2 ⊕ Br(X), so they are different. An explicit order-2 element of Br(Y ) arises naturally in Hosono and Takagi's construction [14,Prop. 3.2.1].
It is worth mentioning that both π 1 and Br are birational invariants, so while birational Calabi-Yau 3-folds are derived equivalent [5], the converse is not true. In addition to the two examples just mentioned, there is the Pfaffian-Grassmannian derived equivalence of Borisov and Cȃldȃraru [4]. In that example X is a complete intersection in a Grassmannian, so H 1 (X, Z) = Br(X) = 0, so from our result we see that H 1 (Y, Z) = Br(Y ) = 0 as well; to show that X and Y are not birational Borisov and Cȃldȃraru use a more sophisticated minimal model program argument.
Before proving our result we fix terminology.
Definition.
A Calabi-Yau 3-fold is a smooth complex projective 3-fold X with ω X ∼ = O X and b 1 (X) = 0. In particular H 1 (X, Z) may be torsion.
This is in contrast to the case of surfaces, where ω X ∼ = O X and b 1 (X) = 0 force π 1 (X) = 0 [17,Thm. 13]. There are several reasons not to require π 1 (X) = 0 for Calabi-Yau 3-folds. As we have just seen, a simply-connected Calabi-Yau 3-fold may be derived equivalent to a non-simply-connected one; it may also be mirror to a non-simply-connected one. Perhaps the best reason is that families of simply-connected and non-simply-connected Calabi-Yau 3-folds can be connected by "extremal transitions," that is, by performing a birational contraction and then smoothing; most known families of Calabi-Yau 3-folds can be connected by extremal transitions [10,18].
Definition. The Brauer group of a smooth complex projective variety X is
Br(X) = tors(H 2 an (X, O * X )),
where tors denotes the torsion subgroup.
This used to be called the cohomological Brauer group until it was shown to coincide with the honest Brauer group [8]. From the exact sequence
H 2 (X, O X ) → H 2 (X, O * X ) → H 3 (X, Z) → H 3 (X, O X ) we see that if X is a Calabi-Yau 3-fold then Br(X) = tors(H 3 (X, Z)).
That is, the Brauer group of a Calabi-Yau 3-fold is entirely topological, in contrast to that of a K3 surface which is entirely analytic.
Proposition. Let X and Y be Calabi-Yau 3-folds with D b (X) ∼ = D b (Y ). Then H 1 (X, Z) ⊕ Br(X) ∼ = H 1 (Y, Z) ⊕ Br(Y ).
Proof. Brunner and Distler [6, §2.5] analyzed the boundary maps in the Atiyah-Hirzebruch spectral sequence and saw that for a Calabi-Yau 3-fold X, or indeed any closed oriented 6-manifold with b 1 (X) = 0, it degenerates at the E 2 page. Thus there is a short exact sequence
0 → H 5 (X, Z) → K 1 top (X) → H 3 (X, Z) → 0,
where K * top (X) is topological K-theory. Since H 5 (X, Z) = H 1 (X, Z) is torsion, this gives an exact sequence
0 → H 1 (X, Z) → tors(K 1 top (X)) → Br(X) → 0.(1)
While it is not strictly necessary for our purposes, they also got an exact sequence
0 → Br(X) * → tors(K 0 top (X)) → H 1 (X, Z) * → 0;(2)
here if A is a finite abelian group then the dual group A * := Hom(A, Q/Z), which is non-canonically isomorphic to A. Doran and Morgan [9, §4] analyzed K * top (X) more carefully using the fact that c 1 (X) = 0 and showed that the sequences (1) and (2)
E, F ∈ D b (X × Y ) such that Φ(−) = π Y * (E ⊗ π * X (−)) Ψ(−) = π X * (F ⊗ π * Y (−)),
and arguing as in [16,Lem. 5.32] we find that the same formulas define inverse isomorphisms K * (X) → K * (Y ) and K * (Y ) → K * (X): use the fact that the pushforward on K * satisfies a projection formula and is compatible with the pushforward on D b .
We conclude with a remark on H 1 and Br in mirror symmetry. Batyrev and Kreuzer [3] predicted that mirror symmetry exchanges H 1 and Br, having calculated both groups for all Calabi-Yau hypersurfaces in 4-dimensional toric varieties. In all their examples the groups are quite small: either H 1 = 0 and Br = Z 2 , Z 3 , or Z 5 , or vice versa. This prediction does not seem to be right in general. On the one hand it is contradicted by a prediction of Gross and Pavanelli [11,Rem. 1.5], based on calculations in Pavanelli's thesis [21], that if X is the abelian fibration above, with H 1 (X) = 0 and Br(X) = (Z 8 ) 2 , then its mirrorX has π 1 (X) = Br(X) = Z 8 . Even more seriously, Hosono and Takagi's X and Y have the same mirror according to [13], but different H 1 and Br as we have discussed. Mirror symmetry is expected to exchange K 0 top and K 1 top , however, so mirror Calabi-Yau 3-folds should have the same H 1 ⊕ Br. I thank P. Aspinwall and A. Cȃldȃraru for helpful discussions, and S. Hosono and H. Takagi for encouraging me to publish this note.
are in fact split. Now the proposition follows from the fact that K 0 top and K 1 top are derived invariants [1, §2.1]. In a bit more detail, if Φ : D b (X) → D b (Y ) and Ψ : D b (Y ) → D b (X) are inverse equivalences, then by [20, Thm. 2.2] there are objects
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The fundamental group is not a derived invariant. C Schnell, arXiv:1112.3586Derived categories in algebraic geometry. Zürich; Box90320Department of Mathematics Duke UniversityEur. Math. Soc.C. Schnell. The fundamental group is not a derived invariant. In Derived categories in algebraic geometry, EMS Ser. Congr. Rep., pages 279-285. Eur. Math. Soc., Zürich, 2012. Also arXiv:1112.3586. Department of Mathematics Duke University, Box 90320
. Durham, 27708-0320Durham, NC 27708-0320
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[
"FUNDAMENTAL PARAMETERS, INTEGRATED RGB MASS LOSS AND DUST PRODUCTION IN THE GALACTIC GLOBULAR CLUSTER 47 TUCANAE",
"FUNDAMENTAL PARAMETERS, INTEGRATED RGB MASS LOSS AND DUST PRODUCTION IN THE GALACTIC GLOBULAR CLUSTER 47 TUCANAE"
]
| [
"I Mcdonald ",
"M L Boyer ",
"J Th Van Loon ",
"A A Zijlstra ",
"J L Hora ",
"B Babler ",
"M Block ",
"K Gordon ",
"M Meade ",
"M Meixner ",
"K Misselt ",
"T Robitaille ",
"M Sewi Lo ",
"B Shiao ",
"B Whitney "
]
| []
| []
| Fundamental parameters and time-evolution of mass loss are investigated for post-main-sequence stars in the Galactic globular cluster 47 Tucanae (NGC 104). This is accomplished by fitting spectral energy distributions (SEDs) to existing optical and infrared photometry and spectroscopy, to produce a true Hertzsprung-Russell diagram. We confirm the cluster's distance as d = 4611 +213−200 pc and age as 12 ± 1 Gyr. Horizontal branch models appear to confirm that no more RGB mass loss occurs in 47 Tuc than in the more-metal-poor ω Centauri, though difficulties arise due to inconsistencies between the models. Using our SEDs, we identify those stars which exhibit infrared excess, finding excess only among the brightest giants: dusty mass loss begins at a luminosity of ∼ 1000 L ⊙ , becoming ubiquitous above L = 2000 L ⊙ . Recent claims of dust production around lower-luminosity giants cannot be reproduced, despite using the same archival Spitzer imagery. | 10.1088/0067-0049/193/2/23 | [
"https://arxiv.org/pdf/1101.1095v1.pdf"
]
| 119,266,025 | 1101.1095 | 26bb583e3b357a411efd66c0fa23c4c84c2cbb8e |
FUNDAMENTAL PARAMETERS, INTEGRATED RGB MASS LOSS AND DUST PRODUCTION IN THE GALACTIC GLOBULAR CLUSTER 47 TUCANAE
5 Jan 2011 January 7, 2011
I Mcdonald
M L Boyer
J Th Van Loon
A A Zijlstra
J L Hora
B Babler
M Block
K Gordon
M Meade
M Meixner
K Misselt
T Robitaille
M Sewi Lo
B Shiao
B Whitney
FUNDAMENTAL PARAMETERS, INTEGRATED RGB MASS LOSS AND DUST PRODUCTION IN THE GALACTIC GLOBULAR CLUSTER 47 TUCANAE
5 Jan 2011 January 7, 2011Draft version January 7, 2011Draft version Preprint typeset using L A T E X style emulateapj v. 11/10/09Subject headings: stars: mass-loss -circumstellar matter -infrared: stars -stars: winds, outflows -globular clusters: individual (NGC 104) -stars: AGB and post-AGB
Fundamental parameters and time-evolution of mass loss are investigated for post-main-sequence stars in the Galactic globular cluster 47 Tucanae (NGC 104). This is accomplished by fitting spectral energy distributions (SEDs) to existing optical and infrared photometry and spectroscopy, to produce a true Hertzsprung-Russell diagram. We confirm the cluster's distance as d = 4611 +213−200 pc and age as 12 ± 1 Gyr. Horizontal branch models appear to confirm that no more RGB mass loss occurs in 47 Tuc than in the more-metal-poor ω Centauri, though difficulties arise due to inconsistencies between the models. Using our SEDs, we identify those stars which exhibit infrared excess, finding excess only among the brightest giants: dusty mass loss begins at a luminosity of ∼ 1000 L ⊙ , becoming ubiquitous above L = 2000 L ⊙ . Recent claims of dust production around lower-luminosity giants cannot be reproduced, despite using the same archival Spitzer imagery.
INTRODUCTION
Circumstellar dust production, and the variation of its occurrence with a star's fundamental parameters, must be understood if we are to gain insight into both Galactic ecology and the history of the chemical enrichment of the Universe. The dusty winds of asymptotic giant branch (AGB) stars dominate the production of interstellar dust at all redshifts at which AGB stars are observed, and have therefore determined the chemical enrichment of Population I stars, including the Sun and Solar System (Gehrz 1989;Sedlmayr 1994;Zinner 2003;Valiante et al. 2009). The integrated mass loss of RGB stars determines the envelope mass of a star leaving the RGB tip. It consequently determines its position on the horizontal branch (HB) and appears to be the major factor causing the 'second parameter' (after metallicity) required to define HB morphology (e.g. Rood 1973;Catelan 2000). Mass loss on the asymptotic giant branch eventually ejects the star's entire hydrogen envelope, creating a post-AGB star and (perhaps) a planetary nebula (PN).
A number of factors hamper the quantitative examination of dust production. These including difficulties in determining outflow velocities in faint, metalpoor stars; the difficulty in determining grain density (or porosity), grain size and grain shape; the difficulty in identifying the mineralogy of grains being produced; even the difficulty in determining whether infrared excess is present at all. Open questions include how instantaneous and integrated RGB mass loss varies as a function of initial mass and metallicity (Fusi Pecci & Bellazzini 1997;Lamers & Cassinelli 1999;Catelan 2000); whether dust can form efficiently around RGB stars; at what evolutionary stage dust production begins; and whether radiation pressure on oxygen-rich dust grains is sufficient to drive a wind, especially at low metallicity (e.g. Lewis 1989), or whether chromospheric (magneto-acoustic) driving or pulsation may also be important (Hartmann & MacGregor 1980;Dupree et al. 1984;Woitke 2006;.
Globular clusters provide an excellent laboratory in which we can examine these questions. They typically host a single or dominant population of stars at identical ages and metallicities, with sufficient number to be statistically useful. Comparisons within globular clusters therefore probe variations with evolution, while comparisons between globular clusters probe variations with metallicity and age. To provide a proper comparison, it is vital to compare fundamental parameters of stars. These can be determined by comparing spectral energy distributions (SEDs) to synthetic spectra from stellar atmosphere models. This approach has the advantage of allowing one to identify infrared excess, characteristic of dust production, in the SED.
In this study, we examine the fundamental parameters of stars in the globular cluster 47 Tucanae (NGC 104), one of the most-massive (6-9 ×10 5 M ⊙ ; Scott & Rose 1975;Meylan & Mayor 1985;Mandushev et al. 1991) and nearest (≈4500 pc; Harris 1996;Percival et al. 2002;McLaughlin et al. 2006;Salaris et al. 2007) Galactic globular clusters, with a metallicity of [Fe/H] ≈ -0.7 (Harris 1996;Carretta & Gratton 1997;McWilliam & Bernstein 2008;Worley et al. 2010) and interstellar reddening of E(B − V ) = 0.04 mag (Harris 1996). Throughout this paper, we will refer to two similar studies: , which covers the cluster ω Centauri (NGC 5139; hereafter Paper I); and Boyer et al. (2009), which covers the cluster NGC 362 (hereafter Paper II).
Dust production in 47 Tuc is of particular interest due to the study of Origlia et al. (2007). This study has claimed observational evidence of dust production along the entire RGB, in contrast to the observations of other clusters (e.g. ω Cen; Paper I). If this claim can be confirmed, it significantly increases the number of stars we know of that are returning dust to the interstellar medium. This would be particularly important at high (z ∼ 6) redshift, where AGB stars remain the dominant dust producers, but cannot account for a galaxy's entire dust budget (Valiante et al. 2009). Such claims must therefore be taken seriously and examined carefully. An analysis of the Origlia et al. claim was undertaken by Boyer et al. (2010), who raise concern that apparentlyred stellar colours may be a result of stellar blending in the densely populated cluster core (∼50 000 M ⊙ pc −3 - Gebhardt & Fischer 1995) and data artifacts due to saturation of the brightest stars. Origlia et al. (2010) defended their original hypothesis, suggesting that this dust is too warm to present considerable reddening of the [3.6]- [8] colour, but does present considerable excess in K s - [8]. One of our aims herein must therefore be to examine this issue in more depth.
This paper follows a similar approach to Paper I. Section 2 details the photometric data we use, its sources and its reduction; determination of stellar parameters for the cluster's stars by fitting their spectral energy distributions (SEDs); and the creation of a Hertzsprung-Russell diagram (HRD). Section 3 covers the fitting of stellar isochrones to the HRD. In Section 4, we derive mid-infrared (mid-IR) excesses for our stars, based on our SEDs, and use these to determine which stars are dusty and compare them to those in Origlia et al. (2007). The mass-loss rates and dust compositions of these stars are analysed in an accompanying work (Paper IV).
DATA REDUCTION
The input data
As 47 Tuc covers a large area on the sky (its tidal radius is 43 ′ ; Harris 1996), a uniform set of photometric data is not available for every star in the cluster. The photometric data used in the SEDs come from a variety of different sources covering different fields of view. These data are summarised in Figure 1 and Table 1. They include data taken with the Spitzer Space Telescope's (Werner et al. 2004) four Infrared Array Camera (IRAC; Fazio et al. 2004) bands, at 3.6, 4.5, 5.8 and 8 µm from the Surveying the Agents of Galaxy Evolution in the Small Magellanic Cloud (SAGE-SMC) program (Gordon et al. 2009). At the distance of 47 Tuc, the 1.66-1.98 ′′ full-width at half-maximum (FWHM) of the point spread function (PSF) of IRAC provides a resolution of 0.036-0.043 pc, with a dynamic range sufficient to detect the cluster's HB stars. Further IRAC data of the cluster core were presented in Origlia et al. (2007) (Spitzer PID20298; PI: R. Rood). Here, we use the reanalysed data from Boyer et al. (2010). For this second IRAC dataset, we used the "deep" photometry when [3.6] > 11 mag, and the "shallow" photometry otherwise.
Longer-wavelength 24-µm data was sourced from the Multiband Imaging Photometer for Spitzer (MIPS; Rieke et al. 2004) data from Barmby et al. (2009), as the SAGE-SMC MIPS data do not cover the cluster.
Near-IR JHK s -band photometry for the entire cluster was taken from the 2-µm All Sky Survey (2MASS; Skrutskie et al. 2006). The 2MASS image near the cluster core is heavily blended, and the recorded photometric fluxes are artificially raised by blending of both partially-resolved and unresolved stars. To solve this issue, we preferentially used the JHK s -band photometry from Salaris et al. (2007) over 2MASS. These data cover a 4. ′ 9 × 4. ′ 9 region around the cluster core. These data are of substantially better resolution (< 0. ′′ 9) and consequently suffer less from source confusion and blending. The "deep" Salaris photometry was used in preference to the "shallow" when K s > 12.5 mag. Some stars, in the southern half of the cluster core, were not covered by the Salaris data: many of these therefore have abnormally high 2MASS fluxes, and therefore scatter to higher luminosities. Here they can be confused with AGB stars, though they typically lie slightly above the AGB branch (their locus lies on the dividing line between regions (1) and (2) in the HRD shown later).
The mid-IR Spitzer photometry is complemented by optical data from the Magellanic Clouds Photometric Survey (MCPS; Zaritsky et al. 2002), which contains Johnson U BV -band and Gunn i-band photometry of the region. Once again, this only covers the south-east of the cluster. To cover the remainder of the cluster, we used the Johnson U BV -band and Cousins I C -band photometry of Salaris et al. (2007), and the Johnson U BV I Jband 7 photometry of Stetson (2000).
Creation of a master catalogue
These data were combined using daomatch/daomaster (Stetson 1993) and objects detected in only one filter (assumed to be bad data) were dropped. The final source list contains photometry for 104 153 stars located within 30 ′ of the cluster core (00 h 24 m 05.2 s -72 • 04 ′ 51 ′′ - Harris 1996).
From this source list, a subset of sources were selected which had enough photometry to determine their tem-
Wavelength
Source Coverage U BV i MCPS; Zaritsky et al. (2002) West of cluster, avoids core U BV I C Stetson (2000) ∼ 20 ′ × 20 ′ around core U BV I J JHKs Salaris et al. (2007) Northern two-thirds of core JHKs 2MASS; Skrutskie et al. (2006) Entire cluster 3.6-8 µm 1 SAGE-SMC; (Gordon et al., in prep.) South-west of cluster, not including core 3.6-8 µm Origlia et al. (2007) 2 Cluster core & immediate west & east 24 µm 3 Barmby et al. (2009) Cluster core & immediate south-west 1 Spitzer IRAC at 3.6, 4.5, 5.8 and 8 µm; Fazio et al. (2004). 2 We use the re-analysed data of Boyer et al. (2010). 3 Spitzer MIPS; Rieke et al. (2004). peratures and luminosities by fitting their SEDs. We selected stars with at least four photometric bands with flux measurements, the bluest of which must be in the optical (I-band or shorter wavelength) and the reddest of which must be in the IR (K s -band or longer wavelength). This reduced the source list to 47 727 stars with usable photometry.
The data reduction process
SEDs were created following the method used in Papers I and II. Briefly, this involves comparing broadband photometry to artificially-dereddened model spectra which have been convolved to the photometric filters of the observations. Here, we again use the marcs models of Paper I (see also Gustafsson et al. 1975Gustafsson et al. , 2008. Interpolating between these models determines the stellar temperature, and the multiplication of the model flux required to match the observed flux determines the stellar luminosity. We initially assume that the distance and reddening to 47 Tuc are 4500 pc and E(B − V ) = 0.04 mag, respectively. We later show by isochrone fitting that these values are indeed appropriate ( §3).
We have revised the process used in Papers I and II to better account for interstellar reddening. This follows the same procedure as in McDonald et al. (2010), namely that we use the absorption profiles of McClure (2009) to obtain A λ /A Ks , assuming that A V = 3.2E(B − V ) and that A V /A Ks = 7.75. While the change in process does not significantly affect the stellar parameters we derive, it does better account for the 10-µm interstellar absorption peak. This can affect the mass-loss rates we derive, though is unlikely to be significant for the small extinction toward the cluster.
Obtaining the correct filter transmissions is important when obtaining accurate stellar parameters. Differences can occur in the transmission efficiency of the filter itself, the CCD, the telescope and (particularly) the atmosphere. We can test whether our filter transmissions and stellar atmosphere models are correct by plotting the ratio of the observed flux to the flux of our stellar model for each star (Figure 2), which is very sensitive to errors. Incorrect zero points for a particular magnitude system manifest themselves as global offsets from unity. Incorrect filter transmissions lead to a very strong temperature dependance in the above ratio. We list the adopted zero points in Table 2, which gives the approximate corresponding wavelengths for each filter for reference.
Errors in one band will propagate themselves in reverse onto neighbouring bands, making identifying sources of error difficult. Such issues exist for the very hottest and coolest (most-luminous) stars (>6500 K and <3900 K, respectively). At low temperatures, dynamic atmospheric processes and incomplete molecular opacities in our model atmospheres introduce variations from the model. Conversely, at high temperatures, our models are limited to 6500 K and do not cover the very hottest stars. Stars offset in Figure 2 in all bands may also be Galactic foreground or background SMC field stars. These are not well modelled as their metallicities and surface gravities are substantially different from those of the cluster.
Several other systematics are visible in Figure 2. These include:
• Quantisation errors among the lowest-luminosity stars, which manifest themselves as fan-shaped spreads visible in the U and 8-µm filters, but are negligible among the hottest stars.
• Poor convergence of models around 6250 K. This only affects the hotter stars which have no mid-IR photometry, for which we cannot therefore determine mid-IR excess.
• Increased scatter in the HB stars around 50 L ⊙ . This is due to unknown processes (though could indicate an abundance spread) and does not affect any giant branch dust producers.
• Small (∼1%) remaining systematic under-or overestimates of flux in several filters. The colour-based · · · · · · · · · · · · detection criterion we use ( §4) are chosen to maximise detection of IR excess in such cases, but the amplitude of these effects is much smaller than the >0.1 mag ( 10%) excesses we are looking for.
We therefore do not expect these systematics to affect our detection of circumstellar dust. Problems caused by unresolved blending in the cluster core are visible in several bands. As noted in the previous section, this is a particular concern in the 2MASS data, where there is considerably more scatter in the fitted parameters and observed:expected flux ratios. Table 3 contains the list of stellar parameters determined for the stars.
2.4. Accuracy of the fits 2.4.1. Analysis of departures from the models As noted in §2.3, the goodness-of-fit of our SED models depends strongly on the zero points and filter transmissions used. While Figure 2 shows that the models agree well with the observations between temperatures of 3900 and 6500 K, stars lying outside of this range are subject to some error. To assess the magnitude of this error, a representative selection of stars was taken and the data from the most-affected filters (U , B and I J ) were removed. This had a negligible (<5 K) effect on the stellar temperatures of the stars in the temperature range 3900-6500 K, with temperatures changing slightly more for stars outside this range. The coolest stars have temperature uncertainties caused by poorlymatching models of order tens of degrees, but these stars are also strongly variable. Any 'instantaneous' measure of temperature such as this will therefore not necessarily be representative of the mean stellar temperature to this degree of accuracy.
A formal estimate of the error may be calculated using 'simulated' observations using the Monte-Carlo method, whereby the flux in each band is set to be the observed value plus-or-minus an error sampled from a Gaussian probability distribution with σ identical to the reported photometric error. This suffers two problems: (1) it would take prohibitively long to perform this test for every star and (2) it only takes into account the photometric error, not the modelling error. We have therefore selected a small sample of stars from across the HRD, and performed Monte-Carlo simulations to determine the errors in their parameters. We do this twice: the first accounts for only the photometric error, as described above; (Table 3), chosen to be representative of their respective parts of the HRD. the second replaces the Gaussian σ with the deviation of the observations from the model.
The stars with which we perform this test are listed in Table 4 and the results are shown in Figure 3. It is firstly clear that the temperature and luminosity we derive are interdependent: spread along the RGB track appears determined primarily by errors in the optical data, while spread perpendicular to the RGB track appears determined primarily by errors in the IR data. It is also clear that the photometric errors are considerably less than the total error as estimated by differences between the observations and stellar atmosphere models. As we note in §4.1, this is probably due to under-reported errors due to image artifacts and blending from sources within a few arcseconds. In the right-hand panels of Figure 3, we note that those stars with no data longward of K s -band have significantly more scatter perpendicular to the giant branch: Spitzer data are therefore crucial in determining accurate stellar parameters, even for warm stars. Varying a star's fundamental parameters can also have an effect on the values we achieve. Likely variations for bona fide 47 Tuc cluster members are somewhat smaller than the corresponding errors in temperature or luminosity caused by inaccuracies in the model, which are of order 1-2% for temperature and 2-9% in luminosity. By varying [Fe/H] between solar and -1.4 we produced differences of similar magnitude, varying the mass from 0.6-1.0 M ⊙ produced differences of order 0.3% in luminosity, and varying E(B − V ) between 0 and 0.12 mag produced luminosity changes of <5%. The largest uncertainty, at least in terms of luminosity, appears to be in the distance to the cluster.
Comparison with other works
In order to determine the accuracy of our SED fits, we compare our temperatures with a number of spectroscopically-derived temperatures. We do not compare the luminosities, as the fitting procedure means that the luminosity is directly determined from the model after the temperature fit has been made. The resulting discrepancies are listed below:
• Carretta et al. (2004) -nine early-RGB stars (≈5100 K), temperatures agree within errors: on average they are 21 K (0.4%) warmer in our SED fits than their spectroscopic temperatures, cf. 108 K for the standard deviation of the temperature differences. (2007) -nine upper RGB/AGB stars (<4000 K, >1100 L ⊙ ), temper-atures also agree within errors, averaging 47 K (1.3%) cooler in our SED fits (st. dev. 311 K). (2008) -eight central RGB stars (4200-4500 K), individual temperatures differ at the 2σ level. As a whole, the stars average 25 K (0.6%) warmer in our SED fits, smaller than the standard deviation of the differences between the two papers (31 K). (2007), individual temperatures differ at the 4σ level, averaging 231 K (6.1%) cooler in our SED fits (st. dev. 108 K).
• McDonald & van Loon
• Koch & McWilliam
The exceptionally low standard deviation of the Koch & McWilliam (2008) temperatures provides a good check of the accuracy of both our measurements and theirs. It is worth noting that the discrepancy between our results and those of Carretta et al. (2004) lies in the accuracy of our photometry, not their spectroscopic determinations. A systematic offset between our SED fitting and conventional spectroscopic techniques is likely limited to <30 K (<0.7%) in temperature and <2.8% in luminosity.
The Hertzsprung-Russell diagram
Having determined the stellar parameters for the above objects, the data were then compiled into an HRD, presented in Figure 4.
The HRD shows a number of features, mostly identifying our target stars as post-main-sequence stars in 47 Tuc. Also visible on the HRD is a background sequence of stars belonging to the SMC (13). This merges with the main sequence (MS) of 47 Tuc at roughly the luminosity where the SMC's HB and red clump would be. Regions (9) and (14) contain foreground and background objects of indeterminate distance. Typically, we would expect region (9) to contain stars in the Galactic disc (47 Tuc lies at Galactic co-ordinates l = 305.90 Harris 1996) and region (14) to contain background galaxies (Boyer et al. 2008;Paper I). In this particular sample, however, region (14) may also contain some SMC young stellar objects (YSOs) and cool main sequence objects. We do not expect many YSOs, however, as our region of interest lies away from the bulk of the galaxy's star-forming regions. The main sequence stars at this temperature are mostly below the sensitivity threshold of our data.
• , b = −44.89 • -
Vertical artifacts are also visible on the diagram. These occur every 250 K and correspond to the temperature grid of our model spectra. They are generally caused by poor-quality optical photometry in low-luminosity stars. Especially important are discrepancies between the observed U -and B-band photometry and those predicted by the marcs model spectra, hence the increased prevalence of artifacts at higher temperatures where more flux is emitted in U and B. As our models only extend to 6500 K, we must assume that stars with temperatures derived to be 6500 K have progressively more-uncertain parameters. We have therefore limited our study to stars with T ≤ 7000 K, of which there are 46 398.
Several important evolutionary points (numbered in Figure 4) can be noted at this point, which we use in the next section to fit stellar isochrone models. These are: the main-sequence turnoff (MSTO; 11 in Figure 4), the start of the RGB (10), the RGB bump (8), the upper RGB (5), the RGB tip (top of (2)), the base of the HB (bottom of (7)) and the start of the AGB (bottom of (4)). A review of these evolutionary processes, with particular emphasis on globular clusters, can be found in .
Circumstellar dust reprocesses optical light into the IR, causing an apparent cooling of the star's effective temperature. This moves the star to the right on the HRD, moving into region (2). Throughout their time on the RGB, stars lose mass through stellar winds. For the majority of the time, these appear to be dustless winds driven from the chromosphere (McDonald & van Loon 2007; Paper I), but if the IR excesses reported by Origlia et al. (2007) are true, we may see some low-luminosity stars in this region by virtue of their circumstellar dust. This issue is confused, however, by the presence of field stars, and of stars with poorly-determined photometry.
Our HRD contains a well-denoted and well-separated early-AGB. By its very nature, however, the AGB asymptotically approaches the RGB, and the two become indistinguishable above ∼300 L ⊙ . More-rapid evolution on the AGB means that there are 4-5 times more stars per unit luminosity on the RGB than AGB.
Helium fusion on the upper AGB also becomes increasingly volatile: its violence can increase a star's luminosity dramatically (up to a factor of several) for a short period of time (10 2 ∼ 10 3 years), termed a thermal pulse (TP; e.g. Zijlstra 1995;Wachter et al. 2008). Such stars may scatter above the main giant branch in the HRD, into region (1), but this area is confused by normal AGB stars with poor photometry.
Stars on the TP-AGB are also unstable to pulsation via the κ-mechanism (Ulmschneider 1998), where harmonic oscillations are stochastically excited. This causes semi-regular variability at first, then more-regular and increasingly-violent pulsation until the AGB tip, where the stellar atmosphere disperses. This variability will cause some horizontal and vertical scatter of these stars in the HRD, as we typically only use one epoch of photometry for every filter.
ISOCHRONE FITTING
Introduction
By fitting stellar isochrones to our data, we can redetermine the basic parameters for the cluster as a whole, namely its distance, reddening and age. To do this, we must make assumptions about the cluster's metallicity, helium fraction and α-element enhancement. The accuracy of this determination depends both on the systematic errors present in our data ( §2.4), and the validity of the assumptions we make. To minimise systematic errors, we have used the 'cleaned' subset of stars presented in the bottom panel of Figure 4, to the exclusion of stars with poor photometry and those in the immediate cluster core.
In this study, we compare our HRD to two sets of isochrones, both of which were used in Paper I. These are namely the Padova isochrones Bertelli et al. 2008) 8 and the Dartmouth isochrones (Dotter et al. 2008)
9 .
The CMD interface for the Padova isochrones does not allow the user to change either the helium content or α-enrichment of the isochrones. The YZVAR interface (Bertelli et al. 2008), however, allows both the helium content and a Reimerslaw mass-loss rate (Reimers 1975) to be varied by the user, though also does not allow the α-enrichment to be varied. In this regard, we have simply adopted a fixed metal content, [Z/H], and assumed that metals scale as solar values. The Dartmouth models allow the variation of both helium and α-enrichment in fixed steps.
Altering the above parameters has the following effects:
• Increasing the helium content of the isochrones steep- ens the slope of the RGB, though the RGB tip stays in roughly the same place. It also causes an increase in the luminosity difference between the MSTO and the start of the RGB.
• Increasing the α-enrichment accelerates MSTO, meaning this point corresponds to lower luminosities. It also shifts the entire isochrone to cooler temperatures and increases the luminosity of the RGB bump.
• Increasing the metallicity of the isochrones will shift them to cooler temperatures, as stronger photospheric lines make the atmosphere more opaque.
• Increasing the fitted distance of the cluster will shift the isochrones toward higher luminosities with respect to the stars.
• If the cluster's reddening is higher, we will have therefore underestimated the de-reddening correction: further correction then requires the stars to move to higher temperatures and luminosities with respect to the isochrones. This does not affect all stars equally, and hotter stars will require larger corrections than cooler stars, with the result that the HRD becomes horizontally stretched and vertically skewed (see Paper I for the way this is treated).
• The cluster's age affects the isochrones in a number of ways, with the largest alterations being in the positioning of the MSTO and base of the RGB.
We initially assumed the following parameters: • Distance: d = 4.5 kpc;
• Reddening: E(B − V ) = 0.04 mag; A V = 0.124 mag; R = 3.1;
• Age: t = 12.5 ± 1.2 1.4 Gyr. Isochrones using these parameters are shown in Figures 5 and 6. It is clear from these figures that the bestfitting isochrone is that of the Dartmouth models, but at the adopted value of [α/Fe] = 0. This provides a remarkably-good fit to the data. The only exception to this is on the upper RGB, above the red clump, where the [α/Fe] = +0.2 model fits better (see also later, in Figure 9). An increase in α-element abundances following first dredge-up is not expected, as the stars have not begun helium burning. The metallicity and abundance of 47 Tuc's stars is better determined from spectroscopic studies of individual stars than from our HRD (see §2.4.2 for such studies). As noted, however, the best-fitting isochrone requires [α/Fe] = 0. To fit the Padova isochrones by changing the αenhancement (which roughly corresponds to moving the isochrone horizontally in the HRD) requires [α/Fe] < 0. For the purposes of isochrone fitting, we have proceeded under the assumption that there is no α-element enrichment compared to solar values.
Reddening
The de-reddening of the cluster we apply can be altered using the relations listed in Paper I. This gives an approximate solution for different values of E(B − V ) without the need to re-analyse the entire dataset. Based again on the Dartmouth [α/Fe] isochrones, we find agreement with the accepted value of E(B − V ) ≈ 0.04 mag (Harris 1996).
Age and distance
Once the metallicity, abundances and reddening of the stars have been set, the age can primarily be determined from the temperature of the MSTO. This is vertically scaled to match the luminosity of the MSTO/RGB base at 5500 K. The parameters of age and distance are thus anti-correlated.
As shown in Figure 7, isochrones for 12 ± 1 Gyr fit the data well. An incomplete stellar catalogue of objects below ∼1 L ⊙ means that the models will fit progressively less well below this luminosity. Isochrones are plotted for distances of 4783, 4611 and 4456 pc for 11, 12 and 13 Gyr, respectively. Adding in the 2.8% systematic luminosity uncertainty in luminosity ( §2.4.2), this equates to a distance of 4611 +213 −200 pc. Both sets (CMD & YZVAR) of Padova isochrones yield similar constraints on age. The distances implied for these isochrones are approximately 4% (180 pc) less than for the Dartmouth isochrones. Figure 8 shows the observed and theoretical luminosity functions of 47 Tuc. The observed luminosity function is derived from all stars below 6000 K and 4200 L ⊙ (to include the dusty long-period variable star 47 Tuc V1, but exclude known field stars) which do not fall in region 13 (SMC) of the HRD. The theoretical isochrone uses the Pisa stellar evolution models for a 0.90 M ⊙ RGB and a 0.65 M ⊙ AGB star. The isochrones have been scaled to one star for every 80 000 years of evolution. It should be noted that the AGB models terminate at 1550 L ⊙ : we have extrapolated them in Figure 8 to estimate that there are some 8±2 AGB stars above the theoretical RGB tip at 2780 L ⊙ .
Luminosity function
Overall, the luminosity function provides a good fit to the data. The departure between 60 and 100 L ⊙ is probably due to contamination from field stars and scatter in the HRD from poor photometry. The fact that we do not find as many HB stars at ≈50 L ⊙ as the models predict suggest that either we do not have a complete sample of all HB stars, or that some of the HB stars have already been ejected from the cluster due to mass segregation (cf. HB lifetime ≈ 190 Myr; core relaxation time ≈ 91 Myr; Harris 1996). This is not readily visible in radial density plots of our sources due to the differing depths of the original surveys we used.
HB mass via isochrone fitting
The amount of mass lost on the RGB can be determined by measuring the difference between the RGB-tip stars' initial masses and the mass of stars on the HB. The age of the stars means we can neglect mass loss during the comparatively short (∼1.8 Myr; Silva Aguirre et al. 2010) evolutionary period between the RGB-tip and zero-age HB. Initial masses for isochrones at 12 ± 1 Gyr are: HB star masses can be estimated from synthetic HB models. The temperature spread of the HB is greater than that of the RGB at the same luminosity (the standard deviations from the mean temperature are ≈ 2.2% vs. ≈ 1.6%, respectively). This implies a moderate variation of stellar parameters among the HB stars. Notably, the HB lies at roughly constant luminosity, though luminosity may rise very slightly towards higher temperatures. Figures 9 & 10 show Dartmouth and Pisa synthetic HB models (Dotter et al. 2008 10 ;Castellani et al. 2003 11 ) for different stellar masses. The Dartmouth models are given at [α/Fe] = 0 and +0.2; the Pisa models at Y = 0.24. Two things are immediately obvious from these figures. Firstly, that the temperature spread of the HB is not due to different stellar masses: the nearconstant-luminosity morphology is more consistent with a spread in α-element enhancement. Secondly, the HB star masses predicted by the two synthetic models varies considerably. Including the 2.6% systematic luminosity uncertainty and 4.5% distance uncertainty, we estimate from Figures 9 & 10 that the mass of the HB stars is 0.90 ± 0.05 and 0.92 ± 0.05 M ⊙ for the Dartmouth models, assuming [α/Fe] ≈ 0 and +0.2, respectively, and 0.68 ± 0.03 M ⊙ using the Pisa HB tracks.
This shows that the two HB models are clearly mutually incompatible, and that significant work in this area must be done before an agreement can be reached. The Dartmouth model suggests that RGB mass loss must be remarkably low, at 0.06 M ⊙ (at the +1-σ level, i.e. for 84% of the stars). The Pisa models, however, predict RGB mass loss to be 0.24 ± 0.04 M ⊙ at the ±1-σ level (in agreement with the 0.242 M ⊙ found by Gratton et al. (2010)).
The small range of modelled HB-star masses provided by each set of models also suggests that mass loss on the HB is low ( 0.1 M ⊙ ) in 47 Tuc, which would indicate that chromospherically-driven winds are not so important in this situation as implied by Dupree et al. (2009): more consistency between the HB models would be necessary to substantiate this statement.
The integrated mass loss and median mass predicted by the Pisa models (≈0.68 M ⊙ ) is in good agreement with that of the stars in ω Cen (Figure 11; data from Paper I). The majority of ω Cen's stars are 8-9 times more metal-poor than those of 47 Tuc, but crucially are of very similar age (hence similar initial masses). Due to the lack of consensus between the models, we use the Pisa models as a rough upper limit to the integrated RGB mass loss occurring in 47 Tuc.
It should also be possible to determine the mass of HB stars directly, by taking our photometrically-derived luminosities and temperatures, and comparing them to literature spectroscopically-derived surface gravities (g), via:
R 2 = L 4πσT 4 = GM g ,(1)
where the symbols take on their usual meanings. Due to the systematic uncertainties (in particular in measuring surface gravity), this can only be done with confidence on a set of stars reduced using identical processes, and preferably with similar temperature and luminosities. Unfortunately, there exist no spectroscopically-derived measurements of surface gravity measurements for a selection of both RGB and AGB stars with the internal accuracy required to do this. If one assumes an integrated RGB mass-loss rate of 0.22 M ⊙ per star, comparable with the 0.20-0.25 M ⊙ derived for ω Centauri in Paper I, an AGB star will typically have a mass of ∼0.67 M ⊙ . This implies that only ∼0.15 M ⊙ of its envelope mass remains, to be subsequently lost towards (or at) the AGB tip. We stress, however, that systematic uncertainties in the HB models mean that this cannot be proven from our data. We return to this subject of integrated RGB mass loss in the discussion.
MASS LOSS AND DUST PRODUCTION
Mass loss during the TP-AGB can give rise to dust production in the stellar outflow. The chemistry of the dust is determined by the amount of dredge-up which occurs: this defines whether carbon or oxygen is more numerous in the stellar atmosphere. Carbon and oxygen combine in the atmosphere to form CO. The remaining carbon or oxygen then goes on to define whether the dust is carbon-rich (primarily in the form of amorphous carbon) or oxygen-rich (primarily in the form of silicates). While a few carbon stars are found in globular clusters (e.g. van , they tend to be rare: their low mass means third dredge-up is not usually sufficient to cause the C/O ratio to exceed unity, thus most stars produce silicate dust. Metallic iron dust can also form around oxygen-rich stars (Kemper et al. 2002;Verhoelst et al. 2009;McDonald et al. 2010). It is not clear whether metallic iron dust forms in carbon-rich stars too, due to the difficulty of spectroscopically separating metallic iron from amorphous carbon dust.
The dust can then drive a dusty wind by absorbing momentum from stellar radiation incident on it. This radiation pressure forces the dust from the star. Dust grains are collisionally-coupled to some extent with the surrounding gas, meaning that the gas is driven away from the star as well. It may be that stellar pulsations provide a net outward velocity to the dust, either as an initial 'kick' velocity, or by the dissipation of acoustic energy in the extended atmosphere (e.g. Bowen 1988;Lewis 1989;van Loon et al. 2008b).
Dust will re-radiate absorbed optical stellar light in the IR. This has the apparent effect of cooling the star and giving it excess in the IR above our model spectrum. Dust emits a modified blackbody spectrum which exists in addition to the stellar photospheric output at those wavelengths. Amorphous carbon, graphite and iron dust have no IR spectral features: such dust will typically give rise to positive values for the colours [5.8]-[8], [4.5]-[5.8] and even [3.6]-[4.5], though the latter depends on the dust's temperature. Particularly warm dust may also produce (K s -[3.6]) > 0 in addition to the above colours. Silicate dust leads to broad emission at 9.5 and 18 µm, which is partly covered by Spitzer's 8-and 24-µm filters, giving rise to an excess in these bands.
The infrared excess stars
Determining infrared excess
Stars will exhibit IR excess for a variety of reasons not attributable to circumstellar dust. These include:
• Increased photometric and astrometric uncertainty due to problems with source separation in the dense cluster core; • Photometric errors due to 'ghost' stars and linear artifacts near bright stars (caused by the bandwidth effect and banding, respectively 12 ); • Artificial brightening due to blending with unresolved objects in mid-IR observations, which are at lower resolution than their near-IR/optical counterpart observations; and • Artificial brightening due to blending with other IRbright objects, such as background galaxies or dusty SMC stars.
Of these, the first two will increase roughly as the square of the stellar density, while the latter two will increase proportionally with the stellar density.
To ensure we only select those stars that harbour circumstellar dust, we wish to perform a visual inspection of the photometry and imagery of each star. Such an inspection is performed to mitigate against data artifacts that may otherwise go undetected. It also allows us to sensibly include photometry from other literature sources (e.g. the AKARI IRC point source catalogue; Ishihara et al. 2010) which improve coverage of dust features in the SED, but where coarser resolution and/or lower signal-to-noise mean flux measurements need to be interpreted with caution on an individual basis.
12
Details can be found in the IRAC handbook: http://ssc.spitzer.caltech.edu/irac/iracinstrumenthandbook/ It would take prohibitively long to do this for our original list of 46 398 stars. We reduce our source list to a manageable number by selecting only stars with L > 31.6 L ⊙ (i.e. the 3299 stars in or above the red clump) which have Spitzer photometry (1993 of those stars) and meet one or more of the following criteria:
• is an evolved star candidate ( §2.5; 58 stars); • has at least two of the (K s -[3.6]), ( When combined, this list gives 258 unique objects. One may wonder why we use colours relative to our model in this case, rather than the absolute colours. In cooler stars, CO bands at 2.3 and 4.5 µm cause substantial deficits in the K s and 4.5-µm bands. In cool stars, around 3500 K, we model the colours of a naked star to be (K s - These colours decrease in warmer stars. By taking into account the expected colours and their variation with stellar parameters, we can probe smaller excesses than possible with conventional colour-magnitude diagrams. The stars we investigate are shown in such a colour-magnitude diagram in Figure 12.
A large number of stars were flagged where the photometry agreed between the 3.6-and 5.8-µm bands and between the 4.5-and 8-µm bands, but were discrepant between these two pairs. This is likely due to the simultaneity of the 3.6-and 5.8-µm, and 4.5-and 8-µm observations. It could either be a direct consequence (intrinsic variability) or indirect consequence (such as differing qualities of photometry due to different coverage in the original mosaics). For this reason, we have ignored the ( At this point, we have a list of potentially dusty objects, using criteria which are qualitatively similar to those applied in Origlia et al. (2007). Astrometry and photometry of the sources in that work have been made available to us (L. Origlia, private communication). Of their 93 sources, only 45 are found to have counterparts with IR excess in our own candidate list (Table 5). This immediately suggests that the primary difference between the two works lies in the original photometric reduction, and that the real photometric errors are considerably larger than those used for the 3σ cutoff in Origlia et al. (2007). Of these 45 matched sources, 23 are over 1000 L ⊙ , while only 22 are below 1000 L ⊙ . We explicitly do not claim one reduction to be more accurate than the other: while several stars known to harbour One comparison we can make is of the magnitudes of Origlia et al.'s dusty stars used in their work, and those used in this work ( Figure 13). It is notable that (particularly for the fainter stars) the K s -band magnitudes are systematically brighter in this work, especially where lower-resolution 2MASS data are available, while the [3.6] and [8] magnitudes are systematically fainter. This is further evidence that the method of photometric reduction and differences in resolution are the major differences between Origlia et al. (2007) and Boyer et al. (2010).
Sample cleaning
Most of the 258 objects that match our criteria above were flagged as a result of incorrectly-matched photometry, either between surveys in our cross-matching routines, or within the individual surveys themselves. The Gunn i-band data from the MCPS and the JHK s data from 2MASS are particularly prone to these errors. We expect dusty stars to have a mid-IR excess, and that the ratio of observed/modelled flux should not appreciably decrease toward longer wavelengths. To remove stars which have been erroneously selected due to bad photometric data or matching, we have removed stars selected with the following cuts:
• To remove objects with poor JHK s -Spitzer matching, we remove objects which show K s -band excesses (compared to our model) more than 0.1 mag greater than the associated 3.6-µm excess. An example is shown in Figure 14, (i). This reduces our source list to 221 objects.
• To remove objects with poor JHK s matching in general, we remove objects which have observed/model flux ratios at J, H and K s -band of <0.8 (i.e. a strong near-IR flux deficit; Figure 14, (ii)). This leaves 188 objects.
• To remove objects with bad photometry in Spitzer data, while retaining stars with only 8-or 24-µm excess, we remove objects which show less than 10% excess above the model at the longest recorded wavelength: 12 sources are discounted because they show no excess at 24 µm and 37 because they show no excess at 8 µm (when 24 and 8 µm are the longest wavelength with data, respectively). These stars mostly have discordant IRAC photometry, but include V16, LW1, LW2 and LW21, which were selected due to their variability, but show no apparent mid-IR excess (Figure 14, (iii)). This leaves 128 objects.
• To remove objects with bad photometric data at Origlia et al. (2007) and this work. Origlia et al. (2007).
ID
2 Offset in arcseconds between Origlia et al. (2007) and this work, limited to <2 ′′ . 3 Variable (V, LW, A) names from Clement (1997); and Lebzelter & Wood (2005). Other names from: Lee -Lee (1977 K s -band, we remove objects which show an observed/model flux ratio at K s -band which is >0.1 mag less than that at both J-band and H-band (Figure 14, (iv); leaving 127 objects).
• To remove objects with mid-IR excess caused by missing near-IR data (leading to unconstrained fits), we remove objects with no K s -band data unless ([3.6]-[8]) > 0.1 mag (Figure 14, (v); leaving 120 objects).
This reduction removes most of the faint giants from the sample, but also a few bright giants. These include Origlia et al. (2007) and Boyer et al. (2010), showing only those stars suggested to harbour circumstellar dust by Origlia et al. V19, LW1, and the blended sources LW7 and LW8. Comparing our reduced sample to Origlia et al. (2007) shows that, out of these 120 sources, 28 match theirs. Of these 28, 18 are above 1000 L ⊙ and ten are below 1000 L ⊙ . The ten objects are listed in Origlia et al. (2007) as IDs 67,81,86,148 (FBV42), 181, 189 (R7), 222 (FBV35), 296, 309 and 373. We note that the study by Forte et al. (2002), whose authors lend their names to the FBV stars, shows no notably-different polarisation (which would be suggestive of additional, circumstellar, line-of-sight dust) in either FBV35 or FBV42, in comparison to the brighter, variable stars.
As both studies find a significant fraction of potentially dust-harbouring stars in the core, we may expect some overlap between the two samples, even if they are randomly distributed. The probability (P ) that two samples of stars (m and n) are randomly drawn from a total of N stars when i stars are found in both samples can be found (in the limit of identical object detection efficiency and independent datasets) by:
P = 1 − i k=0 m! k!(m−k)! (N −m)! (n−k)! ([N −n]−[m−k])! N ! n!(N −n)! .
(2) For the entire sample of stars, N = 3399, n = 103 (not 120, as 17 stars are identified outside the region covered by Origlia et al.), m = 93 and k = 28, giving an answer of P ≪ 10 −10 : i.e. zero chance of the overlap being due to chance. Concentrating on the cluster's inner 2 ′ , however, we find P = 0.01% for the bright giants 13 ([8] < 8, L 1000 L ⊙ ) and P = 8.5% for the faint giants 14 (8 < [8] < 10.874, 35 L 1000 L ⊙ ). Considering only the inner 1 ′ , P = 0.15% for the bright giants 15 and P = 22.2% for the faint giants 16 .
We can therefore state that the detection of infrared excess around the bright giants has a negligibly-low probability of being due to random selection (at a level of around 3-4σ). One may also conclude that the detec- Origlia et al. (2007); green circles mark our own dusty candidates; large magenta dots denote objects found to have IR excess by both parties. Banding is present to the west-north-west of bright stars: the brightest, V1 and V8, are labelled. tion of infrared excess around the faint giants has a nonnegligible probability of being taken from a random selection (at 22.2%, or 1-2σ). However, we remind the reader that the two samples are taken from the same set of Spitzer data and thus not independent: they will therefore have identical noise and image artifacts to contend with. The real probability that the two samples of stars trace objects showing a physical phenomenon is (unquantifiably) lower than the 77.8% that the above probabilities predict. We are therefore not confident that the overlap of 28 stars between our two samples is significant, nor that our two studies are actually tracing a particular sub-population of stars.
Image artifacts
As noted in §4.1.1, the original Spitzer data used both in Origlia et al. (2007) and this work contain a number of known artifacts: notably banding, the bandwidth effect, and blending. Figure 15 shows the central region of the cluster, containing almost all the stars suggested by Origlia et al. (2007) to contain dust, and also the vast majority of our 120 dusty candidates. From this image, the lack of overlap between our two candidate sets is clear. We now investigate how each of the effects affect our data, but stress that we do not discount any individual targets in this section.
Banding is caused by optical scattering and charge transfer within the IRAC array, which causes horizontal and vertical spikes to become manifest around bright stars. In Figure 15, these trails can be seen extending in the "10-o ′ clock" and "4-o ′ clock" directions from the brightest stars, with less-prominent lines perpendicular to this axis.
Banding has clearly affected both selections of dusty candidates: several candidates from Origlia et al. (2007) lie on banding, caused by V1 (IDs 195,222,293 and 379) amongst others; several of our candidates lie on banding caused by V8 (Figure 15). Of the stars both studies find to have IR excess, ID-148 and ID-222 are obviously affected by banding, though several other stars to the west of the cluster may also be affected by this problem (IDs 6,9,10,19,171) as may stars near the center of the cluster (including IDs 3, 26, 36, 67 and 181). While banding affects stars of all magnitudes, the amplitude of the effect scales inversely with received stellar flux. Banding can therefore not reproduce the excess in the brighter giants, whose excess is also confirmed by either their ([3.6]-[8]) colour and/or literature photometry (see below, §4.1.6). Ghost images, caused by the bandwidth effect, affect the brightest stars at 8 µm. While this does not appear to have affected Origlia et al.'s selection, several of our dusty candidates lie on these ghost images, which artificially boosts their 8-µm flux, mimicking excess.
Blending is obviously very significant in the crowded cluster core, affecting a set of stars which include the aforementioned IDs 3, 26, 36, 67 and 181. Every star is blended to some extent, but it is difficult to quantify the amount of blending present. Origlia et al. (2010) present strong evidence from Hubble Space Telescope I-band imagery that very close blends with stars of similar I-band magnitude is not an issue in the majority of cases. What was not addressed are blends with objects of large (I-[3.6]) or (I-[8]) colours, and objects blended with other objects at distances of more than 1 ′′ .
The majority of very red sources, showing large (I-[3.6]) and (I-[8]) colours are galaxies (Paper I). We do not, however, consider this a major source of blending in this case, as these should affect stars randomly, rather than being more prevalent in the inner regions, and are likely to only affect one or two sources in the cluster (cf. Paper I; Matsunaga et al. 2008).
On the other hand, blending with objects outside the central PSF is still highly significant, as in our example in §4.1.4, 2MASS 00234588-7204488. There are two main reasons for this. First of all, these images are limited by diffraction, rather than seeing, meaning that a complex PSF extends a long way from the central source. If the flux ratio between the two blended sources is high enough, a small error removing flux from the brighter object can leave significant flux in the Airy rings it creates. Some of this flux can be attributed to the fainter object, causing it to be artificially brightened, and leading to an obviously non-stellar SED.
A fundamental difference exists in the reduction process between the Origlia et al. papers and ours. Our daomatch/daomaster reduction fits and removes the PSF of the brightest sources, then subsequently removes fainter objects until we have removed all significant flux from the image. Origlia et al.'s romafot-based reduction fits PSFs to the lower-resolution, longer-wavelength Spitzer data. They then sum the flux from all sources within their K-band data which lie inside the Airy radius of each Spitzer source (2.3 ′′ at 8 µm) and use Kurucz model atmospheres to determine the expected flux at longer wavelengths: stars with flux in excess to these values are determined to have infrared excess. Each technique has its benefits. Our more-conventional technique is optimised where the K-band and Spitzer data have similar PSFs (as is the case with the 2MASS data), while Origlia et al.'s more-innovative approach should theoretically work better for their higher-resolution K-band data.
Our concern is whether their method is sophisticated enough to take into account all forms of blending: by summing flux from sources with centroids within 2.3 ′′ , a blended source at 2.2 ′′ from a star is treated the same as one lying much closer to the star, while a source at 2.4 ′′ from a star is not considered to affect the star at all. In the former case, the blending star will not contribute as much to the PSF as is calculated, leading to an overestimation of the expected 8-µm flux; in the latter case, the blending star will contribute some unaccounted flux towards the PSF and lead to an under-estimation of the expected 8-µm flux. This is of particular concern when a source lies on the Airy ring of a bright star, such as Origlia's ID-373, which is affected by the nearby V1. One can calculate the contamination between two stars by convolving the IRAC 8-µm PSF of the measured star, with the PSF of the blending star which has been shifted appropriately in the RA-Dec plane and multiplied by the fractional error in its flux. One can consider a typical example of two stars, where a bright giant is 2.4 ′′ from a star 33× (3.8 mag) fainter than it. The bright giant will have a typical error in flux of 3%, which corresponds to a flux comparable to that received from the fainter star. If one fits a PSF to the fainter star, one will find it 0.12-0.14 mag brighter than it actually is (depending on the position angle of the blended source). It is worth remembering that this is caused by a 1σ error: with ∼20 giants bright enough to cause this effect, one may expect errors up to roughly twice this value. When one takes into account blending by multiple brighter objects (as happens in the cluster core) and any noise inherent in the data, this effect can increase further.
The second problem is that source confusion can lead to astrometric errors. A second fundamental difference exists between our reduction processes: we first fit PSFs to objects detected in each band, then combine the data from different bands together, while Origlia et al. fit PSFs to identical co-ordinates across all bands simultaneously. Again, each technique has its benefits: Origlia et al.'s method mitigates against displacement of the PSF centroid due to blended sources, while our technique avoids problems with image distortion and alignment within surveys, and co-ordinate system alignment among surveys.
While the co-ordinates of most of the brightest stars agree to within ∼0.2 ′′ with the positions provided by Livia Origlia (private communication), LW10 and V26 are misplaced by 0.59 ′′ and 0.74 ′′ , respectively. Both stars are ≈2500 L ⊙ , meaning the poor removal of the stars' PSFs caused by such an error may lead to significant errors in other stars' fluxes. Both studies identify several (different) objects around both of these stars as having IR excess, but it is not clear that this is actually the case.
Despite the differences between our reduction methods, one would expect that if fainter stars had infrared excess created by dust, we would mostly identify the same stars as having infrared excess. Of our remaining 120 objects, several exhibit a 'step' jump of several tenths of a magnitude between K s -band and 3.6 µm, but no additional excess at longer wavelengths. Two possibilities for this are large amounts of hot circumstellar dust, or bad cross-matching between optical/near-IR and Spitzer data due to the very different resolutions between the two. Bad cross-matching is a problem that affects all surveys in crowded regions, but these objects would be retained under simple colour cuts in, say, (K s -[8]).
We can separate bad cross-matching from real dust by examining where these objects lie in a colour-colour diagram (Figure 16,. Strictly speaking, this figure is a colour excess -colour excess diagram, and thus in the following, we denote a colour excess in filter Y with reference filter X as (X-Y ) e . This is equal to the observed colour minus the model atmosphere colour:
(X − Y ) observed − (X − Y ) model(3)
or, identically, the colour excess of the observation with respect to the model in filter X minus the colour excess in filter Y : In Figure 16 (top-right panel), we have taken the SED of a typical star with (K s -[3.6]) e > 0 but ([3.6]-[8]) e ≈ 0: 2MASS 00234588-7204488. We show circumstellar dust models made using metallic iron (optical constants from Ordal et al. 1988) and amorphous carbon (optical constants from Zubko et al. 1996): two of the best grain species for producing flux at short wavelengths (e.g. McDonald et al. 2010). Neither can produce the observed SED as well as the naked photospheric model, multiplied upwards in flux. This is highly suggestive of source blending in the Spitzer images, but not in the K s -band data. Indeed, McLaughlin et al. (2006) report stars with HST magnitudes of F 475W = 14.65 and 15.17 mag at distances of 0.335 ′′ and 1.74 ′′ from this location, respectively, while Salaris et al. (2007) reports the K sband magnitudes of the same two stars as 12.064 and 12.018 mag, respectively. On the other hand, 2MASS reports K s = 11.485 ± 0.073, which compares favorably to [3.6] = 11.539 ± 0.098 mag from Boyer et al. (2010). It would therefore appear that this object is an unresolved blend in both 2MASS and Spitzer data.
(X observed − X model ) − (Y observed − Y model ).(4)
A third test we can do to ensure that these stars are unresolved blends is to look at the radial distribution of these objects (Figure 16, bottom-left panel). Sources with significantly positive ([3.6]-[8]) e colours exhibit more central concentration than the main population of bright stars, falling off roughly as the square of projected source density, as might be expected from blending. Those with near-zero ([3.6]-[8]) e colours are even more centrally condensed, with the entire group located within 2.5 ′ .
One can perform a similar test with the objects found in Origlia et al. (2007). When doing so, we must bear in mind the completeness of our samples: comparing the distribution of their dusty sources to our total source distribution assumes both our datasets achieve the same completeness. The fact we work from the same data, however, would suggest that their completeness at 8 µm is not significantly different to ours. In absolute terms, false star tests retrieve 88% of stars with [8] = 10.9 mag over the image, though this decreases to 75% of stars between 1.5 ′ and 2 ′ and 58% within 1 ′ . These percentages increase to 94%, 88% and 87% for stars of [8] = 10.1 mag. We estimate that we should retrieve ≈90% of stars of similar magnitude to Origlia et al.'s dusty stars (which have an average of [8] = 9.51 mag). Figure 17 shows that the radial distribution of both the bright ([8] < 8 mag) and faint (8 < [8] < 10.874 mag) giants we detect in the original (Origlia et al. 2007) Spitzer images.
The distribution of both the bright and faint giants follow each other within the Poissonian noise ( √ n). The distribution of bright dusty candidates in Origlia et al. (2007) matches the global distribution of bright giants we identify to within the Poissonian noise. The distribution of their faint dusty candidates, however, departs from the global distribution of stars with an identical magnitude range by over 2 √ n, showing considerable central concentration. The lack of sources at radii > 2 ′ corroborates this finding: including stars over 2 ′ from the cluster core increases the number of objects by ≈41%, suggesting there should be ≈41% (27) more faint dusty sources at large radii, whereas Origlia et al. (2007) only find two. Similarly, one expects ≈15 sources over 2.5 ′ from the cluster core, whereas Origlia et al. find none. The differences we find in the radial distribution of their dusty objects and our total detected sample (with or without correction for unresolved targets) cannot be accounted for by differences in detection efficiency. A grossly centrally concentrated distribution of candidates therefore suggests that blending and its associated effects are a significant problem in the sample of Origlia et al. (2007).
Finally, we compare the magnitudes for all stars which are common between the Salaris et al. (2007) and 2MASS catalogues (Figure 16, bottom-right panel). The majority of both groups of stars (with near-zero and with positive ([3.6]-[8]) e colours) scatter generally toward brighter 2MASS magnitudes. The lower-resolution of 2MASS therefore suggests that these stars are blends. We note that the brightest stars are not included in this test, as they are saturated in the data from Salaris et al. (2007).
To summarise, very warm dust does show an appreciable 3.6-µm excess (Figure 16), hence a positive (K s -[3.6]) colour. This would suggest that (K s -[8]) is theoretically a better indicator of the presence of warm dust than ([3.6]-[8]). However, warm dust should still produce a substantial ([3.6]-[8]) colour, meaning it should not be greatly desensitised to the detection of warm dust when compared to (K s -[3.6]). The above four observations indicate that the difference in resolution between the K s and [3.6] images has caused artificially-high (K s -[3.6]) colours in both our data and that of Origlia et al. (2010), while ([3.6]-[8]) colours remain largely unaffected. We therefore conclude that (K s -[8]) may detect warm circumstellar dust more efficiently than ([3.6]-[8]), but with three important caveats: (1) that in dense regions the resolution of the images used to determine colour should be broadly similar, to avoid unresolved blending affecting one band; (2) that stars of 3500 K will naturally show a (K s -[3.6]) colour simply by virtue of being cool; and (3) neither colour can mitigate against the effect of blending with a red object, such as a background galaxy.
Examination of individual stars
With 120 targets remaining, it is now feasible to perform a visual inspection of each SED. We find that many faint stars have been selected merely because of their (K s -[3.6]) colour, which suffers from the problems highlighted in §4.1.4 and Figure 16. This point is quite simple to address: we apply the colour cut shown in the upperleft panel of Figure 16, to exclude stars with ([3.6]-[8]) e <0.1 mag which have no 24-µm data. This removes 40 of our 120 targets, including We now visually identify each of the remaining 80 targets on the Spitzer 8-µm mosaic and remove those objects which lie on bands created by stars, unless the photometric excess observed is significantly greater than the flux in the band. We also remove targets lying on the first Airy ring of bright (>1000 L ⊙ ) stars, according to the same criterion, stars artificially brightened by bandwidth-effect ghosts, and stars on the edge of the image. Banding and blending on the Airy disc removes six stars due to V1 (including ID-222 and ID-373), five stars due to V8, two stars due to V4, two stars due to V27, one star due to LW1, one star due to LW7 and/or LW8, two stars due to LW9, three stars due to LW13 (one of which is also on the Airy disc of A19), and two stars due to LW18. In the cases of V27 and LW18, the effects of banding are compounded by banding from other bright stars in the cluster core. Two stars are here removed as their Spitzer photometry is uncertain, as they lie on the edge of the mosaic. Selected objects which are actually banding-effect ghost images are also removed from around V21, LW3, LW9 and FBV45. This leaves 50 candidates, with the only faint stars in common with Origlia et al. (2007) being ID-67, ID-81 and ID-86.
Our remaining candidates now split into three main groups: 1. 29 bright (>1000 L ⊙ ) giants that are known variables and which (mostly) have known, spectroscopically-confirmed circumstellar dust (van Loon et al. 2006;Lebzelter et al. 2006)). We examine these in §4.1.6.
2. 9 moderately-bright (250-1000 L ⊙ ) giants which are listed due to their positive (K s -[3.6]) e and/or ([8]-[24]) e colours. We examine these immediately after this list.
3. 14 faint giants (<100 L ⊙ ) which have Spitzer IRAC colours consistent with warm dust. We examine these in the remainder of this section.
All nine moderately-bright giant stars lie in the crowded heart of the cluster. Eight of the nine stars (excluding F2, below) have not yet been rejected purely because their 24-µm photometry still suggested the presence of dust: i.e. they had a >10% 24-µm excess. We note, however, that the PSFs of these stars are substantially more blended at 24 µm (FWHM = 6 ′′ ) than at 8 µm (FWHM = 2 ′′ ). Seven of the eight stars (excluding F1, below) share all the following criteria:
• positive (K s -[3.6]) e and/or ([8]-[24]) e colours;
• blended with several other fainter sources within the 24 µm PSF; • blended with much brighter objects outside the 24 µm PSF, including V1, LW10 and LW13 (only six objects, but the seventh only shows a 1.6σ excess at 24-µm); • have (where available) K s -band magnitudes which differ between the 2MASS and Salaris surveys, suggesting image resolution is affecting magnitude determination; • and no other evidence of circumstellar dust.
Given the problems caused by heavy blending which we have already highlighted in this section, we now remove these seven stars, which include ID-67, ID-81 and ID-86. This leaves 43 dusty candidates: 29 of these are above 1000 L ⊙ , and 14 are below.
None of the 14 faint dusty candidates we find are in common with Origlia et al. (2007). We therefore find it unlikely that they are dust enshrouded and are likely merely artifacts and outliers present in our own photometry: from our large number of Spitzer sources (4462) we may expect outliers up to ≈3.6σ. To be rigorous, and because several of these targets lie outside of the field of view covered by Origlia et al. (2007), we explore each target (F1-F14) individually:
• (F1) 00 h 23 m 52.8 s -72 • 04 ′ 33 ′′ , 4305 K, 271 L ⊙ : this star was selected purely on the basis of an 11% excess at 24 µm, however the formal error on the flux is 23%, meaning this is not a significant detection.
• (F2) 00 h 24 m 05.8 s -72 • 04 ′ 44 ′′ , 4058 K, 260 L ⊙ : significant excess is present at 3.6 and 4.5 µm, but not detected at longer wavelengths. It is very close to a star four times its brightness, therefore the excess is likely a result of blend.
• (F3) 00 h 23 m 47.7 s -72 • 04 ′ 35 ′′ , 4939 K, 82 L ⊙ : this object shows some flux excess at 5.8 µm, strong excess at 8µm, but is blended with stars of similar brightness at these wavelengths. It may also be affected by banding from V8, which is expected at this position but not visible on the images due to crowding.
• (F4) 00 h 25 m 39.4 s -72 • 08 ′ 21 ′′ , 4445 K, 73 L ⊙ : identified by an 11% (1.3-1.4σ) excess at both 5.8 and 8 µm, the low significance of this excess means this star is likely a spurious detection.
• (F5) 00 h 23 m 54.1 s -72 • 04 ′ 17 ′′ , 5210 K, 71 L ⊙ : this star lies on the Airy rings of two slightly brighter stars and shows apparent excess at 5.8 and 8 µm as in F3 above. It is also poorly resolved from a nearby, slightly fainter companion, and has poor quality J-band photometry due to nearby companions. We suspect the apparent 29% (3σ) excess is due to blending.
• (F6) 00 h 23 m 59.8 s -72 • 05 ′ 01 ′′ , 5265 K, 59 L ⊙ : this star shows increasing excess at 3.6, 4.5 and 8µm. It is not detected at 5.8 µm. Poorly resolved from nearby bright stars in Spitzer data, and with inconsistent I-band photometry, we suggest that this star shows excess due to blending.
• (F7) 00 h 24 m 25.9 s -72 • 19 ′ 37 ′′ , 5204 K, 51 L ⊙ : flagged for investigation due to a 25% (3σ) excess at 8 µm, photometry at other wavelengths shows considerable scatter above the reported errors. It may be a field star or unusual object, but the excess (if it exists) does not appear to be due to dust.
• (F8) 00 h 23 m 39.2 s -72 • 04 ′ 04 ′′ , 5185 K, 50 L ⊙ : this star's PSF overlaps with the Airy rings of several nearby, unresolved, much brighter objects at 8 µm.
It shows an excess at 3.6 and 8 µm, but a possible deficit at 4.5 µm. We consider the 18% (3σ) excess to probably be due to blending.
• (F9) 00 h 24 m 07.0 s -72 • 05 ′ 46 ′′ , 4807 K, 47 L ⊙ : we have no near-IR photometry for this star, meaning the photospheric contribution to the SED is less constrained. It was selected on the basis of a 16% excess at 8 µm, however this is at a significance of <2σ, so is probably a spurious detection.
• (F10) 00 h 24 m 10.0 s -72 • 03 ′ 25 ′′ , 4765 K, 46 L ⊙ : selected on the basis of a 34% (3σ) excess at 8 µm, this star lies on the Airy ring of an unnamed star, which is 20× brighter at 8 µm, but not luminous enough (725 L ⊙ ) to be selected as a potential bright-star blend earlier in this section.
• (F11) 00 h 24 m 11.4 s -72 • 04 ′ 04 ′′ , 4983 K, 44 L ⊙ : this star was flagged because of its (K s -[3.6]) and ([5.8]-[8]) colours. We note that the same excess persists at 3.6, 4.5 and 8 µm, and that the JHK s fluxes are deficient compared to our model. The apparent excess in this object seems due to the differing resolutions at K s and 3.6 µm (see §4.1.4) and a spurious 5.8-µm flux.
• (F12) 00 h 26 m 04.8 s -72 • 08 ′ 16 ′′ , 5547 K, 42 L ⊙ : identified due to a 34% (2σ) excess at 8 µm, the excess in this object is likely only a spurious detection.
• (F13) 00 h 28 m 35.1 s -71 • 51 ′ 09 ′′ , 4010 K, 39 L ⊙ : this star shows considerable excess at 5.8 and 8 µm (6σ and 4σ, respectively). It is very cool for its luminosity, and is 23 ′ from the cluster core. It seems most likely that this star is a luminous member of the SMC which was marginally brighter than our luminosity cutoff.
• (F14) 00 h 24 m 31.7 s -72 • 06 ′ 00 ′′ , 4576 K, 35 L ⊙ : identified due to apparent excess at 3.6 and 5.8 µm (the only two Spitzer detections), this star has photometry which departs significantly from the models at IJHK s . The SED of this object is of insufficient quality and coverage to determine whether any IR excess truly exists.
In summary, of the above 14 stars, we find that the excess observed in eight is of low statistical significance (< 3.6σ), that two have poor-quality SEDs, that three suffer from blending and that the final star is likely a member of the SMC. We remind the reader that, in any case, none of these stars was found to have mid-IR excess by Origlia et al. (2010). To conclude, we find no compelling evidence that any star in 47 Tuc produces dust until it reaches at least 1000 L ⊙ . We find similar mid-IR excesses as those seen by Origlia et al. (2007), but we find these in different objects. Such objects occur predominantly in heavily-blended environments, and are due to either image artifacts or under-reported photometric errors due to blending.
Excess among the bright giants
We now examine the bright giants which are left as candidates. From these, we remove LW20 and LW17, as they were flagged due to their variability, but have no mid-IR data with which to determine whether they are dusty. Identifying blended objects, we find that LW7 and LW8 are blended with each other and that V26 is blended with a star just over half its brightness. LW7 and LW8 are both identified as having IR excess. It would appear likely that at least one of them harbours dust, however the amount of excess they show and the quality of their photometry prevents us from saying anything more substantial about any such dust. V26 is retained, as the star it is blended with is fainter, and because higher-resolution 8.6-µm photometry (van Loon et al. 2006) confirms the excess seen by Spitzer. Five more targets (LW3, FBV45, LW18, LW15 and LW6) were removed because their excesses are not statistically significant. The remaining 22 bright giants are listed in Table 6.
We again examine the SED and literature data of each star in turn. We include in these SEDs literature mid-IR photometry from Origlia et al. (1997) (ESO-3.6m/TIMMI); Origlia et al. (2002) (ISO ); Ita et al. (2007) (AKARI ) and the AKARI point-source catalogue (Ishihara et al. 2010). The ISO and AKARI data are taken at coarser resolution than that provided by Spitzer IRAC, so caution must be taken when dealing with stars near the cluster core. We also include photometric data and spectra from van Loon et al. (2006) (ESO-3.6m/TIMMI2). Comparison with the Spitzer IRS data of Lebzelter et al. (2006) allows us to more-precisely determine dust compositions.
In the absence of spectroscopy, we have seen that identification of circumstellar dust production based solely on Spitzer photometry can be problematic. Strong IR excess which increases in magnitude (but not necessarily flux) with wavelength is generally a reliable indicator of the presence of circumstellar dust. Spectra are needed to conclusively identify the dust composition, though the presence or absence of silicates or other oxides can sometimes be determined in comparatively-isolated stars with excess between 8 and 24 µm. Based on the quality of the SEDs, literature observations of these stars, and the IR excess we observe, we indicate in Table 6 whether a star is clearly (Y) or probably (?) dusty. Further analysis of the dust minerology and mass-loss rates of these stars can be found in the accompanying Paper IV.
DISCUSSION
RGB mass loss and metallicity
Modelling of horizontal branch star masses ( §3) implies that integrated RGB mass loss per star in 47 Tuc does not greatly exceed that in ω Cen (see also Paper I). This is despite the clusters being of similar age, and ω Cen being (on average) almost a factor of ten more metal-poor. Similar HB stellar masses and integrated RGB mass-loss rates in the two clusters would imply that the dominant RGB mass loss process is metallicity-independent for globular cluster stars.
This would corroborate findings in other clusters (open and globular, including NGC 6791 and ω Cen) that the super-solar metallicity open cluster NGC 6791 that show RGB and AGB dust production is similar to solar-metallicity expectations and that there is no unexpected absence of RGB stars due to super-solar mass loss (van Loon et al. , 2008a. This is not to say that dust production is necessarily metallicity-independent: lower-metallicity stars are warmer at a given luminosity, therefore their chromospheric mass-loss rates are expected to be higher than their higher-metallicity counterparts (Schröder & Cuntz 2005). Better estimation of horizontal branch star masses are required in order to improve models before this can be fully substantiated, however.
RGB dust formation
We find no reliable evidence that dust is being produced by any RGB star below 1000 L ⊙ . This is concurrent with the recent findings of Boyer et al. (2010). We do not confirm the claims of dust production in lowerluminosity RGB stars made in Origlia et al. (2007Origlia et al. ( , 2010.
Fig. 1 .
1-Spatial coverage of the surveys used in this paper. See text for details of each survey. The cross marks the cluster centre.
Fig. 2 .
2-Differences between observed and SED-fitted model fluxes as a function of luminosity (top panels) and temperature (bottom panels, where only stars with L > 5 L ⊙ are shown).
Fig
. 3.-Monte-Carlo fitting results, showing (left) photometric errors only and (right) errors derived from departures from the model. Red bars show the 1-D standard deviations of the distributions. Labels denote sequence numbers of the objects
•
Worley et al. (2010) -five of the spectra used in McDonald & van Loon
Fig
. 4.-HRD of stars in the direction of 47 Tuc, showing evolutionary and spatial components. The top panel contains all stars, the bottom panel contains only stars more than 2 ′ from the cluster core, which have photometric data in the optical, near-IR and mid-IR. Abbreviations are discussed in the text.
Fig. 5 .
5-Top panel: Padova isochrones generated using the CMD utility, overlaid on a density-mapped HRD. Bottom panel: Padova isochrones generated using the YZVAR utility. Arrows show the locations of the RGB bumps of the isochrones. Details of the isochrones' parameters can be found in §3.1.
8Fig. 6 .
6http://stev.oapd.inaf.it/cgi-bin/cmd 9 http://stellar.dartmouth.edu/∼models/webtools.-AsFigure 5for the Dartmouth isochrones.Top panel: isochrones at [α/Fe] = 0. Bottom panel: isochrones at [α/Fe] = +0.2.
Fig. 8 .
8-The luminosity function for 47 Tuc (solid, red line), along with theoretical Pisa isochrones. Lower, short-dashed line (blue): AGB and HB stars only. Upper, long-dashed line (green): AGB/HB + RGB stars.
Fig. 9 .Fig. 10 .
910-The upper RGB, HB and AGB of 47 Tuc (small, black dots). These are overlain with Dartmouth RGB isochrones (blue dashed lines) at [α/Fe] = 0 (left) and +0.2 (right), and with similar HB models (large, coloured dots). In order of increasing luminosity, the HB tracks are for stellar masses of 0.85, 0.90 and 0.95 M ⊙ . -As Fig. 9, but with Pisa HB models at (left-to-right) 0.65, 0.70 and 0.75 M ⊙ .
Fig
. 11.-As Fig. 10, for the cluster ω Cen. • M init = 0.893 +0.023 −0.021 M ⊙ for the Padova isochrones; • M init = 0.874 +0.022 −0.019 M ⊙ for the Dartmouth isochrones at [α/Fe] = 0; • M init = 0.889 +0.025 −0.022 M ⊙ for the Dartmouth isochrones at [α/Fe] = +0.2; • M init = 0.916 M ⊙ for the Pisa evolutionary models (from Gratton et al. 2010).
Fig
. 12.-Absolute (Ks − [8]) and ([3.6] − [8]) colours for all stars in our sample (small, red dots). Circled dots (green) show our initial cut for dusty candidates. Triangles (blue) show our final cut for stars which we scrutinise individually, once bad data have been cleaned.dust do not appear inOriglia et al.'s list (e.g. V13 and V18;van Loon et al. 2006;Lebzelter et al. 2006), they lie outside their survey region.
); FBV -Forte et al. (2002); MV -McDonald & van Loon(2007); R-Feast & Thackeray (1960).
Fig. 13 .
13-Differences in photometric measurements presented in
Fig. 14 .
14-Examples of stars rejected as having poor photometry. See §4.1.2 for details.
Fig. 15 .
15-Spitzer 8-µm imaging of the core of 47 Tuc from the Rood observations. Orange dots mark the stars found to have IR excess by
Two populations separate out in this diagram: ones with positive colours in both (K s -[3.6]) e and ([3.6]-[8]) e and ones with positive colours in only (K s -[3.6]) e .
Fig. 16.-Separation of dusty stars from bad cross-matches and blends. Top-left panel: (Ks − [3.6]) and ([3.6] − [8]) colours for all stars in our sample with respect to our model and our colour cut for separating the two populations. Top-right panel: SED of source 5.9408750-72.0803889 (black points) which is a poorly-cross-matched source, showing: a model photosphere (short-dashed, blue lines) at the fitted luminosity and at 67% brighter than the fitted luminosity, an amorphous carbon dust model at 1500 K (solid, red line), and a metallic iron dust model at 1500 K (long-dashed, green line). Bottom-left panel: radial distribution of (solid, red line) all stars with L > 31.6 L ⊙ with Spitzer photometry, (long-dashed, green line) stars with positive colours for both (Ks − [8]) and ([3.6] − [8]), (short-dashed, blue line) stars with positive (Ks − [8]) colour but zero ([3.6] − [8]) colour. Bottom-right panel: Ks-band magnitudes of stars covered by both Salaris et al. (2007) and 2MASS. Dusty candidates below the line in the upper-left panel are shown as large, black dots, while those above the line are shown as large, blue circles; the line denotes parity between surveys. (Note: the brightest stars are not present in the Salaris data due to saturation problems.)
Fig. 17 .
17-Radial distribution of bright and faint giants in our data, compared to the sample of bright and faint dusty candidates in the works of Origlia et al. All data are normalised to 2 ′ and the number (N ) of stars within that radius is shown for each sample.
TABLE 1
1Input photometric data and their coverage of the cluster.
TABLE 2
2Adopted central wavelengths and zero points for the photometric filters used.Filter Central wavelength Zero point
(nm)
(Jy)
U
350
1659
B
442
4130
V
550
3810
i
786
2427
I C
810
2520
I J
880
2635
J
1240
1602
H
1650
1010
Ks
2160
630
[3.6]
3600
280.9
[4.5]
4500
179.7
[5.8]
5800
115.0
[8]
8000
64.13
[24]
24000
7.14
TABLE 3
3Stellar parameters as determined by SED fitting. A full electronic table is available online.Sequence RA (deg)
Dec (deg)
Temperature Luminosity
Number
(J2000)
(J2000)
(K)
(L ⊙ )
1 4.641571 -72.328035
6147
258.4
2 4.685675 -72.365643
4449
56.53
3 4.706252 -72.387843
5399
1.625
4 4.722603 -72.374671
4500
1.228
5 4.725042 -72.366471
5308
1.057
· · ·
TABLE 4
4Stars chosen for Monte-Carlo simulations. First error encompasses photometric errors, second error encompasses modelling errors.Sequence RA (deg)
Dec (deg)
Temperature 1
Luminosity 1
Coverage Notes
Number
(J2000)
(J2000)
(K)
(L ⊙ )
12515 5.9604583 -72.0725000 3916±4 ±25
982±11 ±35
B − [24] Upper RGB/AGB
9352 5.9282500 -72.1920278 4265±5 ±55
316±3 ±7.2
B − Ks
Central RGB/AGB
4929 5.7577500 -72.1565278 4567±6 ±59
184±1.4 ±16
B − [8]
Lower AGB
8937 5.9237083 -72.0840556 4329±10±65
175±3 ±11
B − [24] RGB
19514 6.0277917 -72.0754167 5283±30±80
60.0±0.7 ±1.7
B − Ks
HB (core)
28472 6.1138333 -72.0774722 5191±27±18
59.8±1.0 ±0.5
B − Ks
HB (not core)
15423 5.9881250 -71.9649722 4782±10±36
45.0±0.4 ±0.6
B − Ks
RGB clump
40122 6.5137500 -72.2503611 6250±13±56
5.05±0.06±0.18 B − [4.5] MSTO
1
Helium fraction: Y = 24% ≈ Y ⊙(Dorman et al. 1989); • α-enrichment:[α/Fe] ≈ +0.2 dex (Dartmouth isochrones only);• Metallicity: [Z/H] = [Fe/H] = -0.7;
•
It could, however, be explained if extra mixing on the RGB occurs around first dredgeup, and if such mixing alters the surface chemistry (e.g. D'Antona & Ventura 2007; Karakas et al. 2010, and references therein). This effect is shown in more detail in §3.2.5. Note that the Padova isochrones, which qualitatively fit the upper giant branches and MSTO, are also for [α/Fe] = 0. 3.2. Cluster parameters 3.2.1. Metallicity and abundances
TABLE 5
5Sources showing IR excess in both
Origlia et al. dusty stars, [8]<8 (N=22) Origlia et al. dusty stars, 8<[8]<10.874 (N=66)0
0.2
0.4
0.6
0.8
1
1.2
1.4
0
50
100
150
200
250
300
Normalised cumulative source count
Radius (arcsec)
Observed stars with 8<[8]<10.874 (N=241)
Observed stars with [8]<8 (N=37)
TABLE 6
6
We refer to the Johnson I band as I J throughout to avoid confusion with the other I-band data in this work.
http://stellar.dartmouth.edu/ models/ 11 http://astro.df.unipi.it/SAA/PEL/Z0.html
N = 44, n = 21, m = 22, k = 16 14 N = 254, n = 29, m = 65, k = 10 15 N = 24, n = 15, m = 10, k = 9 16 N = 131, n = 17, m = 47, k = 7
Clement (1997); andLebzelter & Wood (2005). Other names from:Lee -Lee (1977Our two studies use the same Spitzer data, but differ in the software used to determine photometric magnitudes. Given the strong artifacts present in the original data (Figure 15), we would advise care in any claim of excess among fainter stars in regions affected by these artifacts (including the cluster core). Higher-resolution data, such as those ofMomany et al. (in prep.), are needed before the claim of dust production at low luminosities can be conclusively proven or refuted, and we await their results with anticipation.We instead find that dusty mass loss begins in 47 Tuc at ∼1000 L ⊙ and becomes commonplace by ∼2000 L ⊙ . Between 1000 and 2000 L ⊙ , it is unclear why dust production is not occurring around all stars. In this region, nine out of 56 show possible IR excess, while three (x03, V18 and V13) show strong mass loss. This could represent episodic dust production on either short or long timescales, potentially related to pulsation. Alternatively, it could represent a difference in dust production between RGB and AGB stars (due to, say, gravity or dredge up), which we cannot separate at these luminosities. The luminosity at which dust production first occurs in 47 Tuc is therefore very similar to those we find in ω Cen (Paper I).CONCLUSIONSIn this work, we have used spectral energy distributions to establish physical parameters for stars in the globular cluster 47 Tuc. We have used these to investigate the basic parameters of the cluster as a whole, and the later stages of the evolution of its stars. We summarise our conclusions as follows:• Simple isochrone fits to the cluster's Hertzsprung-Russell diagram corroborate the established distance and age of the cluster. We find d = 4611 +213 −200 pc and t = 12 ± 1 Gyr.• HB models show that mass loss on the RGB is unlikely to greatly exceed that in the similarly-aged but much more metal-poor cluster ω Cen, implying that RGB mass loss does not vary with metallicity. We do not rule out any correlation of dust production with metallicity: our results apply only to integrated mass loss.• We find that apparent IR excess in stars below L = 1000 L ⊙ is almost certainly due to artifacts in the original Spitzer maps and under-reported photometric errors caused by blending, a finding contrary to the results ofOriglia et al. (2007)andOriglia et al. (2010).• Some stars above L = 1000 L ⊙ show IR excess consistent with dust production. The fraction of dustproducing stars approaches unity above L ≈ 2000 L ⊙ .Acknowledgments: We are grateful to Livia Origlia for her co-operation in the production of this paper, for her contribution to the analysis and for sharing her original data with us. This paper uses observations made using the Spitzer Space Telescope (operated by JPL, California Institute of Technology under NASA contract 1407 and supported by NASA through JPL (contract number 1257184)); observations using AKARI, a JAXA project with the participation of ESA; and data products from the Two Microns All Sky Survey, which is a joint project of the University of Massachusetts and IPAC/CIT, funded by NASA and the NSF.
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| []
|
[
"Persistent currents in rings of ultracold fermionic atoms",
"Persistent currents in rings of ultracold fermionic atoms"
]
| [
"Yanping Cai \nDepartment of Physics and Astronomy\nDartmouth College\n6127 Wilder Laboratory03755HanoverNHUSA\n",
"Daniel G Allman \nDepartment of Physics and Astronomy\nDartmouth College\n6127 Wilder Laboratory03755HanoverNHUSA\n",
"Parth Sabharwal \nDepartment of Physics and Astronomy\nDartmouth College\n6127 Wilder Laboratory03755HanoverNHUSA\n",
"Kevin C Wright \nDepartment of Physics and Astronomy\nDartmouth College\n6127 Wilder Laboratory03755HanoverNHUSA\n"
]
| [
"Department of Physics and Astronomy\nDartmouth College\n6127 Wilder Laboratory03755HanoverNHUSA",
"Department of Physics and Astronomy\nDartmouth College\n6127 Wilder Laboratory03755HanoverNHUSA",
"Department of Physics and Astronomy\nDartmouth College\n6127 Wilder Laboratory03755HanoverNHUSA",
"Department of Physics and Astronomy\nDartmouth College\n6127 Wilder Laboratory03755HanoverNHUSA"
]
| []
| We have produced persistent currents of ultracold fermionic atoms trapped in a ring, with lifetimes greater than 10 seconds in the strongly-interacting regime. These currents remain stable well into the BCS regime at sufficiently low temperature. We drive a circulating BCS superfluid into the normal phase and back by changing the interaction strength and find that the probability for quantized superflow to reappear is remarkably insensitive to the time spent in the normal phase and the minimum interaction strength. After ruling out spontaneous current formation for our experimental conditions, we argue that the reappearance of superflow is due to weak damping of normal currents in this limit. These results establish that ultracold fermionic atoms with tunable interactions can be used to create matter-wave circuits similar to those previously created with weakly-interacting bosonic atoms. arXiv:2104.02218v2 [cond-mat.quant-gas] | 10.1103/physrevlett.128.150401 | [
"https://arxiv.org/pdf/2104.02218v2.pdf"
]
| 233,033,728 | 2104.02218 | e5459b600fb9b9b69d72d76472f1cd8ad1b56cae |
Persistent currents in rings of ultracold fermionic atoms
12 Jan 2022
Yanping Cai
Department of Physics and Astronomy
Dartmouth College
6127 Wilder Laboratory03755HanoverNHUSA
Daniel G Allman
Department of Physics and Astronomy
Dartmouth College
6127 Wilder Laboratory03755HanoverNHUSA
Parth Sabharwal
Department of Physics and Astronomy
Dartmouth College
6127 Wilder Laboratory03755HanoverNHUSA
Kevin C Wright
Department of Physics and Astronomy
Dartmouth College
6127 Wilder Laboratory03755HanoverNHUSA
Persistent currents in rings of ultracold fermionic atoms
12 Jan 2022
We have produced persistent currents of ultracold fermionic atoms trapped in a ring, with lifetimes greater than 10 seconds in the strongly-interacting regime. These currents remain stable well into the BCS regime at sufficiently low temperature. We drive a circulating BCS superfluid into the normal phase and back by changing the interaction strength and find that the probability for quantized superflow to reappear is remarkably insensitive to the time spent in the normal phase and the minimum interaction strength. After ruling out spontaneous current formation for our experimental conditions, we argue that the reappearance of superflow is due to weak damping of normal currents in this limit. These results establish that ultracold fermionic atoms with tunable interactions can be used to create matter-wave circuits similar to those previously created with weakly-interacting bosonic atoms. arXiv:2104.02218v2 [cond-mat.quant-gas]
Progress in understanding quantum fluids has often been made by considering spherical, cylindrical, toroidal, or more exotic geometries [1], and circuits built from quantum materials have many important applications including quantum computing. Quantum gases with periodic boundary conditions provide unique opportunities for exploring quantum many-body physics, especially where it is possible to bias a circuit with an external flux [2,3]. One crucial characteristic of such circuits is that they can support quantized currents that flow without being driven by an external power source. Persistent non-equilibrium currents are commonly understood to occur in superconducting [4] and superfluid [5] phases, but equilibrium persistent currents can also appear in normal conducting phases around closed paths shorter than the coherence length [6][7][8]. The current response of such circuits to external flux often conveys important information about the state of the system [9].
Previous experiments with multiply-connected ultracold gases have utilized weakly-interacting atomic Bose-Einstein condensates (BEC) in magnetic and optical traps [10][11][12][13][14][15][16][17]. Experiments on ring BECs have demonstrated the existence of metastable currents [11,13] and quantized phase slips [18,19]. Bosonic superfluid circuits have been constructed by incorporating Josephson junctions [20][21][22], and the experimental usefulness of multiply-connected quantum gases has been demonstrated by studies of collective-mode precession [23], spontaneous currents [24][25][26], quantum turbulence [14], propagation of shock waves [27], the stability of supersonic superfluid flows [28,29], and more [30].
Fermionic quantum gases provide access to a rich variety of physics distinctly different from that of purely bosonic systems. Furthermore, magnetic fields can often be used to continuously tune interactions between fermionic atoms from the weakly attractive limit where BCS pairing can occur to the weakly repulsive limit where the atoms can form a BEC of weakly-bound molecules. Fermionic superfluidity has been extensively studied throughout this BEC-BCS crossover, including experimental observations of an interaction-dependent critical velocity in 3D [31], and very recently in 2D [32]. Josephson junctions and quantum point contacts have also been realized in singly-connected fermionic quantum gases [33,34].
In this work we report the first creation of a multiplyconnected superfluid "circuit" in an ultracold Fermi gas, and show that it is possible to reliably create and detect quantized currents in this system. We demonstrate that currents can survive well into the fragile BCS regime and examine the decay and revival of currents after quenches to the normal phase in this limit, establishing a foundation for other proposed experiments involving quantum gases in rings and ring lattices [2,3,[35][36][37][38][39][40][41].
In these experiments, we use a quantum degenerate gas of 6 Li atoms in an equal mixture of the lowest-energy spin states (|m J = −1/2, m I = 1 and |m J = −1/2, m I = 0 ). Interactions between atoms in these two states are attractive (repulsive) at magnetic fields above (below) a broad (> 10 mT) Feshbach resonance at 83.2 mT. To create a persistent current, we must have a continuous (pair) superfluid around a closed path, which requires a trap with a smooth ring-shaped potential minimum and cooling the system below a critical temperature which depends on the interaction strength and density. We created an optical ring trap with two red-detuned laser beams, a horizontal "sheet" beam (λ = 1068 nm, horizontal waist 290 µm, vertical waist 7 µm) and a vertical ring-pattern beam (λ = 780 nm, average radius 12.0(1) µm, radial 1/e 2 half-width 2.2(1) µm (see Supplemental Materials).
The next critical requirement is to achieve and maintain low enough temperatures to study supercurrents over a wide range of interaction strengths. We loaded the atoms into the ring trap with the sheet beam initially at high power (4 W), and performed final evaporative cooling with the magnetic field near resonance (82.0 mT) by decreasing the sheet beam power to 40 mW while holding the ring beam power at 0.85 mW. Evaporation occurred as molecules fell out of the bottom of the ring FIG. 1. Cross sections of an idealized model of the potential experienced by a 6 Li atom in our "ring-dimple" optical trap in its final configuration (the potential is twice as deep for molecules). (a) Black line is the potential along a horizontal line through the center of symmetry, transverse to the direction of propagation of the "sheet" beam. The radial trap frequency for atoms away from the ring minimum is 37(5) s −1 . b) Vertical cross section of the trap potential at r = 0 (dashed line, blue online) and r = 12.0 µm (dotted line, red online), as indicated by corresponding vertical lines in (a). The vertical trap frequency is 1.5(1) × 10 3 s −1 for atoms near the ring potential minimum, and 1.4(1) × 10 3 s −1 away from the ring beam.The plot vertical range is from Utrap = 0 at the ring minimum to the "trap-off" potential in the midplane of the ring (1.32 µK).
region where the potential barrier was lowest; the gravitational gradient reduced the final evaporation depth to an estimated k B · 0.80(5) µK as shown in Fig. 1(b).
After evaporation there were 1.0(1) × 10 4 atoms in each spin state, paired into weakly-bound molecules with strong repulsive interactions. The chemical potential (µ) was high enough that most of the molecules were not localized to the ring and formed a wide, thin disk in the radially weak, vertically strong harmonic potential of the sheet beam (ν r = 37(5) s −1 , ν z = 1.4(1) × 10 3 s −1 ). The fraction of the population in this "halo" increased if we subsequently tuned interactions to the weakly attractive (BCS) limit where µ ≈ E F (Fermi energy). From a model of our trap we calculated that in this limit E F = h · 16(1) × 10 3 s −1 = k B · 0.77(6) µK (see Supplemental Materials). The radial trap frequency for atoms (and molecules) near the ring potential minimum was ν r = 4.0(2) × 10 3 s −1 .
The low-density halo was hardly visible in absorption images, but easily observed in radial plots of the column density after azimuthal averaging. Fig. 2 shows averaged results from 10 runs where the field was ramped from 82.0 mT to 107.4 mT before imaging the atoms in the ring. Fig. 2(a) has been cropped to show ring density variations (10% peak-to-peak) in more detail. Fig. 2(b) shows the column density, n 2D (r), obtained from the full-frame image by averaging data in radial bins 1 µm wide. Fitting the density profile for r > 20 µm with a model of an ideal Fermi gas in our trap potential indicated that T = 25 (5) nK. Because T < hν z /k B = 72 nK, we accounted for the crossover from 3D to quasi-2D in the outer regions of the halo (see Supplemental Materials). The best fit of this Fig. 1, with 1×10 4 atoms in each spin state. A magnetic field of 107.4 mT has been used to tune the scattering length to −179 nm. The density peak is at r = 12.0(2) µm. This is a 50 × 50 µm region cropped from a 215 × 215 µm image. (b) Radial column density obtained by azimuthal averaging over the full field of view. The plot is shown vertically rescaled ×10 for r > 20 µm to emphasize the broad halo extending to r = 100 µm. The black dotted line is the expected density profile (×10) for an ideal Fermi gas in our trap at T = 25 nK. In region I the system is 3D degenerate, II is quasi-2D degenerate, and III is quasi-2D thermal. Grey band: 2σ variation of n2D(r) when calculated separately for each image in the set.
model to the data for r > 20 µm is shown in Fig. 2(b) as a dotted black line.
When we ramped from 82.0 mT to 68.0 mT (BEC regime) we found that n 2D for r > 20 µm had the Gaussian profile expected for a thermal gas of molecules at 90(3) nK. Temperature changes are expected for isentropic interaction ramps because the temperature dependence of the entropy is different in the BEC and BCS limits [42]. The minimum temperature we observed in the BCS regime was likely limited by heating due to hole creation by collisions with background gas molecules [43]. Fermionic systems are especially sensitive to heating at low temperatures. Retaining a large part of the population in a low-density halo increases the average heat capacity per particle and reduces the heating rate, which was important for the experiments described below.
Our general procedure for creating persistent currents and studying their stability in the BCS regime was the following: We prepared a strongly-interacting molecular BEC at B = 82.0 mT as described above, then initialized the current state by stirring with a blue-detuned laser beam that created a localized repulsive potential. We then changed the interaction strength adiabatically by ramping the magnetic field up to the BCS regime, then ramped back to 82.0 mT. Finally, we ramped to the BEC regime and used a self-interference technique to determine the final current state of the ring.
It was challenging to adapt the supercurrent detection procedures developed with ring BECs [19,22,23,[44][45][46] to rings of strongly interacting light fermionic atoms. Lower condensate fraction in fermionic systems, rapid expansion due to the high chemical potential, and pair breaking in the BCS limit all reduce coherence and the signal to noise ratio in images. These procedures were most effective in the BEC limit after lowering the interaction energy as much as possible. While the field was still at 82.0 mT, we relaxed the radial confinement by lowering the ring beam power to 5% of its initial value over 100 ms, changing the profile of the cloud to that of Fig. 3(a). This transformed a current with winding number into singly-charged vortices in the central region that were too small to detect optically. We then swept the magnetic field from 82.0 mT to 68.3 mT in 20 ms, reducing the scattering length by 96%. (Ramping to even lower field caused the three-body loss rate to become too high during the detection procedure.) Next, we turned off the trap and allowed the atoms to evolve for 5.5 ms in a magnetic field with a weak radial curvature. This caused radial focusing, which increased the signalto-noise ratio in the absorption image taken at the end, using the |m J = −1/2, m I = 1 → |m J = −3/2, m I = 1 transition. Fig. 3 shows typical evolution of the density profile for a system prepared in an | | = 1 current state. The single hole indicates the presence of a single vortex [47].
In this work we initialized the current state by stirring [18,48], but note that phase imprinting is also possible [11,19,49,50] and was demonstrated with fermions by another group while this paper was in review [51]. In our system spontaneous currents often appeared during initial formation of the molecular BEC, and stirring allowed deterministic preparation of a selected current state even when the initial current state was uncertain, which is not possible with phase imprinting. We created a repulsive stirring potential with a steerable blue-detuned beam (λ = 635 nm, radius 6(1) µm). To initialize the system in a zero-current state, we kept the beam stationary at one point on the ring, increased the laser power linearly over 100 ms until the peak of the repulsive potential was around 1.5µ, held for 100 ms, then ramped the beam off in 100 ms. After this procedure the probability of detecting a non-zero current was 0.00 +0.02 −0.00 (Uncertainties are 1σ Bayesian binomial confidence intervals [52]).
To create a current we accelerated the stirring beam around the ring at 100 rad/s 2 up to a maximum angular velocity that we held constant for 300 ms, then ramped the beam power off linearly in the final 100 ms. The angular frequency of a quantized current of pairs with winding number in our ring was Ω 0 ≡ /(m pair R 2 ) = 2π ·5.83(2) rad·s −1 . The probability of creating an = 1 current (P =1 ) became significant for stirring frequencies near 0.5 Ω 0 , increasing to ≈1 above 0.7 Ω 0 . We have created higher current states by stirring at higher angular velocities, but focus here on creation and decay of the = 1 state. In our system T /T c < 0.5 over a significant range of interaction strengths near resonance, and under these conditions persistent currents survived for up to 10 seconds, limited by losses from background gas collisions (1/e lifetime 12 s) or three-body collisions. (When losses reduced the total atom number below 10 4 we could no longer distinguish vortices from thermal density fluctuations using the detection procedure described above.) When ramping into the BCS regime T c falls exponentially, and thermally activated phase slips to lower energy states should occur as T /T c → 1 at the weakest point of the ring [53][54][55]. We estimated the local Fermi temperature at that point to be T F = 0.69(1) µK and the inverse Fermi wavenumber was 1/k F = 0.24(1) µm. Neglecting small corrections due to trap confinement [56], the expected critical temperature for the superfluid transition is T c ≈ 0.277 T F e −π/2k F |a| [57]. Given our measurement that T = 25(5) nK in the BCS limit, the superfluid density should vanish at the weak point of the ring when B = 106(3) mT (−1/k F a = 1.3(1)).
To characterize the decay of the current around this interaction strength, we prepared the system in the = 1 current state at B i = 82.0 mT, swept the magnetic field in 100 ms to a value B max in the BCS regime, then swept back to B i in 100 ms before measuring the final current state. This sweep rate is slow enough to be adiabatic, causing no detectable excitation of collective modes. We found that the current did not decay for B max < 98.0 mT (−1/k F a < 1). The data in Figure 4 show the decreasing probability of detecting an = 1 current (P =1 ) for B max from 98.0 mT up to 107.8 mT (our technical limit). The decrease of P =1 over the range from 98.0 mT to 105 mT is consistent with expectations that the rate of decay via thermally activated phase slips increases as T /T c → 1 [53][54][55]58]. The velocity of the flow (0.44 mm/s) should have a negligible effect on decay of the current since the kinetic energy per pair is less than 0.01 k B T .
Because our detection procedure requires ramping back to the BEC limit, interpretation of the data when any part of the ring is quenched to the normal phase requires consideration of spontaneous current formation, the damping of the normal current, and the effect of thermal phase fluctuations. Spontaneous currents can appear during sufficiently rapid merging of independent superfluid regions [24,59]. This can occur if there is significant azimuthal variation in the ring potential minimum [26], or via the Kibble-Zurek mechanism during a fast quench to the superfluid phase [25,60]. To determine whether spontaneous current formation was significant for our experimental conditions we prepared the atoms in the = 0 state and measured the final current state after similar ramps to the BCS regime. The probability of observing = 0 was 0.04 +0.08 −0.01 , indicating that the non-zero probabilities in Fig. 4 can be attributed to initializing the system in the = 1 current state.
When a normal fluid circulating around a ring is driven into a superfluid phase, it will most likely form in the quantized current state that minimizes the free energy [61,62]. When phase fluctuations are small the distribution of final current states is sharply peaked, with the probability of one state near unity. Large phase fluctuations broaden the distribution and make the result non-deterministic. When B max is high enough that the ring is broken by a region in the normal phase, damping of the current (and excitations in the remaining superfluid) should cause P =1 to fall to zero eventually. For linear ramps up to B max and immediately back down, P =1 did not fall to zero and was nearly the same for the highest three values of B max , with an average value of 0.35 +0.07 −0.06 . To obtain more information about the time scale for damping we repeated the procedure, adding a hold time of either 0.1 or 0.2 s at the highest values of B max (see offset data in Fig. 4). Again, P =1 did not fall to zero, and the dependence on B max was weak (0.04 ± 0.1 /mT for the three highest values of B max , similar for 0.1 s and 0.2 s hold). For a 0.1 s hold, the average value of P =1 for the three highest values of B max was P =1 = 0.20 +0.09 −0.05 . For 0.2 s it was 0.24 +0.09 −0.06 . Fitting to these average values for each hold time, the estimated decay time was 0.5 s, with a 1-σ lower bound of 0.25 s. This is longer than our ramp times, and much longer than the few-millisecond timescale for sound to propagate around the ring. We did not systematically investigate longer hold times because heating was non-negligible and we could no longer treat the temperature as nearly constant. The most plausible explanation for the data at the right of Fig. 4 is that the average total current remained significantly greater than zero even when part of the ring was driven normal, and thermal phase fluctuations and/or long-wavelength excitations in the superfluid broadened the distribution of final current states after the superfluid ring reconnected. It should be possible to study these current decay and reconnection dynamics in detail in future experiments using interferometric techniques in a "target" or doublering trap configuration [22].
In conclusion, we have studied persistent currents in a fermionic matter-wave "circuit" across a range of interaction strengths. We initialized the system in a selected current state and detected single-quantum changes in the current state. We maintained low enough temperatures for supercurrents to survive well into the BCS regime and found that the potential was smooth enough for normal currents to be relatively long-lived. These results also provide a framework enabling future studies of transport and non-equilibrium phenomena in rings of ultracold fermionic atoms. We observed spontaneous currents for faster interaction ramps, indicating an opportunity to study the Kibble-Zurek mechanism with fermions [63] in the annular geometry originally proposed by Zurek [64]. In a spin-imbalanced ring of fermionic atoms it may be possible to create π-Josephson junctions [65] and search for evidence of unconventional spin-polarized superfluid phases [36,66]. Finally, the tight transverse confinement achieved in these experiments could be increased to realize quasi-2D and 1D rings of fermions with tunable interactions, where non-Fermi-liquid behavior is expected and parity effects can be significant [3].
We thank J. Evans, S. Khatry, L. Bezerra, and A. Woronecki for key technical contributions to the experimental apparatus and recognize D. Adams, C. Grant, and D. Collins for critical engineering support. We also thank R. Onofrio for insightful discussions about ultracold fermionic systems. This work was supported by the NSF (Grant No. PHY-1707557). We begin our experimental procedure with successive laser cooling of 6 Li atoms in a 2D magneto-optical trap (MOT) and 3D MOT, followed by D 1 grey molasses cooling and optical pumping of the atoms into the F = 1/2 hyperfine ground states. We capture 10 7 of these atoms in a 40 Watt 1064 nm crossed-beam optical dipole trap and perform forced evaporative cooling in this trap at 33 mT to obtain a balanced spin mixture of 10 6 atoms in the |m J = −1/2, m I = 1 and |m J = −1/2, m I = 0 states at a temperature of 30 µK. We then use a 15 Watt 1064 nm movable optical trap to transport the atoms from the 3DMOT chamber to an octagonal glass cell. We use a matched pair of objectives above and below the cell (f = 30 mm, NA=0.3) for high-resolution projection and imaging in the vertical direction. Water-cooled magnet coils surrounding the cell generate a nearly uniform magnetic field of more than 0.1 T, which we use for tuning the atomic interactions. The vacuum-limited lifetime of atoms in the cell during these experiments was 12(1) seconds.
After transporting atoms to the cell we transfer them into a ring-dimple trap formed by the intersection of a horizontally propagating "sheet" beam and a vertically propagating ring beam. The sheet beam is a λ=1068 nm (red-detuned) asymmetric Gaussian beam with a horizontal (vertical) waist of 290 µm (7 µm). The potential near the focus of this beam is radially symmetric in the horizontal direction, with an asymmetry of less than 3% at a radius of 100 µm. At the maximum beam power of 4 Watts the trap depth (for molecules) is 150 µK, the radial trap frequency is 140 Hz, and the vertical trap frequency is 5.8 kHz. All reported trap frequencies were determined by observations of parametric resonance when modulating the power in the trapping beams. FIG. 1. a) Contour plot of an idealized model of the trap potential in the vertical direction through its center. Potential is shown in µK relative to the lowest point of the ring potential. b) Intensity of the ring-pattern beam as seen when re-imaged onto a camera after it passes through the glass cell. The diffraction fringes are a result of partial apodization of the beam with an adjustable, movable iris that we use to help smooth and control the width of the ring.
For these experiments, the vertically propagating ring beam was a λ=780 nm (red-detuned) laser field with a total power of 0.85 mW. We generated the ring pattern by passing a spatially filtered nearly-Gaussian beam through a positive axicon with a 5 • cone angle, followed by an achromatic doublet which brings the beam to a narrow annular focus. After this focus we directed the beam through another 5 • positive axicon to make the propagating pattern telecentric before re-imaging onto the atoms using a 1 meter focal length tube lens and the objective lens positioned * [email protected] arXiv:2104.02218v2 [cond-mat.quant-gas] 12 Jan 2022 below the cell. We also used an adjustable iris positioned just after the ring focus behind the second axicon to partially apodize the beam (apodization only at the outer radius). We did this to improve the smoothness of the projected beam and provide finer control over its width and azimuthal symmetry. The incomplete apodization results in diffraction of a small amount of light into the region inside the ring, as shown in Fig. 1(b). The only notable effect is a slight increase in the atomic density at the location of the central Poisson spot (visible in Fig. 2), and we do not include these fringes in the model of the trap described below.
We determined the average radius of the ring minimum to be 12.0(1) µm by fitting a cylindrically symmetric 2D ring profile to absorption images (28 averaged) of the atoms in the ring-dimple trap. The dominant contribution to the uncertainty in this measurement is from the magnification of the imaging system. For the data reported in this work the radial trapping frequency of the ring potential was ω r = 2π · 4.0(2) × 10 3 s −1 . Given the power in the beam, the measured radial trapping frequency and the assumption that the radial profile is nearly Gaussian, the calculated 1/e 2 half-width of the ring is 2.2(1) µm. The final stage evaporative cooling is performed at 82.0 mT by decreasing the sheet beam power to 40 mW with the ring beam power held constant at 0.85 mW. After evaporation the radial trap frequency for atoms in the sheet is 37(5) Hz and the vertical trap frequency inside (outside) the ring region is is 1.5(1) kHz (1.4(1) kHz). (e) Azimuthally averaged column density from (a-d) out to larger radii, showing the redistribution of atoms from ring to halo as the interactions are ramped from the BEC to BCS limit. Data for r > 20 is shown ×10 to emphasize the change in the halo population. The data is shown here without correction for blurring due to resolution limits. After correction, the actual peak column densities are higher (See section IV).
In the deep BCS limit the chemical potential, µ, is nearly equal to the Fermi energy, E F , but it decreases as the interactions are ramped across the Feshbach resonance. Fig. 2 shows the changing spatial distribution of atoms in the trap for four different values of the magnetic bias field spanning the BEC and BCS limits. In the weakly-interacting molecular BEC (mBEC) limit at 68.3 mT, shown in Fig. 2(a), around 30% of the atoms are in the spatial region of the ring-dimple. The remainder form a broad halo more readily seen in the plot of azimuthally averaged column density (blue line) shown in Fig. 2(e). In the (weakly-attractive) BCS limit, shown in Fig. 2(d), the increase of the chemical potential to the Fermi energy (E F ) reduces the fraction of atoms in the ring-dimple spatial region to less than 20%, with the balance moving to the halo as shown by the red line in Fig. 2(e). The dash-dotted line shows EF for a harmonic approximation to the ring potential minimum, for comparison.
To calculate the Fermi energy (E F ) of the system in the BCS limit we computed the total (spin up and down) semiclassical density of states in 3D,
g 3D (E) = 2 × 4πm (2π ) 3 V (r)≤E d 3 r 2m(E − V (r)),(1)
where V (r) is a model of the trap incorporating the potential of the sheet beam, the ring beam, and gravity. (Quasi-2D confinement at the edge of the halo has a negligible effect on E F , but not on temperature measurements described below.) From the density of states we then used the defining relation N tot = E F 0 g 3D (E)dE to numerically compute the Fermi energy E F (N tot ). We modelled the ring beam as having a ring-shaped Gaussian intensity profile in its plane of best focus, and numerically propagated it through focus using the angular spectrum method to obtain its full 3D profile. The sheet beam was taken to be an asymmetric Gaussian beam, whose propagation has a known analytic form. Furthermore, to make computations more tractable, we assumed azimuthal symmetry of the potential, since the sheet beam ellipticity is < 3 % in the region of interest. For 1 × 10 4 atoms in each spin state, the estimated Fermi temperature (T F ) is 0.77(6) µK. Uncertainty in E F is due in most part to our uncertainty in the ring width. This is because the dimple capacity depends quite sensitively on the ring width, which in turn affects the estimate of E F . Fig. 3 shows a plot of E F (N ) for the trap configuration used in our experiment. The "ring-dimple" nature of the trap is evident in the rapid increase of E F (N ) up to the dimple capacity of about 1500 atoms, followed by a much slower increase in E F as atoms begin to populate states that spread out into the sheet potential. Here we emphasize that the population of atoms in the dimple, defined via energy considerations, is smaller than the apparent population in the ring-shaped region of higher density seen in absorption images. The red dash-dotted line in Fig. 3 is E F (N ) for a ring of the same radius with idealized harmonic confinement in the transverse direction, given by the expression
E F Ω 0 = 15N tot 16 2/5 ω Ω 0 4/5(2)
whereω = √ ω z ω r is the the geometric mean of the trapping frequencies, and the quantized rotation frequency is
Ω 0 ≡ /(2mR 2 )
. This approximation is reasonably valid for many previous experiments involving ring BECs, but is not appropriate for our experimental conditions.
III. TEMPERATURE MEASUREMENTS
The temperature can be determined from the density profile of the halo and knowledge of the trapping potential for r > 20 µm. In this region, we can approximate the potential by V (r, z) = V 0,s + m 2 (ω 2 s r 2 + ω 2 z z 2 ) where V 0,s is the sheet-only potential at the origin, ω s is the radial trap frequency of the sheet, and ω z is the vertical trap frequency in the sheet. It is important to note that for deeply degenerate Fermi gases, absolute temperature enters into the fit of the data only in the far, dilute wings of the density distribution where the system is potentially quasi-2D. We allow for mixed dimensionality in our description of the density by treating the vertical energies quantum mechanically and the radial energies semi-classically. This approach is similar to those used to model quantum wells in solid state systems, except that our tight vertical confinement is harmonic, not hard-walled. In this way, we may write a hybrid description of the density of states
g j (E) = s (2π ) 2 d 2 rd 2 p δ E − |p| 2 2m − ω z j − V r (r)(3)
which represents the density of available states in the j th axial harmonic oscillator level (j = 0, 1, ...), for a system with s spin degrees of freedom. We have defined V r (r) = V 0,s + ω z /2 + mω 2 s r 2 /2, explicitly accounting for the finite zero-point energy of the axial motion. Integrating over momenta and summing over j, we identify the local density of states
g(r; E) = s m 2π 2 (r) Θ( (r))(4)
where (r) ≡ E−Vr(r) ωz , x is the ceiling function, and Θ is the Heaviside step function. The column density n 2D (r) is found by integrating the Fermi-Dirac-weighted local density of states over E, giving
n 2D (r) = s λ 2 T ∞ 0 dx x/η e x−μ(r) + 1 = s λ 2 T ∞ j=0 F 0 (μ(r) − jη)(5)
where we have further defined λ 2
T ≡ 2π 2 mk B T , η ≡ ωz k B T ,μ(r) ≡ µ−V (r) k B T , and F 0 (x) = log(1 + e x )
, which is a special case of a Fermi-Dirac integral, F ν (x), of order ν. The chemical potential µ is measured above V 0,s + ω z /2. The integral expression looks remarkably similar to the order 1 Fermi-Dirac integral, except for the presence of the ceiling function in the integrand, which accounts for the discrete axial energy levels. This discreteness is blurred out if either η or η/μ(r) is small, which corresponds to the 3D limit. In this case, we can replace x/η with x/η, and the resulting expression gives the proper integrated 3D column density n 2D (r) ≈ s ηλ 2 T F 1 (μ(r)) ; η 1 or η/μ(r) 1
Conversely, if η 1 and η/μ(r) 1, we approach the 2D limit, and we may replace x/η with 1, and the resulting column density gives the proper 2D density n 2D (r) ≈ s λ 2 T F 0 (μ(r)) ; η 1 and η/μ(r) 1
We may writeμ(r;μ 0 , r F ) =μ 0 (1 − r 2 /r 2 F ), with r 2 F ≡ 2µ mω 2 s andμ 0 ≡ µ k B T , and define our column density fit function with fit parameters α = (r F , n max ,μ 0 )
n fit (r, α) = n max C ∞ 0 dx x/η e x−μ(r;μ0,r F ) + 1 = n max C ∞ j=0 F 0 (μ(r;μ 0 , r F ) − jη)(8)
where C is the peak value of the integral or sum, i.e. at r = 0, and depends on the other fit parameters. n max therefore represents the peak density. One may truncate the sum whenever the argument becomes large and negative, which forμ, η 10 requires perhaps 20 terms, (F ν (x) ∼ e x for x 0) making it useful for fitting. Finally, we may eliminate η as a fit parameter as η = ωz k B T = ωzμ0 µ = 2 ωzμ0 mω 2 s r 2 F as long asμ 0 > 0. We pay a modest price for this parameter elimination by introducing uncertainty in α via the vertical and radial sheet trap frequencies ω z and ω s . The largest source of temperature uncertainty is due to uncertainty in ω s , which by itself introduces several nK uncertainty on our reported temperature in the BCS limit.
The temperature of the gas in the molecular BEC limit can be determined in by a similar fit to the halo radial density profile. At 68.3 mT the profile for r > 20 µm is well approximated by a Gaussian, which is expected for a thermal gas in a harmonic potential. The radius of the fit is 68(1) µm, indicating a temperature of 90(3) nK. Since the gas is non-degenerate and the estimated temperature is just above the level spacing in the vertical direction ω z /k B = 70 nK we checked this estimate against one obtained by measuring the rate of ballistic expansion on the halo in the vertical direction. Fitting a series of images taken from a horizontal view, we obtained an estimate of 70 (20) nK, which is in reasonable agreement with that obtained from the Gaussian fit to the radial density profile.
IV. 2D AND 3D NUMBER DENSITY
We obtained and cross-checked two estimates of the peak 3D density of atoms in the ring in the BCS limit. The first estimate was to make the local density approximation and relate the global Fermi energy E F calculated from the total atom number and trap potential (see above) to the peak 3D density n 3D via the T = 0 ideal (scattering length a = 0) fermion equation of state E F = ( 2 /2m)(3π 2 n 3D ) 2/3 , which is accurate to O((T /T F ) 2 ). The Fermi temperature of 0.77 µK obtained as described above corresponds to a local density of 2.8 atoms/µm 3 in the BCS limit. This model also predicts the 3D distribution of atoms in the trap, n 3D (r) = 1 3π 2 2m 2 3/2 (E F − V (r)) 3/2 . Integrating this along the (vertical) imaging direction gives an estimate of the 2D column density as it would appear with perfect imaging resolution. Convolving this calculated 2D density profile with a model of our imaging point spread function (a 1.7 µm airy disk) gives a 2D density profile with a peak value and ring width that are in good agreement with those obtained from in-situ absorption images of atoms taken in the BCS regime (at 107.8 mT).
The variation in density around the ring minimum is large enough to impact estimates of the lowest value of the magnetic field for which T /T c = 1 at some point around the ring. To quantify this we sampled the column density in the absorption images at 200 angular positions around the minimum of the ring potential and applied a multiplicative correction factor to account for the effects of finite-resolution blurring described above. After that correction we found that the 3D density at the "weak point" of the ring was 85% of the average 3D density at the ring minimum, and the local Fermi energy was 90% of the average value around the ring. We used this estimate to determine the critical interaction strength and magnetic field for which T /T c = 1 at that location around the ring. The uncertainty in those calculated values is dominated by uncertainty in the temperature, however, not by the uncertainty in this estimate of the local Fermi energy.
image (10 averaged) of an equal spin mixture of 6 Li atoms in the trap potential of
FIG. 3 .
3Evolution of a molecular BEC during the last part of the procedure for measuring the current state of the ring. (a) Absorption image showing the vertical column density after relaxing the ring confinement and sweeping the magnetic field from 82.0 to 68.3 mT to lower the interaction energy. (b)-(d) Evolution of the density profile after the optical trap is shut off, for 1.5, 3.5 and 5.5 ms time-of-flight. Radial magnetic lensing improves the signal-to-noise ratio in detecting the vortex core associated with the persistent current. Each image is from a separate realization of the experiment.
FIG. 4 .
4Probability of detecting an = 1 current after preparing the superfluid ring in that state near resonance (82 mT) then ramping the interactions into the BCS regime and back. The horizontal axis is the maximum magnetic field (Bmax) used in the ramp (lower scale) and the interaction parameter −1/k F a (upper scale) where a is the scattering length and 2π/k F = 1.49 µm is the local Fermi wavelength at the "weak point" of the ring. Black squares represent 16 runs averaged, where B was ramped up and down in 0.2 ms with no hold time at Bmax. Triangles are data (10 runs each point) obtained when holding at Bmax for 0.1 s (upright, blue online) and 0.2 s (inverted, red online), and are horizontally offset (from squares) for clarity. Uncertainties are 1σ Bayesian binomial confidence intervals[52].
FIG. 2 .
2(a)-(d) Absorption images of 1.0(1)×10 4 6 Li atoms per spin state in the ring-dimple trap showing changes in the column density distribution when the interactions are tuned from the BEC regime to the BCS regime using a Feshbach resonance at 83.2 mT. The magnetic field for each image is: (a) B = 68.3 mT (b) B = 85.7 mT (c) B = 98.0 mT (d) B = 107.8 mT. Each image is an average of 10 experimental runs. The field of view is a 63 µm × 63 µm region of a larger image.
FIG. 3 .
3Calculated Fermi energy EF versus total atom number Ntot for the trap parameters used in our experiment obtained using the model density of states, equation(1). The circle marks the point where Ntot exceeds the capacity of the ring dimple, and the square is the point corresponding to the Fermi energy for the number of trapped atoms in our experiment (Ntot, EF ).
Supplemental Material: Persistent currents in rings of ultracold fermionic atoms: Yanping Cai, Daniel G. Allman, Parth Sabharwal, and Kevin C. Wright * Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover NH 03766, USA I. PREPARING THE DEGENERATE FERMI GAS OF 6 LI
. * Kevin, Wright@dartmouth, * [email protected]
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| []
|
[
"Cryogenic Characerization and Modeling of Standard CMOS down to Liquid Helium Temperature for Quantum Computing Cryogenic Characerization and Modeling of Standard CMOS down to Liquid Helium Temperature for Quantum Computing2",
"Cryogenic Characerization and Modeling of Standard CMOS down to Liquid Helium Temperature for Quantum Computing Cryogenic Characerization and Modeling of Standard CMOS down to Liquid Helium Temperature for Quantum Computing2"
]
| [
"Zhen Li \nKey Laboratory of Quantum Information\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n",
"Chao Luo \nKey Laboratory of Quantum Information\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n",
"Tengteng Lu ",
"Jun Xu \nKey Laboratory of Quantum Information\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n",
"Weicheng Kong \nDepartment of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n\nOrigin Quantum Computing Company Limited\n230088HefeiAnhuiChina\n",
"Guoping Guo [email protected] \nKey Laboratory of Quantum Information\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina\n"
]
| [
"Key Laboratory of Quantum Information\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina",
"Department of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina",
"Key Laboratory of Quantum Information\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina",
"Department of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina",
"Key Laboratory of Quantum Information\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina",
"Department of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina",
"Department of Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina",
"Origin Quantum Computing Company Limited\n230088HefeiAnhuiChina",
"Key Laboratory of Quantum Information\nUniversity of Science and Technology of China\n230026HefeiAnhuiChina"
]
| []
| Cryogenic characterization and modeling of 0.18µm CMOS technology (1.8V and 5V) are presented in this paper. Several PMOS and NMOS transistors with different width to length ratios(W/L) were extensively characterized under various bias conditions at temperatures ranging from 300K down to 4.2K. We extracted their fundamental physical parameters and developed a compact model based on BSIM3V3. In addition to their I-V characteristics, threshold voltage(V th ) values, on/off current ratio, transconductance of the MOS transistors, and resistors on chips are measured at temperatures from 300K down to 4.2K. A simple subcircuit was built to correct the kink effect. This work provides experimental evidence for implementation of cryogenic CMOS technology, a valid industrial tape-out process model, and promotes the application of integrated circuits in cryogenic environments, including quantum measurement and control systems for quantum chips at very low temperatures. | null | [
"https://arxiv.org/pdf/1811.11497v3.pdf"
]
| 119,076,835 | 1811.11497 | f83700b6f644f861c39b7fe12a5c608f97c810f3 |
Cryogenic Characerization and Modeling of Standard CMOS down to Liquid Helium Temperature for Quantum Computing Cryogenic Characerization and Modeling of Standard CMOS down to Liquid Helium Temperature for Quantum Computing2
17 Jan 2019
Zhen Li
Key Laboratory of Quantum Information
University of Science and Technology of China
230026HefeiAnhuiChina
Department of Physics
University of Science and Technology of China
230026HefeiAnhuiChina
Chao Luo
Key Laboratory of Quantum Information
University of Science and Technology of China
230026HefeiAnhuiChina
Department of Physics
University of Science and Technology of China
230026HefeiAnhuiChina
Tengteng Lu
Jun Xu
Key Laboratory of Quantum Information
University of Science and Technology of China
230026HefeiAnhuiChina
Department of Physics
University of Science and Technology of China
230026HefeiAnhuiChina
Weicheng Kong
Department of Physics
University of Science and Technology of China
230026HefeiAnhuiChina
Origin Quantum Computing Company Limited
230088HefeiAnhuiChina
Guoping Guo [email protected]
Key Laboratory of Quantum Information
University of Science and Technology of China
230026HefeiAnhuiChina
Cryogenic Characerization and Modeling of Standard CMOS down to Liquid Helium Temperature for Quantum Computing Cryogenic Characerization and Modeling of Standard CMOS down to Liquid Helium Temperature for Quantum Computing2
17 Jan 2019Cryogenic electronicsMOSFETscharacterizationmodelingthreshold voltagekink effectliquid helium temperature
Cryogenic characterization and modeling of 0.18µm CMOS technology (1.8V and 5V) are presented in this paper. Several PMOS and NMOS transistors with different width to length ratios(W/L) were extensively characterized under various bias conditions at temperatures ranging from 300K down to 4.2K. We extracted their fundamental physical parameters and developed a compact model based on BSIM3V3. In addition to their I-V characteristics, threshold voltage(V th ) values, on/off current ratio, transconductance of the MOS transistors, and resistors on chips are measured at temperatures from 300K down to 4.2K. A simple subcircuit was built to correct the kink effect. This work provides experimental evidence for implementation of cryogenic CMOS technology, a valid industrial tape-out process model, and promotes the application of integrated circuits in cryogenic environments, including quantum measurement and control systems for quantum chips at very low temperatures.
Introduction
Cryogenic electronics have good prospects in application ranging from space exploration to infrared focal plane array [1][2][3], and has been studied for use in quantum computing in recent years [4][5][6][7]. A quantum computer comprises a quantum processor and a classical electronic control system [8]. The quantum processor works in a dilution refrigerator at deep-cryogenic temperatures down to the milikelvin range( Fig. 1), while the electronic readout and control system is implemented using room-temperature (RT) laboratory instruments [9,10]. The requirements for wiring between the cryogenic quantum processor and the RT readout controller are becoming more expensive and less reliable as quantum chips become increasing complexed and highly integrated [5]. Cryogenic complementary metal-oxide-semiconductor (Cryo-CMOS) technology can greatly reduce the thermal noise caused by nonideal long signal lines and improve the signal-to-noise ratio and sensitivity of the quantum chip signals. Obtaining purer quantum control and readout signals with low delay efficiently improves quantum chip performance. Quantum Processors work in dilution refrigerators with maximum effective cooling powers of several hundreds µW. This constraint is relaxed at the liquid helium temperature (LHT) 4.2K, where moderate power dissipation is tolerable. Fig. 2 shows the future quantum interface using Cryo-CMOS technology, we need to implement ADC (Analog Digital Converter), DAC (Digital Analog Converter), oscillator, FPGA (Field Programmable Gate Array), and other integrated circuits at cryogenic temperatures. Unfortunately Cryo-CMOS faces several challenges, including the power limitation of refrigerators, and interconnection, packaging and device modeling [11][12][13].
The first problem to solve when designing Cryo-CMOS circuits is transistor modeling. SPICE model act as a bridge between device characterics and IC design. BSIM3 [14], which is an industry-standard model, is valid from 230K to 430K for submicron processes. However, MOSFET characteristics changes at lower temperatures because of freeze-out effect, which has led to a requirement for SPICE model development for cryogenic temperatures [15][16][17][18][19][20]. Previous work has demonstrated that CMOS technologies have been characterized at temperatures down to 4K [21][22][23]. However, BSIM model parameters have only been extracted down to 77K, and no systematic modeling of PMOS and NMOS devices with different width-to-length ratios are performed under different bias conditions at lower cryogenic temperatures, to the best of our knowledge [1,[24][25][26][27].
In this paper, characterization of SMIC 0.18 µm CMOS transistors and a compact SPICE model based on BSIM3v3 [14] are presented from 300K down to 4.2K. It is an aluminum interconnect process, compared with copper, aluminum has better electrical properties at low temperatures [28,29]. Temperaturedependent parameters are revised at 4.2K and the model shows good agreement with measurement results. The 0.18 µm process V th and resistance of active area are measured from 300K to 4.2K for the first time. This work is the first BSIM SPICE model range down to 4.2K for standard CMOS technology; the model can be applied directly to device and circuit electronic design automation(EDA) simulations.
Measurement Setup
Measurements of CMOS transistors with two different oxide thicknesses and a wide range of device sizes were performed, as shown in Table 1. The sample chips were first pasted and wire bonded to chip-carriers using Al-wire bonds ( Fig. 3(a)). These chip-carriers were then immersed in liquid nitrogen (77 K) and liquid helium (4.2 K) using a dipstick. A schematic of the cross-section of the setup is shown in Fig. 3(c). The dipstick consists of a 1.8m steel pipe with a breakout box for cables placed at the top end and two dual in-line package(DIP) lock sockets at the lower end of the pipe. In total, 36 cables (enamel insulated wire) are used for the DC connections, including four cables for the temperature sensor. The temperature sensor is a Rh-Fe thermometer with a 1.2K-325K measurement range. The cable resistance is 0.3-0.4Ω, it is negligible compared with the resistance of MOSFET which is several hundreds or thousands ohms. Because the pipe can move up and down through a vacuum flange, the temperature can be shifted from 4.2K in the liquid phase to approximately 250K in the helium vapour at the top of the Dewar. All the MOSFET electrical measurements were performed using a Keysight B1500A semiconductor device analyzer, as shown in Fig. 3(b). For the thin-oxide NMOS, we measured transfer characteristics in both linear (drain-source voltage V DS = 50 mV) and the saturation regions (V DS = 1.8V) under various substrate bias voltages, along with the output characteristics under zero substrate bias(bulk-sourse voltageV BS =0V) and reverse bias voltage(V BS =-1.8V) for various gate voltages(V GS ). For the thickoxide MOS, bias condition were increased to 5V, while for PMOS, the bias conditions were reversed. The resistance measurements were performed using a Keysight 3458A Digital Multimeter with 8 1 2 Digit precision.
Characterization
The characteristics of CMOS transistors at different cryogenic temperatures are shown in Fig. 4. As shown in Fig. 4(b) and Fig. 4(c), threshold voltage increases as temperature decreases because of carrier freeze-out in the MOSFET channel region and thus a higher gate drive voltage is required to inject carriers into the channel region. V th varies approximately linearly with temperature, especially in the PMOS. Impurity freeze-out becomes important for temperature lower than 150K for shallow-energy-level dopants. At liquid nitrogen temperature, weak freeze-out takes place, which is mainly annoying for lightly doped drain (LDD) devices. At very low temperatures (<10 K), donor or acceptor impurities currently used to dope the semiconductor are fully frozen-out, at liquid helium temperature, practically no carriers remain in the bands if no field is applied [17][18][19][20]. Fig. 4(a) shows that I DS increases as temperature decreases, because the series resistance decreases as shown in Fig. 4(f), and mobility increase at cryogenic temperatures [12]. In low impurity concentration situations, such as p-well, the resistivity decreases due to an increase of mobility down to liquid nitrogen temperature [24]. However, carrier freeze-out causes a steep increase in n-well resistivity at liquid helium temperature. Other resistances drop with decreasing temperature down to liquid helium temperature. The resistance of the salicide's heavy-doped N/P active area is reduced by 2/3 when compared with the RT value. The resistance of the unsalicide's heavy-doped N active area is reduced by 1/3 when compared with the RT value. In particular, Aluminum resistance drops drastically because of reduced lattice vibrations. This gives a great advantage to reducing noise caused by CMOS switching and power-supply line resistances. This is an advantage of Cryo-CMOS technology compared with RT CMOS technology.
The kink effect at the LHT is shown in Fig.6 and is caused by the LDD freezing out and substrate freeze-out. Impurity ionization will decrease as temperature decreases, especially in the light-doped regions near the source and drain. When the sourcedrain voltage becomes very high, impurity ionization will be activated to restrain the freeze-out effect under strong electric fields, CMOS transistors turn on for the second time [16]. This freezing effect causes the sourcedrain parasitic resistance to decrease and then turn to normal. The second reason is the substrate freezeout, since at very low temperatures the MOS structure has a type of floating substrate potential within the depletion region. Although the applied substrate voltage on the backside of the device is fixed, the depletion region is in a floating state. In this structure, the majority carrier current (substrate current) cannot reach the substrate contact and thus flows through the substrate to the source. Due to the increase of the majority carrier current with increasing drain voltage, flowing through the substrate to the source at increasing drain voltage, this substrate potential within the depletion region increases and causes a decrease of the threshold voltage, for sufficient drain voltage. Therefore, at very low temperature, we obtain an excess drain current which creates a kink in the current-drain voltage characteristics [18]. Additionally, the measured saturation current value is less than the simulated value of BSIM3v3 model because of the channel freeze-out effect( Fig.5(a-d)).
Ion to Ioff ratio(turn-on current and turn-off current ratio) is a typical parameter for MOSFET in digital integrated cirtuits.
Ion to Ioff ratio maintains high value at cryogenic temperatures as shown in Fig. 4(d). The standard process CMOS can work well as a switch at low temperatures for digital circuits with low static power consumption, which is of great significance to the limited cooling power. Gate transconductance(G m ) indicates the gateto-source current control capability. G m max increases when temperature increases (Fig. 4(e)), increase of G m will supply wider bandwidth for the same power budget. Hence, MOS can work in analog circuits if it is modeling accurated.
Modeling
BSIM3 is a semi-empirical but accurate compact model for submicron process. We continue to adopt BSIM's mode and use semi-empirical methods to modeling CMOS device at very low temperatures. Our object is to extend the BSIM model in submicron process to cryogenic temperatures, focus on the accuracy of the model. Fitting errors between simulation with default parameters and LHT measured data are generally higher than 60% under all bias conditions (Fig. 5(ad)), while the BSIM3v3 model achieves a good degree of fitting at RT. First, the extraction procedure is performed using BSIMProPlus. The extraction process performed in the following procedure is based on physical understanding of the model and local optimization. The procedure can be described as follows:1) extract threshold voltage parameters such as V th0 , K1, K2 through large size device (large W&L); 2) extract carrier mobility parameters such as µ0, µa, µb, µc through large size device (large W&L); 3) extract short-channel effect parameters such as Dvt0, Dvt1, Dvt2, Nfactor through one set of devices (large and fixed W different L); 4) extract saturation velocity parameters such as vsat through one set of devices (large and fixed W& different L); 5) extract bulk effect parameters such as a0, ags, Keta through one set of devices (large and fixed W& different L) [14]. The model parameters are shown in Table 2. The rootmean-square (RMS) error is introduced to estimate the deviation between the results from the measurements and simulations. The RMS error is given by equation (1) where N is the number of data, I mi is measured data and I si is simulated data.
RMS ERROR = 1 N n i=1 I mi − I si I th 2 × 100(1)
The value of the threshold current I th can be set appropriately to obtain meaningful results. In this case, I th has been set to the maximum measured value according to BSIMProPlus. After the parameters have been changed appropriately, the RMS error between the simulation results and the test data improves to around 6% (Fig. 5(e-h), Table 3). However, deviations caused by the kink effect still exist in some devices ( Fig. 6(a,b,e,f,i,j)) at LHT. We connected a resistor in series with the substrate to improve the fitting precision as shown in Fig. 6(p), the MOSFET represents the BSIM model with cryogenic parameters and the resistor represents the freeze-out effect in the LDD region and substrate. The resistor value was extracted via Matlab using a polunomial fitting method. The characteristic of the non-linear resistor is shown in Fig.6(m,n,o). Fig. 6(c,d,g,h,k,l) show the corrected sub-curcuit simulation results. Good agreement with the DC measurements was achieved for devices over the entire voltage range at 4.2K as the RMS errors are summarized in Table 3.
Conclusion
A cryogenic study of SMIC 0.18µm 1.8V/5V CMOS technology down to 4.2K has been presented. We performed a relatively simple apparatus for executing the cryogenic measurements.
A compact model based on BSIM3v3 has been proposed to optimize the deviation between measurement results and simulations using default parameters. V th , Ion to Ioff ratio, G m max and resistors on chip were tested down to 4.2K. Using the resulting database and SPICE model, we can design and simulate integrated circuits for cryogenic applications including quantum computer readout and control systems.
Acknowledgment
This work was supported in part by the National Key Research and Development Program of China (Grant No. 2016YFA0301700), In part by the National Natural Science Foundation of China (Grants No.11625419), in part by the Anhui initiative in Quantum information Technologies (Grants No.AHY080000) and this work was partially carried out at the USTC Center for Micro and Nanoscale Research and Fabrication. The authors would like to thank SMIC for devices fabrication and software support.
Figure 1 .
1Superconducting quantum chip readout-control system.
Figure 2 .
2Superconducting quantum chip readout-control system using Cryo-CMOS technology.
Figure 3 .
3(a): Al-bonded sample chip on a chip carrier. (b): Schematic of four-teminal MOS transistor DC measurment. (c): Schematic depiction of our cryogenic test setup. The sample is mounted on a long steel pipe which is shifted into a helium Dewar to reach the liquid helium. The connections to the sample are made with long (standard) cables that are attached to a breakout box on the top. There are 32 shielded connections to two 8×2 DIP lock sockets, 4 DC connections to a resistor therometer.
Figure 4 .
4(a): I DS -V DS curves at different cryogenic temperatures of thin-oxide NMOS,V DS =0V→1.8V, W/L=10µm/10µm, V GS =1.8V, V BS =0V. (b): I DS -V GS curves at different cryogenic temperatures of thin-oxide NMOS,V GS =0V→1.8V, W/L=10µm/10µm, V DS =0.05V, V BS =0V. (c): V th of CMOS transistors at different cryogenic temperatures. (d): Ion to Ioff ratio of CMOS transistors at different temperatures. (e): Gmmax of MOSFETs at different cryogenic temperatures. (f): resistors on chip at different cryogenic temperatures, the six lines from top to bottom are N++ salicide area, W/L=2µm/100µm; P++ salicide area, W/L=2µm/100µm; Al metal square, 0.23µm * 0.23µm * 10000; N+ poly salicide, W/L=0.18µm/100µm; N Well in STI, W/L=20µm/100µm; Unsalicide N+, W/L=2µm/100µm
Figure 5 .
5I DS -V GS curves (a,b,e,f) and I DS -V DS curves (c,d,g,h) of thin-oxide NMOS at LHT. Device size (W/L) is 10µm/10µm.(a)-(d):before extraction, (e)-(h):extraction results, measured data: dashed lines; simulated data: solid lines. (a),(e): V BS =0V→-1.8V, V DS =0.05V; (b),(f): V BS =0V→-1.8V, V DS =1.8V; (c),(g): V GS =0V→1.8V, V BS =0V; (d),(h): V GS =0V→1.8V, V BS =-1.8V.
Figure 6 .
6Cryogenic kink correction.(a)-(l): I DS -V DS curves of CMOS at LHT. Device size (W/L):(a)-(d) thin-oxide NMOS 10µm/0.16µm,(e)-(h) thick-oxide NMOS 10µm/10µm,(i)-(l) thick-oxide NMOS 10µm/10µm. (a)-(b),(e)-(f),(i)-(j):before kink correction, (c)-(d),(g)-(h),(k)-(l):kink correction results. measured data: dashed lines, simulated data: solid lines. (a),(c): V GS =0V→1.8V, V BS =0V; (b),(d): V GS =0V→1.8V, V BS =-1.8V; (e),(g): V GS =0V→5V, V BS =0V; (f),(h): V GS =0V→5V, V BS =-4V; (i),(k): V GS =0V→-5V, V BS =0V; (j),(l): V GS =0V→-5V, V BS =4V; (m) Sub-circuit resistance of thin-oxide MOSFET versus V DS at LHT, W/L=10µm /0.16µm.(n) Sub-circuit resistance of thick-oxide NMOSFET versus V DS at LHT, W/L=10µm /10µm.(o) Sub-circuit resistance of thick-oxide PMOSFET versus V DS at LHT, W/L=10µm /10µm. (p)Schematic representation of sub-circuit model.
Table 1 .
1SUMMARY OF CHARACTERIZED DEVICESTechnology
SMIC 0.18um Bulk CMOS Process
Oxide
Thin(3.6nm)
Thick(11.9nm)
Norminal Voltage
1.8V
5V
Type
NMOS
PMOS
NMOS
PMOS
W/L[um/um]
Table 2 .
2MODEL PARAMETERSOxide
Thin(3.6nm)
Thick(11.9nm)
Norminal Voltage
1.8V
5V
Type
NMOS
PMOS
NMOS
PMOS
Temperature[K] 300K
77K
4.2K
300K
77K
4.2K
300K
77K
4.2K
300K
77K
4.2K
Vth0[V]
0.39
0.4671
0.296
-0.607 -0.5785 -0.4598 0.724
1.1391 0.31124 -0.834
-1.2
-1.0532
K1[V
1
2 ]
0.680104 0.53197 0.6966 0.87354 0.89101 0.8821
0.97
0.84906 1.6667
0.986 0.77845 1.1863
K2
-0.04998 -0.00013 -0.00013 -0.04666 -0.05506 -0.0534 -0.015 -0.00686 -0.01878 0.0626 0.096653 0.035749
u0[cm 2 /V·s]
340
606.51 116.09 0.0085 0.015369 0.000933 0.04415 0.36688 0.00426 0.015 0.076605 0.0035
ua[m/V]
-1E-09 -4.4E-12 -4.5E-12 2.5E-10 2.4E-10 3.52E-10 -3.7E-10 1.11E-10 3.79E-10 2.27E-09 8.07E-09 4.84E-09
ub[(m/V) 2 ]
0.236667 1.12E-17 6.46E-18 9.29E-19 9.29E-19 8.37E-19 2.6E-18 5.55E-18 5.65E-18 2.51E-20 -3.6E-21 -4.5E-19
uc[1/V]
1.2E-10 3.33E-13 3.33E-13 -7.2E-11 -1E-10 -1.7E-10 8.38E-11 8.48E-12 1E-12 -7.6E-11 -3E-10 -4.9E-10
Dvt0
1.3
0.2165
5
1.03
3.0099 4.5143
7.46
7.46
7.46
2.8982 2.0023 26.374
Dvt1
0.577164 2.3925
5
0.35
0.48288 0.14103 0.805
0.805
0.805
0.6
0.21882 0.46447
Dvt2[1/V]
-0.17176 -0.01816
-3
0.052 0.000804 0.000861
0
0
0
0
-0.047
-0.6
Vsat[m/sec]
82500
89809
53820 109320 4202600 26162
77500
77500
77500
88000
47264
98296
a0
0.83
0.70733 1.1883 1.0951 0.87389 0.83893
0.96
0.96 0.013849
1
0.73796 0.5917
ags[1/V]
0.32
0.73655 0.1485 0.3024 0.39778 0.65873
0.15
0.15 0.036746 0.0655 0.006383 0.002047
Keta[1/V]
-0.003 -3.6E-05 -3.6E-05 -0.0389 -0.03197 0.05346 -0.0015 -0.0015 -0.00169 -0.005 -0.00094 -0.013
Table 3. RMS ERROR
IDS(VDS)(VBS=0V)
IDS(VDS)(VBS=VBB a )
Default
Revised Corrected Default Revised Corrected
thin-oxide NMOS W/L=10µm/0.16µm
13.37%
5.74%
1.71%
8.94%
5.71%
2.41%
thick-oxide NMOS W/L=10µm/10µm
25.72%
5.77%
4.66%
27.54%
6.12%
1.67%
thick-oxide PMOS W/L=10µm/10µm
174.15%
4.41%
1.41%
58.19%
3.25%
0.96%
a for thin-oxide NMOS,VBB=-1.8V; for thick-oxide NMOS, VBB=-4V; for thick-oxide PMOS, VBB=4V.
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"Efficient Lineage for SUM Aggregate Queries"
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"Dimitris Fotakis \nSchool of Electrical and Computer Engineering\nNational Technical University of Athens\n15780AthensGreece\n",
"Angelos Vasilakopoulos \nNational Technical University of Athens\n15780AthensGreece\n"
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| AI systems typically make decisions and find patterns in data based on the computation of aggregate and specifically sum functions, expressed as queries, on data's attributes. This computation can become costly or even inefficient when these queries concern the whole or big parts of the data and especially when we are dealing with big data. New types of intelligent analytics require also the explanation of why something happened.In this paper we present a randomised algorithm that constructs a small summary of the data, called Aggregate Lineage, which can approximate well and explain all sums with large values in time that depends only on its size. The size of Aggregate Lineage is practically independent on the size of the original data. Our algorithm does not assume any knowledge on the set of sum queries to be approximated. | 10.3233/aic-140647 | [
"https://arxiv.org/pdf/1312.2990v2.pdf"
]
| 8,055,946 | 1312.2990 | 757381ca03535f6a57b04c3e067794fd0c9e92a7 |
Efficient Lineage for SUM Aggregate Queries
9 Jun 2014
Foto N Afrati
School of Electrical and Computer Engineering
School of Electrical and Computer Engineering
National Technical University of Athens
15780AthensGreece
Dimitris Fotakis
School of Electrical and Computer Engineering
National Technical University of Athens
15780AthensGreece
Angelos Vasilakopoulos
National Technical University of Athens
15780AthensGreece
Efficient Lineage for SUM Aggregate Queries
9 Jun 20141Artificial IntelligenceDatabasesAggregate QueriesDatabase LineageQuery ApproximationRandomised Algorithms
AI systems typically make decisions and find patterns in data based on the computation of aggregate and specifically sum functions, expressed as queries, on data's attributes. This computation can become costly or even inefficient when these queries concern the whole or big parts of the data and especially when we are dealing with big data. New types of intelligent analytics require also the explanation of why something happened.In this paper we present a randomised algorithm that constructs a small summary of the data, called Aggregate Lineage, which can approximate well and explain all sums with large values in time that depends only on its size. The size of Aggregate Lineage is practically independent on the size of the original data. Our algorithm does not assume any knowledge on the set of sum queries to be approximated.
Introduction
Big data poses new challenges not only in storage but in intelligent data analytics as well. Many organisations have the infrastructure to maintain big structured data and need to find methods to efficiently discover patterns and relationships to derive intelligence [1,2]. Thus, it would be desirable to be able to construct out of big data a right representative part that can explain aggregate queries, e.g., why the salaries or the sales of a department are high.
AI systems typically make decisions based on the value of a function computed on data's attributes. Several approaches have in common the computation of aggregates over the whole or large subsets of the data that helps explain patterns and trends of the data. E.g., recommendation systems rank and retrieve items that are more interesting for a specific user by aggregating existing recommendations [25]. For another example, collaborative filtering computes a function which uses aggregates and a sum over the existing ratings from all users for each product in order to predict the preference of a new user [5,19]. User preferences are often described as queries [18], e.g., queries that give constraints on item features that need to be satisfied.
Another reason for which data analytics seek to explain data is for data debugging purposes. Data debugging, which is the the process that allows users to find incorrect data [23,24], is a research direction that is growing fast. Data are collected by various techniques which, moreover, are unknown to and uncontrolled by the user, thus are often erroneous. Finding which part of the data contains errors is essential for companies and affects a large part of their business.
All these applications call for techniques to explain our data. Aggregation is a significant component in all of them. In this paper we offer a technique that constructs a summary of the data with properties that allow it to be used efficiently to explain much of the data behaviour in aggregate for sums. We refer to this summary as Aggregate Lineage, since in most applications it represents the source of an aggregate query 1 .
Lineage (a.k.a. provenance) keeps track of where data comes from. Lineage has been investigated for data debugging purposes [17]. Storing the complete lineage of data can be prohibitively expensive and storage-saving techniques to eliminate or simplify similar patterns in it are studied in [6]. For select-project-join SQL queries, lineage stores the set of all tuples that were used to compute a tuple in the answer of the query [4]. This is natural for select-project-join SQL queries where original attribute values are "copied" in attribute values of the answer. However, in an aggregate query the value of the answer is the result of applying an aggregate function over many numerical attribute values. When we want to understand why we get an aggregate answer it may no longer be important or feasible to have lineage to point to all contributing original tuples and their values. We would rather want to compute few values that can be used to tell us as much as possible about the origin of the result of an aggregate query. However is this at all possible and if it is what are the limitations?
In this paper we initiate an investigation of such questions and, interestingly, we show that useful and practical solutions exist. In particular, we offer a technique that uses randomisation to compute Aggregate Lineage which is a small representative sample (it is more sophisticated than just a simple random sample) of the data. This sample has the property to allow for good approximations of a sum query on ad hoc subsets of data -we call them test queries. Test queries are applied to the Aggregate Lineage -not the whole original data. The test queries which we consider are sum queries with same aggregated attribute conditioned with any grouping attributes depending on which subsets of the data we want to test. We give performance guarantees about the quality of the results of the test queries that show the approximation to be good for test queries with large values (i.e., close to the total sum over the whole set of data). Our performance guarantees hold, with high probability, for any set of queries, even if the number of queries is exponentially large in the size of the lineage. The only restriction is that the queries should be oblivious to the actual Aggregate Lineage. This restriction is standard in all previous work on random representative subsets for the evaluation of aggregate queries and is naturally satisfied in virtually all practical applications. The following example offers a scenario about how Aggregate Lineage can be used in data debugging and demonstrates how some test queries can be defined.
Example 1. Suppose that the accounting department of a big company maintains a database with a relation Salaries with hundreds of attributes and millions of tuples. Each tuple in the relation may contain an identifier of an employee stored in attribute EmplID, his Department stored in attribute Department, his annual salary stored in attribute Sal and many more attribute values. Other relations are extracted from this relation, e.g., a relation which contains aggregated data such as the total sum of salaries of all employees. A user is trying to use the second relation for decision making but he finds that the total sum of salaries is unacceptably high. He does not have easy access to the original relation or he does not want to waste time to pose time-consuming queries on the original big relation. The error could be caused by several reasons (duplication of data in a certain time period, incorrect code that computes salaries in a new department). Thus e.g., if we could find the total sum of salaries for employees in the toy department during 2009, and see that this is unreasonably high, still close to the first total sum of all employees' salaries, then we will be able to detect such errors and narrow them down to small (and controllable) pieces of data.
In order to do that, we need the capability of posing sum queries restricted to certain parts of the data by using combinations of attributes. This will help the user understand which piece of data is incorrect. We do not know in advance, however, which piece of data the user would want to inquire and thus Aggregate Lineage should allow the user to be able to get good approximated answers to whatever queries he wants to try. There are billions of such possible queries and hence billions of subsets of data which we want to compute a good approximation of the summation of salaries. We want Aggregate Lineage to offer this possibility.
We propose to keep as Aggregate Lineage a small relation under the same schema of the original relation. In order to select which tuples to include, we use valued-based sampling with repetition, i.e., weighted random sampling where the probability of selecting each tuple is proportional to its value on the summed attribute. The intuition why this method works is the following. Larger values contribute more to the sum than smaller ones, thus we expect that tuples with larger values should be selected more often than tuples with smaller values. Hence, we could end up with a tuple selected many times in the sample even if it appears only once in the original data. On the other hand, if there are many tuples with values of moderate size, many of them will be selected in the Aggregate Lineage, so that their total contribution to the approximation of the sum remains significant.
Our contribution
In our approach Aggregate Lineage is a small relation with same schema as the original relation and with the property to offer good approximations to test queries posed on it.
To present performance guarantees, we build on Althöfer's Sparsification Lemma [3]. In [3], Althöfer shows that the result of weighted random sampling over a probability vector is a sparse approximation of the original vector with high probability. This technique has found numerous applications e.g., in the efficient approximation of Nash equilibria for (bi)matrix games [21], in the very fast computation of approximate solutions in Linear Programming [22], and in selfish network design [13].
In this paper, we show for the first time that the techniques of [3] are also useful in the context of sum database queries with lineage. Our results show that the Aggregate Lineage that we extract has the following properties (which we describe in technical terms and prove rigorously in Section 4):
-Its size is practically independent of the size of the original data.
-It can be used to approximate well all "large" sums (i.e., with values close to the total sum), of the aggregated attribute in time that depends only on its size, and thus is almost independent of the size of the original data.
Computing Aggregate Lineage
In this section, we present randomised algorithm Comp-Lineage which computes Aggregate Lineage in one pass over the data and in time linear in the size n of the original database relation. In Section 4 we show that the output of Comp-Lineage is useful to approximate arbitrary ad-hoc sum test queries in time independent of n. We note that our algorithm is agnostic of the specific sum queries that will be approximated by using its output.
Suppose that we are given a database with a relation R with n tuples and we are given a positive integer b which is the number of tuples we have decided to include in the Aggregate Lineage (in Section 4 we will explain how we decide b to give good performance and approximation guarantees). Suppose that A is a numerical attribute of R which takes nonnegative values. Let S be the sum of values of attribute A over all n tuples. The algorithm essentially is a biased sampling with repetition that selects b tuples from R. Each tuple t has probability to be selected equal to p t = t[A]/S where t[A] is the value of attribute A in t. It collects initially a bag (a.k.a. multiset and is allowed to have the same element more than once) of tuples (since each tuple may be selected multiple times) which is turned in a set of tuples by adding an extra attribute F r (for Frequency) which shows the number of times this tuple is selected. We denote by L R.A the Aggregate Lineage of relation R with sum attribute A.
Algorithm Comp-Lineage
Input: A relation R with n tuples and positive integer b < n.
Output: An Aggregate Lineage relation LR.A with at most b tuples.
-Randomly select with repetition one out of the n tuples of R in b trials where each tuple t is selected with probability pt. -Form relation LR.A by including all tuples selected above and adding an extra attribute F r to each tuple to record how many times this tuple was selected.
t[A]
value of attribute A in tuple t S sum of all values over attribute A p t probability that Algorithm Comp-Lineage selects tuple t We can use the techniques of [10] for weighted random sampling and efficiently implement our algorithm to run in linear time in the size of the input either in a parallel/distributed environment or over data streams. Table 1 summarizes the main symbols used throughout the paper.
Running Example
Example 2. We illustrate Algorithm Comp-Lineage by applying it to Example 1 with b = 8, 852 and presenting the data and the Aggregate Lineage in Figure 2. Actually Figure 2 only shows the value of the aggregated attribute (Sal in our example), the rest of the tuple is not shown.
The first two columns of Figure 2 present the data in relation Salaries. In order to be able to present many tuples we have chosen a relation with a few values for attribute Sal, actually five (i.e., 10 9 , 10 8 , 10 7 , 10 6 and 10) and their Original Values (O.V.) are shown in the first column. The second column shows how many tuples in Salaries have these values in Sal. Thus, it says, e.g., that there are 100 tuples with value in Sal equal to 10 9 , 1,000 tuples with value in Sal equal to 10 8 and so on.
The third column in Figure 2 shows how many tuples from Salaries with a specific value in Sal are selected by Algorithm Comp-Lineage to be included in the Aggregate Lineage relation. Thus, e.g., all 100 tuples with Sal = 10 9 were chosen, only 681 tuples with Sal = 10 7 were chosen and no tuple with Sal = 10 was chosen.
In order to represent the Aggregate Lineage relation L Salaries.Sal in the most demonstrative way, we have chosen to partition its tuples in blocks (each block further divided in multiple rows in columns 4, 5 and 6), each block corresponding to one value of Sal in Salaries. Thus the first block has 9 rows, the second block has 4 rows and the last three blocks have one row each. This breaking into blocks gives a visualisation of the characteristics of the algorithm.
The fourth column stores the extra attribute frequency F r which tells how many times a certain tuple was selected by the algorithm and the fifth column stores the number of tuples that were selected so many times. Thus, e.g., the first row says that 5 tuples were selected 3 times each. The ninth row says that 4 tuples from Salaries were selected 11 times each.
The blocks give us an intuition of the characteristics of the Aggregate Lineage. The first block corresponds to the largest value of Sal and tuples with this value (i.e., Sal = 10 9 ) contributed quite heavily to the lineage -all 100 tuples with Sal = 10 9 were selected multiple times. In more detail, there are 100 tuples with value Sal = 10 9 . Of those tuples, 5 were added in the bag 3 times each, 10 tuples were added in the bag 4 times each, and so on. Thus, by considering these 100 tuples, the Algorithm Comp-Lineage added in the bag 3· 5 + 4· 10 + 5· 19 + 6· 14 + 7· 13 + 8· 15 + 9· 8 + 10· 12 + 11· 4 = 681 tuples in total. That is to say, each of those 100 tuples contributed on average 6.81 to the bag. When we get a set out of the bag by using frequencies (to avoid repeating a tuple multiple times), then we see that the average frequency per tuple is 6.81. So, from this first block, the 681 tuples in the bag of Algorithm Comp-Lineage are transformed to a set of 100 tuples in Aggregate Lineage with average frequency 6.81. We can compare it with the average frequency in the second block which is 0.681 (this is 1· 347 + 2· 123 + 3· 20 + 4· 7 = 681 divided by 1000 tuples) and see that, in the data of our example, each tuple of the first block contributes more heavily to the lineage.
As we will explain in more detail later, this shows partly why the lineage is useful for discovering almost accurately sub-sums that are large compared to the total sum, whereas when a sub-sum is small in comparison, then the lineage cannot be used to compute it accurately.
The second block did not contribute that heavily but still quite a lot, around half of tuples with Sal = 10 8 were selected at least once and quite a few more than once, in total this block contributed 681 tuples in the bag. The third block contributed moderately. The fourth block is interesting because the value of Sal is very small only 10 6 but it contributed quite a lot due to the fact that there are many tuples in Salaries with Sal = 10 6 , thus it contributed almost 85 percent of the tuples in the Aggregate Lineage. Finally the last column in the figure shows how much each tuple from the Aggregate Lineage contributes to the approximation of sub-sums that are computed by the test queries. The same tuple is added in the Aggregate Lineage several times as recorded in the new attribute F r and thus, in order to calculate the contribution of a certain tuple, we multiply its frequency in F r by S/b. By doing so, some tuples (e.g., the ones in the fourth block) in our example of Figure 2 will contribute much more than their actual value in Sal. But this is to compensate for the tuples with value close to it (same value in our example) that are not selected to be included in the Aggregate Lineage. In the next section we give the technical details on how Aggregate Lineage can be used in order to approximate sub-sums.
Note that Aggregate Lineage does not assume any knowledge of the query set: i.e., we run the random selection of Algorithm Comp-Lineage only once and compute LR.A without assuming anything about the queries. Then, this same relation LR.A can be used to make us understand any sub-sum test query, without requiring that the test queries are given beforehand or requiring that the test queries are chosen in any specific fashion (e.g., they do not have to be chosen uniformly at random), as long as the query choice is oblivious to the actual sample computed by Aggregate Lineage 2 . We first present the theoretical approximation guarantees and then demonstrate how these guarantees play for debugging on our running example.
Approximation Guarantees of Test Queries on Aggregate Lineage
In this section we prove the theoretical guarantees of Aggregate Lineage. Let R be a relation with a nonnegative numerical attribute A. We consider SUM queries that ask for the sum of attribute's A values over arbitrary subsets of the tuples in relation R. We use tuple identifiers in order to succinctly represent subsets of tuples. Thus, any SUM query defines a set of tuple identifiers for tuples that satisfy its predicates, hence the following formal definitions: Definition 1 (Exact SUM Q(R.A)). Let R be a database relation. We attach a tuple identifier on each tuple of R. We denote by IR the set of all identifiers in relation R. Given an attribute A in the schema of R, we denote by ai the value of attribute R.A in the tuple with identifier i in R.
Let Q be a SUM query over R.A. We denote by I Q R the set of tuple identifiers from IR for tuples of R that satisfy Q's predicates.
The result of a SUM query, Q(R.A), is the summation of the values of R.A over the set of tuples with identifiers that appear in I Q R , i.e., Q(R.A) = i∈I Q R ai.
Definition 2 (Approximated SUM Q ′ (LR.A)). Let Q be a SUM query over R.A and let LR.A be an Aggregate Lineage. We attach a tuple identifier on each tuple of LR.A. We denote by IL the set of all identifiers in LR.A. We denote by I Q L the set of tuple identifiers from IL for tuples of LR.A that satisfy Q's predicates (since the set of attributes of R is a subset of the set of attributes of LR.A, we have that the predicates of a SUM query Q, expressed on attributes of R, define I Q L ). We denote by fi the value of attribute LR.A.F r in the tuple with identifier i in LR.A.
The approximated result of SUM query Q, denoted by Q ′ (LR.A), is the summation of the values of LR.A.F r over the set of tuples with identifiers that appear in I Q L multiplied by S/b, i.e., Q ′ (LR.A) = i∈I Q L fi· S/b.
The following theorem provides the performance guarantees for any arbitrary set of m SUM queries computed over the Aggregate Lineage relation in order to serve as an approximation of the corresponding SUM queries over the original data. Theorem 1. Let R be a relation with n tuples having nonnegative values a1, . . . , an on attribute A, and let S = n i=1 ai. Then, for any collection of m SUM queries Q1(R.A), . . . , Qm(R.A) (not known to the algorithm), any p ∈ (0, 1), and any ǫ > 0, the Algorithm Comp-Lineage with input all tuples of R and b = ⌈ln (2m/p)/(2ǫ 2 )⌉ derives an Aggregate Lineage
LR.A such that |Qj(R.A) − Q ′ j (LR.A)| ≤ ǫS, for all j ∈ [m]
, with probability at least 1 − p.
Proof. The proof is an adaptation of the proof of Althöfer's Sparsification Lemma [3]. For simplicity, we assume, without loss of generality, that the set IR of all tuple identifiers of R in Definition 1 is IR = {1, . . . , n}. We define b independent identically distributed random variables X1, . . . , X b , which take each value i ∈ [n] with probability ai/S. Namely, each random variable Xi corresponds to the outcome of the i-th trial of Comp-Lineage. For each tuple i, its frequency in the sample is fi = |{k ∈ [b] : X k = i}|.
Let us fix an arbitrary SUM query Qj(R.A). For each k ∈ [b], we let Y k j be a random variable that is equal to 1, if X k ∈ I Q j R , and 0, otherwise. Since the random variable Y k j is equal to 1 with probability Qj
(R.A)/S, IE[Y k j ] = Qj(R.A)/S. We observe that the random variables {Y k j } k∈[b]
are independent, because the random variables {X k } k∈ [b] are independent. Furthermore, we let Yj be a random variable defined as Example 3. Suppose, in our running example, we want to be able to answer with good approximation m = 10 6 queries. What are the guarantees that the theorem provides? The original data have n ≈ 10 6 tuples. Suppose we select the number of tuples in the Aggregate Lineage to be b ≈ 9000. Then the theorem says that, by setting ǫ = 0.04, we can compute any of 10 6 arbitrary queries within 0.04S of its real value with probability 1 − 10 −6 . Thus, if the real exact value of the query Q1 is equal to Q1(Salaries.Sal) = 0.4S = S1 (remember S is the sum over all tuples of relation R) then the approximation will be 0.04S = 0.04S1/0.4 = 0.1S1. If for another query Q2 we have Q2(Salaries.Sal) = 0.8S = S2 then the approximation will be 0.05S2, so, then with high probability we get an answer that is within a factor of 0.05 of the actual answer. 3 We use the following form of the Chernoff-Hoeffding bound (see [16] Observations on the practical consequences of Theorem 1. Examining closely equation b = ⌈ln (2m/p)/2ǫ 2 ⌉ which gives us an upper bound of the number of tuples in the Aggregate Lineage for m queries and with p and ǫ guarantees as in its statement, we make the following observations:
Yj = 1 b b k=1 Y k j By definition, Yj = i∈I Q j R fi/b = i∈I Q j L fi/b,
-The value of b depends on m as the logarithm, hence if we go from m to m 2 queries, we only need to multiply b by 2 in order to keep the same performance guarantees. Thus it is reasonable to state that, in many practical cases the number m of queries that can be approximated well can be as large as a polynomial on the size of data -even with coefficient in the order of a few hundreds. -The value of b does not depend much on p (again only as in the logarithm) but it depends mainly on ǫ which controls the approximation ratio (the approximation ratio itself is ǫ/ρ if the query to be computed has a sum S ′ = ρS).
A debugging scenario
Here is what a user can do for data debugging when using the Aggregate Lineage we propose.
-He computes sub-sums by filtering some attributes and possibly specific values for these attributes. E.g., what was the sum of salaries of employees in the toy department in Spring 2010 and only for those employees who were hired after 2005. The user devises several such test queries as he sees appropriate and while he computes them and checks that sub-data is ok or suspicious, he devised different test queries to suit the situation. E.g., if he observes an unusually large value, close to the total sum, in the query about employees in the toy department and hired before 2005, then the rest of the queries he devises stay within this department and within the range until 2005, and tries to narrow down further the wrong part of data. E.g., now he narrows down to each month or/and to employees that are hired between 2005 and 2007, etc. On the other hand, if he finds the answer satisfactory, then he announces this part of the data correct, therefore stays outside this sub-data and tries to find some other part of the data that are faulty. The user uses and poses his test queries over the stored small Aggregate Lineage instead of inefficiently use the original big relation.
In the following example we show how using Aggregate Lineage to approximate test queries applies to our running example.
Example 4. We continue our running Example 2 where we computed Aggregate Lineage L Salaries.Sal . Suppose that we have a SUM test query Q1 asking the sum of the salaries of a subset of the employees of the company defined from a subset of EmpID's. Let this subset consist of 50 employees with salary 10 9 , 5, 000 employees with salary 10 7 (so half of them) and of all 10 6 employees with salary 10 6 . We compute the query over Salaries and take the exact answer 1.1 × 10 12 .
In order to use Aggregate Lineage to understand our data we compute I Q 1 L . The Aggregate Lineage has at most 8, 852 tuples. The identifiers of I Q 1 L define the sub-lineage of query Q1 over L Salaries.Sal . The sublineage of Q1 points to 50 of the tuples of L Salaries.Sal with original salaries 10 9 and to all 6, 809 tuples with original Sal values 10 6 (cf. Figure 2). It will also point to some tuples of L Salaries.Sal with Sal values 10 7 : On average query Q1 is applied on half of the 681 selected in Aggregate Lineage tuples, but in extreme cases it may include all or none of them. For this reason, it is a good practice to run the randomised algorithm more than once and compute a few distinct summaries in order to have better results. For instance, we may compute three summaries, use some benchmark subqueries to decide a distance between summaries, toss the summary which is the more distant and keep one of the others arbitrarily. Note that it is easy to compute the benchmark queries in one pass through the original data in parallel with computing the lineage.
We now use the Aggregate Lineage L Salaries.Sal shown in Figure 2 to approximate the value of the sum answer to Q1. In one worst case query Q1 will include: the 50 tuples with salaries 10 9 from L Salaries.Sal tuples with the larger frequencies and all 681 selected tuples with salaries 10 7 . The approximation Q ′ 1 (L Salaries.Sal ) in this case is (4· 11 + 12· 10 + . . . + 681 + 6, 809)S/b = 7, 935· S/b = 1.17· 10 12 . In the other extreme case Q1 includes tuples with the smaller frequencies and none of the selected in Aggregate Lineage tuples with salaries 10 7 , yielding the approximation 6, 995· S/b = 1.03· 10 12 . We see that Q1 is well approximated. Of course the approximation bounds are not the same for every SUM query -we presented the guarantees in Section 4.
Another straw man approach would be to select as lineage the 8, 852 tuples with larger salary values. This method will select all 100 tuples with salaries 10 9 , all 1, 000 tuples with salaries 10 8 and the remaining 7, 752 tuples from tules with salaries 10 7 . With this approach, query Q1 will be on average approximated with the value 50· 10 9 + 3, 876· 10 7 ≈ 8.8 × 10 10 because it loses all the information about all original 10 6 tuples with salaries 10 6 contributing to the sum. On another approach, a simple random sampling of 8, 852 tuples will almost always select all of them from the 10 6 many tu-ples with salaries 10 6 . Query Q1 will then be approximated with the value 8, 852· 10 6 ≈ 8.8 × 10 9 . Note, on the other hand, that if all original tuples had the same salaries then our method would coincide with simple random sampling.
Discussion
We have focused in our exposition only on a single aggregated attribute (e.g., Sal in our example). This is done for simplicity. Our ideas can be easily extended to include more aggregated attributes as long as we are willing to keep a distinct aggregate lineage for each attribute. E..g., suppose we also had a Rev (for Revenue) attribute for each employee. In such a case we keep two lineage relations, one for Sal and one for Rev. The algorithm to compute them can be thought of as a parallel implementation of two copies of the algorithm Comp-Lineage. We need only one pass through the original data. The only difference is that now, a) we need the two total sums S Sal and SRev and b) for each tuple t, we have two probabilities p Sal t and p Rev t , the first to be used for the lineage related to attribute Sal and the second to be used for the lineage related to attribute Rev.
Algorithm Comp-Lineage performs a weighted random sampling which selects with replacement b out of n tuples of R where the weight wi for the tuple with identifier i is equal to the value of attribute A of this tuple. Using b copies (each copy selects a single element) of the weighted random sampling with reservoir algorithm presented in [10], we can implement Comp-Lineage in one-pass over R, in O(bn) time and O(b) space. This implementation can also be applied to data streams and to settings where the values of n and S are not known in advance.
However the technique in [10], does not seem to be efficiently parallelizable, at least not in a direct way. Thus the problem of how to efficiently implement our technique in distributed computational environments such as MapReduce remains open. Issues about how to implement sampling in MapReduce are discussed in [15]. Another open problem is how to apply this technique to evolving data [14]. In data streams, we assume that the sample is to be computed over the entire data. When data continuously evolve with time, the sample may also change considerably with time. The nature of the sample may vary with both the moment at which it is computed and with the time horizon over which the user is interested in. We have not investigated here how to provide this flexibility.
Comparison with Synopses for Data
There has been extensive research on approximation techniques for aggregate queries on large databases and data streams. Previous work considers a variety of techniques including random sampling, histograms, multivalued histograms, wavelets and sketches (see e.g., [7] and the references therein for details and applications of those methods). Most of the previous work on histograms, wavelets, and sketches focuses on approximating aggregate queries on a given attribute A for specific subsets of the data that are known when the synopsis is computed (e.g., the synopsis concerns the entire data stream or a particular subset of the database). Thus, such techniques typically lose the correlation between the approximated A values and the original values of other attributes. For the more general case of multiple queries that can be posed over arbitrary sets of attributes and subsets of the data not specified when the synopsis is computed, those techniques typically lead to an exponential (in the number of other attributes involved) increase in the size of the synopsis (see e.g., [8,9]).
In contrast, our approach is far more general and does not focus on approximating queries over specific attributes or subsets of the data. Our algorithm computes a small sample without assuming any knowledge on the set of queries and keeps the association between the sampled A values and all other attributes. Then, we can use the Aggregate Lineage to approximate large-valued sum queries over arbitrary subsets of the data that can be expressed over any set of attributes. The Aggregate Lineage can approximately answer a number of queries exponential in its size. Of course, the queries should be oblivious to the actual Aggregate Lineage (technically, they should be computed by an oblivious adversary), but this technical condition applies to all previously known randomised synopses constructions (see e.g., [7]).
Conclusions
We have presented a method that computes lineage for aggregate queries by applying weighted sampling. The aggregate lineage can be used to compute arbitrary test aggregate queries on subsets of the original data. However the test queries can be computed with good approximation only if the result of each test query is large enough with respect to the total sum over all the data. The aggregate lineage we compute cannot be used to compute test queries if their result is comparatively small. We give performance guarantees.
Parallel implementation on frameworks such as MapReduce is not studied here. The naive approach of parallelizing [10] would have either to transmit a large amount of data to the several compute nodes, or to have a makespan linear in n.
The idea of getting a single (possibly weighted) random sample from a large data set and using it for repeated estimations of a given quantity has appeared before in the context of machine learning and statistical estimation. Boosting techniques [26] such as bootstrapping [11,12] are used. In [20] BLB is used for the efficient estimation of bootstrap-based quantities in a distributed computational environment.
Fig. 1 .
1F r additional attribute recording the frequency of a tuple in the lineage L R.A the lineage relation computed by Algorithm Comp-Lineage wrto attribute A Q(R.A) (Sec. 4) a sub-sum query computed over original relation R wrto attribute A Q ′ (L R.A ) (Sec. 4) sub-sum query Q computed over the aggregate lineage relation I Q R (Sec. 4) set of identifiers of the tuples in relation R that satisfy the predicates in query Q Main symbols used in the paper.
Fig. 2 .
2Properties of Aggregate Lineage L Salaries.Sal for b = 8, 852. The first two columns describe the data. The next three columns describe the Aggregate Lineage relation. The last column shows how we use this lineage to compute sub-sums.
and thus we have that Yj = Q ′ j (LR.A)/S, i.e., Yj is equal to the approximated result of the SUM query divided by S. Also, by linearity of expectation, IE[Yj] = Qj(R.A)/S. Applying the Chernoff-Hoeffding bound, we obtain that for the particular choice of b, with probability at least 1 − p/m, the actual value of Yj differs from its expectation Qj(R.A)/S by at most ǫ, which implies that Q ′ j (LR.A) differs from Qj(R.A) by at most ǫS. Formally, by the Chernoff-Hoeffding bound 3 , IPr[|Q ′ j (LR.A) − Qj(R.A)| > ǫS] = IPr[|Yj − Qj(R.A)/S| > ǫ] ≤ 2 exp −2ǫ 2 b ≤ p/m , where the last inequality follows from the choice of b. Applying the union bound, we obtain that IPr[∃j ∈ [m] : |Q ′ j (LR.A) − Qj(R.A)| > ǫS] ≤ p which concludes the proof of the lemma.
): Let Y 1 , . . . , Y b be random variables independently distributed in [0, 1], and let Y = 1 b b k=1 Y k . Then, for all ǫ > 0, IPr[|Y − IE[Y ]| > ǫ] ≤ 2 exp −2ǫ 2 b , where exp = 2.71 . . . is the basis of natural logarithms.
Lineage used to be referred to as "explain" in database papers of the late 80's.
In technical terms, the queries are posed by an oblivious adversary, i.e., an adversary that knows how exactly Aggregate Lineage works but does not have access to its random choices. The restriction to oblivious adversaries is standard and unavoidable, since if one knows the actual value of L R.A , he can construct a query that includes only tuples not belonging to L R.A , for which no meaningful approximation guarantee would be possible.
AcknowledgementsThis work was supported by the project Handling Uncertainty in Data Intensive Applications, cofinanced by the European Union (European Social Fund -ESF) and Greek national funds, through the Operational Program "Education and Lifelong Learning", under the program THALES.
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| Healthcare Information Technology (IT) has made great advances over the past few years and while these advances have enable healthcare professionals to provide higher quality healthcare to a larger number of individuals it also provides the criminal element more opportunities to access sensitive information, such as patient protected health information (PHI)and Personal identification Information (PII). Having an Information Assurance (IA) programallows for the protection of information and information systems andensures the organization is in compliance with all requires regulations, laws and directive is essential. While most organizations have such a policy in place, often it is inadequate to ensure the proper protection to prevent security breaches. The increase of data breaches in the last few years demonstrates the importance of an effective IA program.To ensure an effective IA policy, the policy must manage the operational risk, including identifying risks, assessment and mitigation of identified risks and ongoing monitoring to ensure compliance. | 10.5121/ijnsa.2012.4508 | [
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HEALTHCARE IT: IS YOUR INFORMATION AT RISK?
September 2012
Kimmarie Donahue [email protected]
Information Assurance Project Lead
San AntonioTXUSA
Syed
Shawon
PhDM Rahman [email protected]
and Adjunct Faculty
University of Hawaii-Hilo
USA
Capella University
MinneapolisUSA
HEALTHCARE IT: IS YOUR INFORMATION AT RISK?
International Journal of Network Security & Its Applications (IJNSA)
45September 201210.5121/ijnsa.2012.450897Information AssurancePersonal Identification InformationProtected Health InformationandIT Security
Healthcare Information Technology (IT) has made great advances over the past few years and while these advances have enable healthcare professionals to provide higher quality healthcare to a larger number of individuals it also provides the criminal element more opportunities to access sensitive information, such as patient protected health information (PHI)and Personal identification Information (PII). Having an Information Assurance (IA) programallows for the protection of information and information systems andensures the organization is in compliance with all requires regulations, laws and directive is essential. While most organizations have such a policy in place, often it is inadequate to ensure the proper protection to prevent security breaches. The increase of data breaches in the last few years demonstrates the importance of an effective IA program.To ensure an effective IA policy, the policy must manage the operational risk, including identifying risks, assessment and mitigation of identified risks and ongoing monitoring to ensure compliance.
INTRODUCTION
Advances in today's Healthcare Information Technology have allowed healthcare professionals to become highly connected to the information highway which provides them greater access to patients and their healthcare information. In today's society it is becoming more and more common to see healthcare professionals to utilizing mobile devices, such as laptops, to allow them to be better connected, according to a recent survey conducted, over 80 percent of Healthcare IT professionals allow IPads on the enterprise network and 65 percent provide support for iPhones and iPod Touch devices [1].
While these advances have provided great benefits for healthcare professionals and their patients, they also pose a real danger not only to the patients protected heath information (PHI) but also the organizations that are affected by data breaches. Healthcare data breaches are up by ninety seven percent in 2011; this is usually due to malicious attacks such as, theft of laptops, carelessness of an insider threat or hacking [2]. A study from the Ponemon Institute estimated the cost of data breaches has increased for the fifth year in a row to $7.2 million dollars and costs organizations an average of $214 per compromised record [3]. Medical ID theft is becoming big business, the World Privacy Forum found that a social security number has a street value of one dollar and a stolen medical identity goes for fifty dollars [4].
Once the dangers have been identified the next step is to ensure that senior management understands the risk if nothing is done to manage the threats and vulnerabilities. Typically security is an afterthought and organizations are reluctant to budget accordingly, until a security breach occurs.
Information Assurance refers to the measures that organizations take to protect and defend not only information but also information systems by ensuring their availability, integrity, authentication, confidentiality and non-repudiation [5].
Most individuals expect Healthcare professionals and IT professionalsto uphold a higher ethical and legal standard due to their access to sensitive information required in their daily responsibilities and as such should always practice due care and due diligence. It is important that management stay in touch with their personnel, there are personnel that may usually maintain a high ethical standard, but are often easily be influenced if the right opportunity presented itself, such as pending layoff or financial difficulties. Cyber-Ark, a security firm, conducted a survey called "The Global Recession and its Effects on Work Ethics" which revealed that of the individuals interviewed, 56% of workers worried about the loss of their jobs and over half admitted they downloaded sensitive data in order to try to use it at their next position [6].
If an organization fails to practice due care or due diligence,they will be accountable, financially and/or criminally, especially if the threat could have been avoided. A data breach in California resulted in the California Department of Health fining several California hospitals $675,000 for repeatedly failing to adequately secure patient data,and in Louisville a university hospital physician hospital inadvertently exposed the personal information of over 700 patients receiving kidney dialysis treatment when he set up the patient database on an unsecure website [7]. In 2010 CPA Tim W. Kasley was disbarred for failure to exercise due diligence while he was preparing tax returns for a corporation, he failed to determine the right information he received for the tax return [8]. This emphasise the need for all individuals in an organization to exercise due care and due diligence.
• Performing Due care:
o Taking responsibility when identifying a potential threat or risk and having the responsibility to know or find out what actions will correct or eliminate the threat or risk.
• Performing Due diligence:
o Taking the responsibility to put controls in place and properly monitor to mitigate or eliminate the threat or risk, and perform risk analysis as required.
Establishing a comprehensive information assurance program will ensure that all individuals understand the importance to ensuring the security of not only the enterprise but also the sensitive information on accessed on the enterprise. Organizations must ensure that that not only is a Information Assurance program is in place but also that it is adequate enough to address the increased threats to the confidentiality, integrity, and availability of sensitive information, such as patient health information, and stays in compliance with all financial, legal and health care compliance regulations. To ensure the success of any IA program it is essential that senior management fully endorse the program, because if senior management does not support it then no one else will support it.
An important part of an IA program is building the policies that will help provide for an Defense in Depth approach to IA by providing layers of principles and controls that cover not only the individuals but also the various process and technology the organization uses, including roles and responsibilities, acceptable use, etc.
Risk management is a critical part of any IA program because new threats and vulnerabilities are emerging every day. Risk management helps ensures that thesethreats and vulnerabilities are properly identified, and mitigated to reduce risk. It is impossible to eliminate all risk from an enterprise, but mitigated to a level the organization is willing to accept. A risk assessment will provide for the identification of potential threats and vulnerabilities as well as possible mitigation strategies that will bring the identified threats and vulnerabilities to an acceptable level.
Risk management is a continuous process that requires monitoring and updates to ensure that the proper protection and ensure effectiveness and compliance with laws, regulations and directives.
Kingdom Hospital is a factious hospital that is used for this case study and as a hospital has unique requirements, such as medical devices, wireless devices (tablets, blackberries, etc.), Health Insurance Portability and Accountability Act (HIPAA) and Privacy issues that are not currently being fully met. This increases the threat to the confidentiality, integrity, and availability of Kingdom resources and assets, such as electronic protected health information (PHI).
INFORMATION ASSURANCE PROGRAM
When developing and implementing an IA program it is essential that senior management fully supports and is committed to the program, because without their support and direction the program will not be successful. It is important to develop the policies with input from all business owners to ensure that the IA policies and procedures will not only protect IT resources, but also align themselves with the organizations business objectives. The policies should also be developed so they are easily understood, because is the policies are not understood by all individuals they will just be ignored. The IA plan must be based upon the mission and business objectives of the organization and support the future direction of the organization in order to be successful [9].
ETHICAL AND LAWS IMPLICATIONS
Everyone has a responsibility when it comes to information awareness. Senior management has the key responsibility to not only support and promote the IA program to the organization but also to ensure that the organization is in compliance with the industry laws and regulations, such as Privacy act, Health Insurance Portability and Accountability Act (HIPAA), etc., because a data breach can be costly for an organization. Potential lawsuits resulting from data breaches could result in big losses, such as the Emory Healthcare in Atlanta, which has a pending class action lawsuit, resulting from a data breach that compromised the personal information of an estimated 315,000 patients, which could cost the organization an estimated $200 million [10].
ESTABLISH INFORMATION ASSURANCE POLICIES
ACCEPTABLE USE
A key element of any IA policy is an acceptable use policy. This policy will establish what behaviour is appropriate and acceptable by the organization; this includes what the individual is/is not allowed to do, but also covers the consequences for noncompliance with the policy.
Recommended practice is to have each individual sign an acceptable use policy agreement; this not only helps to minimize potential legal action but also helps ensure compliance with industry laws and regulations, such as privacy and HIPAA.
TRAINING AND AWARENESS
An essential element of an IA Policy is the training and awareness program and part if an effective IA program is the comes to protecting information and information systems. Individual actions, intentional or unintentional, greatly contribute to the loss of data breaches. In April 2012 alo breaches resulting in almost 1.1 million records being lost, these breaches were caused by the actions of an insider threat, actions Protecting against this type of threat can be challenging because the individuals have access to the data may not fully understand the impact of their actions. This is why training is essential to educate on potential threats and vulnerabilities that exi react appropriately when faced with the threat. depending on the audience, such as general users, managers or IT/IA staff (Fig 1).
Figure 1: Specialised IA Training
To ensure training is most effective it should the organization and annually as a refresher. This initial training provides a basic understanding of various IA concept and principles to ensure the confidentiality, integrity, authentication and availability of the organizations resources and assets IA training requirement, it is important to establish more focused IA training based on individuals specific roles within the IA program, such as managers and staff.
IA training, depending on role, should include, but not limit principles;
•Training focused on how their privacy and information is protected and their rights and responsibilities.
•Basic: Training on organizational security policy, rules of behavior, thier individual role and responsibilities.
Patient
International Journal of Network Security & Its Applications (IJNSA), Vol.4, No.5, September
WARENESS
An essential element of an IA Policy is the training and awareness program and an e IA program is the personnel; they are usually the first line of defen comes to protecting information and information systems. Individual actions, intentional or unintentional, greatly contribute to the loss of data breaches. In April 2012 alone three data breaches resulting in almost 1.1 million records being lost, these breaches were caused by the actions while unintentional, still had devastating Protecting against this type of threat can be challenging because the individuals have access to may not fully understand the impact of their actions. This is why training is essential threats and vulnerabilities that exist, but also how to recognise them and react appropriately when faced with the threat. IA training should be specialized and focused depending on the audience, such as general users, managers or IT/IA staff (Fig 1). Figure 1: Specialised IA Training Requirements ensure training is most effective it should conducted within 30 days of personnel the organization and annually as a refresher. This initial training provides a basic understanding of various IA concept and principles to ensure the confidentiality, integrity, authentication and availability of the organizations resources and assets. In addition to the basic IA training requirement, it is important to establish more focused IA training based on individuals specific roles within the IA program, such as managers and staff.
IA training, depending on role, should include, but not limited to, the following concepts and •Advanced: Basic training and focus training for individuals with priviledged access to data and systems.
•Intermediate: Basic training and additional training focused on thier role as a manager in ensuring compliance with policy and promoting the IA Policy.
User Manager
User training:
• Basic understanding of IA concepts,
• Physical security of computer hardware and software,
• Proper protection, handling, storage and access of information,
• Privacy Act, HIPAA and PII protection,
• Recognizing and responding to common threats, vulnerabilities and risks,
• Understanding policy for non compliance to the IA rules and regulations,
• Users should understand their role and responsibility in ensuring the organization's security posture.
Manager training;
• User training,
• Intermediate understanding of IA concepts,
• Intermediate understanding of threats, vulnerabilities and risk,
• Intermediate understanding of governing regulations, laws and directives,
• Managers should understand their role in supporting, promoting and ensuring all users comply with organizations security posture.
IT/IA staff training
• User training,
• Intermediate understanding of IA concepts,
• Intermediate understanding of threats, vulnerabilities and risk,
• Intermediate understanding of governing regulations, laws and directives,
• Staff members with privileged access must understand their role in supporting the organizations security posture but also the added responsibility to act ethically and legally when using their privileged access to IT resources and assets.
Patients Training;
• This is usually in the form of a pamphlet or a handout that informs the patient their role and rights regarding privacy, and how the organization protects their privacy and sensitive information.
To develop and maintain the best possible IA workforce it is essential that individuals, according to the role they are in, train, achieve and maintain appropriate certifications. This allows organizations to ensure that individuals will possess the required knowledge and skills to best perform their individual roles. The following table shows the basic recommended certifications and training for IT workforce members.
RISK MANAGEMENT
In order to accomplish the goal of protecting an organizations network and information infrastructure from potential compromise or loss, risk management provides a framework for identifying, assessing and mitigating risk down to an acceptable level [12].
RMF
The National Institute of Standards and Technology (NIST) provides a dynamic approach to risk management by allowing for the ability to effectively manage security risks in environments that deal with complex threats and vulnerabilities and rapidly changing missions [13]. This framework allows and organizations to not only assess the risks to their information resources but also to select the best security controls that protects individuals and information systems and also aligns with the organizations business objectives. This risk management framework consists of six steps; categorize, select, implement, assess, authorize and monitor.
CATEGORIZE
An organizations senior management cannot ensure the protection of their information and information systems unless they have a full understanding of what exists in their organization. This is why it is an important start to the risk management process to perform the categorization process; this provides a way to determine the sensitivity and critical nature of the information that resides on their information systems. This allows a decision to be determined on the systems risk level based on how critical the information system is and what the impact to the organization, such as financial, legal, etc, would be to the organization if the system was lost or compromised.
SELECT
Once the categorization process has been completed the determination can be made as to what security controls would be best implemented on the information system to protect the information systems confidentiality, integrity, authentication and availability. The security controls must be able to mitigate the systems risk and not interfere with the organization's mission and ability to function. The implemented security controls must be cost effective; it is not efficient to implement a security control that costs $10,000 for an information system that was determined to be a non-critical system.
IMPLEMENT
Once the appropriate security controls have been determined, it is necessary to implement them into the organizations security plan. This will address the documentation, including how the security control will be implemented and with what security configuration settings applied.
ASSESS
The assessment of the security controls will to determine if the control is effective. This ensures that the security control was not only implemented properly, meets the security objectives and expectations, and is cost effect for the organization.
AUTHORIZE
The authorization to operate is the approval that the systems documentation, risk assessment and overall system is determined to be at an acceptable level for the organization.
MONITOR
To ensure the continued effectiveness of implemented security controls continuous monitoring is essential. It is essential to monitorand assess the systems security controls to ensure that any changes in the configuration or updates, this ensures that thesystems security is still effective.
MONITOR/UPDATE TO ENSURE COMPLIANCE
The best built IA program can be implemented but is it is not constantly assessed and updated then it will become useless and give an organization a false sense of security. New threats and vulnerabilities are not the only concern, new system updates and/or configuration changes occur that could have an impact on security. This is why constant monitoring and updating as required are essential to maintain security and compliance and provide the organization a way to evaluate the effectiveness of the IA program.
There are several ways that assist an organization in this effort, such as a change configuration management board, automated tools to monitor systems on the enterprise and security assessments to help evaluate changes to the systems or the operations environment.
CASE STUDY: KINGDOM HOSPITAL
Kingdom Hospital is a hospital that realizes the importance of ensuring for the protection of its information resources and assets, including personnel, services and systems. Kingdom hospital has unique requirements, such as medical devices, wireless devices (tablets, blackberries, etc.), Health Insurance Portability and Accountability Act (HIPAA) and Privacy act requirements that not currently fully met. To ensure the protection against threats to the confidentiality, integrity, and availability of Kingdom resources and assets, Kingdom's information assurance policy currently addresses the basic security of the enterprise. It is essential that with new threats and vulnerabilities emerging every daythat the hospitals security posture is routinely assessed and updated to ensure that the hospitals enterprise network is not only secure but also in compliance with all financial, legal and health care compliance regulations. This is why an independent risk assessment was conducted for this case study will cover the entire hospital network including, but not limited to, remote clinics, wireless security, physical security and hospital security policy and compliance.
Kingdom enterprise hospital network security assessment was performed on several critical areas such as physical security (not only the network but also to sensitive areas such as the operating rooms, maternity wards and morgue), security management policies. The ultimate goal is to find the perfect balance between security and hospital operational functionality. Some areas where security weaknesses identified included the following;
• Shared network accounts: There are many network accounts that utilize a common username and password. This increases the security risk of unauthorized individuals gaining access to the network and does not provide for proper authentication. • Weak authentication: the network user accounts only require a username and password with no password complexity required. Weak password policy is a key to network security because it is a common way attackers attack to gain access to a network. • HIPAA Compliance: there is a lack of policy to ensure compliance to HIPAA Regulations. • Acceptable use policy: no policy in place to communicate to the users what the acceptable behaviour is while utilising network resources, such as internet, email, social media and user responsibilities, increases various threats including viruses, malware and data loss.
KINGDOM VULNERABILITIES IDENTIFIED
During the security assessment,several identified threats and vulnerabilities pose could expose the hospital network to potentially dangerous security exposures. These vulnerabilities could open Kingdom hospital with damaging results, such as;
• Financial loss: a successful network attack allows an attacker the opportunity to manipulate data for financial gains.
• Loss of Reputation: Kingdom hospital success relies on it superior reputation with its customers and surrounding community, and a successful Denial of Service (DoS) attack or a breach leading to a compromise of sensitive information could lead to customers losing faith and confidence in Kingdom hospital.
• Legal consequences: A security breach leading to the compromise or loss of sensitive data opens up Kingdom hospital to legal issues.
Kingdom Enterprise Network Risks Identified
Mystical hospital security assessment has identified several as high-risk software and hardware risks to the enterprise network security. Some of these risks identified include;
• Software security: Software updates not implemented in a timely manner. • Firewall/IDS: Mystical does not have a firewall or IDS in place to protect the enterprise network from hostile attacks.
Kingdom Security Requirements
Kingdom's risk assessment has identified several security requirements that need to be address to ensure the protection of the patients, employees and various visitors at the hospital.
Addressing these requirements will also protect the hospital assets, including financial and reputation in the surrounding area. Security requirements identified include;
• Natural Disasters: Being located in Mississippi the threat from hurricanes and tropical storms are a real and constant threat. o Mitigation: Perform quarterly review of Disaster Recovery/Business Continuity Plan with all required personnel on a quarterly basis.
• HIPAA Compliance: Being globally connected the threat from the compromise and/or loss of sensitive patient data is a real threat that requires special attention to ensure the protection of Personal Identification Information (PII), Protected Health Information (PHI) and Electronic Heath Records (EHR) Etc. o Mitigation: Have security measures in place to monitor email and ensure proper security measures are in place.
• Facility: It is essential to maintain constant electrical power and Air conditioning to maintain the critical hospital systems, such as servers and lifesaving medical equipment.
o Mitigation: Ensure proper generators are in place and regularly tested in the event of power/HVAC loss.
• Insider threat: Intentional or unintentional, the insider threat posed a major risk to the network due to the potential for malware, virus or compromise of sensitive information.
o Mitigation: Ensure all users receive proper training and measures are in place to monitor network for abnormalities.
Kingdom Security Training Policy
An important element of Kingdom security is the training and awareness program, because usually the first individual to see potential security issues is the end user. This IA training policy will ensure that all employees receive and maintain the proper level of IA training.Kingdom Hospital requires all individuals to attend an initial Information Training course and have a basic understanding of various concepts and principles to ensure the confidentiality, integrity and availability of Kingdom Hospital resources and assets, since they become our first line of defense. Some items that should be included in training;
• Recognizing unsafe email and/or attachments, • Recognizing and avoiding potentially unsafe websites, • Ensuring users understand the requirement to encrypt all emails containing sensitive information. • Auditing and monitoring policy. sensitive data being releases on social media sites.
Lack of Accountability Depends
Requires users to acknowledge and sign Acceptable user agreement.
usersrepeat failure to comply with IA policy.
CONCLUSION
Healthcare and healthcare IT has been advancing at an amazing rate in the last few years and while this had provided healthcare professional the ability to not only provide better care for their patients, it has also introduced new threats and vulnerabilities to information systems and sensitive patient information. Ensuring for the proper protection is an ongoing and challenging effort with its unique challenges, but with a properly developed and maintained IA program, it is possible. The ultimate goal is to understand the organizations business goals and objectives and find the proper balance between implementation of security controls, patient care and the business goals. These goals can be accomplished by prioritizing risks identified in the risk assessment and reduce those risks to an acceptable level. The U.S. Department of Health and Human Services has a website that gives the public information about medical ID theft and included is the Office of Inspector General's list of Most Wanted Health Care Fugitives (OIG) [14]. It is important that an organization never underestimate the users within the organization and like the Naval Intelligence motto; In God we trust, all others we monitor [15].
•
Social media: Controls are not in place to ensure the users are fully aware of the potential dangers to Kingdom hospital from irresponsible or careless use of various social media sites. • Malware/Virus: Controls are not in place to ensure that users are fully aware of how their irresponsible actions when utilizing internet and email can have dangerous effects on the network due to malware and/or virus infections. • Mobile/Wireless devices: Controls are not in place to protect the wireless network again unauthorized mobile devices and/or individuals. • Router/Switch: Configurations are default setting resulting in routers and switched being in an unsecure status.
Table 1 :
1Recommended IA staff certifications/trainingLEVEL
CERTIFICATION
TRAINING
BASIC TECHNICIAN
A+, OS
COMPTIA, MICROSOFT,
CISCO
SENIOR TECHNICIAN
SECURITY+, MCSE
COMPTIA, MICROSOFT
NETWORK TECHNICIAN
CCNA
CISCO
BASIC IA LEVEL
SECURITY+
COMPTIA
IA MANAGER
CISSP
ISC2
AUDITOR
CISA
ISACA
Table 2 :
2Kingdom Hospital Risk Assessment Summary Ensure mobile/wireless security is included in IA training,implement encryption on all mobile devices, Implement tracking software on all mobile devices, if possible. Dataloss/compromise from loss of devices (mobile devices). Compliance HIPAA/Privacy High Update policy to include HIPAA/Privacy requirements. Add additional training Potential data breaches, sensitive information found accessible or sent through email. System Regular Updates not completed Medium Ensure required updates are tested and completed in a timely manner. Unauthorized access to systems/data. Virus/Malware Medium Ensure users are educated about the dangers and ensure signatures are updated in a timely manner. Social Media High Ensure social media dangers are included in required IA training.Kingdom Risk Assessment Summary
Risk
Impact
Control/Mitigation Action
Early warning signs
Infrastructure
default
passwords on
Routers/Switch
Depends
Ensure all devices have
passwords changed from default.
Authorized systems denied
logons, unauthorised devices
accessing network.
Authors Bio:Kimmarie Donahue is a Senior Information Assurance Project Lead with KSH Solutions that provides Information Assurance, Information Security and Certification & Accreditation support services to various organizations within the federal government. She received her Master's Degree in Information Technology, Information Assurance and Security from Capella University. She currently holds the CISSP certification and NSA/CNSS (INFOSEC) Recognition. Kimmarie is a member of several professional organizations including ISC2, HIMSS, ISACA and ISSA.Syed (Shawon) M. Rahman is an assistant professor in the Department of Computer Science andEngineering at the University of Hawaii-Hilo and an adjunct faculty of information Technology, information assurance and security at the Capella University. Dr. Rahman's research interests include software engineering education, data visualization, information assurance and security, web accessibility, and software testing and quality assurance. He has published more than 75 peer-reviewed papers. He is a member of many professional organizations including ACM, ASEE, ASQ, IEEE, and UPE.
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Information Security Plan for Flight Simulator Applications". Jason Slaughter, Syed Rahman, International Journal of Computer Science & Information Technology (IJCSIT). 33Shawon)Slaughter, Jason and Rahman, Syed (Shawon); " Information Security Plan for Flight Simulator Applications"; International Journal of Computer Science & Information Technology (IJCSIT), Vol. 3, No 3, June 2011
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Shawon). S Dreelin, Gregory, Syed Rahman, International Journal of Computer Networks & Communications. ENTERPRISE SECURITY RISK PLAN FOR A SMALL BUSINESS. IJCNCDreelin, S., Gregory and Rahman, Syed (Shawon);" ENTERPRISE SECURITY RISK PLAN FOR A SMALL BUSINESS"; International Journal of Computer Networks & Communications (IJCNC)
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Shawon); "VIDEO SURVEILLANCE IN THE CLOUD?. David Neal, Syed Rahman, The International Journal of Cryptography and Information Security. IJCISNeal, David and Rahman, Syed (Shawon); "VIDEO SURVEILLANCE IN THE CLOUD?"; The International Journal of Cryptography and Information Security (IJCIS)
Analytic of China Cyberattack. Robert Lai, Syed Rahman, The International Journal of Multimedia & Its Applications (IJMA). 4Shawon)Lai, Robert and Rahman, Syed (Shawon); "Analytic of China Cyberattack"; The International Journal of Multimedia & Its Applications (IJMA), June 2012, Volume 4, Number 3,
| []
|
[
"Four bright eclipsing binaries with γ Doradus pulsating components: CM Lac, MZ Lac, RX Dra and V2077 Cyg",
"Four bright eclipsing binaries with γ Doradus pulsating components: CM Lac, MZ Lac, RX Dra and V2077 Cyg",
"Four bright eclipsing binaries with γ Doradus pulsating components: CM Lac, MZ Lac, RX Dra and V2077 Cyg",
"Four bright eclipsing binaries with γ Doradus pulsating components: CM Lac, MZ Lac, RX Dra and V2077 Cyg"
]
| [
"John Southworth \nAstrophysics Group\nKeele University\nST5 5BGStaffordshireUK\n",
"Timothy Van Reeth \nInstitute of Astronomy\nKU Leuven\nCelestijnenlaan 200DB-3001LeuvenBelgium\n",
"John Southworth \nAstrophysics Group\nKeele University\nST5 5BGStaffordshireUK\n",
"Timothy Van Reeth \nInstitute of Astronomy\nKU Leuven\nCelestijnenlaan 200DB-3001LeuvenBelgium\n"
]
| [
"Astrophysics Group\nKeele University\nST5 5BGStaffordshireUK",
"Institute of Astronomy\nKU Leuven\nCelestijnenlaan 200DB-3001LeuvenBelgium",
"Astrophysics Group\nKeele University\nST5 5BGStaffordshireUK",
"Institute of Astronomy\nKU Leuven\nCelestijnenlaan 200DB-3001LeuvenBelgium"
]
| [
"MNRAS",
"MNRAS"
]
| The study of pulsating stars in eclipsing binaries holds the promise of combining two different ways of measuring the physical properties of a star to obtain improved constraints on stellar theory. Gravity (g) mode pulsations such as those found in γ Doradus stars can be used to probe rotational profiles, mixing and magnetic fields. Until recently few γ Doradus stars in eclipsing binaries were known. We have discovered g-mode pulsations in four detached eclipsing binary systems from light curves obtained by the Transiting Exoplanet Survey Satellite (TESS) and present an analysis of their eclipses and pulsational characteristics. We find unresolved g-mode pulsations at frequencies 1-1.5 d −1 in CM Lac, and measure the masses and radii of the component stars from the TESS data and published radial velocities. MZ Lac shows a much richer frequency spectrum, including pressure modes and tidally-excited g-modes. RX Dra is in the northern continuous viewing zone of TESS so has a light curve covering a full year, but shows relatively few pulsation frequencies. For V2077 Cyg we formally measure four pulsation frequencies, but the available data are inadequate to properly resolve the g-mode pulsations. V2077 Cyg also shows total eclipses, with which we obtain the first measurement of the surface gravity of the faint secondary star. All four systems are bright and good candidates for detailed study. Further TESS observations are scheduled for all four systems, with much improved temporal baselines in the cases of RX Dra and V2077 Cyg. | 10.1093/mnras/stac1993 | [
"https://export.arxiv.org/pdf/2207.09169v1.pdf"
]
| 250,644,335 | 2207.09169 | f21b27ed3ceeff29e5cbe5dc13de4d16f887c8db |
Four bright eclipsing binaries with γ Doradus pulsating components: CM Lac, MZ Lac, RX Dra and V2077 Cyg
19 Jul 2022
John Southworth
Astrophysics Group
Keele University
ST5 5BGStaffordshireUK
Timothy Van Reeth
Institute of Astronomy
KU Leuven
Celestijnenlaan 200DB-3001LeuvenBelgium
Four bright eclipsing binaries with γ Doradus pulsating components: CM Lac, MZ Lac, RX Dra and V2077 Cyg
MNRAS
000000019 Jul 2022Accepted YYYYMMDD. Received YYYYMMDD; in original form YYYYMMDD.Preprint 20 July 2022 Compiled using MNRAS L A T E X style file v3.0stars: fundamental parameters -stars: binaries: eclipsing -stars: oscillations
The study of pulsating stars in eclipsing binaries holds the promise of combining two different ways of measuring the physical properties of a star to obtain improved constraints on stellar theory. Gravity (g) mode pulsations such as those found in γ Doradus stars can be used to probe rotational profiles, mixing and magnetic fields. Until recently few γ Doradus stars in eclipsing binaries were known. We have discovered g-mode pulsations in four detached eclipsing binary systems from light curves obtained by the Transiting Exoplanet Survey Satellite (TESS) and present an analysis of their eclipses and pulsational characteristics. We find unresolved g-mode pulsations at frequencies 1-1.5 d −1 in CM Lac, and measure the masses and radii of the component stars from the TESS data and published radial velocities. MZ Lac shows a much richer frequency spectrum, including pressure modes and tidally-excited g-modes. RX Dra is in the northern continuous viewing zone of TESS so has a light curve covering a full year, but shows relatively few pulsation frequencies. For V2077 Cyg we formally measure four pulsation frequencies, but the available data are inadequate to properly resolve the g-mode pulsations. V2077 Cyg also shows total eclipses, with which we obtain the first measurement of the surface gravity of the faint secondary star. All four systems are bright and good candidates for detailed study. Further TESS observations are scheduled for all four systems, with much improved temporal baselines in the cases of RX Dra and V2077 Cyg.
INTRODUCTION
Eclipsing binary star systems (EBs) are crucial objects for understanding the physics governing stellar structure and evolution, because they are the only stars whose masses and radii can be measured to high precision and accuracy from observational material and geometrical arguments alone. Precise measurements of the masses and radii of stars in EBs were instrumental in the development of stellar theory (e.g. Russell 1914), in the verification of the first modern generation of theoretical stellar models (e.g. Andersen et al. 1990;Pols et al. 1997), and continue to be used to guide theoretical progress (Claret & Torres 2018;Tkachenko et al. 2020). In the current era of échelle spectroscopy and space-based light curves it is possible to measure masses and radii for stars in EB to precisions approaching 0.1% (Maxted et al. 2020;Graczyk et al. 2021). The impact of photometry from space missions has been reviewed in detail by Southworth (2021a).
Another class of stars capable of posing high-quality constraints on stellar theory is that of the pulsating variables (Aerts et al. 2010). Among these, stars showing gravity-mode (g-mode) pulsations are well-suited to studying the interiors of stars as g-modes, which have buoyancy as the dominant restoring force, can travel deep within stars whilst leaving observable signatures on the stellar surface (Bowman 2020). γ Doradus (γ Dor) variables (Kaye et al. 1999) are stars of spectral types A and F that show g-mode pulsations with periods ranging from 0.3 d to 4 d and amplitudes up to 0.1 mag (Henry et al. 2007; Grigahcène et al. 2010).
In recent years, asteroseismic analyses of main-sequence stars with g-mode pulsations have allowed us to place constraints on multiple phenomena. These include near-core stellar rotation (e.g. Bouabid et al. 2013;Van Reeth et al. 2016;Christophe et al. 2018;Takata et al. 2020a,b;Szewczuk et al. 2021), convective core boundary mixing (e.g. Michielsen et al. 2019;Wu & Li 2019;Mombarg et al. 2021), extra envelope mixing (e.g. Mombarg et al. 2020;Pedersen et al. 2021) and magnetic fields (e.g. Prat et al. 2019;Van Beeck et al. 2020;Lecoanet et al. 2022).
The advantages of asteroseismology and binarity can be combined in objects which show both eclipses and pulsations. These systems have the potential to set exacting constraints on stellar theory (e.g. Johnston et al. 2019b). A wide variety of pulsating stars are known in EBs, including δ Scuti da Silva et al. 2014;Lee et al. 2020Southworth 2021d), β Cephei (Southworth et al. , 2021, SPB (Clausen 1996;) and δ Cephei (Pilecki et al. 2018). Over the last decade, multiple γ Dor variables in EBs have been discovered as well. An emblematic case is that of KIC 11285625 (Debosscher et al. 2013). Others were found by, among others, Hambleton et al. (2013); Lee et al. (2014); Sowicka et al. (2017); Hełminiak et al. (2017); Guo et al. (2017b); Lampens et al. (2018); and Van Reeth et al. (2022).
A sample of 115 γ Dor variables in EBs was identified by Gaulme & Guzik (2019) via a systematic search of the Kepler EB catalogue (Kirk et al. 2016). Li et al. (2020) performed an asteroseismic evaluation of these targets and reported the detection of g-mode period-spacing patterns for 34 of them, as well as for one discovered by Colman et al. (2022). An independent systematic search and asteroseismic evaluation of the Kepler EB catalogue was conducted by Sekaran et al. (2020), resulting in a different sample of 93 γ Dor variables in EBs, with detected period-spacing patterns for seven of them. So far, asteroseismic modelling has only been achieved for a small number of targets: KIC 10080943 (Schmid & Aerts 2016;Johnston et al. 2019a), KIC 10486425 (Zhang et al. 2018), KIC 7385478 and KIC 9850387 (Zhang et al. 2020;Sekaran et al. 2021).
The accumulating data from the NASA Transiting Exoplanet Survey Satellite (TESS; Ricker et al. 2015) is enabling previously unattainable analyses of many of the bright and well-known variable stars. The catalogue of objects includes many EBs with a long observational history (e.g. Southworth 2020). In the course of trawling this database we have discovered four bright EBs whose light curves also show γ Dor pulsations. In this work we present these discoveries, studies of their light and radial velocity (RV) curves, and a first analysis of the nature of their pulsations.
OBSERVATIONS
The TESS satellite (Ricker et al. 2015) is in the process of observing almost the entire celestial sphere, split into 69 overlapping sectors. Each sector is a 24 • ×96 • strip of sky and is photometrically monitored for 27.4 d, through a filter with a high throughput between 600 nm and 1000 nm, with a pause near the midpoint for the download of data to Earth. During the nominal two-year mission, fullframe images (FFI) were taken at a default cadence of 1800 s, while a subset of stars were monitored in short cadence, where successive exposures were combined to yield light curves with a cadence of 120 s. In the ongoing extended mission, the FFI cadence has been changed to 600 s.
Reduced light curves are obtained from the data by the Science Processing Operations Center (SPOC; Jenkins et al. 2016) and made available through the MAST portal 1 . We used the simple aperture photometry (SAP) light curves in the current work. We visually inspected the TESS light curves of approximately 2000 EBs present in a bibliography maintained by the first author 2 , and identified four objects that show both eclipses and previously-unrecognised γ Dor pulsations: CM Lac, MZ Lac, RX Dra and V2077 Cyg.
CM Lac was observed only in sector 16 (2019/09/11 to 2019/10/07) and one further sector of observations is scheduled 3 (sector 56, 2022 September). MZ Lac was observed in two consecutive sectors, 16 and 17 (2019/11 to 2019/11/02) and this will recur in sectors 56-57 (2022 Sept-Oct 41 (2021/07/23 to 2021/08/20), and an additional four consecutive sectors (53-56) will be observed in 2022. The light curves for one sector of each of the four targets are shown in Fig. 1.
Following the target selection, we carefully reassessed the quality of the reduced SAP light curves. First, we inspected the aperture masks used in the pixel data and evaluated both the level of captured flux and the level of contamination from nearby stars. For all stars, the standard SPOC aperture mask was accepted, with the exception of RX Dra. Here, we re-extracted the light curve from the pixel data using custom aperture masks, to account for the varying contamination level between sectors by the nearby non-variable star 2MASS J19024829+5844050. Second, we applied additional detrending to the light curves by fitting low-order polynomials. To avoid overfitting, these detrending curves were optimised simultaneously with preliminary binary and pulsation models, built from a sum of sine waves:
f (t) = 10 i=1 ai sin (2πνi (t − t0) + φi) + 10 j=1 aj sin (2πjν orb (t − t0) + φj)
Here, νi and ν orb are estimates for the dominant pulsation frequencies and the orbital frequency, ai and aj are their amplitudes, and φi and φj are their phases, respectively. The mean timestamp of each light curve was used as the zeropoint t0.
ANALYSIS METHODS
Light curve analysis
All four systems are well-detached EBs so are suitable for analysis with the JKTEBOP code 4 (Southworth et al. 2004a;Southworth 2013). In this code, the fractional radii of the stars 5 are parameterised as their sum (rA + rB) and ratio (k = r B r A ), and the orbital eccentricity (e) and argument of periastron are parameterised using the combination terms e cos ω and e sin ω. Other fitted parameters include the orbital period (P ), midpoint of primary eclipse (T0), orbital inclination (i), and the central surface brightness ratio (J). We used the quadratic limb darkening (LD) law, fitted for the linear LD coefficients (uA and uB), and fixed the quadratic LD coefficients (vA and vB) to theoretical values (Claret 2018). We define the primary eclipse to be the deeper of the two types of eclipse, the primary star (the star eclipsed during primary eclipse) to be star A, and the secondary to be star B.
The presence of pulsations complicates the light curve analysis. In our initial analysis we ignored this and fitted the entire light curve with a JKTEBOP model including one low-order polynomial versus time per TESS half-sector to remove slow drifts in brightness due to instrumental or astrophysical effects. Once the residuals of the fit were obtained, these were subsequently used in the pulsation analysis (see below). For CM Lac and V2077 Cyg we then extracted each eclipse from the data, plus half an eclipse duration either side, and then fitted these together with a low-order polynomial for each eclipse representing the pulsation-induced brightness changes around the time of the eclipse. This procedure was successful in the case of CM Lac, but led to small systematic biases for V2077 Cyg due to the amplitude and complexity of the pulsational signature. For RX Dra the surfeit of data mandated a different approach: an initial fit to the full data followed by phase-binning down into a more manageable number of observations (see Section 4.3).
Uncertainties in the fitted parameters were calculated using the Monte Carlo and residual-permutation simulations implemented in JKTEBOP (Southworth 2008). The Monte Carlo simulations account for correlations between photometric parameters, and required the errorbars in the TESS data to be rescaled to force a reduced χ 2 of χ 2 ν = 1. The residual-permutation simulations are sensitive to red noise in data, so are useful in the case of unmodelled pulsations. The two errorbars were checked for every parameter and the larger of the two retained in each case. From recent analyses of the TESS light curves of other EBs (Southworth 2021b,c) we have found that the Monte Carlo and residual-permutation simulations typically agree with each other and with errorbars obtained from splitting up light curves into multiple subsets for analysis in isolation. Additional support for the reliability of the Monte Carlo and residual-permutation errorbars, and of the JKTEBOP model, comes from a recent analysis of AI Phe by multiple researchers working independently and using multiple codes and error estimation methods (Maxted et al. 2020).
The systematics in the modelling of V2077 Cyg have only a small effect on our results because the system is totally-eclipsing and thus possesses a higher intrinsic determinacy of the photometric parameters (Kopal 1959).
Pulsation analysis
After a good fit to the light curve had been obtained, the residuals of the fit were calculated and subjected to a frequency analysis. The data points that were obtained during the eclipses were excluded to minimise the influence of any remaining binary signal in the residuals. Because the available TESS data are in most cases limited to one or two sectors per star, which are generally insufficient to resolve individual g-mode pulsations, this remains a preliminary analysis. Using a Lomb-Scargle periodogram (Scargle 1982), we iteratively prewhitened the light curve (Degroote et al. 2009) to measure the pulsations for which the signal-to-noise ratio S/N 4 (Breger et al. 1993).
The amplitudes, frequencies and phases of these pulsations were subsequently optimised with a non-linear least-squares fit (Bowman Table 1. Parameters measured from the eclipses using the JKTEBOP code. Parameters with a superscripted N were calculated using the nominal physical constants and solar quantities defined by the IAU (Prša et al. 2016
L 3 −0.0043 ± 0.0070 0.377 ± 0.018 0.0176 ± 0.0013 −0.026 ± 0.018 u A 0.237 ± 0.029 0.62 ± 0.13 0.300 ± 0.031 0.248 ± 0.033 u B 0.182 ± 0.040 0.673 ± 0.073 0.195 ± 0.057 0.35 ± 0.11 v A 0.232 (fixed) 0.229 (fixed) 0.081 ± 0.050 0.229 (fixed) v B 0.229 (fixed) 0.229 (fixed) 0.316 ± 0.096 0.229 (fixed) e cos ω 0.000289 ± 0.000082 0.00732 ± 0.00025 0.000030 ± 0.000010 0.0 fixed e sin ω 0.0009 ± 0.0023 0.4116 ± 0.0079 0.00094 ± 0.00062 0.0 fixed P (d) 1.6046879 ± 0.0000046 3.158794 ± 0.00009 3.78639491 ± 0.00000009 5.937226 ± 0.000068 T 0 (BJD/TDB)
2458752.67073 ± 0.00032 2458765.08683 ± 0.00014 2458857.410151 ± 0.000002 2458697.52198 ± 0.00028 & Michielsen 2021) and the S/N ratios were reevaluated using these optimised parameter values. The derived pulsation frequencies and their combinations were then compared with harmonics of the binary orbital frequency to identify possible tidally excited (e.g., Fuller 2017; , tidally perturbed (e.g., Reyniers & Smeyers 2003;Bowman et al. 2019;Steindl et al. 2021) or non-linearly coupled oscillations (e.g., Burkart et al. 2012;Guo 2020). Finally, we searched for combination frequencies
K A ( km s −1 ) 120.0 ± 3.4 54.52 ± 0.82 K B ( km s −1 ) 157.0 ± 3.3 V γ,A ( km s −1 ) 22.7 ± 1.9 −0.23 ± 0.40 V γ,B ( km s −1 ) 25.0 ± 1.8 Derived parameters: r A 0.1860 ± 0.0015 0.1351 ± 0.0032 0.12747 ± 0.00008 0.08238 ± 0.00038 r B 0.1678 ± 0.0021 0.1296 ± 0.0016 0.09107 ± 0.00008 0.04064 ± 0.00034 e 0.0009 ± 0.0015 0.4117 ± 0.0079 0.00094 ± 0.00059 ω ( • ) 71 +213 −14 88.091 ± 0.041 88.2 ± 1.5 Light ratio 0.495 ± 0.018 0.691 ± 0.062 0.4088 ± 0.0008 0.0750 ± 0.0022 Mass of star A (M N ⊙ ) 2.01 ± 0.10 Mass of star B (M N ⊙ ) 1.54 ± 0.09 Radius of star A (R N ⊙ ) 1.636 ± 0.032 Radius of star B (R N ⊙ ) 1.476 ± 0.ν k = niνi + nj νj
where νi, νj and ν k are measured frequencies, and ni and nj are integer numbers such that |ni| + |nj | ≤ 2. Such combination frequency identifications were accepted if they agreed within 3σ.
Physical properties
Published spectroscopic orbits exist for both components of the CM Lac system, allowing us to measure its physical properties directly. To do so we included the published radial velocities (RVs) in the JKTEBOP analysis in order to determine the absolute masses and radii of the stars. In the case of V2077 Cyg RVs are available for the primary but not the secondary component. In this situation the masses and radii of the stars cannot be measured directly (e.g. Hilditch 2001), but it is possible to obtain the surface gravity of the secondary (Southworth et al. 2004b(Southworth et al. , 2007 and thus verify its evolutionary status. The measured properties of the four targets are given in Table 1 Wachmann (1931) and physical properties were measured by Popper (1968). The most detailed study of the system was by Liakos & Niarchos (2012), who determined its physical properties based on light curves observed in the BV RI bands and RVs from a set of 28 mediumresolution spectra. The TESS light curve of CM Lac (Fig. 1) contains 12 primary and 14 secondary eclipses. The pulsations are strong enough to significantly affect the eclipse depths so we cut out the data around each eclipse from the light curve and modelled the out-of-eclipse brightness through each as a quadratic function, as described in Section 3.1. We found that it was possible to constrain one LD coefficient for each star. We also fitted for the amount of third light (L3), obtaining a slightly negative value. This is physically plausible if the background subtraction in the TESS images is too strong. Although the orbit appears circular, a slight eccentricity is preferred for the TESS data. A phased light curve is shown in Fig. 2. Our final solution includes the RVs from Liakos & Niarchos (2012) in order to determine the full properties of the system; we fitted for the velocity amplitudes (KA and KB) and systemic velocities (Vγ,A and Vγ,B) of both stars (Fig. 3). The reduced χ 2 of the fit was forced to unity for each dataset by scaling the data errors. We tried including historical times of primary eclipse as well (Wachmann 1931;Kreiner et al. 2001) but they occurred systematically too late so we rejected them -this may suggest that the orbital period of CM Lac is not constant. The uncertainties from residual permutation are larger than those from Monte Carlo. Our results are in Table 1: they agree well with those of Liakos & Niarchos (2012). Further RVs are needed to refine the mass measurements, which are currently precise to only 5%.
Analysis of the pulsations
Our frequency analysis reveals the presence of γ Dor pulsations. This can easily be seen from the Lomb-Scargle periodogram of the TESS light curve (excluding the eclipses) shown in Fig. 4. There is unresolved variability with frequencies between 1.0 and 1.5 d −1 . This is most likely caused by g-mode pulsations, as the (tentative) dominant frequency that can be obtained with iterative prewhitening (listed in Table 2), differs significantly from the harmonics of the binary orbital frequency. However, because the variability is unresolved, the measured frequency value cannot be assigned to a single g-mode pulsation. The asymmetric shape of the light curve is also typical for g-mode pulsations (Kurtz et al. 2015).
We note that the second harmonic of the binary orbital frequency also lies within the frequency range of the observed variability, as shown in Fig. 4. However, because the available TESS data only cover 27 d, we cannot investigate the nature of the variability at this frequency. It may be caused by a residual signature of the binarity or unresolved nearby g-mode pulsation frequencies, but more highprecision photometric data are needed to resolve this issue. The unresolved variability around the fourth and sixth harmonics of the orbital frequency are likely aliases of the unresolved variability around the second harmonic, as can be deduced from the spectral window shown in the top panel of Fig. 4. From a visual inspection of the TESS light curve, we find that the pulsations remain visible during the primary eclipse, but vanish during the secondary eclipse. This means that the g-mode pulsations belong to the secondary star.
MZ Lacertae
Analysis of the binarity
MZ Lac was found to be an EB by Miller & Wachmann (1971), where it is labelled VV 399, but has been the subject of very little study since. It exhibits apsidal motion (Silhan 1990) with a period of 424 ± 6 yr (Bulut et al. 2016). The TESS light curve shows deep eclipses (Fig. 1) and covers two sectors. The secondary eclipse is at approximately phase 0.5 but is significantly longer than the primary, so e cos ω ≈ 0 but e sin ω is large.
The pulsations in MZ Lac have a much lower amplitude than the eclipses so we treated them as red noise: we fitted the whole dataset with JKTEBOP without attempting to reject out-of-eclipse data or compensate for the pulsations in any way. We fitted for the same parameters as for CM Lac, except for those related to RVs and for the inclusion of only four polynomials for the out-of-eclipse brightness of the system (one for each TESS half-sector). Due to the apsidal motion we did not attempt to use published eclipse timings to help constrain the orbital ephemeris. A substantial amount of third light is required to obtain a good fit to the data.
We were able to find a good fit to the light curve (Table 1). The residual-permutation errorbars are significantly larger than the Monte Carlo errorbars, which suggests it would be possible to improve the light curve analysis in future by either subtracting the pulsations or fitting for them simultaneously. The full physical properties of the system cannot be calculated as no RVs are available. Gaia DR2 (Gaia Collaboration 2018) gives a T eff of the system of 6900 K; if this is appropriate for star A then the T eff of star B is 6700 K based on the value of J we measure. These values are consistent Table 3. An overview of the parameter values for the significant frequencies of MZ Lac. For frequencies that lie close to harmonics of the binary orbital frequency ν orb , the differences between those frequencies and their corresponding harmonics are also given. Possible combination frequencies are listed in the last column.
Analysis of the pulsations
From a detailed analysis of the TESS photometry, we find that MZ Lac exhibits an interesting variety of pulsation behaviour. Following the binary modelling, we excluded the eclipses from the residual light curve and used this clipped light curve for the iterative prewhitening. The prewhitened pulsation frequencies of MZ Lac are listed in Table 3 and shown in Fig. 5. Ten frequencies below 5 d −1 have been measured from the TESS light curve. Five of these lie significantly close to harmonics of the orbital frequency 6 , and four are well within 3σ. A visual inspection of the light curve, phase-folded with the binary orbital period as shown in Fig. 2, reveals that these frequencies are likely related to tidally excited pulsations. An imperfect fit of the binary orbit to the light curve might cause similar residual signal in the Fourier transform at integer multiples of the orbital frequency. However, the dominant variation seen in the phase-folded light curve (in Fig. 2) occurs on a timescale much shorter than the orbital period, which is consistent with tidally excited pulsations, but not with residual signal of the binary orbit itself.
We also obtain five non-harmonic low frequencies. While these have values of typical γ Dor-type pulsations, three pairs of them form combination frequencies that (almost) coincide with harmonics of the orbital frequency: 2ν2 + ν6 = 11.000(3) ν orb , ν5 + ν8 = 11.9966(12) ν orb and ν2 + ν9 = 12.0303(16) ν orb . The first two combinations match the closest orbital harmonics within 3σ, and the last one matches the orbital harmonic within νres. This can be caused by nonlinear mode coupling, whereby the orbital harmonic is the parent frequency and the observed non-harmonic frequencies are the child frequencies (e.g., Burkart et al. 2012;Weinberg et al. 2013;Guo et al. 2017a). Finally, there are four pressure-mode (pmode) pulsations, with frequencies between 46 and 51.5 d −1 .
From a visual inspection of the TESS light curve shown in Fig.2, we find that the pulsations appear to be less visible during both eclipses. Our best explanation for this apparent physical inconsistency is that the (tidally excited) pulsations were partially fitted by the binary model in the eclipses. This could have happened in particular if the pulsations had a similar effect on the eclipse shape as LD, as some LD coefficients were included as fitted parameters. The fitted LD coefficients are indeed larger than expected. Whilst one approach to fixing this problem might be to fix the LD coefficients, this risks biasing the measured radii because theoretical LD coefficients are not completely reliable. An alternative approach in future could be to simultaneously model the eclipses and the pulsations.
RX Draconis
Analysis of the binarity
RX Dra has been known to be eclipsing for over a century. The first photometric analysis was given by Shapley (1913) but it has received very little attention subsequently. It came to the authors' attention by its inclusion in a list of EBs that are candidates for hosting pulsating stars (Soydugan et al. 2006), and inspection of its TESS light curve revealed clear pulsations of the γ Dor type. TESS observations are available for 13 consecutive sectors (14-26) followed by two further sectors (40 and 41). At the time of writing, observations are also ongoing for 14 consecutive sectors (47-60); see Section 2. We base our results in the current work on the data from sectors 14-26.
Because of the huge amount of data (230 766 observations sampled at 120 s cadence for one year) covering a large number of eclipses (approximately 170) we first obtained a preliminary solution with JKTEBOP to establish the orbital ephemeris. We then converted the data into orbital phase and binned them into 1000 points equally spaced in phase. The primary eclipse is annular and the secondary eclipse is total. A good fit to the binned data can be obtained comparatively easily due to the total eclipses. We included orbital eccentricity and third light as fitted parameters and both come out at very small but probably non-zero. It was also possible to fit for both LD coefficients for both stars, an occurrence the first author has not previously encountered despite having recently studied the TESS light curves of over 50 EBs. The errorbars from the Monte Carlo algorithm were adopted as they are significantly larger than those from the residual-permutation approach. Our results are given in Table 1. The uncertainties in the fractional radii are below 0.1%, but we recommend that interested users increase them to at least 0.1% as this is
Analysis of the pulsations
RX Dra is an outlier in our sample. While we only have one or two TESS sectors of data available for most of our stars, RX Dra is located in the northern CVZ and has been observed during 13 con-secutive sectors, spanning 351 days in total. This has allowed us to better resolve the individual stellar pulsation frequencies and measure them with a higher precision, as listed in Table 4 and shown in Fig. 6. Despite the short orbital period, all observed pulsations were found to be g-mode pulsations. Four of the 24 measured frequencies form combinations within 3σ. Additionally, the observed pulsation spectrum is relatively sparse, given that we have almost a full year of photometry at our disposal. The inclusion of additional photometric observations, such as those that are currently being taken during TESS cycle 4, may improve the window functions of the stellar pulsations sufficiently to allow us to measure additional g-mode pulsation frequencies.
Similarly to the pulsations observed for MZ Lac, we cannot assign the pulsations of RX Dra to a specific component. Here, the narrow triangular shapes of the eclipses do not suffice to evaluate the pulsation amplitudes as a function of the light contribution of the individual components, as illustrated in Fig. 2. Additionally, the measured g-mode frequencies are too sparse to detect clear g-mode periodspacing patterns, which would have allowed us to perform pulsation mode identification and select a group or groups of pulsations that originate from the same stellar component in the binary system. There is, however, clear structure present in the observed g-mode pulsation spectrum. The inclusion of the forthcoming TESS observations will be needed to detect period-spacing patterns for RX Dra (e.g. Van Reeth et al. 2015) if they are present in the data.
V2077 Cygni
Analysis of the binarity
V2077 Cyg was identified as an EB using photometry from the Hipparcos satellite, and given its GCVS designation by Kazarovets et al. (1999) as a result. The only study of the object so far was performed by Molenda-Żakowicz et al. (2007), who obtained 29 spectra and measured the spectroscopic orbit of the primary star. The secondary is much fainter and was not detected in the spectra. The TESS light curve (Fig. 1) shows distinctive variability characteristic of γ Dor pulsators, plus primary eclipses that are much deeper than the secondary eclipses. The light curve is strongly reminiscent of the prototypical system KIC 11285625 (Debosscher et al. 2013), except that the eclipses are total. TESS has observed the star during three sectors (14, 26 and 41) and four further sectors of observations (53, 54, 55 and 56) are planned. The four consecutive sectors will be valuable for increasing the precision of the measured pulsation periods. The following analyses only includes sectors 14 and 26, as sector 41 was not available at the time they were performed.
The pulsation amplitudes are comparable to the eclipse depths so we proceeded as for CM Lac (Sect. 4.1): each eclipse was extracted from the light curve and normalised to zero differential magnitude using a polynomial function. We found that a third-order polynomial was necessary in some cases. The light curve was then fitted as for CM Lac, with the exception that we had RVs for only the primary star so did not fit for KB or Vγ,B Fig. 7). The results are given in Table 1. The fractional radii of the stars are well-measured because of the presence of total eclipses in this system. Molenda-Żakowicz et al. (2007) measured a T eff of 7066±201 K for the primary star, in good agreement with the Gaia DR2 value of 6900 K. Our value of J then implies a T eff of 5200 K for the secondary star. Using equation 4 from Southworth et al. (2007) we determined the surface gravity of the secondary star to be log gB = 4.607 ± 0.007, which is appropriate for a K-dwarf. The γ Dor pulsations arise in the primary component: the secondary is too cool to be a γ Dor star and it also does not contribute enough light to the system to produce pulsations with an amplitude of 0.1 mag.
Analysis of the pulsations
Four significant frequencies, listed in Table 5 and illustrated in Fig. 8, can be obtained from iterative prewhitening of the residual TESS light curve. The derived parameter values for these frequencies are all consistent with those expected for γ Dor-type pulsations. However, the available TESS data are again insufficient to ensure that the individual g-mode pulsation frequencies are properly resolved.
More TESS data are needed to properly identify the individual pulsations in this system. Fortunately, it is scheduled to be observed Figure 8. Part of the Lomb-Scargle periodogram (black) of the residual light curve of V2077 Cyg (after the best-fitting binary model has been subtracted). The harmonics of the orbital frequency ν orb (dashed grey lines) and the extracted pulsation frequencies (dashed red lines) are also indicated. in four consecutive sectors (53-56). If TESS continues to operate as expected, and precise radial velocities can be measured for both components, V2077 Cyg could become a useful system for the study of pulsations in a star of known mass and radius. Fig. 2 shows an increase in residuals for the binary fit during the eclipses. This arises because the much smaller secondary star is blocking only part of the surface of the pulsating primary star and thus breaking the spherical symmetry that attenuates the observed amplitudes of pulsations. V2077 Cyg may therefore be a good candidate for elipse mapping to spatially resolve the pulsations on the stellar surface (e.g. Bíró & Nuspl 2011).
SUMMARY AND CONCLUSIONS
The study of pulsating stars in EBs is a promising opportunity to improve our understanding of the interior physics of stars. Gravitymode pulsations are a high priority because they can probe regions deep inside a star and thus help constrain stellar rotation, convective core boundary mixing, envelope mixing, opacity and magnetic fields (see references in Section 1). In pursuit of this goal we have surveyed the TESS light curves of a sample of known EBs and detected g-mode pulsations in four of these. In this work we present a preliminary analysis of all four systems. CM Lac shows very deep eclipses (0.8 mag for the primary and 0.4 mag for the secondary) and unresolved g-modes with frequencies between 1 d −1 and 1.5 d −1 in the one sector of available TESS data. The available data are therefore insufficient for asteroseismology, but remain useful for the determination of the physical properties of stars. We measured the masses and radii of the component stars using the TESS data and the results of a published RV study, but additional RVs are needed to measure their masses to the canonical 2% precision (Andersen 1991).
MZ Lac is a more interesting system due to its large orbital eccentricity (Table 1) and richer pulsation spectrum. Based on the two sectors of TESS data for this object we found evidence for p-modes and for tidally excited or tidally perturbed g-mode pulsations. No RVs are currently available for this object so we are not yet able to determine its physical properties. RX Dra has been observed for 13 consecutive sectors by TESS but this abundance of data yielded only relatively few identified frequencies. The presence of total eclipses allowed the fractional radii of the stars to be measured to exceptional precision so the system is a good candidate for precise measurement of its physical properties. RX Dra is in the process of being observed for a further 14 consecutive sectors by TESS and these additional data may allow the detection of further pulsation frequencies.
V2077 Cyg shows beautiful γ Dor pulsations but the few currently-available TESS light curves are insufficient to perform mode identification. The system is scheduled for observation in three consecutive sectors by TESS in 2022 and these new data should greatly improve the asteroseismic potential of this system. V2077 Cyg is not promising from the viewpoint of determining its physical properties for two reasons. The eclipses are perturbed by the high-amplitude g-mode pulsations, but are total so precise fractional radii are still measurable. The secondary star is much fainter than the primary (light ratio of 0.075 in the TESS passband) so measuring its RVs will require high-quality spectroscopy. A single-lined spectroscopic orbit exists for this system, allowing us to measure the surface gravity of the secondary component and verify that it is a normal low-mass main-sequence star.
All four systems need detailed spectroscopic study to measure their physical properties precisely, which in turn provides context for the pulsation analysis. This work has begun.
DATA AVAILABILITY
All data underlying this article are available in the MAST archive (https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html).
Figure 1 .
1TESS SAP light curves of the four EBs. In each case only one sector is shown.
Figure 2 .
2TESS SAP light curves of the four EBs plotted versus orbital phase. In each case the coloured points give the observations and the black line shows the best fit. The residuals are plotted below each main panel on an inflated scale. Black points show the residuals binned according to orbital phase. The plot for V2077 Cyg has been split into panels concentrating on the eclipses for clarity. The polynomials versus time have been subtracted from the observations in all cases to remove any slow trends in brightness that would otherwise blur the figure in the vertical direction.
Figure 3 .
3Fitted spectroscopic orbit for CM Lac (blue lines) compared to the measured RVs fromLiakos & Niarchos (2012): red circles for star A and red squares for star B. The residuals of the fit are shown in the lower panels.
Figure 4 .
4Spectral window (top panel) and part of the Lomb-Scargle periodogram (black) of the TESS light curve of CM Lac, excluding the eclipses. The harmonics of the orbital frequency ν orb (dotted grey lines) and the extracted pulsation frequency (dashed red line) are also indicated.
Figure 5 .
5Sections of the Lomb-Scargle periodogram (black) of the residual light curve of MZ Lac. The harmonics of the orbital frequency ν orb (dotted grey lines) and the extracted pulsation frequencies are also indicated. The dashed red lines and dash-dotted blue lines mark non-harmonic and harmonic pulsation frequencies, respectively.
Figure 6 .
6Sections of the Lomb-Scargle periodogram (black) of the residual light curve of RX Dra. The harmonics of the orbital frequency ν orb (dotted grey lines) and the extracted pulsation frequencies (dashed red lines) are also indicated.
Table 4 .
4An overview of the parameter values for the significant frequencies of RX Dra. Possible combination frequencies within 3σ are also listed. limit of reliability established by comparing independent analyses of the same data for a similar EB(Maxted et al. 2020).
Figure 7 .
7Fitted spectroscopic orbit for V2077 Cyg (blue lines) compared to the measured RVs from Molenda-Żakowicz et al. (2007) for star A (red circles). The residuals of the fit are shown in the lower panel.
). By contrast, extensive observations exist for RX Dra because it is sited within the northern continuous viewing zone (CVZ) of TESS. It was observed in sectors 14-26 (2019/07/18 to 2020/07/04) and 40-41 (2021/06/08 to 2021/08/20) and observations are currently being obtained in sectors 47-60 (2021/12/30 to 2023/01/18). V2077 Cyg was observed in sectors 14 (2019/07/18 to 2019/08/15), 26 (2020/06/08 to 2020/07/04) 1 https://mast.stsci.edu/portal/Mashup/Clients/ Mast/Portal.html 2 A hobby during Covid-19 lockdowns. 3 https://heasarc.gsfc.nasa.gov/cgi-bin/tess/ webtess/wtv.py
). The T eff values are fromLiakos & Niarchos (2012) for CM Lac and fromMolenda-Żakowicz et al. (2007) for V2077 Cyg.CM Lac
MZ Lac
RX Dra
V2077 Cyg
Fitted parameters:
r A + r B
0.35383 ± 0.00078
0.2557 ± 0.0024
0.21854 ± 0.000099
0.12303 ± 0.00022
k
0.902 ± 0.018
0.893 ± 0.027
0.7144 ± 0.0010
0.4933 ± 0.0063
i ( • )
87.573 ± 0.082
88.53 ± 0.41
88.589 ± 0.018
88.668 ± 0.048
J
0.5956 ± 0.0074
0.888 ± 0.048
0.8063 ± 0.0055
0.321 ± 0.013
, including published effective temperature (T eff ) values where available.CM Lac was discovered to be an eclipsing binary by4 DISCUSSION OF INDIVIDUAL SYSTEMS
4.1 CM Lacertae
4.1.1 Analysis of the binarity
Table 2 .
2The parameter values for the significant prewhitened frequency of CM Lac. Bracketed quantities indicate the uncertainty in the final digit of the preceding number.Frequency (d −1 )
Amplitude (mmag)
Phase (rad)
S/N
1.1799 (2)
12.5 (1)
0.4276 (12)
6.0
0
1
2
3
4
5
0
1
2
amplitude (mmag)
46
48
50
52
frequency (d 1 )
0.0
0.2
0.4
0.6
amplitude (mmag)
Table 5 .
5An overview of the parameter values for the significant frequencies of V2077 Cyg. The identification of possible combination frequencies within 3σ is also given.Frequency
Amplitude
Phase
S/N Combinations
(d −1 )
(mmag)
(rad)
ν 1
0.48209 (1)
6.97 (10)
−0.244 (2)
4.1
ν 2
0.65013 (1)
8.5 (1)
−0.163 (2)
5.3
ν 3 0.708696 (3)
33.4 (1)
0.4049 (5) 21.1
ν 4
1.41741 (1)
8.72 (10)
0.075 (2)
6.7 ν 4 ≈ 2ν 3
© 0000 The Authors
http://www.astro.keele.ac.uk/jkt/codes/jktebop.html 5 r A = R A a and r B = R B a where R A and R B are the true radii and a is the semimajor axis of the relative orbit.
MNRAS 000, 1-11 (0000)
MZ Lac experiences apsidal motion, so its sidereal and anomalistic periods are not identical. However, they differ by only 0.000035 d so our analysis is not affected by which we choose to represent the orbital frequency.
ACKNOWLEDGEMENTSTVR gratefully acknowledges support from the Research Foundation Flanders (FWO) under grant agreement N°12ZB620N. We thank Johanna Molenda-Żakowicz for sending us her RV measurements for V2077 Cyg, and Dominic Bowman and the anonymous referee for discussions and comments. The TESS data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute (STScI). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support to MAST for these data is provided by the NASA Office of Space Science via grant NAG5-7584 and by other grants and contracts. Funding for the TESS mission is provided by the NASA Explorer Program. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France; the SAO/NASA Astrophysics Data System; and the VizieR catalogue access tool, CDS, Strasbourg, France.
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| []
|
[
"The muonic hydrogen Lamb shift and the proton radius",
"The muonic hydrogen Lamb shift and the proton radius"
]
| [
"Clara Peset \nGrup de Física Teòrica and IFAE\nUniversitat Autònoma de Barcelona\n08193Bellaterra, Barcelona\n"
]
| [
"Grup de Física Teòrica and IFAE\nUniversitat Autònoma de Barcelona\n08193Bellaterra, Barcelona"
]
| []
| We obtain a model independent expression for the muonic hydrogen Lamb shift up to O(m µ α 6 , m µ α 5 m 2 µ m 2 ρ ). The hadronic effects are controlled by the chiral theory, which allows for their model independent determination. We give their complete expression including the pion and Delta particles. Out of this analysis and the experimental measurement of the muonic hydrogen Lamb shift we determine the electromagnetic proton radius: r p =0.8412(15) fm. This number is at 6.8σ variance with respect to the CODATA value. The parametric control of the uncertainties allows us to obtain a model independent determination of the error, which is dominated by hadronic effects. | 10.1016/j.nuclphysbps.2015.01.049 | [
"https://arxiv.org/pdf/1411.3931v1.pdf"
]
| 119,212,529 | 1411.3931 | 361878a758abe64d2289bc71e845b05033a48677 |
The muonic hydrogen Lamb shift and the proton radius
Clara Peset
Grup de Física Teòrica and IFAE
Universitat Autònoma de Barcelona
08193Bellaterra, Barcelona
The muonic hydrogen Lamb shift and the proton radius
Chiral LagrangiansBound statesHeavy quark effective theorySpecific calculations
We obtain a model independent expression for the muonic hydrogen Lamb shift up to O(m µ α 6 , m µ α 5 m 2 µ m 2 ρ ). The hadronic effects are controlled by the chiral theory, which allows for their model independent determination. We give their complete expression including the pion and Delta particles. Out of this analysis and the experimental measurement of the muonic hydrogen Lamb shift we determine the electromagnetic proton radius: r p =0.8412(15) fm. This number is at 6.8σ variance with respect to the CODATA value. The parametric control of the uncertainties allows us to obtain a model independent determination of the error, which is dominated by hadronic effects.
Introduction
The recent measurement of the muonic hydrogen (µp) Lamb shift, E(2P 3/2 ) − E(2S 1/2 ) [1,2], ∆E exp L = 202.3706 (23) meV (1) and the associated determination of the electromagnetic proton radius: r p = 0.84087(39) fm has led to a lot of controversy. The reason is that this number is 4-7σ away from previous determinations of this quantity coming from hydrogen and electron-proton (ep) scattering [3,4]. In order to asses the significance of this discrepancy it is of fundamental importance to perform the computation of this quantity in a model independent way. In this respect, the use of effective field theories (EFTs) is specially useful. They help organizing the computation by providing with power counting rules that asses the importance of the different contributions. This becomes increasingly necessary as higher order effects are included. Even more important, these power counting rules allow to parametrically control the size of the higher order non-computed terms and, thus, give an estimate of the error. The EFT approach is specially convenient in the case of bound states where there are different, well separated scales, namely, the hard scale or reduced mass (m r ), the * Speaker Email address: [email protected] (Clara Peset) soft scale or typical momentum (m r v ∼ m r α) and the ultrasoft scale or typical binding energy (m r v 2 ∼ m r α 2 ).
In the case of µp we need to deal with several scales:
m p ∼ m ρ , m µ ∼ m π ∼ m r ≡ m µ m p m p + m µ , m r α ∼ m e .
from which we obtain the main expansion parameters by considering ratios of them
m π m p ∼ m µ m p ≈ 1 9 m e m r ∼ m r α m r ∼ m r α 2 m r α ∼ α ≈ 1 137 .(2)
These, together with the counting rules given by the EFT provide the necessary tools to perform the full analysis of the Lamb shift in µp up to leading-log O(m µ α 6 ) terms and leading O(m µ α 5 m 2 µ m 2 ρ ) hadronic effects. In our approach we combine the use of Heavy Baryon Effective Theory (HBET) [5,6], Non-Relativistic QED (NRQED) [7] and, specially, potential NRQED (pN-RQED) [8][9][10]. Partial results following this approach can be found in [11][12][13] (see [14] for a review). In Ref. [15] we computed the n = 2 Lamb shift with accuracy O(m µ α 6 , m µ α 5 m 2 µ m 2 ρ ). A more detailed account of the hadronic part can be found in [16]. These proceedings are based on the work carried out in Refs. [15,16].
Lamb shift and extraction of the proton radius
All contributions to the Lamb shift up to the order of our interest are summarized in Table 1
which compared to Eq. (1), gives a value of the proton radius r p = 0.8412 (15) fm. The first term of Eq. (3) corresponds to the first ten entries of Table 1, which provide the QED-like contribution up to O(m r α 5 ), plus the leading logs at O(m r α 6 ) which allow us to estimate the error of this number. The main contribution is the electron vacuum polarization at O(m r α 3 ). The remaining amount corresponds to higher order effects such as higher loops, relativistic corrections, ultrasoft photons or perturbation theory effects. A more detailed description (with comprehensive references) of this contribution can be found in [15]. The lower part of Table 1 summarizes all the hadronic contributions to the Lamb shift up to the order of our interest, which we explain here in more detail. All the hadronic contributions are encoded in the 1/m 2 potential in pNRQED:
D had d ≡ −c had 3 − 16παd had 2 + 2πα 3 r 2 p m 2 p ,(4)δV (2) had (r) ≡ 1 m 2 p D had d δ 3 (r) → ∆E = − D had d m 2 p (m r α) 3 n 3 π δ l0 . (5)
Entries 11 and 12 correspond to the r p -dependent term in Eq. (3) (i.e. the Wilson coefficient c had D ), and the 13th entry allows us to estimate the uncertainty of this number. The last term of Eq. (3) comes from the two last entries of Table 1. The 14th entry of the table corresponds to the hadronic vacuum polarization (encoded in the matching coefficient d had 2 ), which can be determined from dispersion relations (DR) [17] with a small error for our purposes, and this is the number we quote here. The last entry of Table 1 corresponds to the two photon exchange (TPE) and deserves more care since it generates most of the uncertainty in the Lamb shift. This contribution is encoded in the Wilson coefficient c had 3 , which is unique from an EFT point of view, although it is customary to split it into Born and polarizability pieces so that c had
3 = c Born 3 + c pol 3 .
We have computed both of them separately, in the pure chiral limit and also including the contribution due to the ∆(1232), which could give the largest subleading contribution not only for being the closest resonance to the proton, but also because both of them are degenerate in the large-N c limit [18]. When going from HBET to NRQED, we integrate out the pions and the Delta and we can write
c had 3 ∼ α 2 m µ m π F(m π /∆)+O α 2 m µ m ρ ,
where no counterterms are needed to compute the leading order of the contribution, as it is argued in Refs. [12,15]. The Born contribution at leading order in the NR expansion (which guarantees that only the low energy modes contribute to the integral) reads
1 O(m r α 3 ) V (0) VP 205.00745 2 O(m r α 4 ) V (0) VP 1.50795 3 O(m r α 4 ) V (0) VP 0.15090 4 O(m r α 5 ) V (0) VP 0.00752 5 O(m r α 5 ) V (0) LbL −0.00089(2) 6 O(m r α 4 × m 2 µ m 2 p ) V (2) + V (3) 0.05747 7 O(m r α 5 ) V (2) soft /ultrasoft −0.71902 8 O(m r α 5 ) V (2) VP 0.01876 9 O(m µ α 6 × ln( m µ m e )) V (2) ; c (µ) D −0.00127 10 O(m µ α 6 × ln α) V (2) V P ; c (µ) D −0.00454 11 O(m r α 4 × m 2 r r 2 p ) V (2) ; c (p) D ; r 2 p −5.1975 r 2 p fm 2 12 O(m r α 5 × m 2 r r 2 p ) V (2) VP ; c (p) D ; r 2 p −0.0283 r 2 p fm 2 13 O(m r α 6 ln α × m 2 r r 2 p ) V (2) ; c (p) D ; r 2 p −0.0014 r 2 p fm 2 14 O(m r α 5 × m 2 r m 2 ρ ) V (2) VP had ; d had 2 0.0111(2) 15 O(m r α 5 × m 2 r m 2 ρ m µ m π ) V (2) ; c had 3 ; r 3 0.0344(125)c pl i 3,Born = 4(4πα) 2 M 2 p m l i d D−1 q (2π) D−1 1 q 6 G (0) E G (2) E (−q 2 ) ,(6)
where G (0) E = 1 and G (2) E (q 2 ) together with an analytic expression for c pl i 3,Born can be found in [16]. This coefficient can also be related with (one of) the Zemach moments:
c pl i 3,Born = π 3 α 2 M 2 p m l i r 3 (2) ,(7)r 3 (2) = 48 π ∞ 0 dQ Q 4 G 2 E (−Q 2 ) − 1 + Q 2 3 r 2 . (8)
The Zemach moments can be determined in a similar way as the moments of the charge distribution of the proton. We have studied some and compared them to their values obtained applying DR techniques. A set of these results is summarized in Table 2 (a more complete discussion on this can be found in [16]). One would expect the chiral prediction to give the dominant contribution of r n for n ≥ 3 and the leading chiral log for n = 2. Nevertheless we observe large differences (bigger than the errors) with different determinations fitting experimental data to different functions [19][20][21]. In this respect, the chiral result could help shaping the appropriate fit function and thus, resolving the differences between the fitted results as well as assessing their uncertainties. This difference in the fit functions has an impact on the determination of the proton radius, as can be clearly seen in Ref. [26] v.s. Refs. [4,27] for direct fits to the ep scattering data, where the determination differs in about 3-σ. In any case, the reason for such large discrepancies should be further investigated. Note that for all n ≥ 3, the chiral expressions give the leading (non-analytic) dependence in the light quark mass as well as in 1/N c . This is a valuable information for eventual lattice simulations of these quantities where one can tune these parameters. We can extract the contribution of the Born term to the energy shift from Eq. (5), and this is what we quote in the last two entries of Table 3. The first two entries correspond to two different DR-analyses. Note that in the HBET computation the addition of the Delta has a good convergence. On the other hand, our result is much smaller than the standard ones obtained from DR. Whether this discrepancy is due to relativistic corrections or to a need for refining the fitting procedure should be further investigated. The polarizability contribution is computed through the diagrams represented in Fig. 1 for the pure chiral case and in Fig. 2 both for the tree level Delta exchange (top diagram) and for the one-loop Delta contribution. These diagrams are summed up in the polarizability tensor:
T µν pol = −g µν + q µ q ν q 2 S 1 (ρ, q 2 ) + 1 M 2 p p µ − M p ρ q 2 q µ p ν − M p ρ q 2 q ν S 2 (ρ, q 2 ) . (9)
The polarizability energy shift cannot be fully obtained from DR and thus, needs some subtractions. This fact In Table 4, we compare our HBET results to others obtained by a combination of DR for the inelastic term and different modelling functions for the subtraction term, and also to the result obtained using BχPT. This last one is carried out within a theory that treats the baryon relativistically. The result incorporates some subleading effects, which are sometimes used to give an estimate of higher order effects in HBχPT, but it also assumes that a theory with only baryons and pions is appropriate at the proton mass scale, which should be taken with due caution. Still, it would be desirable to have a deeper theoretical understanding of this difference, which may signal that relativistic corrections are important for the polarizability correction. In any case, the BχPT computation differs from our chiral result by around 50% which we consider reasonable. For the total TPE energy shift we obtain:
∆E TPE = ∆E Born + ∆E pol = 28.59(π) + 5.86(π&∆) = 34.4(12.5) µeV ,(11)
which, however is in good agreement with the total result [28] used for the determination of the proton radius in [2]. This result is a pure prediction of the EFT, and it is also the most precise result that can be obtained in a model independent way since O(m µ α 5 m 3 µ Λ QCD ) effects are not controlled by the chiral theory and would require new counterterms.
Conclusions
We have computed in a completely model independent way the Lamb shift for n = 2 in muonic hydrogen, which allows for the extraction of the proton radius, focusing on the hadronic contributions (mainly the TPE). Our result of the proton radius is 6.8σ away from the CODATA value and has much larger uncertainties. We have computed the pure chiral contribution to the TPE, and also the contribution due to the ∆(1232). This computation of the TPE gives a similar result to the one obtained by the combination of DR plus the use of different models. However, the partial computations (Born and polarizability) differ from the partial results obtained in these frameworks, fact that should be further understood.
+0.0633(144) meV,
Figure 1 :
1Diagrams corresponding to the pure chiral contribution (only pions) of the TPE.
Figure 2 :
2Diagrams corresponding to the Delta tree level and loop contribution (pions & Deltas) of the TPE.
Table 1 :
1The different contributions to the µp Lamb shift in meV units.
Table 2 :
2Values of r n in fermi units. The first two rows give the prediction from the EFT at LO and LO+NLO. The third row in the standard dipole fit. The last two rows are different DR analyses.
Table 3 :
3Predictions for the Born contribution to the n = 2 Lamb shift. The first two entries correspond to DR analyses. The last two entries are the predictions of HBET: at LO and at LO+NLO.makes our model independent computation even more
relevant. The Lamb shift obtained in HBET is:
∆E pol =
c
pl µ
3,pol
M 2
p
1
π
m r α
2
3
= 18.51(π)−1.58(∆)+9.25(π∆)
= 26.2(10.0) µeV
(10)
(µeV)
[22]
[23]
[24]
∆E pol
12(2)
7.4(2.4)
15.3(5.6)
(µeV) BχPT[25](π) HBET[13](π) [15](π&∆)
∆E pol
8.2( +1.2
−2.5 )
18.5(9.3)
26.2(10.0)
Table 4 :
4Predictionsfor the polarizability contribution to the n =
2 Lamb shift. The first 3 entries use DR for the inelastic term and
different modeling functions for the subtraction term.
AcknowledgementsThe author thanks Antonio Pineda for his collaboration in the development of this work. This work was supported by the Spanish grant FPA2011-25948 and the Catalan grant SGR2009-00894.
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| []
|
[
"Topological state transfers in cavity-magnon system",
"Topological state transfers in cavity-magnon system"
]
| [
"Xi-Xi Bao \nLanzhou Center for Theoretical Physics\nKey Laboratory of Theoretical Physics of Gansu Province\nLanzhou University\n730000LanzhouGansuChina\n",
"Gang-Feng Guo \nLanzhou Center for Theoretical Physics\nKey Laboratory of Theoretical Physics of Gansu Province\nLanzhou University\n730000LanzhouGansuChina\n",
"Lei Tan \nLanzhou Center for Theoretical Physics\nKey Laboratory of Theoretical Physics of Gansu Province\nLanzhou University\n730000LanzhouGansuChina\n\nKey Laboratory for Magnetism and Magnetic Materials of the Ministry of Education\nLanzhou University\n730000LanzhouPeople's Republic of China\n"
]
| [
"Lanzhou Center for Theoretical Physics\nKey Laboratory of Theoretical Physics of Gansu Province\nLanzhou University\n730000LanzhouGansuChina",
"Lanzhou Center for Theoretical Physics\nKey Laboratory of Theoretical Physics of Gansu Province\nLanzhou University\n730000LanzhouGansuChina",
"Lanzhou Center for Theoretical Physics\nKey Laboratory of Theoretical Physics of Gansu Province\nLanzhou University\n730000LanzhouGansuChina",
"Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education\nLanzhou University\n730000LanzhouPeople's Republic of China"
]
| []
| We propose an experimentally feasible scheme for realizing quantum state transfer via the topological edge states in a one-dimensional cavity-magnon lattice. We find that the cavity-magnon system can be mapped analytically into the generalized Su-Schrieffer-Heeger model with tunable cavity-magnon coupling. It can be shown that the edge state can be served as a quantum channel to realize the photonic and magnonic state transfers by adjusting the cavity-cavity coupling strength. Further, our scheme can realize the quantum state transfer between photonic state and magnonic state by changing the amplitude of the intracell hopping. With a numerical simulation, we quantitatively show that the photonic, magnonic and magnon-to-photon state transfers can be achieved with high fidelity in the cavity-magnon lattice. Spectacularly, the three different types of quantum state transfer schemes can be even transformed to each other in a controllable fashion. This system provides a novel way of realizing quantum state transfer and can be implemented in quantum computing platforms. | null | [
"https://arxiv.org/pdf/2203.13727v1.pdf"
]
| 247,748,905 | 2203.13727 | e592011bdb097e2004e99c98f3a8873969beaf2d |
Topological state transfers in cavity-magnon system
Xi-Xi Bao
Lanzhou Center for Theoretical Physics
Key Laboratory of Theoretical Physics of Gansu Province
Lanzhou University
730000LanzhouGansuChina
Gang-Feng Guo
Lanzhou Center for Theoretical Physics
Key Laboratory of Theoretical Physics of Gansu Province
Lanzhou University
730000LanzhouGansuChina
Lei Tan
Lanzhou Center for Theoretical Physics
Key Laboratory of Theoretical Physics of Gansu Province
Lanzhou University
730000LanzhouGansuChina
Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education
Lanzhou University
730000LanzhouPeople's Republic of China
Topological state transfers in cavity-magnon system
We propose an experimentally feasible scheme for realizing quantum state transfer via the topological edge states in a one-dimensional cavity-magnon lattice. We find that the cavity-magnon system can be mapped analytically into the generalized Su-Schrieffer-Heeger model with tunable cavity-magnon coupling. It can be shown that the edge state can be served as a quantum channel to realize the photonic and magnonic state transfers by adjusting the cavity-cavity coupling strength. Further, our scheme can realize the quantum state transfer between photonic state and magnonic state by changing the amplitude of the intracell hopping. With a numerical simulation, we quantitatively show that the photonic, magnonic and magnon-to-photon state transfers can be achieved with high fidelity in the cavity-magnon lattice. Spectacularly, the three different types of quantum state transfer schemes can be even transformed to each other in a controllable fashion. This system provides a novel way of realizing quantum state transfer and can be implemented in quantum computing platforms.
I. INTRODUCTION
Quantum state transfer (QST) is of major interest in physics with potential applications in large-scale quantum information processing . Several schemes have been proposed to discuss QST theoretically and experimentally. Such as, cavity electromechanical system [21][22][23], the spin chain [5,6,18,24,25], quantum dots [26,27] and the trapped ions system [2], etc. Among these studies, it can be found that the major requirement of QST protocols is robustness. To this regard, the topological QST has attracted much attention because the topology can provide invariance against sizable imperfections in quantum information processing [28][29][30][31][32][33][34][35][36][37]. The Su-Schrieffer-Heeger (SSH) model is a promising platform to analyze the topological QST due to its structural simplicity and abundant physical insight [38][39][40][41]. Concretely, the robust QST assisted by Landau-Zener tunneling has been reported [42]. In addition, Ref. [43] has displayed the robust QST via topologically protected edge pumping in two-dimensional topological spin system. Subsequently, in superconducting qubit chain [44] and superconducting flux qubit chain [45], robust QST via topological edge states has been realized. Especially, Qi et al. [46] studied the one-dimensional modulated SSH chain composed of small optomechanical lattice to implement photonic and phononic topological state transfer.
On the other hand, cavity optomagnonic systems have been extensively investigated recently, which provide a new platform for studying quantum effect [47][48][49][50][51][52][53][54][55][56]. Magnon, the collective spin excitation for ferromagnetic material, can realize the strong (even ultrastrong) couplings with cavity photons [47]. Further, the yttrium iron garnet (YIG) sphere is often used as the ferromagnetic material for the experiment, and this has been characterized by high collective spin excitation density and low dissipation [57,58]. In addition, the strong coupling between cavity photons and magnons has been observed at both low, room and high temperature experimentally [59,60]. Moreover, compared to cavity-optomechanical systems, the cavity optomagnonic system possesses a long coherence time and intrinsically good tunability [47]. Based on the merits of the cavity optomagnonic system, it is also expected to implement the potential application in quantum information networks [56], quantum sensing [61,62] and magnon dark modes in magnon-gradient memory [63].
Here, in terms of these excellent properties of cavity magnetic systems, a new protocol for implementing QST can be proposed. By putting YIG spheres into the cavity, a one-dimensional cavity magnetic chain is constructed. We analytically find that the system is equivalent to the generalized SSH model. Then, through adiabatical ramping of the cavity-magnon couplings, we find that the topological edge states can be obtained and used as topologically protected quantum channels to realize the photonic, magnonic and magnon-to-photon state transfers. Our results numerically demonstrate that the process of QST is accomplished with high fidelity via the gap state because of the topological protection. More spectacularly, we unveil that the different kinds of transition channels can be switched to each other. Our results provide an experimentally feasible scheme for implementing the QST in a cavity-magnon system, which will offer potential application for quantum information processing.
The paper is organized as follows. Sec. II presents the theoretical model of cavity-magnon chain which can be mapped into a tight-binding SSH model. Sec. III is devoted to research the photonic state transfer in the cavity-magnon chain. Sec. IV explores the magnonic state transfer and magnon-to-photon state transfer. Finally, the conclusions are presented in Sec. V. The diagrammatic sketch of a onedimension cavity-magnon system consisting of N + 1 cavity modes and N YIG spheres. a1, a2... and aN+1 represent cavity modes which is pumped by the laser shown as purple arrows. YIG spheres (red spheres) are placed at the cavity and in a uniform bias magnetic field. The coupling between magnon mn and the two adjacent cavity is gn and g n , respectively. In addition, the two adjacent cavities have the coupling with the coupling strength J.
II. THE CAVITY-MAGNON CHAIN
We consider a one-dimensional cavity-magnon system composed N +1 cavity modes [64][65][66] and N YIG spheres (red spheres), as depicted in Fig. 1. Each cavity field is driven by a laser with strength Ω n (n = 1, 2, ..., N + 1). Specifically, the system can be described by
H 1 = N +1 n=1
(ω a,n a † n a n + Ω n a † n e −iω d t + Ω * n a n e iω d t )
+ N n=1 ω m,n m † n m n + N n=1 [g n (a † n m n + a n m † n ) + g n (a † n+1 m n + a n+1 m † n ) + J(a † n+1 a n + a † n a n+1 )],(1)
where a † n (a n ) and m † n (m n ) are the photonic and magnonic creation (annihilation) operators. The first term is the free energy of the cavity fields with frequencies ω a,n and driving laser with frequency ω d . The second term is the free energy of the magnons with frequencies ω m,n . The last term represents the coupling between the cavity fields and the magnons through magnetic dipole coupling and cavity-cavity coupling, in which the coupling strengths are g n , g n and J, respectively. Then, using a rotating transformation with the external driving frequency ω d , the total Hamiltonian H 1 becomes
H 2 = N +1 n=1
(∆ a,n a † n a n + Ω n a † n + Ω * n a n )
+ N n=1 ∆ m,n m † n m n + N n=1 [g n (a † n m n + a n m † n ) + g n (a † n+1 m n + a n+1 m † n ) + J(a † n+1 a n + a † n a n+1 )],(2)
where ∆ a,n = ω a,n − ω d (∆ m,n = ω m,n − ω d ) is the detuning between the cavity fields (magnon modes) and the external lasers. Subsequently, we utilize the mean field approximation method to analyse the steady-state dynamics of the cavity-magnon lattice. In other words, the operators a n and m n are replaced by a n = a n + δa n = α n + δa n and m n = m n + δm n = β n + δm n , respectively [67]. After dropping the notation "δ" for all the fluctuation operators δa n (δm n ) [46], the Hamiltonian is given by
H 3 = N +1 n=1 ∆ a,n a † n a n + N n=1 ∆ m,n m † n m n + N n=1 g n [(a † n m n + a n m † n ) + g n (a † n+1 m n + a n+1 m † n ) + J(a † n+1 a n + a † n a n+1 )].(3)
We further implement another rotating transformation with respect to ∆ a,n a † n a n and ∆ m,n m † n m n , i.e.,
U = exp [−i( N +1 n=1 ∆ a,n a † n a n + N n=1 ∆ m,n m † n m n )]. (4)
Then, the Hamiltonian becomes
H 4 = N n=1
[g n (a † n m n e i(∆a,n−∆m,n)t + a n m † n e −i(∆a,n−∆m,n)t )
+ g n (a † n+1 m n e i(∆a,n−∆m,n)t + a n+1 m † n e −i(∆a,n−∆m,n)t ) + J(a † n+1 a n + a † n a n+1 )].(5)
When the parameters satisfy ∆ a,n = ∆ m,n , the Hamiltonian ultimately becomes
H 5 = N n=1 [g n (a † n m n + a n m † n ) + g n (a † n+1 m n + a n+1 m † n ) + J(a † n+1 a n + a † n a n+1 )].(6)
Remarkably, if J = 0, the Hamiltonian H 5 only possesses the interactions between the adjacent cavity field and the magnonic mode. It means that the onedimensional cavity-magnon lattice can be transformed into a generalized SSH model physically.
III. PHOTONIC STATE TRANSFER IN CAVITY-MAGNON CHAIN
We first consider the standard SSH model, i.e., J = 0. The Hamiltonian becomes
H 6 = N n=1 [g n (a † n m n + a n m † n ) + g n (a † n+1 m n + a n+1 m † n ),(7)
where we take g n = g and g n = g and both g and g in a periodic way as g = g 0 (1 − cos θ) and g = g 0 (1 + cos θ). The edge state are exponentially localized at the boundaries. Specifically, in single excitation subspace, the wave function has the form |Ψ (θ) = n λ n (αa † n + βm † n )|G [44], in which |G = |0, 0, 0, ...0 and λ denotes the localization indexes. Then, the eigenvalue equation can be acquired as
gλ n βa † n |G + g λ n+1 αm † n |G + gλ n αm † n |G + g λ n−1 βa † n |G = Eλ n (αa † n + βm † n ).(8)
Therefore, the zero-energy edge state wavefunction can be derived as
|Ψ = n (− g g ) n a † n |G ,(9)
Eq. (9) analytically demonstrates that the edge state occupies near the left boundary when θ ∈ (0, π 2 ) and θ ∈ ( 3π 2 , 2π), while θ ∈ ( π 2 , 3π 2 ), it is localized near the right boundary. Note that both the left and right sites belong to the cavity fields. To further visualize the above analytical expression, we plot the distribution of the zeroenergy mode, as shown in Fig. 2(b). One can find that the numerical results are consistent well with the analytical calculation. Then, the result above provides some important information that we can achieve a photonic topological state transfer between the first and the last cavity fields assisted by the zero-energy mode via varying the periodic parameter θ from 0 to π. Concretely, the cavity-magnon coupling strength becomes g = 0 (g = 0) when θ = 0 (θ = π), which induces that the leftmost (rightmost) cavity mode is decoupled from the rest of the cavity-magnon chain. Therefore, the edge states become
|L = |1, 0, 0...0 ,(10)|R = |0, 0, 0...1 .(11)
Here, the photon state can be transferred adiabatically via the channel of the zero-energy edge state. To do this, the periodic parameter θ is set as θ(t) = Ωt, with Ω being the ramping frequency [44]. If the initial state is prepared in the photonic left edge state |L = |1, 0, 0, ..., 0 , the photonic right edge state can be obtained through the evolution of the time-dependent Hamiltonian i ∂ ∂t |Ψ t = H(θ t )|Ψ t . Further, the fidelity of this evolution process F = | R|Ψ f | can be numerically calculated in Fig. 2(c). Affirmatively, the fidelity is maintained as the value of unity over a large range. For example, we can choose Ω = 0.03g 0 to ensure the photonic state transfer from |L to |R with high fidelity. The coupling strength of cavity-magnon can reach g/2π = g /π = 2 GHz [55]. Namely, this strong coupling strength corresponds to the time of QST being t = π/Ω = 8.3 ns, which is fast. Further, we consider the case of J = 0. In Figs. 3(a) and 3(b), the energy spectrum and the corresponding gap state distribution can be displayed with J = 0.125. Clearly, this gap state is not the zero mode, but the energy gap is still nonzero. In other words, this state also can be viewed as a channel to realize QST. Correspondingly, as shown in Fig. 3(b), the photon prepared initially in the first cavity mode can be transferred into the last cavity mode finally. In Fig. 3(c), we exhibit the fidelity of this evolution process once again, in which the red line stands for J = 0.125. Obviously, the fidelity F > 99% when −4 < log 10 Ω < −2.5, i.e., the photon state can be transferred almost perfectly. In addition, we also calculate the fidelity for some values of J. It can be shown that the situation becomes different when J continues to increase but less than g. Concretely, with J = 0.5 (the black line), the fidelity is close to zero. Physically, it can be explained as that the large J completely destroys the original state transfer channel, as shown in Fig. 3(d), with the gap states coming into the bulk. Up until we found that the photonic state can be transferred across over the chain with high fidelity. Next, we will explore that what new phenomena will occur in our cavity-magnon system.
IV. MAGNONIC STATE TRANSFER AND MAGNON-TO-PHOTON STATE TRANSFER
In this section, we now discuss other state transfer protocols in cavity-magnon system by changing the cavitycavity coupling strength (J) and the intracell hopping strength (g 0 ).
M agnonic state transf er As we all know, the closure of the energy gap marks that the gap state is not existence, i.e., the choice is not suitable for QST. In Fig. 4 (a), we plot the energy gap ∆ E versus the length of the chain with g 0 = g 0 = 1 and J = 8. One can find that the energy gap tends to be narrow as increasing the number of the unit cell. Therefore, in order to better elucidate the state transition process, the length of the following cavity-magnon lattice can be taken as L = 5.
To explore the QST, as shown in Fig. 4(b), the energy spectrum is obtained. It can be shown that the shape of the gap state is wavy-like, which implies that a new channel realizing another type of state transfer may be apparent. In order to demonstrate this statement, we plot the state distribution in Fig. 4(c). Obviously, corresponding θ = 0, the gap state is localized near the first YIG sphere, while it is located near the second YIG sphere when θ = π. Those results show that the gap state becomes a new state transfer channel between the magnonic state of |ψ 0 = |0, 1, 0, 0, 0 and |ψ π = |0, 0, 0, 1, 0 . The numerical results mentioned above can be understood physically as follows. The two adjacent cavities coupling strength J is larger, which makes the original three cavity modes can be regarded as a cavity field [46]. Therefore, this system can realize state transfer between magnons. To further confirm it, we numerically simulate the fidelity of the magnetic state transfer in Fig. 4(d). We find that, when the ramping speed log 10 Ω < −2.3, the state transfer between |ψ 0 = |0, 1, 0, 0, 0 and |ψ π = |0, 0, 0, 1, 0 can be realized with high fidelity. We also depict the state transfer process when Ω = 3 × 10 −4 in Fig. 4(e). Obviously, the state transfer between |ψ 0 = |0, 1, 0, 0, 0 and |ψ π = |0, 0, 0, 1, 0 can indeed be implemented. can find that the gap state is occupied near the second site when θ = 0, whereas it is localized near the last site when θ = π. In other word, it can realize the transfer from magnonic state |ψ 0 = |0, 1, 0, 0, 0 to photon state |ψ π = |0, 0, 0, 0, 1 by dent of this channel. Similarly, the fidelity of the state transfer versus the ramping speed Ω can be shown in Fig. 5(c). It can be found that when −4 < log 10 Ω < −3, the fidelity F > 90%. A small Ω is required to meet the adiabatic evolution condition since the energy gap is narrow at this point. Moreover, to make the results more intuitive, in Fig. 5(d), we choose an appropriate value of Ω = 0.001. The numerical result reveals exactly that the state transfer between |ψ 0 = |0, 1, 0, 0, 0 and the photon state |ψ π = |0, 0, 0, 0, 1 .
From above all, it seems that although photonic, magnonic and magnon-to-photon state transfers are different state transfer processes. However, all of those processes are fulfilled by the gap state. The information from Sec. III and IV tell us that the photonic state and magnonic state transfer processes only adjust the cavity-cavity strength J appropriately. Analogously, the magnonic and magnon-to-photon state transfer processes can regulate the intracell hopping g 0 . In other words, the three different types of quantum state transfer schemes can be switched to each other by designing parameters suitably for our system.
V. CONCLUSION
In this work, based on a new platform of the cavitymagnon system, we explore different types of quantum state transfer in terms of the topologically protected edge state. We analytically find that the cavity-magnon lattice is equivalent to the generalized SSH model. It can be found that this edge state can be employed as a quantum channel to realize the photonic, magnonic state transfers by adjusting the cavity-cavity coupling strength. Further, by changing the intracell hopping amplitude, our scheme can also realize the transfer between photonic state and magnetic state. In addition, one can find that the photonic, magnonic and magnon-to-photon state transfer can be achieved with high fidelity. The results obtained here provide an experimentally feasible scheme for realizing the QST in a cavity magnonic system, which will offer valuable insight for quantum information processing.
Experimentally, it is a mature technology that a strong coupling magnon-photon system can be engineered in experiments [54,[68][69][70]. Moreover, the linear array of 3D cavities for experiments has been developed, which leads that many YIG spheres couple the cavity fields [71,72]. In addition, a multimode cavity couples atom has been realized [65]. Therefor, the theoretical model our proposed may be experimentally realized.
VI. ACKNOWLEDGMENTS
FIG. 1 .
1(Color online)
FIG. 2 .
2(Color online) (a) The energy spectra of the cavitymagnon system with g0 = g 0 = 1 and J = 0. A zero-energy state always exists. (b) The corresponding state distribution of zero-energy state. (c) The fidelity of the state transfer between |L and |R versus the varying rate of Ω.
FIG. 3 .
3(Color online) (a) The energy spectra of the cavitymagnon system with g0 = g 0 = 1 and J = 0.125. A gap state always exists. (b) The corresponding state distribution. (c) The fidelity with J = 0.125(the red line), J = 0.2(the blue line), J = 0.3(the green line), J = 0.4(the pink line) and J = 0.5(the black line). (d) The energy spectrum with J = 0.5. Here, the lattice size is L = 21.
FIG. 4 .
4(Color online) (a) The energy gap (the gap state and the low band) ∆E versus the length of the chain. (b) The energy spectra of the cavity-magnon system. It always exist a gap state. (c) The corresponding state distribution of gap state. (d) The fidelity between |ψ0 = |0, 1, 0, 0, 0 and |ψπ = |0, 0, 0, 1, 0 . (e) The state transfer process corresponding to Ω = 3 × 10 −4 . Other parameters are g0 = g 0 = 1 and J = 8.M agnon − to − photon state transf er We have seen that the magnonic state transfer is applicable when g 0 = g 0 = 1 and J = 8. Then, what will happen if we change the value of g 0 ? We can now take g 0 = 16. The energy spectrum and the corresponding gap state distribution are plotted in Figs. 5(a) and 5(b). One
FIG. 5 .
5(Color online) (a) The energy spectra of the cavitymagnon system with g0 = 16, g 0 = 1 and J = 8. (b) The corresponding state distribution of gap state. (c) The fidelity between |ψ0 = |0, 1, 0, 0, 0 and |ψπ = |0, 0, 0, 0, 1 . (d) The state transfer process corresponding to Ω = 0.001.
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| []
|
[
"Automatic discovery of cell types and microcircuitry from neural connectomics",
"Automatic discovery of cell types and microcircuitry from neural connectomics"
]
| [
"Eric Jonas \nDepartment of Electrical Engineering and Computer Science\nUniversity of California\nBerkeleyBerkeleyUnited States\n",
"Konrad Kording \nDepartment of Physical Medicine and Rehabilitation\nNorthwestern University\nChicagoUnited States\n\nDepartment of Physical Medicine and Rehabilitation\nRehabilitation Institute of Chicago\nChicagoUnited States\n\nDepartment of Physiology\nNorthwestern University\nChicagoUnited States\n"
]
| [
"Department of Electrical Engineering and Computer Science\nUniversity of California\nBerkeleyBerkeleyUnited States",
"Department of Physical Medicine and Rehabilitation\nNorthwestern University\nChicagoUnited States",
"Department of Physical Medicine and Rehabilitation\nRehabilitation Institute of Chicago\nChicagoUnited States",
"Department of Physiology\nNorthwestern University\nChicagoUnited States"
]
| []
| Neural connectomics has begun producing massive amounts of data, necessitating new analysis methods to discover the biological and computational structure. It has long been assumed that discovering neuron types and their relation to microcircuitry is crucial to understanding neural function. Here we developed a non-parametric Bayesian technique that identifies neuron types and microcircuitry patterns in connectomics data. It combines the information traditionally used by biologists in a principled and probabilistically coherent manner, including connectivity, cell body location, and the spatial distribution of synapses. We show that the approach recovers known neuron types in the retina and enables predictions of connectivity, better than simpler algorithms. It also can reveal interesting structure in the nervous system of Caenorhabditis elegans and an old man-made microprocessor. Our approach extracts structural meaning from connectomics, enabling new approaches of automatically deriving anatomical insights from these emerging datasets. | 10.7554/elife.04250 | null | 15,685,708 | 1407.4137 | 7f2225f8523b4912ce49ced6cd2b20e7b7f13a60 |
Automatic discovery of cell types and microcircuitry from neural connectomics
Eric Jonas
Department of Electrical Engineering and Computer Science
University of California
BerkeleyBerkeleyUnited States
Konrad Kording
Department of Physical Medicine and Rehabilitation
Northwestern University
ChicagoUnited States
Department of Physical Medicine and Rehabilitation
Rehabilitation Institute of Chicago
ChicagoUnited States
Department of Physiology
Northwestern University
ChicagoUnited States
Automatic discovery of cell types and microcircuitry from neural connectomics
elifesciences.org TOOLS AND RESOURCES
Neural connectomics has begun producing massive amounts of data, necessitating new analysis methods to discover the biological and computational structure. It has long been assumed that discovering neuron types and their relation to microcircuitry is crucial to understanding neural function. Here we developed a non-parametric Bayesian technique that identifies neuron types and microcircuitry patterns in connectomics data. It combines the information traditionally used by biologists in a principled and probabilistically coherent manner, including connectivity, cell body location, and the spatial distribution of synapses. We show that the approach recovers known neuron types in the retina and enables predictions of connectivity, better than simpler algorithms. It also can reveal interesting structure in the nervous system of Caenorhabditis elegans and an old man-made microprocessor. Our approach extracts structural meaning from connectomics, enabling new approaches of automatically deriving anatomical insights from these emerging datasets.
Introduction
Emerging connectomics techniques (Zador et al., 2012;Morgan and Lichtman, 2013) promise to quantify the location and connectivity of each neuron within a tissue volume. These massive datasets will far exceed the capacity of neuroanatomists to manually trace small circuits, thus necessitating computational, quantitative, and automatic methods for understanding neural circuit structure. The impact of this kind of high-throughput transition has been seen before-the rise of sequencing techniques necessitated the development of novel computational methods to understand genomic structure, ushering in bioinformatics as an independent discipline (Koboldt et al., 2013).
The brain consists of multiple kinds of neurons, each of which is hypothesized to have a specific role in overall computation. Neuron types differ in many ways, for example, chemical or morphological, but they also differ in the way they connect to one another (Seung and Sümbül, 2014). In fact, the idea of well defined, type-dependent local connectivity patterns (microcircuits) has a long history (Passingham, 2002), and is prominent in many areas, from sensory (e.g., retina; Masland, 2001) to processing (e.g., neocortex; Mountcastle, 1997) to movement (e.g., spinal cord; Grillner et al., 2005). These types of repeated computing patterns are a common feature of computing systems, even occurring in man-made computing circuits. It remains an important challenge to develop algorithms to use connectivity-based anatomical data (connectomics) to automatically back out underlying microcircuitry.
The discovery of structure is a crucial aspect of network science. Early approaches focused on global graph properties, such as the types of scaling present in the network (Watts and Strogatz, 1998).
While this approach leads to an understanding of the global network, more recent work aims at identifying very small-scale repeat patterns, or motifs, in networks (Milo et al., 2002). These motifs are defined not between different node types, but rather represent repeated patterns of topology.
The discovery of structure in probabilistic graphs is a well-known problem in machine learning. Commonly used algorithms include community-based detection methods (Girvan and Newman, 2002) and stochastic block models (Nowicki and Snijders, 2001). While these approaches can incorporate the probabilistic nature of neural connections (Hill et al., 2012), they do not incorporate the additional richer structure present in connectomics data-the location of cell bodies, the spatial distribution of synapses, and the distances between neurons. It is of particular importance that the probability of connections has a strong spatial component, a factor that is hard to reconcile with many other methods. A model attempting to fully capture the variation in the nervous system should take into account the broad set of available features.
When it comes to neuroscience and other computing systems, we expect patterns of connectivity much more complex than traditional motifs, exhibiting a strong spatial dependence arising from the complex genetic, chemical, and activity-based neural development processes.
To address these challenges, here we describe a Bayesian non-parametric model that can discover circuit structure automatically from connectomics data: the cell types, their spatial patterns of interconnection, and the locations of somata and synapses. We show that by incorporating this additional information, our model both accurately predicts the connection as well as agrees with human neuroanatomists as to the identification of cell types. We take as inspiration previous work on identifying cell types automatically from morphology (Guerra et al., 2011) and electrophysiology (Druckmann et al., 2013).
We primarily focus on the recently released mouse retina connectome (Helmstaedter et al., 2013), but additionally examine the Caenorhabditis elegans connectome (White et al., 1986). Comparing the cell types discovered by the algorithms with those obtained manually by human anatomists reveals a high degree of agreement. We thus present a scalable probabilistic approach to infer microcircuitry from connectomics data available today and in the future. eLife digest The human brain is made up of billions of neurons, which are organised into networks via trillions of connections. The study of the nature of these connections will be central to understanding how the brain works. In recent years, a number of new methods for imaging the brain have made it possible to visualise and map these connections, generating striking images and creating an additional field of neuroscience known as 'connectomics'.
However, the sheer volume of data generated by connectomics is now beginning to exceed the capacity of researchers to analyse it. Just as the advent of genome sequencing required the development of statistical techniques to analyse the resulting data, so the emergence of connectomics has created a need for similarly powerful mathematical models in neuroscience.
Jonas and Kording have developed one such algorithm that can classify the component units of circuits, both biological and man-made, and identify the connections between them. When applied to connectomics data for 950 neurons in the mouse retina, the algorithm generated predictions regarding cell types and patterns of connectivity. The predicted cell types agreed closely with those identified by human neuroanatomists. Results were similarly convincing when the algorithm was applied to the nervous system of the nematode worm and genetic model organism, Caenorhabditis elegans, and even when it was asked to classify electronic components and connectivity patterns in a man-made microprocessor.
Algorithms such as that developed by Jonas and Kording will soon be essential for making sense of the vast quantities of data generated by connectomic studies of the human brain. At present, an analysis of 950 neurons requires several hours, thus refinements that make the process faster will likely be required prior to the analysis of larger human datasets. Such algorithms will open up a range of possibilities for examining the structure of the healthy brain, as well as the changes triggered by developmental abnormalities and disease.
Model
We build a structured probabilistic model which begins with the generic notion of a cell being a member of a single type-and these types affect soma depth, distribution of synapses, as well as a cell type and distance-dependent connection probability. For example, retinal ganglion cells may synapse on nearby, but not far away, amacrine cells, with bipolar cells clearly tessellating space and synapsing on both. In machine learning parlance, our method is unsupervised-it seeks to discover structure in data and make predictions in the absence of training data. Rather than taking in examples of types annotated by human neuroanatomists, we instead start with the weakest possible assumption in an attempt to algorithmically discover this structure. We contrast this with the supervised approaches taken in Guerra et al. (2011), where there is high confidence in the (morphologically defined) types and then a supervised classifier is built, as our goal here is explicit discovery of types.
From these assumptions (priors) we develop a generative Bayesian model that estimates the underlying cell types and how they connect. We take as input ( Figure 1A) the connectivity matrix of cells ( Figure 1B), a matrix of the distance between cells ( Figure 1C), the per-cell soma depth ( Figure 1D), and the depth profile of the cell's synapses ( Figure 1E). We perform joint probabilistic inference to automatically learn the number of cell types, which cells belong to which type, their typespecific connectivity, and how connections between types vary with distance. We also simultaneously learn the soma depth associated with each type and the typical synaptic density profile ( Figure 1F-H).
We start with a model for connectivity, the iSBM (Kemp et al., 2006;Xu et al., 2006), which has been shown to meaningfully cluster connection graphs while learning the number of hidden groups, or types. We extend this approach by adding distance dependence to model salient aspects of microcircuitry via logistic and exponential distance-link functions. We form a unimodial model of cell body depth and a multimodal synapse density profile model (see 'Materials and methods' for mathematical details).
As an illustrative example, consider a network with only three cell types, labeled A, B, and C. Assume these cells are uniformly distributed in space, and that the probability of connection between any two cells, c i and c j , depends only on their type and their distance, according to a logistic (sigmoidal) function. Let A connect only to nearby B and C cells, but B connect to any C regardless of distance. This is the prior intuition our model is designed to capture.
For the basic link-distance model, we take as input a connectivity matrix R defining the connections between cell e i and e j , as well as a distance function d(e i , e j ) representing a (physical) distance between adjacent cells. See the supplemental material for extension to multiple connectivity matrices. We assume there exist an unknown number K of latent (unobserved) cell types, k ∈ f1; 2; 3; …; K g, and that each cell e i belongs to a single cell type. We indicate a cell e i is of type k using the assignment vector (c), so c i = k. The observed connectivity between two cells R(e i , e j ) then depends only on their latent type and their distance through a link function f(·, d(e i , e j )). We assume f is parameterized based on the latent type, c i = m and c j = n, via a parameter η mn , as well as a set of global hyperparameters θ, such that the link function is f(d(e i , e j )|η mn , θ).
We then jointly infer the posterior distribution of the class assignment vector (c) = {c i }, the parameter matrix η mn , and the global model hyperparameters θ:
pðc; η; θ|RÞ ∝ ∏ i;j p R À e i ; e j Á |f d À e i ; e j Á |η c i c j ; θ ∏ m;n pðη mn |θÞpðθÞpðc|αÞpðαÞpðθÞ:(1)
Our subsequent analysis uses both the full posterior distribution over these parameters as well as the most probable, or maximum a posteriori (MAP), estimate.
For the retina data, we then extend the model with the additional features indicated. Cell soma depth is modeled as a cell-type-dependent Gaussian distribution with latent (unknown) per-type mean and variance. Similarly, each cell has some number N i of synapses, each of which is drawn from a celltype-specific density profile with up to three modes.
Inference is performed via MCMC via three composable transition kernels-one for structural, one for per-type parameters, and one for global parameters and hyperparameters. Details of data preprocessing, inference parameters, and runtime can be found in the 'Methods section.
Metrics
To evaluate the quality of the model fit, we need to use information that quantifies aspects of the data for which we have ground truth information. We focus on two aspects of performance. First, if the Figure 1. Deriving circuitry and cell types from connectomics data. (A) As input we take the connectivity between cells (B), the distance between them (C), the depth of the cell bodies (D), and the depth profile of the synapses (E). (F) Our algorithm discovers hidden cell types in this connectivity data by assuming all cells of a type share a distance- Figure 1. continued on next page model works well, then the probability that a pair of neurons is of the same type should be high if the neurons actually are of the same type. Second, the model should assign a high probability of connection between two cells if they have a connection in the underlying data. We term these two factors clustering accuracy and link-prediction accuracy.
To assess the accuracy of a clustering compared to that determined by neuroanatomists, we employ three metrics-clustering homogeneity, clustering completeness, and the ARI. All metrics equal 1.0 when two clusterings completely agree. Homogeneity reflects the degree to which a found cluster or type contains only a single true type. Completeness measures how much of a true type is contained within a single identified type-a completeness of 1.0 means no true type is split into multiple subtypes. ARI is a metric that reflects both measures (see the supplemental material for more information).
To assess the accuracy of the model for connections, we use link prediction accuracy. If our model accurately captures the true structure of the data, it should be good at predicting if a link exists. We thus train the model on the data with a subset of the links marked as unobserved and thus compute our predictive accuracy. We perform 10-way cross-validation on a given dataset (Guerra et al., 2011), learn the resulting model, and use that model to predict the missing synapses. Each potential link between cells is assigned a probability, and we compute the AUC for the resulting ROC curve. An AUC of 1.0 means that we perfectly predict the presence and absence of the missing synapses. We use link prediction accuracy to quantify how good the model is at discovering the underlying connectivity.
Results
We will first establish that our algorithm works properly and try to understand its properties using simulated data. Subsequently, we will analyze in detail a dataset on the retina. Lastly, we will briefly discuss the analysis of data from the worm C. elegans and from an old man-made microprocessor.
Validation with simulated data where ground truth is known
To validate our model, we performed a series of simulations to test if the model can accurately recover the true underlying network structure and cell type identity. We thus simulate data for which we know the correct structure and compare the estimated structure based on the algorithm (see 'Materials and methods') with the one we used for simulation. We find that the model does a good job of recovering the correct number of cell types (Figure 2A), the cell identities ( Figure 2B), and the spatial extent of each type ( Figure 2C). For comparison, we show the results using the infinite stochastic block model (iSBM) instead ( Figure 2A-C, black line) which assumes that only cell type matters, and thus finds small neighborhoods of connected nodes (instead of global connectivity patterns). This contrast shows that while the regular block model can not correctly deal with distance-dependent connectivity, our model can. Our model converges relatively quickly (see 'Mixing of Markov chains') to an estimate of the most probable values for the cell types, which is enabled by using a combination of simulated annealing and parallelized Markov-chain Monte Carlo (MCMC) (see 'Materials and methods' for details). Thus our model at least is promising for application to biological datasets.
Model mismatch
We next analyze how our model performs in cases where the data are generated with assumptions different from ours. To understand the properties of our model, we attempt connectivity inference on four sets of synthetic data. This helps us understand what our model would do if the data do not obey our assumptions. This results in a clustering of the cells by those hidden types. (F) Shows the cell connectivity matrix with cells of the same type grouped together. (G) Shows the learned probability of connection (p(conn)) between our different types at various distances-in this case, the cells are likely to connect when they are close. (H) Shows the probability of connection (p(conn)) between two cell types that very rarely connect-there is a background 'base' connection rate to account for errors in data, but the probability is very low. (I) Shows that we also recover the expected laminarity of types and the depth-specific (J) synaptic connectivity. (K) We then plot how the connectivity between these types changes as a function of distance between the cell bodies to better understand short-range and long-range connectivity patterns. DOI: 10.7554/eLife.04250.010
We thus generate 10 sets of synthetic data from each of four existing models. The distancedependent stochastic block model assumes type depends on distance, the traditional stochastic block model has no notion of distance, the mixed membership block model assumes type is combinatorial, and the latent position cluster model assumes that type is clustered-but-continuous.
If the data are sampled from our model, inference according to our model, unsurprisingly, is good by all measures. It correctly estimates the number of cell types, it is good at predicting connectivity (high area under the curve, AUC), it agrees with human classification (Rand index), it discovers all types, and leads to homogeneous estimates ( Figure 3, first row). If the data come from a block model without distance dependence, we see that it still does well on all meaningful measures ( Figure 3, second row). This is unsurprising, as our model learns the distance dependence, even its absence. For the mixed membership model (Figure 3, third row), the model grossly overestimates the number of types, by basically allocating a type for each combination of memberships. Otherwise, it still performs relatively well. Lastly, for the latent position clustering model ( Figure 3, fourth row), the model does poorly. If type is continuous instead of discrete, then our model is basically trying to cover a continuous set with a discrete scenario leading to rather poor performance. However, as we do expect cell types to have a discrete biological basis, we might expect our model to do well with real data.
Sensitivity to edge effects
Connectomic efforts so far have reconstructed only small sections of neural tissue. Consequently, many connections to cells outside that tissue volume will be lost. We are concerned that this selective The infinite stochastic block model (which only uses connectivity information) over-estimates the number of classes as it fails to take distance into account, whereas our modeling of the combination of distance and connectivity finds close to the true number of classes. Conn: connectivity; dist: distance. (B) As we increase the true number of types, our method continues to find the correct clustering (as measured by the adjusted Rand index, ARI) whereas the infinite stochastic block model (iSBM) overclusters and thus poorly matches ground truth. (C) We examine the spatial extent (size) of the discovered types (clusters) by measuring the two-dimensional standard deviation of the cell locations. The y-axis indicates what fraction of the discovered types had a given spatial extent. Without incorporating distance, we identify a large number of small, spatially-localized types. With distance, we see a correct recovery of the spatial extent of each type. DOI: 10.7554/eLife.04250.003 elimination of connectivity along the boundary might give the appearance of distance-dependent connectivity when there is none. We thus performed simulations to check if edge effects could destroy spatial structure and if edge effects could introduce artificial, spurious spatial structure. We measure the degree to which distance-dependent effects can arise from selecting regions that are smaller than the 'scale' of connectivity ( Figure 4). We do this by generating two collections of synthetic datasets-one with distance-dependent connectivity and one without. We then in each dataset randomly examine contiguous circular regions with area varying from zero to the entire volume, and empirically calculate the spatial variance in type-dependent connectivity. We find that, if there is no distance dependence, edge effects do not artificially introduce distance dependence. However, if the section we are examining is too small, our model can miss the distance dependence. Thus with respect to distance-dependent connectivity inference, our model errs on the side of caution. But we also find that for spatial extent that is similar to the currently available datasets, the effects of this are quite limited.
Learning types and circuitry in the retina
The mouse retina (Masland, 2001) is a neural circuit which we expect to have connectivity patterns that are well approximated by our generative model. It is known that there are multiple classes of cells that can be broadly grouped into: ganglion cells that transmit information to the rest of the brain; bipolar cells that connect between different cells; and amacrine cells that feed into the ganglion cells. Recent research (Helmstaedter et al., 2013) has produced a large dataset containing both the types of cells from orthogonal approaches, and also the connectivity matrix between all reconstructed cells ( Figure 5A).
The algorithm took 8 hr to perform inference, dividing neurons into a set of cell types which reflect known neuroanatomical distinctions ( Figure 5 shows the MAP result). For each pair of neurons there is a specific distance-dependent connection probability ( Figure 5D), which is well approximated by the model fit. Moreover, each type of cell is rather isotropically distributed across space ( Figure 5C) as should be expected for true cell types.
Comparing the results of the algorithm to other information sources allows evaluation of the quality of the type determination. Our types closely reflect the (anatomist-determined) segmentation of cells into retinal ganglion, narrow amacrine, medium/ wide amacrine, and bipolar cells ( Figure 6B). We find that the types we find tend to reflect the known laminar distribution in the retina ( Figure 6C) as well as the known synaptic density profiles.
The algorithm yields a separation of neurons into a smaller number of types than the fully granular listing of 71 types found by the original authors of the paper (Helmstaedter et al., 2013), although it is still highly correlated with those finer type distinctions (see section 'Mouse retina'). It is our expectation that, with larger datasets, even closer agreement would be found. Our fully Bayesian model produces a distribution over probable clusterings. Figure 6 shows this posterior distribution as a cell-cell coassignment matrix, sorted to find maximum block structure. Each large, dark block represents a collection of cells believed with strong probability to be of the same type. When we plot ( Figure 6B) the anatomist-derived cell types along the left, we can see that each block consists of a roughly homogeneous collection of types. We evaluate our model along three sets of parameters ( Figure 6): how closely does our clustering agree with neuroanatomists' knowledge? Given two cells, how accurately can our model predict the link between them? And how closely does the spatial extent (within a layer) of our identified types agree with the spatial extent of types identified by neuroanatomists?
For our model we show the receiver operating characteristic (ROC) curve ( Figure 6D) which shows how the true and false positive rates trade off. We plot the posterior distribution of the area under this curve in Figure 6E. We then plot the posterior distribution for cluster agreement metrics-completeness, homogeneity, and adjusted Rand index (ARI) ( Figure 6F). We see that our model tends to over-cluster-cells which are of distinct type (at the finest granularity of neuroanatomist-identified type) are grouped as a single type by our model.
We compare link-prediction accuracy across the methods, including our own ( Figure 6G, AUC, red). We find that given the dataset, many techniques allow for good link-predictive accuracy. All the methods allow decent link prediction with an AUC in the 0.9 range. However, our algorithm clearly outperforms the simple statistical models that only use connectivity.
As a second measure we compare link-prediction accuracy across the methods ( Figure 6G, ARI, blue). We find that our algorithm far outperforms the controls. We also find that when it is based on more of the same information used by anatomists, then it gets better at agreeing with these anatomists. In particular, using connectivity, distance, synapse distribution, and soma depth leads to the highest ARI. When using the available information, the algorithm produces a good fit to human anatomist judgments.
Finally we look at the spatial extent of the discovered types both within a layer and between layers ( Figure 6H). We see that, in the absence of distance information, mere connectivity information Figure 4. Two sets of generated synthetic data, one with spatially dependent connectivity and one without. We measure the variance in the connectivity-distance plot for randomly selected regions of each dataset, ranging from single cells to the entire volume. We see that while selecting too small a region can destroy the appearance of distance-dependent connectivity, it does not create it in non-spatial data. DOI: 10.7554/eLife.04250.005 results in types which only span a small region of space-essentially local cliques. Incorporation of distance information results in types which span the entire extent of the layer. The depth variance of all models continues to be substantially larger than that predicted by human anatomists-future directions of work include attempting to more strongly encode this prior belief of laminarity.
Recovering spatial connectivity in multiple graphs simultaneously
Having shown our model to work on the repeating tessellated, laminar structure of the mammalian retina, we then apply our model to a structurally very different connectome-the whole body of a small roundworm: C. elegans is a model system in developmental neuroscience (White et al., 1986), with the location and connectivity of each of 302 neurons developmentally determined, leading to early measurement of the connectome. Unlike the retina, only the motor neurons in C. elegans exhibit regular distribution in space-along the body axis. Most interneurons are concentrated in various ganglia that project throughout the entire animal, and the sensory neurons are primarily located in a small number of anterior ganglia. C. elegans also differs from the retina in that the measured connectome is actually two separate graphs-one of directed chemical synapses and another of undirected electrical synapses. As this is a very different connectome, it allows an interesting generalization test: how well will our model work on such a distinct dataset?
Using both the chemical and electrical connectivity (see 'Materials and methods'), we determined the underlying cell types explained by connectivity and distance ( Figure 7A). A superficial inspection of the results shows clustering into groups consisting roughly homogeneously of motor neurons, sensory neurons, and interneurons. Closer examination reveals agreement with the classifications originally outlined by White in 1986(White et al., 1986. Note our clustering does not perfectly reflect known divisions-several combinations of head and sensory neurons are combined, and a difficult-to-explain group of mostly VB and DB motor neuron types, with VC split between various groups. Our identified cell types thus reflect a 'coarsening' of known types, based entirely on connectivity and distance information, even when the organism exhibits substantially less spatial regularity than the retina.
Types and connectivity in artificial structures
To show the applicability of our method to other connectome-style datasets, we obtained the spatial location and interconnectivity of the transistors in a classic microprocessor, the MOS Technology 6502 (used in the Apple II) (James et al., 2010). Computer architects use common patterns of transistors when designing circuits, with each transistor having a 'type' in the circuit. We identified a region of the processor with complex but known structure containing the primary 8-bit registers X, Y, and S ( Figure 8).
Our algorithm identifies areas of spatial homogeneity that mirror the known structure in the underlying architecture of the circuit, segmenting transistor types recognizable to computer architects. Using the original schematics, we see that one identified type contains the 'clocked' transistors, which retain digital state. Two other types contain transistors with pins C1 or C2 connected to ground, mostly serving as inverters. An additional identified type controls the behavior of the three registers of interest (X, Y, and S) with respect to the SB data bus, either allowing them to latch or drive data from the bus. The repeat patterns of spatial connectivity are visible in Figure 8C, showing the man-made horizontal and vertical layout of the same types of transistors.
Discussion
We have presented a machine learning technique that allows cell types and microcircuitry to be discovered from connectomics data. We have shown its applicability to regularly structured laminar neural circuits like the retina, as well as a less structured whole neuronal organism (C. elegans) and a classic processor. When compared to existing methods, we show how the incorporation of all of this data yields results that combine both high link-prediction accuracy and high agreement with human anatomists. We have found that combining the available data types allows us to discover cell types and microcircuitry that were known to exist in the systems based on decades of previous research and allows good prediction of connectivity.
For our probabilistic models, no known solution exists to exactly find the most probable parsing of the neurons into cell types and connectivity patterns. We employ a collection of MCMC techniques (see 'Materials and methods'), but while different initializations converge to similar ultimate values, we can never realistically obtain the global optimum. There are a broad range of techniques that may offer good approximations to the global optimum and future work could adapt them to find more precise solutions to our problem. For our probabilistic model, inference becomes slower as the amount of data increases. Our algorithm required several hours for 1000 neurons. Scaling this class of probabilistic model is an active area of research, and recent results in both variational methods (Hoffman et al., 2013) and spectral learning (Anandkumar et al., 2012) and future work could adapt them to find faster approximate solutions to our problem.
Larger datasets will allow algorithms to distinguish more distinct types and we expect closer agreement with existing anatomical knowledge as more data become available. Moreover, in general, for such problems precision increases with the size of the dataset and the cells that we have are not sufficient to statistically distinguish all the cell types known in anatomy (such as the ∼70 in the retina). Still, using only connectivity and distance, it is possible to meaningfully divide neurons into types. Figure 6. Continued operating characteristic (ROC) curves from 10-fold cross-validation when predicting connectivity, as well as (E) the area under the curve (AUC) and (F) the type agreements with known neuroanatomist types. ARI: adjusted Rand index. Model comparison, showing using human-discovered types with and without distance information, as well as our model incorporating just connectivity, connectivity and distance, or connectivity, distance, and synaptic depth (as well as the alternative latent position cluster model, see text). (G) A comparison of the predictive accuracy (AUC) for hand-labeled anatomical data, versus inclusion of additional sources of information, as well as the clustering accuracy. Note that our model sacrifices very little predictive accuracy for additional clustering accuracy. By comparison, conventional methods fail at one or both. ARI: adjusted Rand index. (H) The spatial extent (in depth and area) of the types identified by humans and our various algorithmic approaches. DOI: 10.7554/eLife.04250.007
Our small collection of hand-selected distance-dependent likelihood functions is clearly nonexhaustive, and assumes monotonicity of connectivity probability-for a given class, closer cells are never less likely to connect. This is known to be insufficient for various neural systems. Future models could incorporate a wider variety of likelihood functions, or even learn the global functional form from the data. There are a range of previous approaches to the discovery of neural microcircuitry (Mountcastle, 1957;Douglas and Martin, 1991;Freund and Buzsáki, 1998;Barthó et al., 2004). These generally involve a great deal of manual labor and ad hoc determination of what constitutes a type of cell-to this day there are disagreements in the literature as to the true types in the mammalian retina. Much as phylogenomics has changed our understanding of animal ontologies, modern large scale data will allow the efficient unbiased discovery of cell types and circuits. The sheer amount of available data demands the introduction of algorithmic approaches.
The development of automatic identification and quantification of cell type may also provide a new computational phenotype for quantifying the effect of disease, genetic interventions, and developmentally experienced neural activity. Our method can in principle identify neuron types across non-connected graphs, for example, across animals. For example, the types of neurons in one animal can be associated with the types of neurons in another animal, in the same way as this is already possible through molecular markers (Brown and Hestrin, 2009). This could be particularly important if cell types appear that are due to properties of the stimuli and experience as opposed to just the molecular properties of cells, such as color and orientation selective types in primary visual cortex (Lennie and Movshon, 2005;Sincich and Horton, 2005). This would allow comparative quantitative anatomy across animals, and aid the search for the ultimate causes of connectivity. Our model combines connectivity, cellular and synaptic properties, and suggests the way towards combining even richer data. Distinct cell types differ in morphology, connectivity, transcriptomics, relation to behavior or stimuli, and many other ways. Algorithms combining these data and type information may allow us to synthesize all the available information from one experiment or even across experiments into a joint model of brain structure and function.
Our work shows how rich probabilistic models can contribute to computational neuroanatomy. Eventually, algorithms will have to become a central tool for anatomists, as it will progressively become impossible for humans to parse the huge datasets. This transition may follow a similar transition to that of molecular biology (with gene-finding algorithms) and evolutionary biology (with computational phylogenetics). Ultimately, computational approaches may help resolve the significant disagreements across human anatomists.
Methods
Probabilistic model
Our model is a extension of the iSBM (Kemp et al., 2006;Xu et al., 2006) to incorporate spatial relations between entities, inspired by attempts to extend these models with arbitrary discriminative functions (Murphy, 2012).
We take as input a connectivity matrix R defining the connections between cell e i and e j , as well as a distance function d(e i , e j ) representing a (physical) distance between adjacent cells. See the supplemental material for extension to multiple connectivity matrices. We assume there exist an unknown number K of latent (unobserved) cell types, k ∈ f1; 2; 3; …; K g, and that each cell e i belongs to a single cell type. We indicate a cell e i is of type k using the assignment vector (c), so c i = k. The observed connectivity between two cells R(e i , e j ) then depends only on their latent type and their distance through a link function f(·, d(e i , e j )). We assume f is parameterized based on the latent type, c i = m and c j = n, via a parameter η mn , as well as a set of global hyperparameters θ, such that the link function is f(d(e i , e j )|η mn , θ).
We then jointly infer the MAP estimate of the class assignment vector (c) = {c i }, the parameter matrix η mn , and the global model hyperparameters θ:
i;j p R À e i ; e j Á |f d À e i ; e j Á |η c i c j ; θ ∏ m;n pðη mn |θÞpðθÞpðc|αÞpðαÞpðθÞ:(2)
We describe the spatial 'logistic-distance Bernoulli' function here, and others in the supplemental material.
The 'logistic-distance Bernoulli' spatial model assumes that, if cell e i is of type m and cell e j is of type n, then η mn = (μ mn , λ mn ), and the probability that two cells e i and e j are connected is given by
p * = 1:0 1 + exp dðei ;ej Þ − μ mn λmn ;
(3)
p = p * · ðp max − p min Þ + p min ;(4)
where p max and p min are global per-graph parameters. We place exponential priors on the latent parameters:
μ mn ∼ exp μ|μ hp ; (5) λ mn ∼ exp λ|λ hp ;(6)
using λ hp and μ hp as global per-graph hyperparameters. We use a Dirichlet-process prior on class assignments, which allows the number of classes to be determined automatically. In brief, for N total cells, the probability of a cell belonging to a class is proportional to the number of data points already in that class, N k , such that pðc i = kÞ ∝ m k N + α and the probability of the cell belonging to a new class k′ is pðc i = k′Þ ∝ α N + α . α is the global concentration parameter-larger values of α make the model more likely to propose new classes. We grid the parameter α and allow the best value to be learned from the data.
Where we model cell depth, we assume that each cell type has a typical depth, and thus a Gaussian distribution of s i . We assume s i ∼ Nðμ We model synapse depth profile that each cell type has a characteristic depth distribution of synaptic contact points, and mixture of Gaussian distributions over cell is N i contact points, g i . We do this by assuming the g i j are drawn from an M = 3-component mixture of Gaussians. Thus associated with each cell type k is a vector of M Gaussian means ðμ g k;1 ; ⋯; μ g k;M Þ, and a mixture vector π k . This representation can thus model depth distributions of contact points that have up to three modes, an assumption that is well matched in the bulk of anatomical studies of cell-type-dependent connectivity.
Inference
We perform posterior inference via MCMC, annealing on the global likelihood during the traditional burnin phase. MCMC transition kernels for different parts of the state space can be chained together to construct a kernel whose ergodic distribution is the target ergodic distribution over the entire state space.
Our first transition kernel ('structural') performs Gibbs sampling of the assignment vector pðc|η; θ; αÞ. The lack of conjugacy in our likelihood model makes an explicit evaluation of the conditional assignment probabilities impossible, motivating us to use an auxiliary variable method (Neal, 2000) in which a collection of ephemeral classes is explicitly represented for the duration of the Gibbs scan.
We then employ a transition kernel to update the per-component parameter values η mn . Conditioned on the assignment vector c and the model hyperparameters θ, α the individual η mn are independent. We slice sample (Neal, 2003) each component's parameters, choosing the slice width as a function of the global hyperparameter range.
The global hyperparameters, both α and θ, are allowed to take on a discrete set of possible values. As θ is often a tuple of possible values, we explore the Cartesian product of all possible values. We then Gibbs sample (our final transition kernel), which is always possible in a small, finite, discrete state space.
We chain these three kernels together, and then globally anneal on the likelihood from a temperature of T = 64 down to T = 1 over 900 iterations unless otherwise indicated, and then run the chain for another 100 iterations. We then generate at least 20 samples, each taken from the end of a single Markov chain initialized from different random initial points in the state space. For visualization we pick the chain with the highest log likelihood, but for all numerical comparisons (including link probability and cluster accuracy) we use this full collection of samples from the posterior distribution to estimate the resulting statistics.
Link prediction
To compute link-prediction accuracy, we compute the probability of a link between two cells using each model, trained via 10-fold cross-validation. We use a full collection of posterior samples when computing the link probability, and then compute the area under the ROC curve for each. We compare our model with a standard network clustering model, the latent-position clustering model. This model assumes each cell belongs to one of K clusters, and each cluster is associated with a d-dimensional Gaussian distribution. The probability of a link is then a function Figure 11. Type agreement evaluation metrics as a function of splitting types, merging types, and randomly distributing cells between types. DOI: 10.7554/eLife.04250.013 of the distance between the data points in this continuous space. We use a variational implementation provided in R (Salter-Townshend and Murphy, 2013), parametrically varying the number of latent dimensions and the number of requested groups. While this model provides reasonable link-predictive accuracy, the clusterings dramatically disagree with those from human anatomists.
Parameters
Hierarchical generative models can be sensitive to hyperparameter settings, thus for most hyperparameters we perform inference. In cases where we cannot, we run separate collections of Markov chains at separate settings and show the results across all pooled parameters. For the case of the mouse retina data, we consider maximum link probability p max ∈ f0:95; 0:9; 0:7g, variance scales for the synapse density profile of σ 2 ∈ f0:01; 0:1; 1:0g (of normalized depth), and K ∈ f2; 3g possible synapse density profile mixture components. For the connectivity-distance-only model we actually perform inference over both p max and p min .
Mixing of our Markov chains
Evaluating whether or not approximate inference methods, such as MCMC, produce samples which are valid approximations of the posterior distribution is an ongoing area of research in the computational statistics community. We use a rough proxy here-synthetic likelihood evaluation. For synthetic datasets of sizes comparable to our real data size, do we recover known ground truth information after running our Markov chains for the appropriate amount of time? Figures 9 and 10 show the cluster accuracy (ARI) to ground truth and the total log score as a function of runtime. We see dramatic changes in log score initially as we vary the temperature, stabilizing as runtime progresses, for each chain. Then we see the characteristic jumps between nearby modes towards the end of the run, in both log score and ARI. Importantly, regardless of whether our model over-or under-estimates the exact posterior variance about the network, we find points in the latent variable space that are both predictive and parsimonious, largely agreeing with the human anatomists and predicting existing connections.
Dataset details
Mouse retina
Dense serial electron microscopy of a 114 μm × 80 μm area in the mouse retina by Helmstaedter et al. (2013) yielded a listing of places where neurons come into contact. There were over 1000 cells originally, and we selected the 950 for which the location of the soma could be reconstructed from the provided cell plots (soma locations were not provided by the study's authors in machine-readable form). The result was a matrix of the total synapse-like contact area between all pairs of 950 cells. Area was thresholded at 0.1 μm, determined by hand, to yield a 950 × 950 entry matrix that served as input to our algorithm. We measured the distance between cells using the reconstructed soma centers, and used the logistic-distance spatial relation. Hyperprior griddings are shown in the 'Hyperprior grids and hyperprior inference' section.
C. elegans
We obtained the connectome of C. elegans from data published previously (Varshney et al., 2011), and isolated the 279 non-pharyngeal neurons, with a total of 6393 chemical synapses and 890 gap junctions originally cleaned up in Chen et al. (2006). A cell's position was its distance along the anterior-posterior axis normalized between 0 and 1. We used both networks, the chemical network as a directed graph and the electrical network as an undirected graph. We use the synapse counts with the logistic-distance Poisson likelihood, scaling the counts by 4.0 to compensate for the Poisson's overdispersion.
Microprocessor
We extracted the connection graph for the transistors in the MOS 6502 (James et al., 2010). Each transistor has three terminals (gate, source, drain), but the methods of the original dataset were unable to consistently resolve which of the C1 and C2 terminals were source and drain, leading to ambiguity in our encoding. We identified a region consisting of three registers X, Y, and S via visual inspection and focused our efforts there. We created a total of six connectivity matrices by examining possible terminal pairings. For example, one graph encodes the connectivity between pins g and c 1 : We then have, R gc 1 ðe i ; e j Þ = 1 if transistor e j and e j are connected via pins g and c 1 .
Other likelihoods
We reparameterized the logistic-distance Bernoulli likelihood to better capture the microprocessor data structure. We are explicitly setting the maximum probability p of the logistic function on a percomponent basis, drawing from a global p ∼ Betaðα hp ; β hp Þ. Then λ is set for each component as a global hyperparameter, λ.
The 'logistic-distance Poisson' spatial model is used to explicitly model the count of synapses, c, between two neurons. The probability of c synapses between two neurons is distributed c ∼ Poissonðc|rÞ, where r (the 'rate') is generated by a scaled logistic function (the logistic function has range [0, 1]). For each component η mn we learn both the threshold μ mn and the rate scaling factor r mn . Thus if cells m and n are likely to have on average 20 synapses if they are closer than 5 μm, then μ mn = 5 and r mn = 20.
Thus the probability of R(e i , e j ) = c synapses between two cells e i and e j is given by: r * = 1:0 1 + exp dðe i ;e j Þ − μ mn λ ;
(7) r = r * · ðr mn − r min Þ + r min ;
R À e i ; e j Á ∼ Poissonðc|rÞ;
where λ and r min are per-graph parameters and we have per-component parameters μ mn ∼ Expðμ|μ hp Þ and r mn ∼ Expðr mn |r hp scale Þ.
Source code and data
All source code and materials for running experiments can be obtained from the project website, at http:// ericmjonas.github.io/connectodiscovery/. All preprocessed data has been made publically available as well.
Extension to multiple graphs
The model can handle multiple graphs R q simultaneously with a shared clustering by extending the likelihood to include the product of the likelihoods of the individual graphs.
pðc; η q ; θ q |R q Þ ∝ ∏ q ∏ i;j pðR q À e i ; e j Á |f ðd À e i ; e j Á |η q c i c j ; θ q Þ ∏ m;n p À η q mn |θ q Á pðθ q Þ ! pðc|αÞpðαÞ:
Hyperprior grids and hyperprior inference
For the mouse retina logistic-distance Bernoulli model, we gridded μ hp and λ hp into 40 log 10 -spaced points 1.0 and 80. For the C. elegans data with the logistic-distance Poisson model, we gridded μ hp and λ into 20 log 10 -spaced points between 0.2 and 2.0, and the ratescale hp parameter into 20 log 10 -spaced points between 2.0 and 20.0. We globally set rate min = 0.01.
For the microprocessor with the logistic-distance with fixed lambda parameter and Bernoulli likelihood, we gridded mu hp into 50 log 10 -spaced points between 10 and 500 and set λ = μ hp /10. p min ∈ f0:001; 0:01; 0:02g and both p α and p β ∈ f0:1; 1:0; 2:0g.
Measuring clustering similarity
We compare discovered types to known types via cluster comparison metrics: cluster homogeneity, cluster completeness, and the ARI. Homogeneity measures how many true types are in a given found type. If every cell type is split into two types, each subtype is still completely homogeneous. Completeness measures how many members of a given true type are split across found types.
ARI takes into account both effects (Hubert and Arabie, 1985)-two identical clusterings have an ARI of 1.0, while progressively more dissimilar clusters have lower ARIs, becoming slightly negative as the clustering gets anti-correlated. Figure 11 shows the result of taking 20 different clusters and moving data points between them according to the following operations.
•distribute: take a class and distribute its elements uniformly among the remaining types.
•merge: take a type and merge it into another existing type.
•split: take a type and split it into two distinct types.
We can see the impact on ARI, completeness, and homogeneity as we perform these operations on more of the original 20 types. In all cases, 'distribution' of one type among the others is detrimental to the metric. Splitting impacts completeness but not homogeneity, and merging impacts homogeneity but not completeness.
Figure 1 .
1Continued dependent connectivity profile, similar depth, and a similar synaptic density profile, with cells of other types.
Figure 2 .
2Correct recovery of true numbers of hidden types in synthetic data when incorporating spatial information. (A)
Figure 3 .
3Model inferences when the true generating model differs from our distance-block-model prior. Horizontal columns show results with synthetic data generated according to the distance-dependent stochastic block model, the non-distance-dependent stochastic block model, the mixed membership block model, and the latent position cluster model. In all cases histograms represent posterior distribution over the indicated metric. (A) The number of types found by the model; the vertical dashed line indicates the 'true' type number (not applicable to the mixed membership model). (B) The area under the receiver operating characteristic (ROC) curve, indicating link prediction accuracy. (C, D, E) Clustering metrics quantifying degree of type agreement with known ground truth. DOI: 10.7554/eLife.04250.004
Figure 5 . 006 Figure 6 .
50066Discovering cell classes in the mouse retina connectome. Here we show the maximum a posteriori (MAP) estimate for the types in the mouse retina data. (A) Input connectivity data for 950 cells for which soma positions were known. (B) Clustered connectivity matrix; each arbitrary color corresponds to a single type and will be used to identify that type in the remainder of the plot. (C) The spatial distribution of our cell types-each cell type tessellates space. Colors correspond to those in (B). (D) Connectivity between our clusters as a function of distance-the cluster consisting primarily of retinal ganglion cells (brown nodes on the graph) exhibits the expected near and far connectivity. Conn prob: probability of connection. DOI: 10.7554/eLife.04250.Visualizing type inference uncertainty. Our fully Bayesian model gives a confidence estimate (posterior probability) that any two given cells are of the same type. In (A) we visualize that cell-cell coassignment matrix, showing the probability that cell i is of the same type as cell j on a range from 0.0 to 1.0. The block structure shows subsets of cells which are believed to all belong to the same type. For comparison, (B) shows the anatomist-defined type for each cell, grouped broadly into the coarse types identified in the previous panel. (C) Link versus cluster accuracy. (D) The posterior distribution of receiverFigure 6. continued on next page
Figure 7 .
7Discovering connectivity and type in C. elegans. (A) Posterior distribution on cell connectivity as a function of discovered type, similar toFigure 6. In (B) we plot neuroanatomist-derived types along with their labels. Our model shows a high probability of motor neurons, sensory neurons, and various interneuron classes being of the same type. Soma positions along the body axis are plotted in (C) where we see that we cluster spatially distributed motor neurons together, whereas head sensory neurons are more likely to be grouped together as well. (D) The receiver operating characteristic (ROC) curves for held-out link probability for both the electrical synapses (gap junctions) and chemical synapses in C. elegans. (E) The posterior distribution of the area under the ROC curve (AUC) for the curves in (D). (F) Measurements of the agreement of our identified cell types compared to neuroanatomists. The high completeness but low homogeneity (and corresponding low adjusted Rand index, ARI) reflects our model's tendency to group multiple types into a single type. DOI: 10.7554/eLife.04250.008
Figure 8 .
8Discovering connectivity and type in the MOS 6502 microprocessor. (A) The micrograph of the original microprocessor, with the region containing the registers under study highlighted. (B) Our graph consists of the interconnections of MOS field-effect transistors with three terminals, Gate, C1, and C2. The reconstruction technique did not permit resolution of C1 and C2 into source and drain. (C) The spatial distribution of the transistors in each cluster show a clear pattern. (D) The clusters and connectivity versus distance for connections between Gate and C1, Gate and C2, and C1 and C2 terminals on a transistor. Purple and yellow types have a terminal pulled down to ground and mostly function as inverters. The blue types are clocked, stateful transistors, green control the ALU and orange control the special data bus (SDB). DOI: 10.7554/eLife.04250.009
Figure 9 .
9Adjusted Rand index (ARI) for synthetic data as a function of run iteration. DOI: 10.7554/eLife.04250.011 pðc; η; θ|RÞ ∝ ∏
k
Þ, where the (s) superscript indicates these model parameters are associated with the soma-depth portion of our model. We use a conjugate prior for ðμ ðsÞ k ; σ 2ðsÞ k Þ with μ ðsÞ k ∼ Nðμ ðsÞ hp ; σ 2ðsÞ k =κ ðsÞ hp Þ and σ 2ðsÞ k ∼ χ −1 ðσ 2ðsÞ hp ; νðsÞhp . The use of conjugacy simplifies inference while allowing for each cell type to have its own depth mean and distribution.
Figure 10 .
10Total model score (log score) versus wall clock time. DOI: 10.7554/eLife.04250.012
AcknowledgementsWe thank Josh Vogelstein for discussions and reading of the manuscript, Finale Doshi-Velez for early discussions on the model, and Erica Peterson, Jonathan Glidden, and Yarden Katz for extensive manuscript review. Funding for compute time was provided by Amazon Web Services 'AWS in Education' grants. The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.Additional informationAdditional filesMajor datasetThe following dataset was generated:
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"Tell Me What They're Holding: Weakly-Supervised Object Detection with Transferable Knowledge from Human-Object Interaction",
"Tell Me What They're Holding: Weakly-Supervised Object Detection with Transferable Knowledge from Human-Object Interaction"
]
| [
"Daesik Kim [email protected] \nSeoul National University\n\n\nNAVER WEBTOON Corp\n\n",
"Gyujeong Lee \nSeoul National University\n\n",
"Jisoo Jeong \nSeoul National University\n\n",
"Nojun Kwak [email protected] \nSeoul National University\n\n"
]
| [
"Seoul National University\n",
"NAVER WEBTOON Corp\n",
"Seoul National University\n",
"Seoul National University\n",
"Seoul National University\n"
]
| []
| In this work, we introduce a novel weakly supervised object detection (WSOD) paradigm to detect objects belonging to rare classes that have not many examples using transferable knowledge from human-object interactions (HOI). While WSOD shows lower performance than full supervision, we mainly focus on HOI as the main context which can strongly supervise complex semantics in images. Therefore, we propose a novel module called RRPN (relational region proposal network) which outputs an object-localizing attention map only with human poses and action verbs. In the source domain, we fully train an object detector and the RRPN with full supervision of HOI. With transferred knowledge about localization map from the trained RRPN, a new object detector can learn unseen objects with weak verbal supervision of HOI without bounding box annotations in the target domain. Because the RRPN is designed as an add-on type, we can apply it not only to the object detection but also to other domains such as semantic segmentation. The experimental results on HICO-DET dataset show the possibility that the proposed method can be a cheap alternative for the current supervised object detection paradigm. Moreover, qualitative results demonstrate that our model can properly localize unseen objects on HICO-DET and V-COCO datasets. | 10.1609/aaai.v34i07.6784 | [
"https://ojs.aaai.org/index.php/AAAI/article/download/6784/6638"
]
| 208,158,215 | 1911.08141 | 30b8286d14b00922885bdfa1862b4ed171310f77 |
Tell Me What They're Holding: Weakly-Supervised Object Detection with Transferable Knowledge from Human-Object Interaction
Daesik Kim [email protected]
Seoul National University
NAVER WEBTOON Corp
Gyujeong Lee
Seoul National University
Jisoo Jeong
Seoul National University
Nojun Kwak [email protected]
Seoul National University
Tell Me What They're Holding: Weakly-Supervised Object Detection with Transferable Knowledge from Human-Object Interaction
In this work, we introduce a novel weakly supervised object detection (WSOD) paradigm to detect objects belonging to rare classes that have not many examples using transferable knowledge from human-object interactions (HOI). While WSOD shows lower performance than full supervision, we mainly focus on HOI as the main context which can strongly supervise complex semantics in images. Therefore, we propose a novel module called RRPN (relational region proposal network) which outputs an object-localizing attention map only with human poses and action verbs. In the source domain, we fully train an object detector and the RRPN with full supervision of HOI. With transferred knowledge about localization map from the trained RRPN, a new object detector can learn unseen objects with weak verbal supervision of HOI without bounding box annotations in the target domain. Because the RRPN is designed as an add-on type, we can apply it not only to the object detection but also to other domains such as semantic segmentation. The experimental results on HICO-DET dataset show the possibility that the proposed method can be a cheap alternative for the current supervised object detection paradigm. Moreover, qualitative results demonstrate that our model can properly localize unseen objects on HICO-DET and V-COCO datasets.
Introduction
In a decade, object detection has become one of the most successful fields in computer vision with various applications (Ren et al. 2015;Dai et al. 2016;Redmon et al. 2016;Liu et al. 2016). Most of the successful models have emerged after the release of large scale datasets (e.g. PASCAL VOC, MS-COCO (Everingham et al. 2010;Lin et al. 2014)) with bounding box annotations. Given input images, conventional object detection models can localize boxes with the corresponding class scores. Thus, they normally require manually annotated bounding boxes containing accurate coordinate values and object labels for training.
However, annotating bounding boxes is time-consuming and labor-intensive. It can also be difficult to expand the volume of a dataset by adding more object classes or adding Figure 1: 1) Two different types of description of an object. A is human's way of identifying an object while B is for machines. 2) Manually annotating time for three tasks. Bearman et al. (Bearman et al. 2016) estimated annotation times for image level, bounding box and pixel level. At rightmost, annotating time of relation sentence can be similar to that of image-level since only action verb "hold" is added. more images. Therefore, researches to reduce those costs in various ways have drawn attentions these days.
Weakly supervised object detection (WSOD) has been proposed to tackle the aforementioned problems (Zhu et al. 2017;Shi, Caesar, and Ferrari 2017;Jie et al. 2017). It is to detect objects within images by weak supervision such as image-level labels. At the cost of lowered annotation cost, WSOD performs worse than full supervision.
To overcome a limitation of weak supervision, some approaches (Shi, Caesar, and Ferrari 2017) rely on another type of full supervision with transfer learning. Transferred knowledge from a source domain could support weak super-vision in a target domain. However, annotating other types of labels such as segmentation mask is also expensive.
Our main intuition is that supervision of machines is totally different from that of humans. For example, Fig. 1(1) shows different ways of identifying objects between humans and machines. While we should provide accurate coordinate values of object boxes for machines, humans usually recognize new objects from contexts. Contexts also can reinforce supervision without much additional efforts.
Especially, how objects are related to human actions can be practical and advantageous since information about a human can be a proper evidence for recognizing contexts in an image. Moreover, humans can easily express contexts with sentences as shown in Fig. 1(1), so that linguistic labels can be a key to reduce annotation cost for humans as shown in Fig. 1(2). Compared to other annotating costs (Bearman et al. 2016), the cost of annotating a relation sentence such as "person, hold, bottle" can be almost similar to that of imagelevel annotation. Thus, we propose a novel paradigm to learn unseen objects based on human-object interaction (HOI).
Our key idea is to exploit transferable knowledge from HOI contexts annotated by language as is in (Chao et al. 2018). Specifically, we propose a novel module that predicts object locations from HOI. Since the actual coordinate values can not be specified, we use an attention map as localization results to connect it with a bounding box. Moreover, in order to train a full object detector (e.g. Faster-RCNN (Ren et al. 2015)) in an end-to-end fashion, we design a new module as an add-on type.
The objective of this paper is to make our model learn additional rare classes with weak verbal supervision annotated easily by human. During the first stage, strong supervision on non-rare classes teach our model to localize a proper location with a human pose and an action verb. In the next stage, only weak supervision with transferred knowledge keeps training an object detector for unseen rare classes.
Our main contributions can be summarized as follows:
• We define a new weakly-supervised object detection scheme which mainly relies on interactions between a human and objects without box annotations.
• We propose a novel module called RRPN (Relational Region Proposal Network) to localize boxes by using the location of a human and verb embeddings.
• The proposed RRPN is designed as a universal add-on type that can be easily adapted into existing models such as Faster-RCNN.
Our experiments validate that our model outperforms baselines on the HICO-DET (Chao et al. 2018) dataset and can effectively transfer the knowledge to other dataset such as V-COCO (Gupta and Malik 2015).
Related works Weakly Supervised Object Localization and Detection
Most of the weakly supervised object localization and detection methods have been proposed based on an imagelevel supervision. With cheaper but weaker annotations, studies (Bilen and Vedaldi 2016; Diba et al. 2017;Kantorov et al. 2016;Oquab et al. 2015;Tang et al. 2017;Jie et al. 2017) mainly tried to enhance performance by multiple instance learning (MIL). In MIL, a bag is defined as a collection of regions in an image. It is labeled as positive if at least one object is positive and labeled as negative if all the objects are negative. Tang et al. 2018 have proposed proposal cluster learning algorithm to learn refined instance classifier. Yang et al. 2019 have proposed activity-driven WSOD, which also exploits action classes as contextual information to localize objects without box annotations. However, those works has generated box proposals using Selective Search (Uijlings et al. 2013), which is a rule-based algorithm. Since the box proposal method cannot be trained, there is a fundamental limitation which proper proposals hardly exist in novel data. Uijlings, Popov, and Ferrari 2018 have addressed a new WSOD framework that revisits knowledge transfer for training object detectors on target classes. Since this work has optimized a box proposal network for target classes by MIL, box generators can be insufficiently trained with rare classes due to a lack of contextual information. Our method resolves the aforementioned issues with a novel box proposal module that can transfer knowledge using an HOI dataset. Human-object interaction Visual recognition of HOI is crucial for comprehending a scene in an image. Early work studied mutual context of human pose and objects (Yao and Fei-Fei 2010) and Bayesian model (Gupta and Davis 2007;Gupta, Kembhavi, and Davis 2009) with handcraft features. Recently, with success of deep learning, Chao et al. 2015 introduced a new large-scale benchmark, "Humans Interacting with Common Objects" (HICO), for HOI recognition, which was expanded for detection problems in HICO-DET (Chao et al. 2018). In order to solve HICO-DET datasets, various approaches have been proposed. In (Chao et al. 2018), combined features from human proposals and object regions were used to solve HOI detection. Gkioxari et al. 2018 proposed a human and object detector-based approach estimating a density map based on Faster-RCNN architecture. A recent approach (Qi et al. 2018) generates the HOI graph and propagates message between nodes to infer relationships in a parsing graph. In this paper, rather than directly solving HOI problems, we exploit contextual information in HICO-DET to construct a weakly-supervised object detector.
Algorithm Overview
An overview of our algorithm is illustrated in Fig. 2
. Let D = {(I i , y i )} N i=1
is the data set, where y i is the label of the image I i and N is the number of images. The image label y i is organized as a tuple as shown below:
y i = {(H j verb , O j bbox , O j cls )} M j=1(1)
where, O j bbox , O j cls are the bounding box and the class of an object in the image, H j verb is the action verb corresponding to the object, and M is the number of tuples in the image I i . To evaluate the proposed method, we divided D into two sub-categories based on the number of objects in a class: non-rare (source classes) D S and rare (target classes) D T . Figure 2: Overview of our algorithm. 1) During the training phase for source classes, RRPN is also trained to predict attention map from human-object interaction. 2) In the targetclass training phase, an object detector is trained using the ground truth class label, and the box label provided by the trained RRPN. In other words, our problem focuses solely on solving the weakly supervised object detection problem on the target classes. 3) As a result, the trained object detector for target classes can infer box coordinates and object classes with only an image input.
Note that, there is no object class duplicates but all action verbs are overlapped between two subcategory datasets.
For D S , we normally train the first object detector (blue circle in Fig. 2) with full supervision using (O j bbox , O j cls ). Along with training of the object detector, we also train an object localizer (red circle) called RRPN with newly defined inputs. Since the RRPN should learn how to localize an object only with the information on a human and an action verb, we use the image I i , the verb H j verb , and the pose of the human H j pose as inputs. The H j verb simply comes from y i but the human pose H j pose is extracted from an image I i with an existing human pose estimation method. As a results, the RRPN predicts an attention mapà j i of an object location in the i-th image from human's action and appearance. We optimize losses regarding the object class and the location using O j bbox and O j cls for the object detector, but create a Gaussian map of O j bbox and use it as a ground truth in the training of the attention map of the RRPN. In this phase, since the ground truth bounding box location is available, the RRPN can learn common knowledge between objects and human actions.
For D T , we assume that only object class information O j cls and the action verb H j verb are available but the bounding box information is not. To fill the absence of O j bbox , we exploit learned knowledge inferred by the RRPN with the same kinds of inputs as the training phase for the source classes. Since the output of RRPN is an attention map, we extract a coordinate by thresholding it and generate pseudo bounding boxÔ j bbox . Then, we normally train the second object detector (green rectangle in Fig. 2) for D T . Since we already have used all action verbs to train the RRPN in the previous phase and transfer the same parameters in the training phase for the target classes, it can infer an object location with a human pose and an action verb. In Fig. 2, after the RRPN already learned to localize unseen object "Apple" with verb "EAT" and grabbing pose in the training phase of D S , it can infer a proper location as a pseudo ground trutĥ O j bbox . In conclusion, we use weak supervision by human actions to train a full object detector.
Eventually, the trained object detector in the second phase can predict objects in D T only with an input image I i as shown in Fig. 2. Although we have not shown the real location of "Apple", it is possible to predict the class score and the coordinate of an "Apple" object.
In this scheme, we can additionally train new object detector for unseen rare classes without bounding box annotations. Moreover, since we already trained the RRPN with strong supervisions, we need smaller amount of data in target classes compared to other WSOD algorithms. Our experiments validate that our target domain which contains extremely rare object classes is trained successfully by our method. Fig. 3 depicts the overall architecture of the proposed algorithm. The proposed algorithm consists of two modules, including the RRPN and the object detector. More precisely, it means that RRPN can be combined with the conventional architecture such as the Faster-RCNN. RRPN is a multistage encoder-decoder network, which is responsible for predicting an object-location-centric attention map A j i from a multi-domain integrated feature map. The other module, object detector, is a conventional object detector which is trained for a given input image using the ground truth label O cls and the bounding box O bbox .
Architecture
In order to exploit the knowledge of interaction, we train D T after the training of D S is done. Since, however, we do not account for the continual learning, D S and D T do not share parameters for the object detector. While the object detector is trained for D S with supervision, at the same time, RRPN is also trained to learn the knowledge from interactions between a human and an object through action verbs. Then, the object detector is trained for D T without object bounding boxes, i.e. in a weakly supervised way, using the transferred knowledge from D S .
Training on the Source classes D S D S are object classes on which data can be easily acquired. Training on D S is a standard supervised object detection procedure by using ground truth class and box labels for all the objects. The main purpose of training on D S is to predict an object location from a human-object interaction. Therefore, RRPN is also trained at the same time as the training of the object detector. The detailed training procedure for Faster-RCNN is applied in the same way as the original paper. The training procedure of RRPN is as follows.
Relational Region Proposal Network (RRPN) RRPN is designed to be universally applicable to various task's models, including other object detectors, in an add-on manner, In RRPN, a combined feature F int from verb, pose and image produces an attention map through a network which has four blocks. With source classes, the RRPN is trained with a Gaussian mask from the ground truth bounding box. However, with target classes, the RRPN generates a pseudo ground truth bounding box so that Faster-RCNN can be optimized. and can share the backbone network with other model for image features to improve memory efficiency.
As mentioned above, RRPN predicts attention mapà j i for a given image I i using a multi-domain feature map F i,j int ∈ R C×H×W as an input, where C, W, H are the depth, width and height of the feature map. F int is obtained by
F i int = {F i,j int } M j=1 = {(F j img ⊕ F j pose ⊕ F j verb )} M j=1 ,
(2) where, F img , F pose , F word are the image feature, pose feature, and verb feature obtained by their corresponding models f img (I i , θ img ), f pose (I i , θ pose ), f word (H j act , θ verb ), and ⊕ is the matrix concatenation. Here, θ x is the corresponding model parameters. The convolution operation for F i int computes the object existence probability for a combination of F j img , F j pose , and F j verb at a specific location on I i . As in (2), we used three feature maps to utilize contexts from various domains in a given dataset, and each feature map has its own contribution. The pose and word feature are responsible for the visual context of the human's location and action, and the distinguishable linguistic context for the human's action, respectively. The image feature is responsible for representing the whole scene as well as the object of interest. The details for each feature maps are as follows: Pose feature We use the well-known human pose estimation model, OPENPOSE (Cao et al. 2017), to extract pose features. OPENPOSE predicts the location of human body joints using image or video as an input. The output consists of channels corresponding to each joint and a channel representing background information. In this paper, we used a pose estimation model with 19 channels including 18 joints and 1 background. In order to feed distinct information of human pose to the RRPN, we exploit the 18 channels except for the background channel as the pose feature. Verb feature The widely used GloVe-twitter-27B-25d model (J. Pennington and Manning 2014) is applied as the word embedding model for the verb. Since a word is embedded into a vector, one needs to convert it into a tensor form for integration with other features. While F j img and F j pose may have different spatial-wise activations depending on I i , F j verb must have the same value regardless of positions. In designing F j verb , we also take this consideration into account. In order to match the spatial dimension with others, the verb feature is copied to every spatial position. So that dimension of F j verb is converted from R 25 to R 25×H×W . By stacking a depth-wise word vector at all spatial positions, we can conduct a convolution operation using the same verbal information at all position of F j img . Note that, among the HICO-DET datasets, tuples with 'No interaction' verb labels were excluded from training and validation phases for accurate evaluation of the proposed algorithm.
Image feature The proposed algorithm makes use of the representative two-stage object detection model, Faster-RCNN. It consists of a feature extractor, a back-bone network, and a region proposal network (RPN). The output feature map of the back-bone network of the Faster-RCNN is used as the image feature for the RRPN. When training on the target classes, the parameters of the backbone network are reused, but the parameters of RPN are reset.
Multi-domain feature map F i,j int is then fed into the network to predict attention map A j i . In order to robustly detect objects in various sizes, we designed the network architecture which has four blocks as in : an Encoder block f en , two decoder blocks f de1 , f de2 and an attention block f att . f en takes F i,j int as an input and outputs two feature maps with different spatial dimensions. Then, each output feature map feeds into f de1 and f de2 , respectively. The output feature maps of f de1 and f de2 having the same spatial dimension are concatenated and inputted to the attention block resulting in an attention mapà j i as
A j i = f att [f de1 {f 1 en (F i,j int )} ⊕ f de2 {f 2 en (F i,j int )}].(3)
The output of RRPN is an attention map which emphasizes the location where the object is likely to be located. To train attention maps, we create a Gaussian map A j i , as a ground truth attention map, using O j bbox . RRPN is trained using A j i as the label. We use pixel-wise binary cross entropy loss (BCE) L att between (A j i ,Ã j i ). The total loss for training on the source classes including RRPN and object detector is shown below:
L total = L det + λL att , L det = L cls + L loc(4)
where, λ is a hyper-parameter balancing between the two losses and L det is the loss for the Faster-RCNN. In the object detector point of view, the proposed algorithm on D S is trained in the same way as the conventional supervised objected detection algorithms.
Training On the Target classes D T
The object detector for D T should be trained without O bbox . Therefore, we define this problem as a weakly supervised object detection (WSOD) problem. We useÕ bbox as an alternative to the missing O bbox utilizing RRPN learned in the source classes training phase. It is expected that the trained RRPN can predict locations of unseen objects i.e. D T , since it is trained to predict the object location using a human pose, an action (verb) and an image feature. The training process on D T using the trained RRPN is as follows:
The F pose , F verb , and F img are fed into the trained RRPN. We apply a threshold to obtain a pseudo bounding box from the output attention map as
A j i = 1, if,Ã j i > δ 0, otherwise(5)
where, δ is a pre-defined threshold. The largest bounding box containing a valid value inà j i is calledÕ j bbox . The pseudo ground truth bounding boxes {Õ j bbox } M j=1 obtained from the attention maps {à j i } M j=1 of all tuples in the image I i are collected together and used as bounding box labels O bbox for training an object detector. In this step, a different type of object detector from the one trained in D S training phase can be used for training. The object detector is trained to minimize detection loss using O cls andÕ bbox .
Experiment
In this section, we evaluate the performance of the proposed WSOD algorithm. To the best of our knowledge, no previous studies have been conducted on the relationship between object detection and HOI. Nevertheless, we conducted the performance comparison with prior works on HICO-DET.
Dataset and Pre-processing
HICO-DET dataset consists of 47,776 images (38,118 training and 9,658 testing) classified into 117 actions (verb) and 80 object classes, and the object classes are the same as MS-COCO dataset. The ground truth labels consist of a tuple of (H act , O bbox , O cls ) as in (1). Note that, the RRPN is trained based on tuples, so images containing multiple tuples are fed multiple times. The total number of tuples is 151,276 (117,871 training and 33,405 testing), and we use 131,560 tuples (102,450 training and 29,110 testing) excluding the tuples corresponding to the action label 'no-interaction'.
In order to construct the problem environment, the whole dataset is divided into source and target datasets according to the frequency of the object class. Our basic experiment is set up with 116 verbs excluding 'no interaction'. The number of object classes are 70 for D S and 10 for D T . In order to more clearly show the effectiveness of RRPN, in the experiment for qualitative result, we use 5 verbs and 10 D S and 70 D T . Other hyperparameters remain the same as the basic set up.
We also verified the proposed algorithm on V-COCO dataset for qualitative analysis. The purpose of evaluation on the V-COCO dataset is to show that the knowledge can be transferred from one dataset to other. Details on both datasets are described in the supplementary material.
Metrics
We use mean Average Precision (mAP) and Recall as evaluation metrics. Because RRPN produces one bounding box for one tuple (action), Recall is used to measure how accurate the location of an object corresponding to an action is. In other words, Recall evaluates the objectness ofà predicted by RRPN, and is calculated as the ratio of tuples for which IoU > 0.5. On the other hand, the object detector detects all the objects in an image at once. Therefore, we use the mAP in measuring the performance of Faster-RCNN which are the standard metrics for object detectors. D S and D T in Recall are the performance of the RRPN's agent after the training on D S . When training on D T , RRPN is fixed and not trained. Note that Recall is measured on test set for D S and on both training and test set for D T .
Comparison with prior works
We conduct experiments to compare with prior works on HICO-DET as shown in Table 1. First two columns represent overall results of original algorithms in AD (Yang et al. 2019) and PCL (Tang et al. 2018). However, both results are only able to show performance of all object classes with an entire dataset. Since our method is designed for transfer learning, we experiment to validate PCL on each of source and target domains. As a result, our best model with image, pose and verb has 17.19% which is 4 times better than the result of PCL on D T . Moreover, our model only with the image feature outperforms PCL on D T . Although a direct apple-to-apple comparison is difficult, we can see that our method is far better than the compared methods.
Comparison with different feature combination
We experiment to verify the performance of different feature combinations. We train and test the RRPN using the same types of feature for both D S and D T in each experiment. Table 2 shows the performances of RRPN and Faster-RCNN as Table 1: Comparison of the mAP with other WSOD algorithms on HICO-DET. (PCL* is tested by ourselves, § is trained on the entire dataset and † is trained on D S and D T separately. I : F img , P : F pose , V: F verb , W : Weakly supervised object detection, λ = 10, δ = 0.1)
Methods AD PCL PCL* Ours (I) Ours (I+P+V)
D S (W ) - -4.80 § 5.01 † - - D T (W )
--0.01 § 4.75 † 9.57 17.19 Total 5.39 3.62 4.42 § --- Table 2: Performance comparison of different feature combination. (Notations, λ, and δ are the same as Table 1) Recall and mAP, respectively, using different combinations of features. D T (W ) in mAP is the results of our WSOD, and D S and D T (S) are the results of full supervision.
I P V [email protected] (RRPN) [email protected] (Faster-RCNN) D S (%) D T (%) D S D T (W) D T(
Our full model combining all three features in the top of Table 2 shows the highest performance in both Recall and mAP among all combinations. The mAP of D T (W ) has 17.19%, which is 7.62% better than image-only model and the Recall of the D T is 28.64% which is about 5% higher than other combinations. Moreover, the mAP score of our full model is only 4.88% lower than image-only fully supervised model in D T (S). Compared to models of full supervision D T (S), we believe that the mAP score of our full model can meaningfully show that it can be trained despite weak supervision of rare classes.
In the middle, using F img alone, Recall for D T has 22.75% and mAP (D T (W )) are much lower than mAP (D T (S)). It means that RRPN could not be trained solely by F img . The two results in the bottom are the performance for combined features. When F pose is combined with F img , Recall degrades and mAP increases slightly. It can show that F pose that is extracted from an image is redundant unless it interacts with a verb. Combining F verb with F img , however, Recall and mAP significantly increase and mAP(D T (S)) also increases. It is interesting that it might be more effective for not only RRPN but also Faster-RCNN when using combination of features from other domain.
Comparison with different λ and δ
In experiments in Table 3, we focus on verifying the effect of shared parameters such as λ in (4) and δ in (5).
In top of Table 3, according to the change of the λ in (4), the ratio of the loss weight in RRPN is determined. When λ is zero, due to untrained RRPN, Recall and mAP for D T (W ) have the lowest score while mAP for D S has the highest score. On the other hand, Recall and mAP are the highest at λ = 10 with performance improvements of 17.45% and 15.58% compared to λ = 0, respectively. On the contrary, on some levels of λ, we can see that the performance degradation for not only D T (W ) but also D S . This can be understood as an effect of parameter sharing for image feature extractor between RRPN and an object detector. As mentioned earlier, RRPN is a universal add-on type module which can be adapted to various computer vision tasks. To effectively utilize these advantages, we share the backbone network of RRPN and the object detector in consideration of memory efficiency. Therefore, the RRPN and the object detector affect each other through the backbone network during training.
In bottom of Table 3, according to the δ in (5), the size of the pseudo bounding box is determined. A small δ makes the size of the boxes increase, while a large δ makes the box small or disappear. As δ increases, partial information of the object is trained. For example, in the case of an apple, only the central part of the apple is trained with high δ, which causes many false positive. On the other hand, lowering the threshold of the box, Faster-RCNN is trained not only with an object but also with backgrounds. It is interesting that δ affect differently to both metrics where Recall gets higher when δ gets smaller but mAP get the highest score when δ = 0.1. We believe that the RRPN can easily learn objectness with a larger box due to small δ, but classification of objects could be more difficult due to inaccurate localization. Therefore, too small or too large δ causes a degradation of mAP, and we have found the suitable value, δ = 0.1, through the experiments by selecting the value with the highest performance in D T (W ). Fig. 4 shows the qualitative results of the proposed algorithm on D T . The first column indicates input images and the last column indicates output attention maps inferred by the corresponding actions. We can see that RRPN predicts an accurate attention map on unseen object classes in D T . Furthermore, it can be seen that the pattern of the predicted attention map differs depending on the verb. For example, while 'hold' shows a strong activation value near the human hand, 'ride' tends to activate at the bottom of a person. Based on this, we can confirm that the object location can be estimated based on the interaction between the verb and the pose. The role of the pose can be found in the example of [Truck, Ride]. Despite that two trucks exist in an image, the activation of a truck on which the human is riding shows stronger than the other. This can be seen as a contribution of F pose to the object localization. We also verified the performance of RRPN on the object from a different dataset, V-COCO. The RRPN is trained using D S of HICO-DET and predicts the attention map of D T of V-COCO. The bottom row shows the predicted attention map on V-COCO. We can see that the proposed algorithm can also predict the object location accurately on images even from other datasets. Fig. 5 depicts the comparison of predicted attention map between different feature combinations on [glove, hold]. As described in section 5, the pattern of the resulting attention map can be changed by the combination of features. Since "Glove" is an unseen class, backbone has no information to extract reliable feature, so that RRPN cannot predict the location of an object accurately using only F img . However, if RRPN is trained using more than two features including F img , RRPN can infer the location of an object based either on F pose or on F verb . Specifically, F img + F verb predicted a more distinguishable attention map for an object, compared to F img + F pose feature map. Since F img and F pose are extracted from the same image, some of the information can be redundant between two features. On the other hand, F verb is able to provide useful information to F int because it is extracted from a different domain, language. Consequently, the location of an object can be predicted precisely when we use all three features. On the contrary, if RRPN trained using only F pose and F verb without F img , the output attention map only activates around the human. Thus, it can be understood that F verb plays a role of providing supplementary information to F img about the object of interest.
Qualitative results
Conclusion
In this paper, we proposed a novel weakly-supervised scheme for object detection problems. We introduced the RRPN which can universally localize objects in an image with information on human poses and action verbs. Using transferable knowledge from the RRPN, we can continuously train any object detector for unseen objects with weak verbal supervision describing HOI. We validated our method based on the results on HICO-DET dataset and the performances show the possibility of our method for a new WSOD training scheme. Our work shows sufficient potentials to overcome the inefficiency of the supervised training scheme in recent deep learning. Also, we can develop our method in the direction to the continual learning since we already suggested a novel method to transfer common knowledge to localize objects with HOI.
Figure 3 :
3Overall network architecture of the proposed algorithm. Relational Region Proposal Network (RRPN) at the top is mounted on a basic Faster-RCNN model at the bottom.
Figure 4 :Figure 5 :
45(Left) Input image with pose, (middle) ground truth Gaussian attention mask (A) in yellow, and (Right) predicted attention map (Ã). Red box is pseudo object box , blue box is ground truth and white box indicates the human in action. (V) the last row is the result on the V-COCO dataset. Note that a white box is used solely for visually representing an acting human in an image and is not used in training on D T . Comparison of predicted attention maps trained only by the image feature and by various integrated features with [glove, hold]. The predicted attention maps show different activations depending on the role of each feature map.
Table 3 :
3Comparison of quantitative result of λ and δ (Notations are the same asTable 1)parameter [email protected] (RRPN) [email protected] (Faster-RCNN)
λ
δ
D S (%) D T (%)
D S
D T (W) D T (S)
0
0.1
11.64
11.19
31.06
1.61
25.57
1
0.1
41.51
24.46
23.87
9.27
25.43
5
0.1
46.37
23.37
30.10 14.38
26.11
10 0.1
47.69
28.64
30.34 17.19
29.37
15 0.1
46.65
23.07
23.32 15.85
25.45
20 0.1
43.17
26.00
22.68 15.75
22.92
10 0.05
48.22
29.41
30.27
9.41
25.04
10 0.10
47.69
28.64
30.34 17.19
29.37
10 0.15
39.67
17.96
30.40 14.01
26.66
10 0.20
34.14
16.72
30.22 13.24
30.39
AcknowledgmentsThis work was supported by Next-Generation Information Computing Development Program through the NRF of Korea (2017M3C4A7077582) and Promising-Pioneering Researcher Program through Seoul National University(SNU) in 2015.
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| []
|
[
"Entanglement, Flow and Classical Simulatability in Measurement Based Quantum Computation",
"Entanglement, Flow and Classical Simulatability in Measurement Based Quantum Computation"
]
| [
"Damian Markham \nDépartement Informatique et Réseaux\nCNRS LTCI\nTelecom ParisTech\n23 avenue d'Italie51327, 75214Paris CEDEX 13CSFrance\n",
"Elham Kashefi \nSchool of Informatics\nUniversity of Edinburgh\n10 Crichton StreetEH8 9ABEdinburghUK\n"
]
| [
"Département Informatique et Réseaux\nCNRS LTCI\nTelecom ParisTech\n23 avenue d'Italie51327, 75214Paris CEDEX 13CSFrance",
"School of Informatics\nUniversity of Edinburgh\n10 Crichton StreetEH8 9ABEdinburghUK"
]
| []
| In measurement-based quantum computing one starts with a large entangled resource state |Ψ RES on n qubits. We identify a two sets of qubits, I which will represent the inputs, and O which will represent the outputs of the computation, with n ≥ |O| ≥ |I|. Generally one can consider three types of computation using this resource, one with a classical input and a classical output (let's call this CC), one with a quantum input and a classical output QC and one with a quantum input and a quantum output QQ. Clearly QQ is the most general, since one can always encode classical information onto quantum states. In this work we focus on QQ. When considering a quantum input |ψ S (on a system S of |I| qubits) the first step is to teleport the input system qubits S onto I on the resource state, by some global map on I and S. This can be done for example by entangling I with S (using, say, a control-Z gate) and performing Pauli X measurements on S then appropriate corrections (see e.g.[25] for graph state resources). The computation then proceeds by a series of measurements on individual qubits, followed by corrections, then further measurements and corrections and so on until the computation is complete. We call the sequence of measurements and corrections the measurement pattern (see[6]for formal definitions). The outputs qubits, labelled O are those qubits which at the end of the computation are not measured. In this way the computation uses the resource state to transfer the input from I to O, in a kind of involved teleportation, at the same time performing some unitary over the input.We begin with the following definitions. | 10.1007/978-3-319-06880-0_22 | [
"https://arxiv.org/pdf/1311.3610v1.pdf"
]
| 34,644,012 | 1311.3610 | 3d70ec01b96c26563fbb0d7c82df686c9a657ea9 |
Entanglement, Flow and Classical Simulatability in Measurement Based Quantum Computation
14 Nov 2013
Damian Markham
Département Informatique et Réseaux
CNRS LTCI
Telecom ParisTech
23 avenue d'Italie51327, 75214Paris CEDEX 13CSFrance
Elham Kashefi
School of Informatics
University of Edinburgh
10 Crichton StreetEH8 9ABEdinburghUK
Entanglement, Flow and Classical Simulatability in Measurement Based Quantum Computation
14 Nov 2013
In measurement-based quantum computing one starts with a large entangled resource state |Ψ RES on n qubits. We identify a two sets of qubits, I which will represent the inputs, and O which will represent the outputs of the computation, with n ≥ |O| ≥ |I|. Generally one can consider three types of computation using this resource, one with a classical input and a classical output (let's call this CC), one with a quantum input and a classical output QC and one with a quantum input and a quantum output QQ. Clearly QQ is the most general, since one can always encode classical information onto quantum states. In this work we focus on QQ. When considering a quantum input |ψ S (on a system S of |I| qubits) the first step is to teleport the input system qubits S onto I on the resource state, by some global map on I and S. This can be done for example by entangling I with S (using, say, a control-Z gate) and performing Pauli X measurements on S then appropriate corrections (see e.g.[25] for graph state resources). The computation then proceeds by a series of measurements on individual qubits, followed by corrections, then further measurements and corrections and so on until the computation is complete. We call the sequence of measurements and corrections the measurement pattern (see[6]for formal definitions). The outputs qubits, labelled O are those qubits which at the end of the computation are not measured. In this way the computation uses the resource state to transfer the input from I to O, in a kind of involved teleportation, at the same time performing some unitary over the input.We begin with the following definitions.
Abstract. The question of which and how a particular class of entangled resource states (known as graph states) can be used for measurement based quantum computation (MBQC) recently gave rise to the notion of Flow and its generalisation gFlow. That is a causal structure for measurements guaranteeing deterministic computation. Furthermore, gFlow has proven itself to be a powerful tool in studying the difference between the measurement-based and circuit models for quantum computing, as well as analysing cryptographic protocols. On the other hand, entanglement is known to play a crucial role in MBQC. In this paper we first show how gFlow can be used to directly give a bound on the classical simulation of an MBQC. Our method offers an interpretation of the gFlow as showing how information flows through a computation, giving rise to an information light cone. We then establish a link between entanglement and the existence of gFlow for a graph state. We show that the gFlow can be used to bound the entanglement width and what we call the structural entanglement of a graph state. In turn this gives another method relating the gFlow to bounds on how efficiently a computation can be simulated classically. These two methods of getting bounds on the difficulty of classical simulation are different and complementary and several known results follow. In particular known relations between the MBQC and the circuit model allow these results to be translated across models.
Measurement Based Quantum Computing (MBQC) [1] has attracted attention recently for its potential towards the realisation of a quantum computer, its role in understanding the power and significance of entanglement for computation [2,3], and that it plays a key role in the development of cryptographic protocols [4,5]. In MBQC one starts off with a large multiparty entangled resource state and the computation is driven by a series of local measurements, the choice of which can depend on the result of previous measurements in the series. The formal language for MBQC was jointly developed by Prakash Panangaden in [6]. In this work we are interested in the question of how to recognise or characterise a 'good' resource for measurement based quantum computing. Given the fact that after the generation of the state, all operations are local, it is natural to expect entanglement to play a key role. Indeed it has been shown that the entanglement of a resource state must be sufficiently high for it to be universal and not classically simulatable (we note that these two properties are currently not known to be equivalent, though it is broadly expected that they are) [7,8,9,10,11].
A related question to universality is that of the ability, or not, for a resource to allow any unitary MBQC computation on it at all. This question is addressed by what is called Flow or its generalisation referred to hereinafter as gFlow [12,13], for a particular class of resource states, called graph states [14] and its extension open graph state [12] (see also below). There exist efficient algorithms [15,16] to find gFlow if it exists, and once gFlow is found, it gives an explicit measurement pattern which gives a unitary computation across the resource graph state in hand. Subsequently gFlow has been a useful tool for exploring many aspects of MBQC such as efficient translation between MBQC and the circuit model [12,17], analysing cryptographic protocols [18], direct pattern design in MBQC [19], proving bounds on depth complexity [13,20] and from a more fundamental perspective, the arrival of causal order in MBQC [20,21,22,23].
In this paper we show that gFlow also gives a bound on the difficulty in classically simulating MBQC, and how it can be interpreted as a flow of information. This leads to the observation that the causal forward cone (the 'forward cone' of a qubit is given by the qubits who's corrections directly or indirectly depend on that qubit's measurement results) is equal to the information cone (the cone of qubits where the information spreads to through the computation). We then establish an intuitive link between gFlow on the one hand, and entanglement of a resource state on the other. We further make this connection explicit by showing how gFlow can be used to give bounds on the entanglement of a graph state. In this way we will see that properties of simulateability of MBQC on a resource in terms of entanglement can be translated to conditions in terms of gFlow. Via a known relationship between the circuit model and MBQC these results can also lead to conditions on simulateability of circuits. One such example is a rederivation of the result by Jozsa [24].
The organisation of this papers is as follows. In Section 1 we mention basic observations about entanglement conditions for any good resource state for MBQC which will be then linked to gFlow. In Section 2 we introduce graph states and review the notion gFlow and several preliminary notions necessary for the rest of the paper. In Section 3 we prove that gFlow can be used to give bounds on direct simulation of a MBQC. In Section 4 we discuss how gFlow can be used to see how information flows through a resource in an MBQC, giving in particular an information light cone which coincides with the causal cone as defined in [22,23]. In Section 5 we show how gFlow can be used to bound the entanglement of a resource state which gives a new route to bounding simulatability of a MBQC, which is different and complementary to the direct simulation in Section 3. We finish with discussions.
Definition 1 A resource state |Ψ RES on n qubits, with defined input qubits I and output qubits O is D-Happy if for all bi-partitions A, B such that I ∈ A and O ∈ B we have
E A,B (|Ψ RES ) ≥ |I|,(1)
where E A,B (|Ψ RES ) = S(ρ A ) = S(ρ B ) is the entropy of entanglement across partition A, B, where S(ρ A ) = − ln T r(ρ A ) is the von Neumann entropy of the reduced state ρ A = T r B (ρ AB ).
Definition 2 A MBQC pattern is called unitary if for all inputs the returned state of the output is an encoding of a unitary acting on the input.
A similar notion was defined in [26] as information preserving pattern. We present a simple but important observation about the link between the above two definitions.
Observation 1 There exists a unitary MBQC pattern on a resource state |Ψ Res on n qubits, with input qubits I and output qubits O only if it is D-Happy.
Proof. We start by noting that in order to teleport a state |ψ ∈ C ⊗|I| 2 perfectly across |Ψ Res to its output space, it is necessary to have E A,B ≥ |I|, for all bi-partitions A, B such that I ∈ A and O ∈ B. To see this is true one can consider the state to be teleported as half a maximally entangled state. After the teleportation one would end up with a state with entanglement E A,B = |I|. Since all operations are local, and it is not possible to increase entanglement in the process, this implies that we started with E A,B ≥ |I|. We then note that any measurement based computation can be considered as a teleportation across any cut which divides the inputs from the outputs -since all operations are local to each qubit, they are certainly local to any cut.
Recall that a MBQC computation evolves through various branches, depending on the measurement outcomes. In a unitary MBQC pattern as defined above, it is possible that different branch implements different unitary operators. A weaker notion of unitary computation is given below.
Definition 3 A measurement based quantum computation is called deterministic if for all inputs the returned state of the output is an encoding of a fixed unitary acting on the input independent of the branch of the computation.
Other types of determinism and their connections can be found in [13,26]. In this paper we only consider the above central notions as they can be directly linked to the concept of structural entanglement as we present later. Moreover it is known that for graph states with |I| = |O| the two definitions of unitary and determinism, defined above, are equivalent [26]. This will allow us to link the concept of gFlow to D-Happiness as we discuss later.
Preliminaries: Graph states, Flow and gFlow
In the previous section we presented a necessary condition for computation across a resource state based on entanglement. The simple idea there was that if information can be transferred across a resource state, that state must be maximally entangled across each cut. We did not say anything about how this can be done however. This is where the ideas of Flow [12] and its generalisation gFlow [13] play a role, where a constructive definition together with efficient algorithm could be obtained for particular class of resource states of many qubits -graph states (defined below), with chosen input I and output O. If a graph state has gFlow, it implies that a unitary computation can be carried out across it [12,13]. Not only that, gFlow also gives instructions how to do it, and tells you what class of computations will be carried out (which unitaries). We will show that we can further use gFlow to give a simple bound on classical simulation of the computation based on the size of the forward cones implied by the measurement patters. This gives rise to an interpretation of the gFlow as showing us how information is 'spread' across the resource state throughout the computation, in an information light cone (which coincides with the causal forward cone in MBQC [20,22,23]). In this section we review the definitions of graph states [14], open graph states, Flow [12] and gFlow [13] and related concepts.
We start by defining the resource states considered, graph states [14]. A graph state is a multipartite state |G of n qubits, in one to one correspondence with a simple undirected graph G, with vertices V and edges E. Every vertex is associated to a qubit, and every edge can be understood as an entangling operation between qubits which have been initialised in the state
|+ := (|0 + |1 )/ √ 2.
We then have
|G V := i,j∈E CZ i,j | + + · · · + V ,(2)
where CZ i,j is the control-Z operation between qubits i and j. It is clear from this definition that the entanglement across a cut A, B is bounded by the number of edges cutting it, denoted
C A,B , i.e. E A,B ≤ C A,B .
Graph states can equivalently be defined by their stabiliser operators [14], a set of n operators, each associated to one vertex defined as
K i := X i ⊗ j∈N (i) Z j ,(3)
where X and Z are the Pauli operators (and Y = iZX). The graph state |G is the unique state satisfying all the eigenvalue equations (also called stabiliser relations or equations)
K i |G V = |G V .
The above relation is the key to how gFlow works -gFlow tells us how to apply the stabilisers to correct for measurements. When used as a resource state for MBQC we assign some vertices as inputs I ∈ V and some as outputs O ∈ V . In order to preserve the space we have that the size of the input set |I| ≤ |O|. We call the graph, with these assignments an open graph denoted as G(I, O, V ). The associated state is slightly different, the input vertices are no longer prepared in the |+ state, but can be arbitrary input qubits |ψ I . The rest of the vertices are prepared as normal, and again, every edge corresponds to a control-Z operation. We denote such a state as |G(ψ)
|G(ψ) V := i,j∈E CZ i,j |ψ I | + · · · + V /I(4)
where the state only depends on the inputs so different open graphs may have the same open graph state if they share the same set I, and graph G even if they have different assigned outputs O. The stabilisers are now reduced to those only on the non-inputs (we denote this set I c )
K i |G(ψ) V = |G(ψ) ∀i ∈ I c .(5)
Here the stabilisers define a space (of dimension 2 |I| ) of states such that this equation holds. The open graph state defined in Equation 4 is equivalent to starting in the standard graph state Equation 2 and teleporting an input |ψ S over system S (of |I| qubits) onto the input vertices I by performing control-Zs between S and I, followed by Pauli X measurements on the S qubits and corrections (see e.g. [25]).
In the standard model of MBQC [6,27] measurements are performed in one of the equatorial planes defined by the X − Y , X − Z or X − Y planes, and correction operations are local Pauli operators. By the end of the computation all vertices will be measured except the outputs (we denote this O C ). The gFlow assigns a set of correction operators for each of these measurements.
Before giving the definition of gFlow, we give the intuition to how it works for measurements in the X −Y plane. This corresponds to measuring in the basis |± θ := (|0 ± e iθ |1 )/ √ 2. We denote the projections associated to results ±1 as P ±,θ := |± θ ± θ |. For later use we denote the results in binary form as r i = 0 for +1 and r i = 1 for −1 outcomes. When measureing a state |ψ , in quantum mechanics the result is random (in fact normally in MBQC the probabilities are 1/2 and 1/2), which takes the resulting state to one of two branches, either the positive branch P +,θ |ψ /p + with probability p + = ψ|P +,θ |ψ , or the negative branch P −,θ |ψ /p − with probability p − = ψ|P −,θ |ψ .
Clearly to perform a deterministic computation U , we need to recover a deterministic evolution, hence corrections need to be applied. By convention we take the positive branch to be the ideal branch (note that of course P +,θ |ψ /p + is not in general a unitary embedding, this is an additional requirement which is also satisfied for our case). The task is then to find a correction operator to take the state when projected onto the −1 result to that of the +1 result (possibly ignoring the state of the measured qubit, since it is no longer used). The starting point is to notice that for all measurements in the X − Y plane, the projections are related to each other by a Pauli Z operator (for the other planes it is similarly the orthogonal Pauli operator) P +,θ = ZP −,θ Z. Imagine if it were possible to know the outcome of the measurement before it was performed (for example by traveling back in time after the measurement was performed and telling yourself), instead of correcting after the event, if we knew that we were about to get −1, we could cheat and apply a Pauli Z operator -then the 'measurement' (projection) would take us onto the projection we wanted, the positive branch. Obviously this is not possible without time travel since in quantum mechanics the results of measurements are random and cannot be known beforehand (we can only predict probabilities). However, we can use the stabilisers to simulate this strategy.
Imagine we applied the measurement on qubit i, then our time-travelling correction strategy for the −1 result would be to perform a Pauli Z operator on qubit i. Now, if we take a neighbour j / ∈ I, the stabiliser condition (Equation 5) tells us that
Z i |G(ψ) = Z i K j=N (i) |G(ψ) (6) = I 1 i ⊗ X j ⊗ k∈N (j) =i Z k |G(ψ) .(7)
Since X j ⊗ k∈N (j) =i Z k are on different systems from the measured qubit i, it does not matter when they are performed (they commute with the measurement). In this way, applying X j ⊗ k∈N (j) =i Z k correction operator after the measurement, is the same as applying a Z correction before the measurement -so that it has exactly the same effect. The later is sometimes called an 'anachronical correction', since it is as if we could go back in time and correct the measurement before it happened. The same works if a product of stabilisers is used in Equation 6 as long as their product results in one Pauli Z operator on qubit i, and we call the vertices associated to these stabilisers as the correcting set. Graphically this condition is ensured if the total number of edges between the correcting set and the vertex being measured is 1 modulo 2. This motivates the definition of the odd neighbourhood of a set of vertices K, denoted Odd(K) := {µ | |N (µ) ∩ K| = 1 mod 2}, which will be used in the definition of gFlow below. Using this idea, gFlow plays the role of making sure it is possible to make a good choice of which neighbour (or set of neighbours) to choose in a consistent way -so that corrections do not somehow contradict or interfere with one another. Indeed, gFlow is composed of a time order ≺ (partial order over vertices) and a choice of neighbouring sets (correcting sets) for each measured vertex i, denoted g(i) with this in mind. Firstly the time order should be consistent, so that corrections happen after the assigned measurements -this appears as (g1) in the Definition 4 below. Secondly, the correction should not invalidate or effect earlier corrections. This is true if no Pauli Z operators appear in the past when applying the stabiliser corrections, i.e. the correcting set is not oddly connected to the past -this appears as (g2) in Definition 4. Finally the correcting set should correct for the measurement it is assigned to. For measurements in the X − Y plane this corresponds to the application of a Pauli Z operator when the correcting stabilisers are applied, which means the correcting set should be oddly connected to the measured vertex -which appears as (g3) in the definition below (the analogous corrections for the other planes appear after).
Definition 4 An open graph state G(I, O, V ) has gFlow if there exists a map g : O c → P I c (from measured qubits to a subset of prepared qubits) and a partial order ≺ over V such that for all i ∈ O c (g1) if j ∈ g(i) and i = j then i ≺ j (g2) if j ≺ i and i = j then j / ∈ Odd(g(i)) (g3) for measurements in the X − Y plane, i / ∈ g(i) and i ∈ Odd(g(i)) (g4) for measurements in the X − Z plane, i ∈ g(i) and i ∈ Odd(g(i)) (g5) for measurements in the Y − Z plane, i ∈ g(i) and i / ∈ Odd(g(i)) Flow is a special case of gFlow, when all measurements are performed on the X − Y plane, and the correction sets g(i) have only one element.
In this way the product of j∈g(i) K j applies the appropriate 'anachronical' correction on vertex i, whilst not effecting other previous corrections. The associated computation can be carried out as follows. First generate the open graph state, then go through round by round (in the order given by ≺), measureing each qubit i, denoting the binary form of the outcome r i , followed by the correction given by
σ i j∈g(i) K j ri (8) where σ i is the pauli Z i , Y i or X i for measurement on qubit i done on the X − Y , X − Z or Y − Z planes respectively, so that Equation 8
is trivial over i and non-trivial only on future qubits of i i.e. on j such that i ≺ j. Following the convention in [12,13] inputs are identified by vertices with squares around them, and outputs are identified as hollow vertices (hence all non-hollow vertices will be measured in the computation). The choice of gFlow for a given vertex is indicated by red dotted arrows from the vertex to its gFlow (these are called gFlow paths, see Definition 5. Note that gFlow paths need not follow graph edges, as in Figure 3b). The induced measurement rounds are highlighted in grey, (see Definition 8).
In [13] it is shown that gFlow is a necessary and sufficient condition for an open graph state to allow a uniform unitary, deterministic computation to be performed across it, where uniform means that each qubit can be measured at an arbitrary angle on one of the planes. Hence the existence of gFlow implies the resource is also D-Happy. Intuitively on can think that the existence of gFlow guarantees that the entanglement of the graph state is such that the random effects of local measurements can be absorbed and countered by yet unmeasured qubits. The following definitions will be used to discuss how information travels throughout the computation [20].
Definition 5 A gFlow path starting from a vertex µ, denoted as gP ath(µ), is an ordered set of vertices such that for each pair (i, j) we have j ∈ g(i) and the first element of the set is µ.
Definition 6 An influencing path starting from a vertex µ, denoted as gInf (µ), is an ordered set of vertices such that each pair (i, j) is on a gFlow path or is preceded immediately by a pair on a gFlow path.
Definition 7
The forward cone F C (µ) of a vertex µ is the set of all vertices touched by all influencing paths from µ.
The concept of the forward cone appears in [20,22,23] and can be understood as a causal light cone, as described in [22,23]. The partial order ≺ in a gFlow defines time order for the rounds of measurements. We say a vertex µ is in a round R x if it is measured in round x.
Definition 8
The set R x , denotes the set of vertices which are measured in the xth round of measurements according to the gF low.
The best way to understand these definitions is through some examples. The gFlow (which is also a Flow in this case) is illustrated for the 2D cluster state in Figure 1. In Figure 2 we show examples of influencing paths and their union, which make up the forward cone for the 2D cluster state. , for the gFlow paths given by the red dotted arrows. An influencing path is path which follows gFlow paths and no more than one edge between gFlow paths. b) The collection of all influencing paths identifies the set of vertices (in red) in the forward cone (see Definition 7). The maximum size of forward cone for the 2D cluster state is indicated by the red shaded region (for the same gFlow). For an n × m 2D cluster state the maximum forward cone is of size |F Cmax | = nm − n 2 /4. This gives a bound on classical simulation for a computation, in Observation 2. The same region has an interpretation as an information light cone (see Section 4).
Before moving on to the interpretations of gFlow with respect to simulation and information flow, we review some examples which illustrate its power as a tool for analysing entanglement (as potential resources for MBQC), and in accessing the tradeoff between classical processing and number of measurement rounds (depth [20]). We start with an example of an open graph for which there is no gFlow in Figure 3 a). It can easily be seen that there is no possible assignment of correction sets g(i) and time order satisfying the conditions in gFlow for any measurement axes. Indeed its inability to act as a resource for computation across it follows directly from the fact that the entanglement across it is less then the number of inputs (hence it is not D-happy). We note however that there are examples of graph states which are D-happy, but do not allow a gFlow [13,26]. All such known examples still do allow computation across them. The second example is one where there exists a gFlow, but it necessarily has some correction sets which have more than one member -i.e. there is no Flow, as shown in Figure 3 b). The associated gFlow is give by assignments g(1) = 4, g(2) = 5 and g(3) = 4, 5, 6, with partial order given by the ordered measurement rounds R 1 = 1, R 2 = 2 and R 3 = 3, and all measurements in the X − Y plane. Note here that a gFlow path need not lie on a graph edge as for the gFlow path (3,4). The third example is the simple linear cluster state in Figure 3 c), where gFlow follows along the line.
A final example illustrates how gFlow can be used to find advantages in the number of rounds needed in a computation (taken from [13]). In Figure 4 the same open graph can have different gFlows. In the first case, Figure 4 a), the gFlow has correcting sets of size one, hence it is a Flow (g(i) = i + 4), and the number of rounds is the number of inputs (in the example this is four, but it easily extends to arbitrary size). More complicated gFlows can be found by increasing the size of some correcting sets, with the benefit of reducing the number of rounds. Figure 4 b), we set the correcting sets as g(1) = 5, 6, 7, 8, g(2) = 6, 7, 8 , g(3) = 7, 8 and g(4) = 8. It can easily be checked that this assigment allows all measurements to be done in the same round since for every vertex i, the correcting set g(i) is oddly connected only to i.
The above example illustrates a general scheme that could be understood as a tradeoff between rounds of computation and the amount of classical processing needed, but we have not yet talked about classical processing. To see how it works, we should think back again at what the gFlow does. Recall that gFlow tells us on which sets of vertices we should apply corrections (Equation 8). In Fig. 4: This open graph state has several possible gFlows, and illustrates how gFlow can be used to find advantages in terms of the number of rounds needed (depth) in a computation. a) is a gFlow with one correcting vertex per qubit, hence it is also a flow. This requires a number of rounds scaling with the number of inputs. b) is a gFlow which has largest size scaling with the number of inputs, but all measurements can be done in one round. Indeed all intermediary tradeoffs are also possible.This exemplifies the tradeoff between classical computation required and the number of measurement rounds needed. particular, for a vertex i, the correction associated to its measurement result (r i , where r i = 0 corresponds to the ideal branch and r i = 1 to that which needs to be corrected) is the application of the product of the stabilisers of all the vertices in g(i) (minus the Z i ) -i.e. the correction is (Z i j∈g(i) K j ) ri . Thus, if a vertex l is in the gFlow (or is a neighbour to a gFlow vertex) of another vertex i, then it will receive an X ri l (or Z ri l ) correction. The total number of corrections for a vertex depends on how many gFlow set (or neighbourhoods of gFlow set) to which that vertex belongs to. In the example Figure 4 b), vertex 8 has corrections from all inputs -hence it must receive the correction X r1⊕r2⊕r3⊕r4 8 (where ⊕ is the sum mod 2). In general, to calculate the Pauli X correction that should be applied on qubit j requires calculating the parity of all the r i s where j ∈ g(i) and for the Pauli Z correction the parity of all the r i s where j is a neighbour of g(i). We assume this is done classically (since it is a simple calculation), however, by increasing the size of the gFlows (in order to reduce the depth), we necessarily increase the size of this classical computation. This tradeoff has recently been translated to a tradeoff between the degree of the initial Hamiltonian and time of computation in the adiabatic model [28].
This tradeoff, a particular feature of the measurement based model, gave rise to a distinction in the power of measurement based quantum computation compared to the circuit model with respect to the number of time steps required [29]. The first example of a depth separation between quantum circuit and MBQC was proven for the calculation of parity function (where depth is defined to be the minimum number of rounds for a computation) [20]. Indeed this is a general feature that the depth of MBQC can be logarithmically better than the circuit model, where the difference is absorbed into the classical processing. More concretely it was shown that the depth complexity of MBQC is equal to the depth complexity for the circuit model with the addition of unbounded fan out gates [29].
Direct Simulation from gFlow
We will now see how we can derive a simple classical simulation, by tracking the stabilisers and logical operators. This idea is exactly how one can understand the Gottesman Knill theorem for the efficient classical simulatability of computations including only Clifford operations [30]. The proof follows from tracking stabilisers operator since they are an efficient way to describe a stabiliser state (such as a graph state), and Clifford operations, by definition transform stabiliser states to other stabiliser states, so computations can be simply tracked and described [30].
In what follows we will represent the computation in terms of the evolution of a set of logical operators. In physics there are two main, equivalent, ways that one views quantum evolution. One method (more common in quantum information) is where we look at how a state develops, and keep track of it as it evolves. This is known as the Schrödinger representation. Equivalently, one can view the state as having not altered, but the operators defining measurements having changed. This picture is known as the Heisenberg representation of evolution. In between these two pictures lies another way of representing evolution, which has been developed for quantum information -the so called 'logical Heisenberg' representation [30,31]. In this method we track the evolution of a complete set of logical operators -in this case the Pauli operators. To recover the Shrödinger representation, we remember that any state density matrix can be decomposed into Pauli operators (see Equations (13 and 14 in the next section). The logical Heisenberg representation has proved a very instructive way to view the evolution of MBQC [32,27], and as we will see leads to a simple bound on the cost of classical simulation.
Our simulation will follow the main treatment of [32,27], with the addition that we will consider rotated operators and their decomposition into Pauli operators, and we will use gFlow to instruct our procedure for updating the operators, which eventually leads to our main Observation. Our main tool will be the stabiliser formalism [30]. As mentioned in Section 2, for an open graph state the stabilisers define a space. Generally we talk in terms of a stabiliser group S, which is a subgroup of the Pauli operators. In the case of the open graph states, the generators of the stabiliser group are given by the operators K i (Equation 3), so that S = {K i } n i=1 . These are not the unique generators, indeed multiplying each of these by any one generator gives a new set of generators. The stabiliser group defines a space (the stabiliser space, or 'code' in error correction terminology) by a set of eigen equations -it is the space of states which are unchanged by the group. For the open graph states this is given by Equation 5 that is for all i ∈ I c : K i |G(ψ) = |G(ψ) , which implies all products of K i (i.e. all elements of S) leave the states unchanged. We say the states |G(ψ) are stabilised by the group S. In general if the stabiliser group for n qubits is generated by k elements, then the stabiliser space is of dimension 2 n−k . Essentially the stabilisers act like the identity over the this space, defining the space itself. In addition to tracking the logical operators, we will also track the stabiliser operators -indeed this will be a key tool for the former.
One can picture the whole of the computation in a high level as follows. The attaching of the input to the graph state (forming the open graph state), encodes the input space onto the many qubit state. The information is in some sense 'spread' over the large entangled state (we will talk more about computation as spreading of information in the next section). We call this encoded space the logical space. During the computation the information is pushed forward through the measurements towards the outputs, so that after the final measurements the logical space sits on only the output qubits. During this push the logical space is also rotated around, resulting in unitary computation. One can think of the stabilisers as keeping track of where the logical space is sitting, and the logical operators as telling you how the space has been rotated (in a sense the logical operators track both).
If state |ψ in the stabiliser space, with stabiliser group S = {S i } evolves under unitary U , the new state U |ψ is clearly stabilised by {U S i U † }, giving the updated stabiliser group. Under measurement things are slightly more complicated. In this work we use only single qubit projective measurements, which we write as two outcome measurements of the form
A i = P + i − P − i where P ± i
are the projectors onto the ±1 outcomes where i indicates the qubit measured. As usual we denote r i as the binary representation of the measurement outcome with r i = 0 when the outcome is +1 and r i = 1 when the outcome is −1. If it is possible to find a set of generators such that only one anticommutes with the measurement, call it S i , and the rest commute, the update simply replaces S i with −1 ri A i . It is not hard to see that this group will stabilise the state after measurement [30]. The projection from the measurement will not change the eigenvalue relation of commuting operators, and the projected state is clearly a +1 eigen state of the operator −1 ri A i . The trick is to find a suitable set of generators allowing for such an update (i.e. such that one and only one anticommutes with the measurement) -which is where the gFlow comes in.
So how should we describe the evolution of our logical operators? We want them to describe the information as it evolves. Talking in terms of pure states (which suffices for our discussion) if |ψ → |ψ , we want that our logical operators evolve L →L so that their expectation is preserved, that is we demand ψ|L|ψ = ψ |L|ψ . In this way, the new operatorsL genuinely reflect the information of the evolved space (see [31,32,27] for more details). Under a unitary evolution |ψ → U |ψ , we then have L → U LU † , clearly satisfying our requirement. For measurements, the trick will be to ensure that the logical operators commute with the measurement operators, in which case, they remain unchanged (measuring commuting observables cannot effect their expectation). The way of doing this will be to multiply by stabilisers -which act as identity on the logical space, so can be introduced without effecting the validity of the logical operators.
We will now see how we can track the evolution of the stabilisers and logical operators through the computation. This will be done in three steps. Note that our procedure does not exactly reflect the step by step process of the computation, as we do not consider corrections, rather it reflects the update as if all measurements had the outcome +1 -which is indeed the role of the corrections in the first place. In our discussion below we focus on measurements on the X − Y plane, similar arguments simply apply to the other planes.
Step 1: The first step is to prepare the stabilisers in a form that will allow us to simulate the measurements through the computation more easily. Physically it corresponds to the unitary process of applying the control-Z operators generating the open graph states (Equation 4), followed by simplifying the measurement operator by applying first the appropriate local rotation. The stabilisers of the open graph state are already given in Equation 3. For each input i an informationally complete set of operators is given by the Pauli operators X i , Z i and Y i = iZ i X i . If we know X i and Z i we can calculate Y i , hence we concentrate only on these two, and denote them as L Xi and L Zi as we trace them through the computation. The control-Z operators generating the open graph state is unitary, thus after being attached to the graph the logical operators become L Xi = X i ⊗ j∈N (i) Z j and L Zi = Z i (using the relation L → U LU † where U is the control-Z operator, see also [6]). Now we want to put these in a form ready to simulate measurements. The idea is based on the fact that a measurement in the X − Y plane is equivalent to first rotating around the Z axis, followed by measurement in the X basis (similar relations are true for the other two planes used). We initialise all the stabilisers and logical operators by doing this rotation, and consider Pauli X measurements afterwards. The resulting state is sometimes called a rotated graph state. At the same time we replace the individual stabilisers by products given by the gFlow. We thus start with stabilisers
S = S i := j∈g(i) K θj j i∈OC , {G i } i ∈ O ,(9)
where K θi i := e iθi/2Zi K i e −iθi/2Zi = cos θ i X i ⊗ j∈N (i) Z j + i sin θ i Z i X i ⊗ j∈N (i) Z j are the rotated graph state stabilisers and θ i is the angle of the measurement for qubit i. The set {G i } i ∈ O are there simply to complete the set of generators in the case that |I| < |O|, chosen such that [G i , X j ] = 0, ∀j / ∈ O. Such a set can always be found as follows, take an arbitrary set of operators which complete a generating set (note that the operators S i above are by definition all independent and so can form part of a generating set, then there is always some set of operators in S which complete this set of generators). To ensure commutation relation, we go round by round, starting from R 1 , we go through each vertex ν in the round, and check if it commutes with X ν -and if not we multiply it by S i . These operators are still valid generators and they commute with all the X ν measurements. At the same time, by applying the local unitary Phase rotations, the logical operators are initialised to
L Xi = e iθi/2Zi X i e −iθi/2Zi ⊗ j∈N (i) Z j , = cos θ i X i ⊗ j∈N (i) Z j + i sin θ i Z i X i ⊗ j∈N (i) Z j , L Zi = Z i .(10)
Step 2: The second step is to take the logical operators to a form which is convenient for measuring X ν on all the non-outputs -by making sure that they commute with X ν ∀ν ∈ O C . This update does not actually reflect any physical operation, rather it is just rewriting by multiplication of logical identities, i.e. stabilisers. However it is this step where the cost of the simulation arises, both in time and space of simulation. Although this is not a physical update we will trace through what would happen in the computation to see how our update can be carried out to ensure consistency in maintaining commutation.
We first expand logical operators in terms of products of Pauli operators
L α = i a i M α i ,(11)
where M α i is some product of Pauli operators, this is always possible since the Pauli operators forms a complete operator basis. Then, starting in R 1 with the stabilisers (Equation 9) and logical operators (Equation 10), we proceed with each round as follows, going from the first to the final round in sequence. In round R x we update each Pauli term M α i in each L α as follows:
∀µ ∈ R x : If [M α i , X µ ] = 0, M α i → M α i If {M α i , X µ } = 0, M α i → S µ M α i
After this step is complete, by the properties of gFlow it is easy to see that all the L α will commute with all X ν , i.e. [L α , X ν ] = 0 ∀α, ∀ν ∈ O c .
Step 3: The third step reflects the measurement of the computation, however with the unphysical condition that all outcomes are plus one. Although this does not really reflect measurement, it reflects the computation, since corrections are made so that this is always the final state. We first update the stabilisers and then use these to update the logical operators so that they are trivial (identity) everywhere except the outputs. The stabilisers are replaced with
S = {X i } i∈OC , {G i } i ∈ O .(12)
One can picture this as measurements with fixed +1 outcome as follows. We first notice that {S i , X i } = 0 and [S i>j , X j ] = 0, as can easily be seen from the definition of gFlow. To update the stabiliser operators to arrive at Equation 12 from Equation 9, again we start in R 1 and proceed with each round, going from the first to the final round, and in each round R x , we replace all the stabilisers S i∈Rx with X i (corresponding to measuring X i and getting result +1). Because of the condition {S i , X i } = 0 and [S i>j , X j ] = 0, this reflects exactly measurement with the +1 outcomes, and finally we end up with the stabilisers (Equation 12).
The next part is to use these new stabilisers to update the logical operators one final time. Again we do so term by term in the decomposition into Pauli operators. If a term M α i has an X µ for µ / ∈ O, it is multiplied by X µ (which is now a stabiliser, hence a logical identity). The remaining logical operators are trivial (i.e. identity) on everything except the outputs, and they encode the unitary evolution of the computation L α → U † L α U . This completes the classical simulation.
Fig. 5: Illustration of
Step 2 the update procedure for the cluster state. The red vertices represent the qubits where the logical operators L X1 and L Z1 are non-trivial. a) is the point directly after X 1 has been considered. b) is the point directly after qubits in the first and second round have been measured. c) is the point after qubits in the third round have been considered. See text for details. The number of qubits touched in the update procedure is equal to the size of the forward cone F C , which gives an upper bound to the size of the final logical operators, and hence the cost of directly simulating the computation (see observation 2). The F C also acts as a light cone for the information spread throughout the computation.
The efficiency of this procedure is dominated by the size of the logical operators (the number of terms occurring in the expansion). The stabilisers are updated efficiently (nothing in the initialisation or the update scales larger than O(n) where n is the size of the pattern). Similarly the initialisation of the logical operators is efficient, however, during each update step, each term in the expansion into Pauli operators must be checked and possibly updated. When an S µ is added to the term in the second step, the size increases by 2 |g(µ)| , where |g(µ)| is the number of vertices in the correcting set. This is necessary for every Pauli Z operator introduced by previous updates. Starting from R 1 these Pauli Z operators are introduced on all the neighbours of the correcting sets -that is along the gFlow path and one graph edge further. Thus they follow along all possible influencing paths. Some terms may cancel out, so the total number of terms is less than equal to 2 |FC (ν)| . From this we get the following observation.
Observation 2 An
MBQC over an open graph state with gFlow can be simulated classically in O(n exp(|F Cmax |)) where F Cmax is the maximum forward cone over all the inputs. More explicitly the logical operators L α associated to vertex µ can be updated with O(exp(|F C (ν)|)).
As mentioned, the above simulation does not take into account correction (since it is unnecessary in terms of simulating the computation). One may wonder given the simulation above where would the corrections come in at all. The answer is in the last step -when measureing X i , and getting result r i , instead of replacing by X i , we should replace by −1 ri X i . This would add signs throughout the logical operators which in general could not be undone by simply products of Pauli operators. With the exception being the case when each logical operator only has one M α in expansion (Equation 11), i.e. is just a product of Pauli operators, which occurs when the angles θ i = 0, π, i.e. measurements onto Pauli operators only. Then the minus signs can be all flipped coherently by multiplying by stabilisers. This is another way of seeing that if only Pauli measurements are made, all corrections can be made at the end. In such a case one can also see that the size of the logical operators becomes small -only one term eachso that this simulation itself is efficient. This simple observation will allow us to derive the equivalent of Gottesman-Knill Theorem directly in MBQC, as all the Clifford operates can be implemented in MBQC using only Pauli operators. Having removed any dependency as described above will lead to an efficient classical simulation of any MBQC pattern implementing Clifford operators and Pauli measurements. This interplay between efficiency and the angles of measurement is something not taken into account in the above observation, and offers more potential for better bounds. We leave this to future work for now.
To see the updating which truly corresponds to a computation, i.e. including corrections, one can combine steps 2 and 3 to get rid of the X i s round by round by applying the post measurement stabilisers −1 ri X i and in addition perform the correction operation (given by the gFlow) to remove the −1 ri . The effect is that one can simply remove the measured X i s whilst tracing through the computation.
For clarity we go through the example for the first few rounds on the 2D cluster state. For input of qubit 1 before being attached to the graph it is described entirely by two logical operators L X1 = X 1 and L Z1 = Z 1 . After Step 1 initialisation (joining to the open graph state and 'rotating' each qubit according to the measurement basis), these become
L X1 = e iθ1Z1 X 1 Z 3 L Z1 = Z 1 .
Here we have abbreviated the terms coming from the rotated basis into the exponent e θ1Z1 = cos θ 1 I 1 + i sin θ 1 Z 1 , and for ease of notation we remove the tensor product symbol.
We next consider Step 2, starting with round R 1 and operator X 1 that anticommutes with Z 1 s, hence for those terms in the L α where this occurs we are required to multiply by S 1 = K θ3
3 = Z 1 ⊗ Z 2 ⊗ e iθ3Z3 X 3 ⊗ Z 4 ⊗ Z 7
. This is equivalent to putting it up into the exponent, so that the logical operators become
L X1 = X 1 e iθ1Z2e iθ 3 Z 3 X3Z4Z7 Z 3 L Z1 = Z 2 e iθ3Z3 X 3 Z 4 Z 7 .
In Step 3 the X 1 s are removed (since after measurement and correction the X 1 are a logical identity), and the logical operators are thus non-trivial on qubits 2, 3, 4, 7 after R 1 , as illustrated in Figure 5a). In the second round R 2 , X ν on qubits 2, 3 and 4 are considered. We update the logical operators by considering these one by one, starting with X 2 (any order in the same round gives the same final result). This anti commutes with Z 2 -which comes from the application of S 1 = K θ3 3 in the previous round. Indeed this is how the updates are effected along all influencing paths. When the Z 2 occurs we are forced to multiply the term by S 2 = K θ6 6 = Z 2 ⊗ Z 5 ⊗ e θ6Z6 X 6 ⊗ Z 7 ⊗ Z 11 . This takes the logical operators to
L X1 = e iθ1e iθ 3 Z 3 X3Z4Z5e θ 6 Z 6 X6Z11 Z 3 L Z1 = e iθ3Z3 X 3 Z 4 Z 5 e θ6Z6 X 6 Z 11 .
Note that here, two Z 7 operators have cancelled out -they came from two occasions where qubit 7 was a neighbour of one of the correcting sets. In this way, it is possible that some qubits in the set of influencing paths cancel out -this happens if the number of influencing paths they sit in as gFlow paths is even, and the number arriving from non-gFlow paths is also even (at this point in our calculation the number of times it is on a gFlow path is zero, and it is in 2 influencing paths not as a gFlow).
After qubits X 3 and X 4 are also considered, we have to do the same trick to get rid of the Z 3 and Z 4 s, by multiplying the terms where they occur by S 3 = K θ7 7 and S 4 = K θ8 8 respectively. Finally we end up with logical operators
L X1 = e iθ1e iθ 3 Z 6 e iθ 7 Z 7 X 7 Z 8 Z 12 X3Z5e θ 6 Z 6 X6Z7e iθ 8 Z 8 X8Z9Z11Z13 Z 6 e iθ7Z7 X 7 Z 8 Z 12 L Z1 = X 3 e iθ3Z6e iθ 7 Z 7 X7Z8Z12 Z 5 e θ6Z6 X 6 Z 7 e iθ8Z8 X 8 Z 11 Z 12 .
Again, in Step 3 we get rid of the X 2 , X 3 , X 4 s, hence after the measurements in round R 2 the logical operators are non-trivial only on qubits 6,7,8,9,11,12,13, as indicated in Figure 5b). It is then clear how after the third round of measurements we will be left with logical operators that sit on the highlighted qubits 10, 11, 12, 13 and 14, as indicated in Figure 5c).
For any graph and any measurement pattern with gFlow, each time a Pauli Z operator is added, unless it is in the output set, we will have to multiply that term by a stabiliser -which will add a splitting of two. During the update procedure, Pauli Z operators are added along every influencing path. Sometimes these will cancel out, depending on the graph, but sometimes not, so that this gives an upper bound to the complexity for the direct update procedure which is the content of Observation 2.
We thus see an initial way to go from a gFlow to a classical simulation. However, for certain examples this bound can be very bad. We have already mentioned that this is the case where all of the measurement angles are zero or π/2 -i.e. measuring the Pauli operators -there is no splitting of the logical operators, and only one term is needed for each logical operator, hence this simulation becomes efficient, which is not captured by Observation 2 (where we effectively assume the worst case for the angles). Another example is a computation across along a 1D graph state, with one input, say on the left, and an output on the right (see Fig. 3c). There the gFlow simply follows the line, thus the influencing volume is big, however, this is always a simple one qubit computation, and indeed all computations on a 1D cluster state are easy to simulate classically [33]. In Section 5 we will see how connections to entanglement allow us to make tighter bounds on classical simulatability which will work well for this example and many others. Before we do that however, in the next section we will discuss how the update above can be interpreted as information flow, in tern giving the interpretation of the forward cone F C as a light cone for the information.
Flow of Information and F C as Information Light Cones
The gFlow gives a causal structure on top of a graph state induced by the correction procedure, called the forward cone F C (Definition 7). In this section we will also look at how the same cone can be understood as a forward cone of information, and moreover a light cone (so that information cannot travel beyond this cone).
The forward cone can be viewed as an information forward cone directly from the simulation procedure described in the previous section, and the interpretation of the logical Heisenberg representation as showing us where information sits (see for example [34]). Consider a density matrix of some input i
ρ i = 1 2 (I 1 + η x X i + η y Y i + η z Z i )(13)
The state is totally described by the coefficients η i . The logical Heisenberg representation ensures that at any time the evolved state, denoted asρ, which now can be sitting over many systems, is described as
ρ = 1 2 I 1 + η xLXi + η yLYi + η zLZi ,(14)
where theL α are the updated logical operators of α corresponding to the evolution. The information is then preserved, but 'spread' over to different qubits in the following sense. To recover the information encoded on the original system i (i.e. recover the η i ), we should measure the logical operatorsL Xi ,L Yi andL Zi . Thus the information can be said to have spread over the range of the logical operators. From the simulation in the previous section, it is clear that the logical operators, and hence the information of input qubit i spreads out over the causal forward cone F C defined by the gFlow (see Figure 5).
One may then ask if this is all that is allowed, or could we understand the information as having spread further than the influencing cone (after all, this is not the only way one may simulate a computation)? The answer (at least for patterns where we wish uniform determinism, i.e. that all measurements on a Pauli equator are allowed) is no, in that the spread must be balanced by consistency amongst all measurements, which is the function of the gFlow, which defines the cone F C .
To see how this works, let us first return back to Step 2 in the simulation above, which is where this spread of information occurs in the simulation. The trick is simply multiplying the logical operators by a logical identity (i.e. the stabilisers). This part however is clearly not restricted to the cone. One could easily expand a logical operator to cover practically all qubits in this way. The reason we do not allow this is because we want to do measurements, and we want to do them over all qubits not outputs so that all logical operators are preserved (this is what we mean by consistency). Say one did this for operator L Xi , so that its extent was over many qubits. Taking its expansion into products of Pauli operators (as in Equation 11), one would have a sum of many terms, including Pauli Z and X operators on any given qubit in its range. When measuring qubits, to ensure the survival of the information, we asked that the logical operators be taken to a form which commutes with the measurement -this was the role of Step 2 in the above. If one did not have this, information may be lost. This can only be the case if in each term M Xi j of the expansion of L Xi , the part of M Xi j on vertex µ is the same (say σ µ ) or identity for all terms.
One could have, for example, that this is indeed the case, i.e. for a particular L Xi , extended so that it touched many qubits, that over each such qubit µ all the terms in the expansion of L Xi were either the same Pauli σ µ or identity. In such a case, one could happily measure those qubits in the Pauli basis σ µ , the information would be preserved, and the logical operators could be calculated (if we wanted to consider the information over the outputs we would then follow
Step 3 to leave them as identity everywhere else, though one would have potentially different evolutions for different branches). In this way its final spread may indeed be beyond the light cone given by gFlow. The problem with this would be that we want to transfer the logical operators not just of one input i, but of all the inputs. It is shown in [27] that to achieve this, in such a way that every measured vertex one can choose amongst a set of measurements across one of the planes, the only way to do it is via a gFlow. Hence, for an input i, not only is the forward cone F C (i) also an information cone, but to transfer all the information at the same time, it is a light cone for the information contained i -that is, the information can not spread beyond it, and the computation be consistent for all inputs.
From the perspective of information flow, observation 2 says, unsurprisingly, that the more information is spread through a computation, the more costly it is to simulate. However, again we should be careful to note that the true cost of simulation depends on the angles of the measurements, which is not captured by the size of F C , hence not by observation 2. As we saw, for angles 0, π, the simulation is simple, however the spread of information is still large.
Bounds on Entanglement from gFlow
In this section we will show a connection between gFlow on the one hand, and entanglement conditions for both the universality of a resource state and the classical simulatability of a computation on the given resource, on the other. More precisely we will show that the Flow and gFlow can be used to bound the entanglement of the graph state, in terms of the entanglement width [9] and what we will call the structural entanglement (though not explicitly defined, it can be understood from [8], see also [7]). Conditions of universal family of resource states, and for classical simulatability are known for both these measures, which can be translated to conditions about the gFlow through our bounds [9,10,11]. Several known results can then be derived for both the measurement based model and circuit based model (through the known maps between the two models [12,17]). For example we reproduce the result by Josza stating that a circuit which any wire are touched by at most logarithmic (in the size of the input) many number of two qubit gates can be classically simulated efficiently [24].
Let us first define the entanglement measures we are interested in. The entanglement width [9] of a pure state |ψ is defined as
χ wd (|ψ ) := min T max e χ bi T,e (|ψ ),(15)
where χ bi T,e (|ψ ) is the the log-Schmidt rank across the bipartite cut defined by T and e where T is a sub-cubic graph with n leaves and e is an edge of T . Each leaf corresponds to a qubit of the state |ψ . The bipartite cut is defined by removing edge e to give two separate trees. The leaves of one tree correspond to one side of the cut, and the other tree the other side of the cut. It was shown that if the entanglement of a family of resource state does not scale polynomially with the size of its input space then, that family cannot be a universal [9,11] (in the case of QQ computations, note that this is not the same as asking for universality in the CC case). It was also shown that any MBQC can be simulated in O(npoly(2 χ wd )) [10].
Motivated by the proofs in [8], we define the structural entanglement as
χ AB (|ψ ),(16)
where the minimum is taken over all orderings (labelings) of the qubits 1, . . . , n, and the max is taken over a cut defined for a given ordering by taking all qubits 1, . . . , k on partition A and the rest on partition B and χ AB (|ψ ) being the log-Schmidt rank over cut AB. Although not explicitly stated in terms of this measure, in [8] it is shown that any MBQC pattern can be simulated classically in O(n 2 poly(2 Estruc )).
It is easy to see that the tree in Figure 6 defines a set of cuts such that any cut either splits the graph in two with all leaves below or equal to a value k on the left, and above k on the right (as per the optimisation for E struc ), or else it just identifies one leaf. This clearly implies that Fig. 6: This tree defines a set of cuts showing that E struc ≥ χ wd . Any cut either sits in the same set of cuts as that optimised for E struc -effectively choosing a k such that all qubits of number less than or equal to k are in partition A and all higher qubits are in partition B (cut a)), or else it singles out one qubit (cut b)), which can never be the unique maximum.
E struc ≥ χ wd .(17)
We will now see how the Flow and gFlow can be used to bound E struc , and in turn χ wd . We start by considering Flow, which is simpler to picture, but the ideas easily extend to gFlow. The idea is that they both can be used to define a natural order, which gives a simple bound to E struc which comes from induced disjoint input-output paths. Indeed, if an open graph state has Flow, following the image of the Flow function, f , (Definition 4) from each of the inputs leads to disjoint lines to the outputs, which cover all the non outputs [15,17] (called Flow wires).
We consider first the case where |I| = |O|. The numbering goes as follows. We start with an arbitrary input going along the image of the Flow function of that vertex till we reach an output qubit. Next we choose another not selected input qubit and carry on in this fashion, till we cover all the inputs, see Figure 7 and Figure 8. Note that based on the definition of Flow, no input qubit could belong to the image of the Flow function of another input qubits hence on each such Flow wire there will sit only one input qubit and hence we have |I| such wires. To calculate the entanglement we note the fact that the entanglement across a cut C for a graph state can be bounded by the number of edges crossing the cut Fig. 7: Flow defines a natural ordering from top left to right across each Flow wire from top to bottom as shown. This is used to define cuts by a number k where partition A consists of all qubits below qubit k in the ordering and partition B consists of all qubits above k in the ordering. The entanglement across any cut is upper bounded by the number of edges cut (in this case k = 10 and the entanglement is exact). E A,B ≤ C. This is clear since in preparation of the state each edge corresponds to a control-Z operator, and C such operators can create at most C e-bits.
Definition 9
For an open graph with flow, we denote C F the maximum number of edges crossing between Flow wires.
We easily see that a cut between two Flow wires gives entanglement at most C F (see Figure 7). This can be at most doubled (plus one) by choosing a lower number to cut at (thus potentially increasing the number of edges cut) (see Figure 8). We thus have that E struc ≤ 1 + 2C F .
To extend this to the case where |O| > |I| we must consider the worst case, for which each extra qubit adds one unit of entanglement. In this general setting we now call C F the maximum number of edges crossing between Flow wires, when the output qubits not in a Flow wire are ignored along with their edges. We also call ∆ := |O| − |I|. We then have the following observation.
Observation 3 A graph state with Flow has structural entanglement
E struc ≤ 1 + 2C F + ∆.(18)
Thus any computation can be simulated in at least O(n 2 poly(2 2CF +∆ )).
We note that any computation which can be done with a number of outputs greater than |I| can be done with |I| = |O| without changing the Flow or C F by simply removing the extra ∆ output qubits from the graph resource. Thus ∆ = 0 for most interesting cases. This is clear since the existence of Flow is robust against losing the extra outputs, and this guarantees the computation. This result can be extended to open graphs with gFlow by using gFlow to find disjoint input-output lines as follows. As we saw earlier, it is clear that if an open graph state has gFlow, then it is necessarily D-happy (from Observation 1 and the fact that gFlow implies unitary computation). This in turn means that any cut which separates the input and the output goes through at least C ≥ |I| edges. Taking a result from graph theory, Menger's theorem [35] says that this implies there are at least n parallel paths going from inputs to outputs. Furthermore it can be shown that there are parallel paths which sit along gFlow paths and can be found systematically also [36]. This can be used to give a natural order to the graph as for Flow, but with the possibility that non-output qubits do not sit in the disjoint paths, and so should be added to the ∆ term. Again the size of ∆ may be reduced or removed by considering equivalent smaller graphs, but this is less well understood for gFlow.
This result covers examples not covered by the direct simulation from Section 3, for example the 1D cluster state. The statement of Observation 3 is a very similar sounding statement to Jozsa's [24] which states that a quantum computation on a circuit can be simulated in O(npoly(2 D )) where D is the maximum number of gates that touch or cross a circuit wire.
Conclusions
We have seen that gFlow can be used for two complementary approaches for giving bounds on the efficiency of classical simulation for MBQC. In Observation 2 we saw that classical simulation is possible with resources scaling as exponential in the size of the largest causal future cone defined by the gFlow. On the other hand in Observation 3 gFlow can be used to bound the entanglement, and hence give bounds on resources for classical simulation in terms of the number of edges crossing gFlow wires (parallel wires from input to output induced by gFlow). Simple and straightforward, but illuminating conditions for entanglement of general resource states are described in Observation 1. Furthermore the causal future cone induced by gFlow is seen to be at the same time a light cone for information spreading.
The results on classical simulation combine two of the main approaches for bounding the cost of classical simulation for quantum computation -explicit tracking of the computation using an efficient form (used for example in the Gottesman Knill [30] theorem and related results (e.g. [37])) and bounds coming from entanglement (used for simulating computation [7,8,9,10,11] as well as many body physics (e.g. [38])). This offers the perspective of bridging these two approaches via gFlow. A natural question is the interplay between the angle of measurement and efficiency of simulation via the gFlow update procedure presented here. Setting all angles to zero or π makes the simulation efficient (as per the Gottesman Knill theorem), however for general angles it is not efficient(indeed Observation 2 represents this worst case situation). The in between ground, combined with bounds by entanglement may present new classes of computation admitting efficient simulation for example. Furthermore, we may gain more insight into how efficiency of classical simulation is related to other features of computation illuminated by the study of gFlow.
It is also interesting in itself that from Flow and gFlow one can derive bounds on the entanglement of a graph state. Since there exist efficient algorithms to calculate the Flow and gFlow of graphs [15,16], and given Flow and gFlow one can easily bound the entanglement, one may use this to bound the entanglement of general graph states. This is both important for recognising good resources (since the existence of Flow does not talk about universality, whereas the entanglement gives bounds on this also [9,11]), and more generally as entanglement represents important resource in other areas of quantum information.
The two approaches to classical simulateability can also be understood as arising from two notions of 'spreading' of information. We have seen in Section seen in Sections 3 and 4, a big 'spread' of information is however not enough to imply that a computation is difficult to simulate -MBQC with only Pauli measurements is efficient to simulate, but the spread of information is large (in both senses -the future cone is large, and the entanglement is large). To capture the difficulty in simulation, one must also include something about how this 'spread' of information is used. In the case of MBQC studied here, universality (and presumably the difficulty in simulation) is given by the use of arbitrary angles for the measurements, using the spread of information in the most universal way. This balance between spread of information through entanglement and how it can be used also plays a key role in analogies between MBQC and thermodynamics and in particular phase transitions [39,40]. It is an exciting prospect that these pictures may be unified from the perspective of gFlow or similar notions.
Fig. 1 :
1An example of gFlow for the two dimensional clusters state as an open graph state.
Fig. 2
2: a) The bold red lines are examples of two possible influencing paths from the central input vertex (see Definition 6)
Fig. 3 :
3Examples of open graph states with and without gFlow. The gFlow paths are red dotted lines, and the induced measurement rounds are highlighted in grey (seeDefinition 8). The graph inFigure a)does not have gFlow. This can be seen since the entanglement across the cut input/output is lower than the number of inputs. The graph inFigure b) has gFlow but no Flow[13]. The graph inFigure c)is the linear cluster state which has a gFlow that is also Flow.
Fig. 8 :
8In the worst case the number of edges cut by a line equals 1 + 2C F . This implies a bound on the structural entanglement (see text).
that the forward cone given by gFlow bounding the cost of classical simulation can also be interpreted as a spread of information -so that the more spread the information is, the more costly the simulation. The bounds arising from entanglement ([7],[8] e.t.c) which lead to Observation 3, can also be understood as assigning a cost to the spread of information as follows. The entanglement measure key to these results is a bipartite measure, the Schmidt measure of entanglement, which counts the minimum number of product states (with respect to a particular cut) needed to describe the state. This may be interpreted as saying how 'spread' across product bases the state is. Indeed it is exactly the rank of the reduced density matrix of one cut, so in a sense says the size of the space in which it must be understood to sit (in this sense the 'spread' is over the state space rather than precisely the parties). The trick of[7] and subsequent work is to find an efficient form to describe the state and its updating through a computation based on this minimum decomposition. Again, the smaller the 'spread' in this sense, the smaller the cost of this simulation. As we have also
ACKNOWLEDGEMENTSThe authors would like to thank Bobby Antonio, Simon Perdrix and Einar Puis for useful discussions and feedback. We are particularly grateful to Vincent Danos and Prakash Panangaden for many discussions on the topics of this paper which gave rise to many of the ideas mentioned here directly and indirectly. DM is funded by the FREQUENCY (ANR-09-BLAN-0410), HIPERCOM (2011-CHRI-006) projects, and by the Ville de Paris Emergences program, project CiQWii. EK is funded by UK Engineering and Physical Sciences Research Council grant EP/E059600/1.
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[
"A Gelfand Transform for Spinor Fields on Embedded Riemannian Manifolds",
"A Gelfand Transform for Spinor Fields on Embedded Riemannian Manifolds"
]
| [
"Colin Roberts \nColorado State University\nFort Collins80523Colorado\n"
]
| [
"Colorado State University\nFort Collins80523Colorado"
]
| []
| A classical result of Gelfand shows that the topologized spectrum of characters on commutative Banach algebra is homeomorphic to the underlying space. This fact is used in solving the Calderón problem in dimension 2 via the boundary control (BC) method. To apply the BC method in dimension 3, the algebra of complex holomorphic functions can be replaced by the space of harmonic quaternion fields, but this space is no longer an algebra and is not commutative. Nonetheless, it has been shown that a suitable notion of a spectrum exists and in the case when the underlying space is convex, the spectrum is homeomorphic to the ball. Our goal is to generalize this result to more general manifolds in arbitrary dimension. To do so, we use Clifford algebras of multivector fields and the set of functions we care about is the space monogenic spinor fields. The spectrum consists of spinor valued functionals that respect the module and subalgebra structure of the space of monogenic spinor fields. We prove that for compact regions in R n , the spectrum is homeomorphic to the manifold itself. Furthermore, some essential lemmas and propositions are proven for arbitrary compact Riemannian n-manifolds. Finally, we end with a Stone-Weierstrass theorem which shows the algebra generated by the monogenic spinor fields on arbitrary compact M with boundary is dense in the algebra of continuous spinor fields. | null | [
"https://arxiv.org/pdf/2203.00118v1.pdf"
]
| 247,187,727 | 2203.00118 | 7efe063967466909cff49958d0e1418d694b7935 |
A Gelfand Transform for Spinor Fields on Embedded Riemannian Manifolds
28 Feb 2022 March 2, 2022
Colin Roberts
Colorado State University
Fort Collins80523Colorado
A Gelfand Transform for Spinor Fields on Embedded Riemannian Manifolds
28 Feb 2022 March 2, 2022
A classical result of Gelfand shows that the topologized spectrum of characters on commutative Banach algebra is homeomorphic to the underlying space. This fact is used in solving the Calderón problem in dimension 2 via the boundary control (BC) method. To apply the BC method in dimension 3, the algebra of complex holomorphic functions can be replaced by the space of harmonic quaternion fields, but this space is no longer an algebra and is not commutative. Nonetheless, it has been shown that a suitable notion of a spectrum exists and in the case when the underlying space is convex, the spectrum is homeomorphic to the ball. Our goal is to generalize this result to more general manifolds in arbitrary dimension. To do so, we use Clifford algebras of multivector fields and the set of functions we care about is the space monogenic spinor fields. The spectrum consists of spinor valued functionals that respect the module and subalgebra structure of the space of monogenic spinor fields. We prove that for compact regions in R n , the spectrum is homeomorphic to the manifold itself. Furthermore, some essential lemmas and propositions are proven for arbitrary compact Riemannian n-manifolds. Finally, we end with a Stone-Weierstrass theorem which shows the algebra generated by the monogenic spinor fields on arbitrary compact M with boundary is dense in the algebra of continuous spinor fields.
Introduction
This paper is motivated by the inverse tomography (or Calderón) problem for Riemannian manifolds [1,2,3,4,5,6,7]. First, let us explain an important related problem called the Electrical Impedance Tomography (EIT) problem and note that it is equivalent to the classical Calderón problem in dimension n = 3. Let M be a body free of interior charges and made from a (possibly anisotropic) Ohmic material with conductivity γ. Applying a voltage φ on the boundary ∂M induces a voltage u in the interior and since the interior is free of charges, u satisfies div(γ grad(u)) = 0. We are not allowed access to the interior, which leaves us solely with the ability to making measurements along the boundary ∂M . Given φ, we can measure the corresponding current flux ∂u ∂ν on ∂M where ν is the outward normal field. Hence, we have a map Λ called the voltage-to-current map defined by Λφ := ∂u ∂ν . Is it possible to determine the conductivity γ from the voltage-to-current map Λ?
One avenue of active research seeks to solve a more general problem called the Calderón problem for Riemannian manifolds [5,6,7,8,9]. In this setting, one attempts to reconstruct an n-dimensional smooth manifold M and metric g from the Dirichlet-to-Neumman (DN) map Λ defined on the boundary traces of harmonic differential forms, i.e., forms in the kernel of the Laplace-Beltrami operator ∆. Given the Dirichlet boundary condition φ, we have a unique harmonic u. The DN map is defined by Λφ := ι * (⋆du) which is valid for any k-form φ. Fixing φ to be a 0-form then yields the classical problem discussed previously and in dimension 3, this is equivalent to the EIT problem. From complete knowledge of pairs of Dirichlet and Neumann data (φ, ι * (⋆du)) for forms of all grade, the geometric inverse problem is to determine the pair (M, g) up to isometry. If you would like, please see Uhlmann's article [2] for an excellent explanation of the relationship between the EIT and Calderón problem.
Solutions to the Calderón problem exist in a handful of special cases, but the C ∞ -smooth problem remains open in dimensions greater than two. Belishev [10] and Lassas-Uhlmann [3] both show that for surfaces S with one boundary component, the classical DN map determines S up to conformal class. A better result cannot be achieved since the Laplacian is conformally invariant in dimension 2. Belishev, Badanin, and Korikov recently achieved a similar result for surfaces with multiple boundary components [11]. It is also known that the DN map for forms recovers partial topological information on arbitrary M such as the Betti numbers [6], but it is not known whether the DN operator can recover M up to homeomorphism. The extension of the DN operator to the complete DN operator in Sharafutdinov and Shonkwiler's paper [8] is able to recover the absolute and relative cohomologies as well.
We will mostly concern ourselves with the technique used by Belishev in his 2003 paper [10] called the Boundary Control (BC) method. This technique utilizes the classical Gelfand representation for commutative Banach algebras. That is, for a surface S, the spectrum (maximal ideal space) of the commutative Banach algebra of holomorphic functions is homeomorphic to S. Alongside this result, Belishev then used the complex structure of the algebra to determine the metric up to conformal equivalence. Furthermore, it is the Gelfand transform that allows for all of this information to be obtained from the boundary. This led to a solution to the Calderón problem in dimension 2. For more on the boundary control method, see [12].
To solve the problem in dimension three using the BC method, it is natural to consider replacing complex functions with quaternion-valued functions. Belishev and Vakulenko wrote a series of papers on this topic [13,14,15]. In those papers, the authors work with the space of harmonic quaternion fields using the language of differential forms. Their goal was to complete one portion of the BC method by realizing a Gelfand theory for quaternion fields. The first hurdle in defining a Gelfand spectrum on the space of harmonic quaternion fields is that the space fails to be an algebra. Moreover, it is not commutative. However, the space does contain commutative subalgebras and by carefully defining a meaningful notion of a spectrum that is multiplicative over these subalgebras, they find that for convex regions in R 3 , the spectrum is homeomorphic to the ball.
Is a similar result true for a more general class of manifolds?
In this paper, we seek to faithfully capture the necessary Gelfand theory in dimension 2 and 3 as well as generalize it to arbitrary dimension by replacing the exterior algebra of differential forms with Clifford algebras of multivector fields. Note that we do not lose any information with this substitution since the exterior algebra naturally embeds into in any Clifford algebra. Our focus will be on nondegenerate (Euclidean) Clifford algebras G which we call (Euclidean) geometric algebras since they capture both complex and quaternion algebras as spinor subalgebras G + ⊂ G when defined on R 2 and R 3 resectively. Many of the nice properties of complex and quaternion valued functions can be generalized to higher dimensions by replacing them with spinor fields f + ∈ C ∞ (M ; G + ) which are sections of the bundle of the spinor subalgebras.
A complex function on a surface can be written as a sum of even grade forms, specifically, a 0-form ω and 2-form η. This function ω + η is holomorphic if and only if the generalized Cauchy-Riemann equation dω = −δη where d is the exterior derivative and δ is the codifferential is satisfied. Such ω and η are called a conjugate pair. The same is true for a harmonic quaternion fields. Instead of looking at pairs of conjugate forms, we consider spinor fields in the kernel of the Hodge-Dirac operator ∇. This operator is fundamental in Clifford analysis and it is equivalent to operator d + δ on forms. Spinor fields in the kernel of ∇ are not restricted to being only pairwise conjugate and the associated Cauchy-Riemann equations are a bit more general in dimensions higher than three. We refer to such fields in the kernel of the Hodge-Dirac operator as monogenic.
As a side note, if we consider fields of only a single grade, if they are monogenic then they correspond directly to harmonic fields. This lands us in the realm of Hodge theory where it is a central result that for each grade k, the space of harmonic fields is finite dimensional. Fundamentally, by considering fields consisting of sums of even grades (i.e., a spinor), we find the kernel to be far more rich in content. For example, in R 2 , monogenic spinor fields are equivalent to holomorphic functions and in R 3 , they are equivalent to harmonic quaternion fields and both of these spaces are infinite in dimension. The real benefit of using Clifford algebras is that it allows us to be dimension-agnostic like we can with differential forms, but recover many of the staple complex analysis theorems in any given dimension as well.
Within this framework, we prove a Gelfand representation for n-dimensional compact regions of R n and define the corresponding Gelfand transform. Here, the spectrum consists of a certain type of functionals which we call G-currents that act on the space of monogenic spinor fields. These functionals are valued in the base geometric algebra and respect the structure of certain subalgebras nested inside the space of monogenic spinor fields. Furthermore, we report a Stone-Weierstrass theorem showing that the algebra generated by the closure of the monogenic fields is dense in the space of continuous fields. Both of the above results answer questions posed by Belishev and Vakulenko in [14].
The construction we use for the spectrum bootstraps from our knowledge of surfaces: given a surface S, the monogenic spinor fields on the surface M + (S) are a commutative Banach algebra isomorphic to the algebra of holomorphic functions. When the dimension of the manifold M exceeds two, the space M + (M )
is not an algebra since products of monogenic fields need not be monogenic. At a local scale, a monogenic spinor f + ∈ M + (M ) is, in essence, built out of a monogenic fields propagated off of surfaces embedded in M .
In fact, we can think of these fields as a local set of variables and this lets us write power series for f + locally in these variables. We find that these variables are direct analogs to the variable z in complex analysis.
The key challenge we face in this paper is to define a notion of a spectrum for the space M + (M ).
In the case of a surface S, characters δ in the spectrum sp M + (S) are the continuous algebra morphisms sp M + (S) → C. Since M + (M ) is not an algebra but is a G + -module, we define characters sp M + (M ) to be continuous module morphisms M + (M ) → G + . Furthermore, we require that characters are also algebra morphisms on subalgebras A B (which are akin to M + (S) for S a subsurface in M ) to a an algebra A B (which is isomorphic to C). Applying such characters to the power series representation for f + in terms of the z variables yields our main theorem. Furthermore, we find that the closure M + (M ) separates points which follows from the fact from Calderbank's thesis [16]. Namely, if we know f + ∈ M + (M ) locally, we can uniformly extend this to a unique field on all of M . Given this and a theorem from Laville and Ramadanoff's paper [17], we prove the following Stone-Weierstass result. Lastly, I hope that others find that this work warrants further investigation on the utility of C * -and Banach-algebras of Clifford algebra-valued functions and to this end I will finish the paper with a discussion of other questions addressed by Belishev and Vakulenko and their relation to this work and the Calderón problem. It is certainly a worthy endeavor to consider what extent these theorems can be used when the information is extracted solely from Dirichlet-to-Neumann operators.
Acknlowedgements:
Without the unending guidance and input from my advisor Dr. Clayton Shonkwiler, none of this work would be possible. I deeply appreciate your help and your willingness to follow me on my own mathematical journey over these past few years. Thank you.
Preliminaries
We address only the necessary preliminaries here, though more detail can be found in various sources, for instance from Doran and Lasenby [18], and Hestenes and Sobczyk [19]. Our preliminaries will include the basics of Clifford algebras with a large motivating example, foundations of the Clifford analysis of the Hodge-Dirac operator, and the notion of algebraic currents.
Clifford Algebras
To construct any Clifford algebra, take an n-dimensional vector space V over a field F with quadratic form Q. By the polarization identity, given a Q there is a unique corresponding symmetric bilinear form g. For more information on vector spaces with quadratic or bilinear forms, please see [20].
The tensor algebra is given by n∈N V ⊗ n and we form the Clifford algebra Cℓ(V, Q) by the quotient
Cℓ(V, Q) := n∈N V ⊗ n / v ⊗ v − Q(v)(1)
with the induced addition and multiplication from the tensor algebra. For sake of clarity, we will think of Cℓ(V, Q) = Cℓ(V, g) as needed, though most define Clifford algebras only using quadratic forms. Elements of Cℓ(V, Q) are referred to as multivectors.
When we take the totally singular form Q = 0, the corresponding Clifford algebra is the exterior algebra (V ) = Cℓ(V, 0), and otherwise (V ) ⊆ Cℓ(V, Q) as a subalgebra. Note that Cℓ(V, Q) is a F-vector space of dimension 2 n . For the remainder of this paper, we take F = R.
Given linearly independent vectors v 1 , . . . , v k and the exterior product ∧, an element of the form
A k = v 1 ∧ · · · ∧ v k(2)
is a k-blade, which are the simplest multivectors. Using the ∧ inherently removes lower grade elements and keeps only the highest grade element (grade-k) of the product A = v 1 v 2 · · · v k since, at the very least, the product of two vectors yields eq. (6). This multivector A, often called a versor since it is a product of vectors, is a sum of different graded elements called k-vectors. A k-vector is a linear combination of k-blades and we denote this subspace of grade-k elements by Cℓ k (V, Q). Therefore, we have the direct sum decomposition
Cℓ(V, Q) = n k=0 Cℓ k (V, Q).(3)
Some may refer to k-blades as simple or decomposable k-vectors as they correspond to rank-1 tensors in the tensor algebra. Moreover, they are also representative of subspaces which we discuss later. We write Cℓ k⊕ℓ (V, Q) to represent the direct sum space of k-and ℓ-vectors. A basis e i of V induces a k-blade basis by taking I = {i 1 , . . . , i k } to be a list of increasing indices i 1 < · · · < i k and putting
E I := e i1 ∧ · · · ∧ e i k .(4)
Some graded elements have special names. We say grade-0 objects are scalars, grade-1 are vectors, grade-2 are bivectors, and grade-n objects are pseudoscalars. Typically, objects of grade-(n − k) receive the prefix "pseudo", for example, we have pseudoscalars of grade-(n − 0) and pseudovectors of grade-(n − 1). Given the direct sum decomposition in eq. (3), a multivector A ∈ Cℓ(V, Q) is given by
A = n k=0 A k (5) where A k ∈ Cℓ(V, Q) k .
Multivectors also split into even and odd grades. We will work with the even-graded elements called spinors as they form their own subalgebra Cℓ + (V, Q). Due to this, some refer to Cℓ(V, Q) as a superalgebra since there is this Z/2Z-splitting. Spinors may also be defined slightly more generally (see [16]), but this notion suffices for this paper.
One purpose of using a Clifford algebra Cℓ(V, Q) is to extend the exterior algebra (V ) to include a useful interior multiplication. Given v, w ∈ Cℓ(V, Q) 1 , their product splits as
vw = v · w grade-0 + v ∧ w grade-2 ,(6)
where · is the interior product. However, for general Clifford algebras, we may have degenerate vectors v such that for any other vector w, v · w = 0. We will want to rid of this case.
To remove degenerate vectors from the algebra, we can force the quadratic form Q to be completely nonsingular. We refer to such a Clifford algebra with nonsingular Q as a geometric algebra and we write G = Cℓ(V, Q) to denote such an algebra. To see more reasoning of calling such algebras "geometric", we again refer the reader to the chapter [20]. Using the vector space basis, we can determine the coefficients of the bilinear form by
g ij = g(e i , e j ) = e i · e j .(7)
Let us quickly remark that if we were considering a Clifford algebra that was not a geometric algebra, then the matrix g ij of the bilinear form would be singular. We will carry on the rest of this paper working solely with geometric algebras for this reason, so the reader can assume that we take g whose matrix representations g ij are full rank and symmetric.
Given a k-vector A k and an ℓ-vector B ℓ , the product is
A k B ℓ = A k B ℓ |k−ℓ| + A k B ℓ |k−ℓ|+2 + · · · + A k B ℓ k+ℓ ,(8)
where the brackets − k : G → G k denote projection into the grade-k subspace by
A k = A k (9)
when A is given by eq. (5). We define the (left) contraction A k B ℓ := A k B ℓ ℓ−k . In general, the lowest grade term of A k B ℓ is the interior product A k · B ℓ = A k B ℓ |k−ℓ| and the exterior product ∧ is the highest grade term of the product so that A k ∧ B ℓ = A k B ℓ k+ℓ . For a vector v we have v · A = v A so many equations can be written with either · or . Most will be written with as it is algebraically and geometrically more convenient. For notational simplicity, we also remove the subscript when projecting into the scalar subspace, − = − 0 , but this should not be confused with the notation for the ideal generated by a relation used only in eq. (1).
The reciprocal basis vectors e i are those that satisfy e i · e j = δ i j . Reciprocal basis elements allow us to use the Riesz representation in order to avoid extraneous use of dual space elements since we are able to capture this functionality through the interior product. For sake of clarity, e i · e j = g ij is the matrix inverse to g ij and e i = g ij e j are just the "raised up" indices. For a basis blade E I , the reciprocal blade is E I and it satisfies the equation E I · E J = δ I J where δ I J = 1 only when the sets of indices I and J are identical. Geometric algebras have a bilinear product G × G → R called the multivector inner product which is given by
(A, B) := A † B .(10)
This equation is given in terms of the reverse operator † which for λ ∈ R satisfies
(A + B) † = A † + B † , (λA) † = λ † A † = λA † , A † † = A, (AB) † = B † A † ,(11)
and on a versor we have
(v 1 v 2 · · · v k ) † := v k · · · v 2 v 1 .(12)
Note that † acts as the adjoint in the product (−, −) which follows from the cyclic property of the scalar grade projection [21, eq. (138)]. To see this, we take another multivector C and note
(CA, B) = (CA) † B = A † C † B = (A, C † B),(13)
We define a semi-norm | − | 2 := (−, −) called the multivector norm. If |A| = ±1 we say that A is unit. It is worth saying that for a multivector field written in terms of basis blades f = I f I E I that
f I = (f, E I ),(14)
so long as the quadratic form is definite (e.g., the quadratic form is the Euclidean norm). We also have that
E I = (e i1 ∧ · · · ∧ e i k ) † .(15)
There exists a vector basis for V where p vectors square to −1 and q vectors square to +1 and p + q = n.
The corresponding geometric algebra is often written as G p,q . We will focus most on the the case where Q is positive definite and to distinguish this, we can write G n if it is necessary. For G n , the multivector inner product and multivector norm are both positive definite. One can see that the multivector inner product treats the space G n as a 2 n -dimensional inner product space with a basis given by the blades E I .
Equation (14) just specifies that we have chosen a set of blades orthonormal with respect to the multivector inner product.
If the basis e i is orthonormal in V , then the set of basis blades E I are orthonormal versors in G n since
E I = e i1 ∧ · · · ∧ e i k = e i1 e i2 · · · e i k .(16)
Their products become much clearer to compute. We have
E I E J = ±E I△J ,(17)
where △ is the symmetric difference of the sets I and J and the ± is used solely due to the fact that vectors e i comprising the versors E I may need to be swapped and
−E I = e i1 e i2 · · · e ij+1 e ij · · · e i k .(18)
For a concrete example, take E 123 = e 1 e 2 e 3 and E 124 both in G n , then
E 123 E 124 = e 1 e 2
Using an orthonormal vector basis shows how nicely versors act algebraically. Multiplication is just reduction of words in the characters e i subject to the relations e 2 i = 1 and e i e j = −e j e i when i = j.
Remark 2.1. Though we will not cover the content here, it is worth mentioning that versors in a geometric algebra G p,q form a group under multiplication called the Clifford group and the unit versors define the spin group Spin(p, q). The algebra of bivectors in G p,q with the commutator [−, −] (often written as × as well) is the Lie algebra spin(p, q).
Geometric algebras also have a unique isomorphism ⊥ : G k → G n−k and this ⊥ is equivalent to the Hodge star ⋆ in (V ). To construct this isomorphisms, we first take an arbitrary basis for V and build the volume
element µI := e 1 ∧ · · · ∧ e n ,(20)
where the unit n-blade I is the unit pseudoscalar and the scalar µ represents the volume scaling. Note that I represents the subspace V and for geometric algebras it defines the dual
A ⊥ := AI −1 .(21)
For example, if v is a vector then we have v ⊥ is a pseudovector representing a scaled copy of the hyperplane perpendicular to v.
Moreover, the dual allows for exchanging products
(A B) ⊥ = A ∧ B ⊥(22)
and the exterior product
(A ∧ B) ⊥ = A B ⊥(23)
and helps elucidate the geometrical meaning of and ∧ (see [21]). Finally, it is worth saying that in G n we
have I −1 = I † .
Given a k-dimensional subspace U ⊂ V in a space with nonsingular Q, we can put V = U ⊕ U ⊥ where U ⊥ is an n − k-dimensional subspace. In much the same way, given a unit k-blade U k we can find a decomposition of I by I = U k ∧ U k ⊥ . Actually, multiplication in G allows for projection onto subspaces using this identification.
Definition 2.2. Given an multivector B and unit k-blade U k , the projection onto the subspace U k is
P Uk (B) := (B U k )U k −1 .(24)
The projection preserves grades P Uk (B ℓ ) ∈ G ℓ . A specific application of the projection is to view spinors along planes or, eventually, surfaces. This will be our key methodology to bootstrap from complex analysis.
Definition 2.3. Let B be a unit 2-blade, then the space of plane spinors are the elements
A B := R ⊕ Span(B).(25)
As we will see in the following example, G + 2 ∼ = C and G + 3 ∼ = H. Furthermore, for any B ∈ G n we also have
A B ∼ = C.
Hence, just like the Grassmannian of planes in R 3 , Gr(2, 3), parameterizes copies of C in H by choice of imaginary unit, plane spinors A B as a subalgebra of G + n are parameterized by the Grassmannian Gr(2, n). If you find the following example lacking, then Doran & Lasenby's text [18] is very insightful and filled with great intuition.
Example 2.4. Rather than a sequence of multiple examples, it will prove to be far more illuminating to construct one large example for which most of the preliminaries to this point can be used in a meaningful way.
As such, we shall not rule out the utility that other researchers may gain out of using geometric algebras with pseudo inner products even though this paper is predominantly concerned with the positive definite case.
The classical example is the spacetime algebra defined by taking V = R 4 with a vector basis e 0 , e 1 , e 2 , e 3
satisfying e 0 · e 0 = −1 (26a) e 0 · e i = 0 i = 1, 2, 3 (26b) e i · e j = δ ij , i, j = 1, 2, 3.(26c)
We refer to e 0 as temporal since its square is negative and e i for i = 1, 2, 3 are spatial since their squares are positive. For this basis, the matrix for this inner product in this basis assumes the form η = diag(−1, +1, +1, +1)
(often called the Minkowski metric). The associated quadratic form Q can be found from η by polarization.
For a spacetime vector v = v 0 e 0 + v 1 e 1 + v 2 e 2 + v 3 e 3 , |v| 2 = (v, v) = v · v = −v 2 0 + 3 i=1 v 2 i .(27)
It is clear that the norm is definite when all vectors are spatial, but in the case of spacetime there are null vectors c such that |c| = 0. For example, c = e 0 + e 1 . The collection of null vectors define the light cone in Minkowski space. Also, it is important to distinguish these null vectors from degenerate vectors. Though c is null, it is not true that for any c and all other vector v that c · v = 0. This is only true for other v on the light cone and the light cone is not a subspace.
As the notation above suggests, the geometric algebra of Euclidean space R 3 , G 3 , should naturally appear inside of the spacetime algebra. The spatial trivector e 1 e 2 e 3 is unit |e 1 e 2 e 3 | = (e 1 e 2 e 3 ) † e 1 e 2 e 3 = e 3 e 2 e 1 e 1 e 2 e 3 = 1 (28) and represents the spatial subspace Span(e 1 , e 2 , e 3 ) ⊂ R 4 . With slight abuse of notation, the projection of G 1,3 onto this subspace yields
P e1e2e3 (G 1,3 ) = G 3 .(29)
In G 3 , we can specify an arbitrary multivector A by A = a 0 + a 1 e 1 + a 2 e 2 + a 3 e 3 + a 12 e 1 e 2 + a 13 e 1 e 3 + a 23 e 2 e 3 + a 123 e 1 e 2 e 3 .
It will be pertinent later to define B ij := e i e j for i = j. Using this substitution, the grade projections read
A = a 0 (31a) A 1 = a 1 e 1 + a 2 e 2 + a 3 e 3 (31b) A 2 = a 12 B 12 + a 13 B 13 + a 23 B 23 (31c) A 3 = a 123 e 1 e 2 e 3 .(31d)
Hence, we can write a spinor as
A + = a 0 + a 12 B 12 + a 13 B 13 + a 23 B 23 .(32)
Note as well that the spatial unit 2-blades always satisfy
B 2 23 = B 2 13 = B 2 12 = −1(33)
and we find that
B 23 B 13 B 12 = −1.(34)
Hence, the even subalgebra G + 3 isomorphic to the quaternion algebra H by
i ↔ B 23 , j ↔ B 13 , k ↔ B 12(35)
Given a quaternion, there is an equivalent spinor A + ; the imaginary part of the quaternion corresponds to the grade two part of the spinor A + 2 .
We can project down one dimension further by P B12 (G 3 ) = G 2 and we can verify quickly that
P B 12 (A) = a 0 + a 1 e 1 + a 2 e 2 + a 12 B 12 .(36a)
Given that B 2 12 = −1 we can put z := x + yB 12 ∈ G + 2 for x, y ∈ R which is exactly a representation of the complex number ζ = x + iy in C and i here can be thought of as the unit pseudoscalar in the plane. Again, the imaginary part is z 2 .
But, the above work is not special to the starting point of G 1,3 or G 3 . In fact, if we take G n for n ≥ 2, then there are natural copies of C contained inside of G n . In particular, we have the isomorphism
C ∼ = {x + yB | x, y ∈ R, B ∈ Gr(2, n).},(37)
which shows that complex numbers arise as plane spinors via the representation ζ = x + yB. Given the standard basis e 1 , . . . , e n we have the n 2 unit bivectors B ij for j = 1, . . . , n and i < j. The plane spinors A B ij are each isomorphic to C.
Clifford Analysis
Given a smooth semi-Riemannian manifold M with metric tensor field g and boundary ∂M , each tangent space T x M can be made into a geometric algebra G x M := Cℓ(T x M, g x ) and we call G x M the geometric tangent space. The geometric tangent spaces are glued together to form the geometric algebra bundle GM := x∈M G x M . We will call the C ∞ sections of this bundle C ∞ (M ; G) the smooth multivector fields and the continuous sections C 0 (M ; G) the continuous multivector fields. Let ∇ be the Levi-Civita connection on M .
Then for a vector field v we have the covariant derivative ∇ v which is extended to multivector fields, e.g., by [22]. Given local coordinates x i on M we have the induced (gradient) basis e i (x) ∈ G x M . Suppressing the pointwise notation, e i · e j = g ij . Also in the tangent space are the reciprocal vectors e i which are Riesz representatives corresponding to the dual basis dx i since dx i (e j ) = e i · e j . We will not get rid of the dual basis entirely since we still find use for the elements as coordinate measures in integration. Basis blades E I and their reciprocals E I carry over to each geometric tangent space as well.
The Hodge-Dirac operator (or gradient ) ∇ in these coordinates is
∇ := n i=1 e i ∇ ei .(38)
This derivative acts algebraically as a vector and
∇f = ∇ f + ∇ ∧ f.(39)
In particular, ∇∧ is equivalent to the exterior derivative d on differential forms Ω(M ), ∇ is equivalent to the codifferential δ, and ∇ 2 = ∆ is the Laplace-Beltrami operator. The kernel of d + δ on Ω k (M ) is coupled to the (co)homology of M and so ∇ retains this property as well. This relationship of the analysis of d + δ
(and equivalently ∇) to the absolute and relative (co)homology of M is formalized in Hodge theory and is useful for proving existence and uniqueness for boundary value problems [23]. However, in Clifford analysis, it is commonplace to consider mixed grade multivectors also in the kernel of ∇. This space is far bigger since we find that grades can "mix" together. For example, take M to be the unit disk D ⊂ R 2 and
f + = f 0 + f 2 B with f 0 , f 2 ∈ C ∞ (D; R)
where B is the unit 2-blade field. Then we have
∇f + = ∇ ∧ f 0 + ∇ (f 2 B).(40)
If we considered only singly graded elements such as the scalar fields f 0 or bivector fields f 2 B on their own, then the only elements in the kernel of ∇ are constant fields. On the other hand, when we combine them together into a spinor field f + ∈ C ∞ (D; G + ) then this f + now be any holomorphic function such as z = x 1 + x 2 B or even any power series in z which we will revisit later. Since this space is far bigger, it can be used to extract more topological data about M , namely the homeomorphism type, as we see in theorem 3.2.
We can note that each of the graded components of f + ∈ M + (M ) are also harmonic ∆ f + 2k = 0. Also, the fact that a product of spinor fields is again a spinor field will make it more convenient to work with M + (M )
instead of the whole of M(M ). Not only is it more convenient, but to prove the theorem 3.2, we only need
M + (M ).
A key piece of the proof for theorem 3.2 will be the ability to uniformly approximate monogenic spinor fields on open subsets U by fields defined on M . We will cite [16, theorem 11.7] from Calderbank's thesis. This will be used repeatedly. Other supplemental information can be found in [24] as well.
From Ryan's [25, theorem 4], we have the fact that for B a ball in Euclidean space, there exists a power series representation. In fact, one can think of this as an explicit realization of the approximation given by
Calderbank which uses the embedding B ⊂ R n ⊂ S n where S n is the n-sphere. We will construct the power series in section 3.2. The coefficients, which we define in eq. (82), will require us to compute integrals of multivector fields which we define next.
In order to integrate, we build differential forms from k-vector fields by attaching a measure. Take the coordinate measures (dual basis in the cotangent space) dx i in local coordinates and multiply by the corresponding reciprocal vector to get basic directed measures dx i = e i dx i (no summation implied), which determine the k-dimensional directed measures
dX k := 1 k! i1<···<i k dx i1 ∧ · · · ∧ dx i k = 1 k! I E I † dx I(43)
An arbitrary differential k-form α k is given by taking a corresponding k-vector A k and contracting along the
k-dimensional directed measure α k = A k dX † k .(44)
Specifically, A k = I α I E I is called the multivector equivalent of α k . This is a realization of the isomorphism the equivalent to µ is simply I. On ∂M , we have the boundary pseuodoscalar I ∂ and dual to this the boundary normal vector field ν = I ⊥ ∂ . As on M , the boundary pseudoscalar I ∂ is the multivector equivalent of the boundary area form µ ∂ . From this point forward, we work solely with multivector fields and contract with directed measures to integrate. We now realize the action of ∇∧ and ∇ as the equivalents of d and δ by
dα k = (∇ ∧ A k ) dX † k+1 , δα k = (∇ A k ) dX † k−1 .(45)
One beautiful result in Clifford analysis is the generalization of the Cauchy integral formula. Details for various cases are in our standard sources [18,19,16,24]. For R n with n ≥ 2, we define the vector-valued
function G(x) := 1 ω n x |x| n(46)
where ω n is the area of the unit sphere in R n and the use of the bold x indicates that we treat this point in space as a vector. For this work, the bold variable x will be a clear way to distinguish when we are embedded in R n , so it is important to keep track of this. The function G is the Green's function of the Dirac operator since
∇G(x) = δ x ,(47)
where δ x is the Dirac mass located at x ∈ R n . There is also a solution in the case n = 1, but it assumes a different form and we do not need it here.
Define the Cauchy kernel G(x ′ − x)ν(x ′ ) by translation and multiplication by the outward normal ν on the boundary ∂M . For compact M ⊂ R n and monogenic field f ∈ M(M ), we convolve the boundary value of f with the Cauchy kernel to arrive at the Cauchy integral formula
f (x) = (−1) n ∂M G(x ′ − x)ν(x ′ )f (x ′ )µ ∂ (x ′ ).(48)
Hence, we have a method for uniquely determining a monogenic field f from the boundary values f | ∂M . A more general notion of the Cauchy integral exists for arbitrary manifolds with boundary (see [16]), but it seems this also depends on a choice of embedding into a closed manifold. In the case of our Green's function G, this corresponds to the embedding M ⊂ R n ⊂ S n [16, proposition 9.10].
Currents
Recall that since g is positive definite, at any point x ∈ M we have that |f (x)| 2 = (f (x), f (x)) is nothing but the Euclidean vector norm on R 2n , so f really is a norm on C 0 (M ; G). Furthermore:
Proposition 2.7. If M is a compact Riemannian manifold, then the space C 0 (M ; G) is a (real) C * -algebra with involution †.
Proof. Note that G is a real 2 n dimensional Banach space with the multivector inner product. Since M is a compact Hausdorff space, it follows that the space C 0 (M ; G) is a Banach space (see [27]). Taking f, g ∈ C 0 (M ; G), at each point
(f g, f g) = (gg † , f † f ),(50)
since † is the adjoint. Using the Cauchy-Schwarz inequality
|f g| 2 = (f g, f g) ≤ (f † f, f † f )(gg † , gg † ) = |f | 2 |g| 2 .(51)
The last equality follows from taking an orthonormal basis e i at any G x M and forming the orthonormal vector basis blades (versors) E I and putting f = I f I E I . Then we have
f † f = I J f I f J E † I E J(52)
from which we see that
(f † f, f † f ) = |f † f | 2 = I J f I f J 2(53)
and finally
|f | 2 2 = I f I 2 2 = I J f I 2 f J 2(54)
which implies that |f † f | = |f | 2 . Taking suprema, it follows that f g ≤ f g which shows C 0 (M ; G) is a real Banach algebra.
For f, g ∈ C 0 (M ; G) and λ ∈ R we have that
(f + g) † = f † + g † , (λf ) † = λ † f † = λf † , f † † = f, (f g) † = g † f † ,(55)
by definition and at each point
|f † f | = |f | 2(56)
as shown before. By taking suprema, f † f = f 2 which shows C 0 (M ; G) is a real C * -algebra.
We topologize the C * -algebra C 0 (M ; G) with the uniform norm topology. Dually, we construct G valued functionals on this space which we call currents à la de Rham.
Definition 2.8. The space of G-currents is C 0 G (M ; G) ′ := {T : C 0 (M ; G) → G | T is continuous} (57)
Given a subalgebra A ⊂ G, we have the A-currents
C 0 A (M ; G) ′ = {T : C 0 (M ; G) → G | T is continuous}.(58)
The G-currents are given the weak- * topology, i.e., the coarsest topology on C 0 G (M ; G) ′ where point evaluation of fields is continuous. Specifically, for x ∈ M , the Dirac mass G-current δ
x ∈ C 0 G (M ; G) ′ defined by δ x [f ] = f (x) for f ∈ C 0 (M ; G) is continuous. The A-currents inherit the subspace topology.
Since the target G of the G-currents is itself a C * -algebra and a G-module, we expect some currents to respect these algebraic structures. For example, C 0 (M ; G + ) is a G + -bimodule. Given a A ⊂ G + , G + and C 0 (M ; G + ) are both A-modules and Banach algebras. Definition 2.9. Let A ⊂ G be a subalgebra and let T ∈ C 0 G (M ; G) ′ be a G-current. We say that T is right A-linear if it is a right A-module homomorphism
T [f α + g] = T [f ]α + T [g](59)
for α ∈ A and f, g ∈ C 0 (M ; G). Furthermore, we say that T is multiplicative on A if it is an R-algebra homomorphism
T [pq] = T [p]T [q](60)for p, q ∈ C 0 (M ; A). Finally, a current T is grade preserving if for h ∈ C 0 (M ; G k ) we have T [h] ∈ G k .
The set of grade preserving linear multiplicative currents are the most useful for us. It is worth remarking that currents as defined here provide an ample setting for further study. There are plenty of tweaks that could be interesting. One such example would be the subset of the de Rham currents (dual to the C ∞ -smooth forms) given by the R-currents C 0 R (M ; G) ′ . Here, we will make choices that allow us to generalize the classical Gelfand result.
Subsurface fields
The algebra C 0 (M ; G) is not commutative in general and the space M(M ) is not an algebra. This poses a direct issue for a straightforward generalization of the Gelfand representation and Belishev's 2-dimensional boundary control method [10], but it does not thwart the effort completely since Belishev and Vakulenko manage to build a 3-dimensional version [14]. Extending this approach, we will use insight on axial fields from those two authors but make the change to think not of an axis, but of a plane. Of course, in R 3 a plane and axis are dual, but when we extend beyond dimension-3, we will be required to use planes. If S is dimension 2, then M + (S) is a copy of the commutative algebra of holomorphic functions. Intuitively, we can build commutative Banach algebras of monogenic fields for surfaces in M .
Let U ⊂ M be a geodesically convex region, i.e., that all points x ∈ U are connected with unique shortest paths. Let B(x) be a unit 2-blade in G x U for some x ∈ U . Since U is convex, there exists a shortest geodesic between all points in U which allows us to parallel transport B(x) to build a unit 2-blade field B ∈ C ∞ (U ; G 2 ). Then, at all points in U , we have a projection P B onto B(y) in each geometric tangent space G y U . Definition 2.10. Let U and B be as before, then a continuous spinor field f ∈ C 0 (U ; G + ) satisfying
f + = P B • f +(61)
is a subsurface spinor field on U .
The definition for a subsurface spinor field on U requires that f + = P B • f + which means that we can
put f + = f 0 + f 2 B where f 0 , f 2 ∈ C 0 (U ; R).
Definition 2.11. Let U and B be as before, then the space of monogenic subsurface spinors on U is
A B (U ) = {f + ∈ C 0 (U ; G + ) | f + = P B • f + , ∇f + = 0}.(62)
The collection of all monogenic subsurface spinors on U is
A(U ) = {f + ∈ A B (U ) | B parallel transported from B(x) ∈ G x U , ∀B(x) ∈ G x U }.(63)
Proposition 2.12. Let U and B be as before, then the space A B (U ) is a commutative Banach algebra.
Proof. Note that multiplication of two fields f = f 0 + f 2 B and g = g 0 + g 2 B (dropping the subscripted + on f and g momentarily for clarity) in A B (U ) is commutative and given pointwise by the familiar complex
multiplication f g = f 0 g 0 − f 2 g 2 + B(f 0 g 2 + f 2 g 0 ) = gf.(64)
Using the overdot notation to say which field we are taking derivatives of, we find commutivity gives us
It is immediately clear that z = P B • z and in applying the Hodge-Dirac operator
∇z = ∇(x · v) + ∇(x · w)B (69) = 0.(70)
We can define such a function z for any choice of B and construct new functions from polynomials in these variables. The notation z should serve as a reminder of the connection to complex analysis and one may consider v as the real axis and w as the imaginary axis. The behavior of fields on an arbitrary convex U inside an arbitrary compact M is identical.
A Clifford-Algebraic Gelfand Theorem
Spinor Spectrum
Through Belishev's generalization of Gelfand's classical result, surfaces are determined up to conformal equivalence by the spectrum (or maximal ideal space) of M + (S). The naive generalization would be to seek this out in M + (M ), but, again, this space is not an algebra! Maximal ideals can also be identified with multiplicative functionals and this allowed Belishev and Vakulenko to achieve their 3-dimensional result. We follow suit with multiplicative linear currents.
= {δ = 0 ∈ C 0 G + (M ; G + ) ′ | δ grade preserving, δ(f g + hα) = δ(f )δ(g) + δ(h)α, ∀f, g, h ∈ A(U ), α ∈ G + },(71)
and we refer to the elements as spin characters. We prove theorem 3.2 in the following steps:
i. Utilize a power series representation for elements in a ball B which shows that the monogenic polynomials
M P (B) are dense in M + (M ).
ii. Build the elements of this series from homogeneous polynomials in variables of the form z (i.e., eq. (68)).
Using the fact that the spin characters are multiplicative over the collection A(M ), continuous, and G + -linear we show that it suffices to determine the action δ[z] for δ ∈ sp M + (M ).
iii. Determine that the action δ[z] is point evaluation at some point x δ ∈ R n by looking at the algebraic relationships between the variables z and combining this with the multiplicativity of δ.
iv. Construct a carefully selected sequence of monogenic fields on M and use continuity of δ to show that
x δ ∈ M .
The correspondence between δ ∈ sp M + (M ) is then clear and the homeomorphism follows by choice of the weak- * topology. The fact that the Gelfand transform is an isometry follows directly from the fact that each character corresponds to point evaluation.
Power series
Take the standard orthonormal basis fields e 1 , . . . , e n for R n and define the functions
z ij = x j − x i B ij where
x i are the coordinate functions corresponding to our basis vector fields and where B ij := e i e j . This takes eq. (68) and multiplies by B ij to match Ryan [25] and, in effect, this is just replacing z with iz. Note that Ryan's use of e −1 i become unnecessary due to our choice of a positive definite quadratic form. Identifying B ij with its plane, note that for any compact region M ⊂ R n each z ij ∈ A B ij (M ). Fix a natural number k ≥ 0 and natural numbers k j to form the tuple k = (k 2 , . . . , k n ) such that k 2 + · · · + k n = k with k j ≥ 0.
This is often called a multi-index with absolute value | k| = k. The set of all multi-indices of absolute value k is of size n−2+k n−2 . For example, we can write down a degree-k polynomial in terms of the monomial variables z ij based on a multi-index k by
z k2 12 (x)z k3 13 (x) · · · z kn 1n (x).(72)
But, ordering does matter. To build the homogeneous monogenic degree k polynomials we sum over permutations σ which rearrange the order in which we write the monomials but keep the total powers of each monomial the same throughout
p k (x) = 1 k! σ z 1σ(1) (x) · · · z 1σ(k) (x),(73)
where σ(j) ∈ {2, . . . , n} permutes the monomials without rearrangement. To reiterate, monomials that appear in the summand of eq. (73) with the powers given by k are just reordered from what we see in eq. (72) based on σ which is why we do not see k explicitly appear on the right hand side of eq. (73). Note that this is necessary since the monomials do not commute with one another. Ryan [25,Proposition 1] shows each of these polynomials is monogenic and linearly independent. We remark that the polynomials p k are homogeneous in the elements z ij ∈ A B ij (M ).
As examples, take n = 3 and k = 2 with k 2 = 2 and k 3 = 0 so that the multi-index is k = (2, 0). Then in
coordinates x = (x 1 , x 2 , x 3 ) p (2,0) (x) = 1 2! σ z 1σ(1) z 1σ(2) (74) = 1 2! z 12 (x)z 12 (x) (75) = 1 2! (x 2 − x 1 e 1 e 2 ) 2 .(76)
We can see that given our choice of k, there is only one choice of σ, i.e., the σ such that σ(1) = 2 and σ(2) = 2. On the other hand if we take the multi-index k = (1, 1), then
p (1,1) (x 1 , x 2 , x 3 ) = 1 2! σ z 1σ(1) z σ(2) (77) = 1 2! (z 12 (x)z 13 (x) + z 13 (x)z 12 (x)) (78) = 1 2! (x 2 − x 1 e 1 e 2 )(x 3 − x 1 e 1 e 3 ) + (x 3 − x 1 e 1 e 3 )(x 2 − x 1 e 1 e 2 ) .(79)
Again, our choice of k allowed for two choices of σ that were not repetitive: first σ(j) = j and the other σ(1) = 2 and σ(2) = 1. Furthermore, working out the details of ∇p (k2,k3) shows the necessity of summing over permutations in order to ensure that each is monogenic. The collection of all such polynomials for all multi-indices is the set of monogenic polynomials
M P (M ) = N k=0 k | k|=k p k (x)a k N ∈ N, a k ∈ G n (80)
The use of multi-index notation is also to facilitate taking higher order partial derivatives by defining
∇ k := ∂ k ∂x k2 2 ∂x k3 3 · · · ∂x kn n .(81)
In the case of a smooth manifold, the partial derivatives can be replaced with their corresponding covariant derivatives if the need should arise. Next, lemma 3. Proof. Without loss of generality, suppose B is centered at the origin. Then let f ∈ M(B) and define the coefficients a k ∈ G n by
a k = ∂B ∇ k Gνf µ ∂ ,(82)
where G is the Green's function for the Hodge-Dirac operator in R n . By [25, theorem 4], we have
f (x) = ∞ k=0 k | k|=k p k (x)a k ,(83)
which converges uniformly to f for points x ∈ int B.
But, as stated previously, we have theorem 2.6 which tells us that for open subsets in B we can uniformly approximate monogenic fields on those subsets. We apply this fact to get the following corollary.
(z) = ∞ k=0 z k a k where a k ∈ A B .
Remark 3.6. It is important to note that if f + ∈ M + (M ), the local power series has coefficients a k ∈ G + which you can see by eq. (82).
Characters
For M a compact region embedded in R n and f + ∈ M + (M ), we can see that for δ ∈ sp M + (M )
δ [f ] = ∞ k=0 k δ[p k ]a k (84)
by continuity and right G + -linearity of δ since a k ∈ G + . On each monogenic polynomial,
δ(p k ) = 1 k! σ δ z 1σ(1) · · · δ z 1σ(k) ,(85)
by multiplicativity over A(M ). Hence, the action of δ is completely determined by the action on each z ij . Proof. Since δ is grade preserving, it must be the case that δ[z] ∈ G 0⊕2 . Since δ is an algebra morphism, Proof. Take δ ∈ sp M + (M ) and the coordinate planes B ij and the corresponding z ij . Applying δ to z ij yields δ[z ij ] = α ij + β ij B ij with α ij , β ij ∈ R by proposition 3.7 and we will collect all α ij and β ij into matrices α and β respectively. Then, since
z ij B ji = (x j − x i e i e j )e j e i = −z ji (86) it follows that δ[z ij B ji ] = δ[z ij ]B ji = −δ[z ji ],(87)
and hence
(α ij + β ij B ij )B ji = β ij + α ij B ji = −α ji − β ji B ji .(88)
Therefore, α ij = −β ji for all i = j. Similarly, for arbitrary ℓ = i and ℓ = j we have
z ij = z ℓj + z iℓ B ℓj (89) so δ[z ij ] = δ[z ℓj + z iℓ B ℓj ] = δ[z ℓj ] + δ[z iℓ ]B ℓj .(90)
Expanding this yields the relationships α ij = α ℓj and β ij = β iℓ for all i, j, ℓ.
The relationships α ij = α kj and β ij = β ik show that both sets of constants α and β are given by n numbers since they are constant along one index. Taking this with the relationship α ji = −β ij shows that both are determined by the same n numbers which we call x i δ = α ji = −β ij for i = 1, . . . , n, just with swapped index and magnitude. Hence there exists some
x δ = (x 1 δ , . . . , x n δ ) ∈ R n so that δ[z ij ] = z ij (x δ ) since z ij (x δ ) = x j δ − x i δ B ij .(91)
To see that the corresponding point x δ lies in the given region M for any δ, we use continuity and a singular monogenic spinor field.
which does not converge due to the singularity at x δ which contradicts the fact that the limit converges to a monogenic function. Hence, it must be that x δ ∈ M .
One practical reason behind working with regions of R n is that there are clear choices of functions to use to probe whether a point is in the region or not. For an arbitrary n-dimensional manifold we cannot guarantee an embedding into R n and the technique using the Cauchy kernel G fails and a version of lemma 3.9
for arbitrary compact manifolds is not obvious. Likewise, lemma 3.8 could be viewed as a local result for characters on a coordinate patch, but it is not clear how the restriction of a character to local coordinate patches behaves. Nonetheless, we arrive at the proof for the main theorem. To see that the Gelfand transform M + (M ) → C 0 (sp M + (M ); G + ) is an isometry, we note that
f = sup δ∈sp M + (M) |f (δ)| = sup δ∈sp M + (M) |δ[f ]| = sup x δ ∈M |f (x δ )| = f .(94)
Hence, we have our theorem.
Further results and discussion
Though the behavior of characters on regions has been determined, it is still an open question whether theorem 3.2 can be extended to arbitrary n-dimensional compact Riemannian manifolds with boundary.
This extension is not immediately obvious, but there is more to be said that may assist the general case in the future. Again using motivation from complex analysis, M + (M ) retains some necessary features but others are missing.
Stone-Weierstrass
Firstly, let us prove a Stone-Weierstrass result showing the density of closure of the monogenic spinor fields in the space of continuous spinor fields. The proof of the theorem will require the following lemma. Proof. Let x, y ∈ int M be distinct points. We want to construct some field f ∈ M + (M ) such that f (x) = f (y). Since M is compact, there exists a shortest path γ : [0, 1] → M between x and y and moreover by [28] this path is C 1 since both M and ∂M are C ∞ . Since γ must always be C 1 , γ has a well-defined tangent vector at each t and a well-defined normal space N γ(t) γ which is orthogonal to the tangent vectorγ(t).
Since M is compact, for all t the injectivity radius of the exponential map at γ(t) is bounded from below by some ǫ > 0. Hence, we can construct a tube T γ about γ by taking T γ := γ × D ǫ where D ǫ (t) is the image under the exponential map of the disk of radius ǫ in the normal space at N γ(t) γ. Any pointx ∈ T γ is given The space M + (M ) is not an algebra, but we can consider the minimal algebra that the space generates.
uniquely by coordinates (t, v) where v ∈ N γ(t) γ. Given a unit 2-blade B(x) ∈ G x M ,
Let ∨M + (M ) represent the minimal algebra generated by M + (M ), then using the previous lemma and a result from Laville and Ramadanoff in their paper on the Stone-Weierstrass theorem for Clifford-valued functions [17], we will get the following theorem.
Theorem 3.11. ∨M + (M ) is dense in C 0 (M ; G + ).
Proof. Since M + (M ) contains 1 and separates points, it is a candidate for the use of Laville and Ramadanoff [17, theorem 3]. In order to use their result in all dimensions, we must have that f ∈ M + (M ) is invariant with respect to the principal involution f * . Since f is a spinor field, f = n 2k=0 f 2k and
f * = n 2k=0 (−1) 2k f 2k = n 2k=0 f 2k = f * ,(95)
so f is invariant under the principal involution.
Matching our notation to Laville's, take a basis 2k-blade E I (i.e., |I| = 2k is an ordered list of indices and E I is given by eq. (4)), then given f ∈ ∨M + (M ) we can take f → f I = (f, E I ) (see eq. (14)) which produces a dense subset ∨C 0 (M ; R) I ⊂ C 0 (M ; R) by the classical Stone-Weierstrass theorem. Hence, since
C 0 (M ; G + ) = 2k C 0 (M ; R)E I we conclude that ∨M + (M ) is dense in C 0 (M ; G + ).
Tomography
As discussed earlier, the work in this paper is heavily motivated by the Boundary Control (BC) method for the inverse tomography problem. The BC method was used in Belishev's proof for the 2-dimensional Calderón problem in [10]. Our paper manages to show that we can determine the homeomorphism type of an embedded manifold from the spinor spectrum, but we are missing other key facts that would lead to a solution for the tomography problem. Essentially, we need the following additional facts to use the BC method:
i. The Dirichlet-to-Neumann (DN) map determines the space tr M + (M ).
ii. The boundary trace map tr : ∨ M + (M ) → tr ∨M + (M ) where f + → f + | ∂M is an isometric isomorphism of algebras.
iii. The space M + (M ) determines the metric structure of M up to isometry.
Given the results of this paper alongside items (i) and (ii), we would be able to determine a compact embedded M up to homeomorphism. We can view (iii) as an extension of our result here. Namely, we have determined the homeomorphism type of M from the space M + (M ), but have yet to gain any metric data.
Let us discuss each of the points above.
i. Using the Dirichlet-to-Neumann operator Λ on differential forms, Belishev and Sharafutdinov in [6] describe an ancillary boundary operator called the Hilbert transform T . The Hilbert transform is a classical operator in complex analysis and it also appears in Clifford analysis as an operator on the L 2 -completion of the space tr C 0 (M ; G). Two good sources include Brackx and De Schepper's paper [29] (which specifically concentrates on compact regions of R n ) and Calderbank's thesis [16].
Given a function on the boundary φ ∈ tr C 0 (M ; G), the Hilbert transform in Clifford analysis allows one to uniquely recover the complete boundary data of a monogenic field f with φ as a component of ii. This point is essentially given as an open question by Belishev and Vakulenko [13]. iii. There is likely geometric content inside the spinor spectrum and this could lead to determining, at the very least, the conformal class of the metric. First, it is widely known that the Hodge-Dirac operator is conformally invariant [30]. Hence, it may be possible to construct a metric g up to conformal equivalence from the spinor spectrum given that the spinor spectrum is already homeomorphic to M . This should not be shocking; Belishev in [10] was able to do this for surfaces S with single boundary component, as he proved that the topologized spectrum is conformally equivalent to S (and in fact was determined by the classical DN operator). It could be that an procedure analogous to Belishev's technique for finding a conformal metric in [10] can be performed with the spinor spectrum.
It could be that we can do better than extracting just the conformal class for dimension n ≥ 3. As a reminder, the 2-dimensional problem cannot determine more than the conformal class of g since ∆ is conformally invariant in dimension 2. But, if we consider subsurfaces S inside of M , we can collect conformal copies of the metric restricted to the surface, vary the over a collection of surfaces passing through a point, and perhaps the combined data from all surfaces passing through each point could produce a metric in the isometry class of M .
Characters on arbitrary compact M
Aside from the above points, we want the results of this paper to hold true for arbitrary compact M , not just compact regions of R n . We briefly discussed the issue with our proof technique just before the proof of theorem 3.2, but the core issue is that our proof hinged on a global power series representation which was valid since M was embedded in R n . If the power series is only local, then we must, in some sense, understand the restriction of spin characters to local coordinate patches, but this is not understood.
To view the spin characters from a different perspective, it could be interesting to take δ ∈ sp M + (M ) and consider ker δ. In the case for a surface S, the kernel of a character is in one-to-one correspondence with the set of maximal ideals of the algebra of holomorphic functions. Succinctly, we can match a character δ x with the class of holomorphic functions [f ] who vanish at the point x. On a different note, it could also be useful to identify the G-currents C 0 G (M ; G) ′ with G-valued Radon measures. Additivity of measures over subsets and the regularity of Radon measures may allow for characters to be applied to local power series representations of the monogenic spinor fields. If this is the case, compactness of M would mean that a spin character corresponds to evaluation at finitely many points. As a final step, we could possibly use the fact that M + (M ) separates points to conclude that a character δ ∈ sp M + (M ) is evaluation at only a single x δ ∈ M .
Conclusion
The heart of this paper is to extend the theory Gelfand on commutative Banach algebras of C-valued functions to noncommutative Banach algebras of G-valued functions. In essence, we can find copies of C as subalgebras of plane spinors A B ⊂ G + and copies of complex holomorphic functions as the monogenic surface spinors A B .
Using the fact we can also locally construct a power series for monogenic spinor fields in terms of monogenic variables z with coefficients in G + , we define the characters accordingly. Hence, we have a meaningful notion of a spectrum and achieve our main theorem.
This theory, when restricted to a 2-dimensional surface, yields the Gelfand representation, but allows us to achieve new results in higher dimensions with the same process. More or less, this hinges on the Cauchy integral formula for multivector fields in the same way the Cauchy integral acts as a linchpin in many of the classical complex analysis theorems. Hopefully this entices others to consider the role of the Banach algebra of G-valued functions and the dual space of G-currents.
We expect that more results will follow in the future. For example, we suspect theorem 3.2 can be generalized to arbitrary compact Riemannian manifolds. If this is to help towards a solution of the Calderón problem, this information would all need to be extracted from the boundary and, in particular, from the Dirichlet-to-Neumann operator. Experts in Clifford analysis and elliptic theory may have useful insights into this problem.
Theorem 1 . 1 .
11Let M be a compact region in R n . For any δ ∈ sp M + (M ), there is a point x δ ∈ M such that δ(f ) = f (x δ ) for any f + ∈ M + (M ) a monogenic field. Given the weak- * topology on the space of G-currents, the map Γ : sp M + (M ) → M, δ → x δ is a homeomorphism. The Gelfand transform M + (M ) → C 0 (sp M + (M ); G + ) given by f + (δ) = δ[f + ] is an isometric isomorphism onto its image so that M + (M ) ∼ = M + (M ).
Theorem 1 . 2 .
12Let ∨M + (M ) represent the minimal algebra generated by M + (M ). Then ∨M + (M ) is dense in C 0 (M ; G + ).
e 3 e 1 e 2 e 4 = e 1 e 2 e 1 e 2 e 3 e 4 = −e 1 e 2 e 2 e 1 e 3 e 4 = −e 1 e 1 e 3 e 4 = −E 34 .
Definition 2. 5 .
5Let f ∈ C ∞ (M ; G), then we say that f is monogenic if ∇f = 0. We denote the space of monogenic fields by M(M ). Monogenic fields are the emphasis of Clifford analysis and many of the theorems of holomorphic functions in complex analysis apply to these fields. Once again, on the unit disk D take f + = f 0 + f 2 B then if ∇f + = 0 we can use eq. (39) to derive the Cauchythe function z is indeed monogenic. For more detail on the relationship of monogenic spinor fields to complex holomorphic functions see Doran and Lasenby [18, §6.3.1]. Since ∇ is grade-1, we have ∇ : C ∞ (M ; G ± ) → C ∞ (M ; G ∓ ) which yields the direct sum decomposition M(M ) = M + (M ) ⊕ M − (M ).
Theorem 2. 6 (
6Calderbank). Let U be an open subset of M . Then any monogenic fields on U may be approximated (locally uniformly in all derivatives) by restrictions of monogenic fields on M .
between C ∞ (M ; G) and Ω(M ) as C ∞ (M )-modules and it can be viewed as an extension of the musical isomorphisms between vectors and 1-forms[26, chapter 13]. The multivector equivalent of the Riemannian volume form µ is I −1 † and I(x) represents the tangent space T x M . Of course, in the case where g is definite,
the remainder, let M be a smooth oriented compact Riemannian manifold of dimension n with boundary ∂M and positive definite metric tensor g. Let I be the unit pseudoscalar on M which is the multivector equivalent of the Riemannian volume form µ, I ∂ the boundary pseuodoscalar which is the multivector equivalent of the Riemannian boundary area form µ ∂ and dual to ν the boundary normal field. The space C 0 (M ; G) comes with the uniform norm f := sup x∈M |f (x)|.
since f is monogenic and f g = gf (66) = 0 since g is monogenic.(67)Since A B (U ) is a subalgebra of C 0 (U ; G), it is a commutative Banach algebra.This construction provides a notion of complex functions that are nested in multivector fields on any manifold of dimension n ≥ 2. In the case n = 1, no such fields exist and it is exactly in the 2-dimensional Euclidean case that the complex-valued functions are just the spinor fields themselves and the unit 2-blade field is the tangent pseudoscalar to the surface. The special case of monogenic subsurface spinor fields serve as a realization of complex holomorphic functions inside the more general spinor fields. If we take B = e 1 e 2 , then we have the Cauchy-Riemann equations from ∇f + = 0 via eq. (41).For example, take the case where M is a compact region of R n with the Euclidean metric. Then M itself is compactly contained inside of some ball B which is convex. The set of bivectors is parameterized by B ∈ Gr(2, n) (i.e., the possible coordinate planes) and for each such B we can consider A B (M ) as a restriction of A B (B) via theorem 2.6. Each unit 2-blade decomposes into two orthogonal unit vectors. Let B = vw where v and w are a pair of orthogonal unit vectors and consider the monogenic subsurface field z : M → A B ⊂ G + defined by z(x) := P B (vx).
Definition 3. 1 .
1The spinor spectrum sp M + (M ) ⊂ C 0 G + (M ; G + ) ′ is the set of nonzero grade preserving right linear currents that are multiplicative over the collection of all subsurface spinor algebras A(U ), sp M + (M ) :
One choice of spin character is point evaluation: if δ is defined on f + ∈ M + (M ) by δ(f + ) = f + (x δ ) for some x δ ∈ M , then it follows that δ ∈ sp M + (M ). This shows that M injects into sp M + (M ) by the map x → δ x where δ x [f ] = f (x). We will find that (at least for embedded M ) characters defined by point evaluation are the only elements of sp M + (M ). This shows the inclusion is surjective. In fact, the main result is that this map is a homeomorphism.
Theorem 3. 2 .
2Let M be a compact region in R n . For any δ ∈ sp M + (M ), there is a point x δ ∈ M such that δ(f ) = f (x δ ) for any f + ∈ M + (M ) a monogenic field. Given the weak- * topology on the space of G-currents, the map Γ : sp M + (M ) → M, δ → x δ is a homeomorphism. The Gelfand transform M + (M ) → C 0 (sp M + (M ); G + ) given by f + (δ) = δ[f + ] is an isometric isomorphism onto its image so that M + (M ) ∼ = M + (M ).
3, corollary 3.4, and proposition 3.5 show that for arbitrary M , M(M ) are locally uniformly approximated by monogenic polynomials.
Lemma 3. 3 .
3Let B be a compact ball in R n , then the space M P (B) is dense in M(B).
Corollary 3. 4 .
4Let M ⊂ R n be a compact region. Then there exists B such that M(B) are dense in M(M ). Proof. Since M is compact in R n there exists a ball B such that the closure of M is contained in B. Then by theorem 2.6, any monogenic fields on M can be uniformly approximated by monogenic fields in M(B), and we have our result by lemma 3.3, Proposition 3.5. Let M be an n-dimensional compact Riemannian manifold and let f ∈ M(M ). Then f admits a local power series representation over finitely many open subsets. Proof. Take f ∈ M(M ), let (U, ϕ) a local coordinate chart such that U ⊂ M is an open convex region and ϕ(U ) ⊂ B ⊂ R n where B is some closed ball in R n . Then f • ϕ ∈ M(ϕ(U )) and by lemma 3.3 and corollary 3.4, f • ϕ admits a power series representation. Since M is Riemannian there exists a finite covering of M by convex regions and this gives us a local power series representation over finitely many open sets.Following the details of the above proofs for a surface S yields the fact that holomorphic functions on a surface admit local power series representations. In this case, take x 1 , x 2 as local isothermal coordinates and define z = x 2 − x 1 B. Then for f + ∈ M + (S), we have the local power series f +
Proposition 3. 7 .
7Let M be a compact manifold embedded in R n and B a unit 2-blade field that is a parallel translation of a coordinate plane. Then for any δ ∈ sp M + (M ) we have δ(A B (M )) = A B .
δ[A B (M )] ⊂ G 0⊕2 is commutative subalgebra. Using linearity as well, δ[α+βB] = δ[1](α+βB) = α+βB for α, β ∈ R. Hence A B ⊂ δ[A B (M )]. IfB ∈ δ[A B (M )] commutes with B, these bivectors must not intersect as subspaces except at zero which yields the 4-vector BB / ∈ G 0⊕2 . This contradicts the grade preservation of δ and thus δ[A B (M )] = A B .Next, we show that the characters sp M + (M ) correspond to evaluation at some point in R n .Lemma 3.8. Let M be a compact region in R n and δ ∈ sp M + (M ). Then δ(z) = z(x δ ) for some x δ ∈ R n .
Lemma 3. 9 .
9Let M ⊂ R n be a compact region and let f ∈ M + (M ) and δ ∈ sp M + (M ). Then δ(f ) = f (x δ ) for some x δ ∈ M .Proof. To see that x δ ∈ M , take f 0 (x) := G(x − x 0 )e 1 with x 0 ∈ M . Again, G is the Green's function for the Hodge-Dirac operator. Then f 0 | M ∈ M + (M ). By lemma 3.8 we have some x δ ∈ R n such thatδ(f 0 | M ) = f 0 | M (x δ ).(92)Take a sequence x n → x δ and suppose for a contradiction thatx δ / ∈ M and each x n / ∈ M . Then this defines a sequence of functions f n (x) := E(x − x n )e 1 | M ∈ M + (M ) and the sequence converges uniformly to a monogenic function lim n→∞ f n (x) = G(x − x δ )e 1 . By continuity of δ, lim n→∞ δ(f n ) = lim n→∞ f n (x δ ),
Proof of theorem 3 . 2 .
32Fix M a compact region of R n . It is clear that the map M → sp M + (M ) is an embedding by mapping a point x ∈ M to δ x ∈ sp M + (M ) (inverse of Γ). Then, by lemma 3.9, any δ ∈ sp M + (M ) corresponds to x δ ∈ M showing the reverse inclusion. Hence the sets are in bijection via Γ and under the weak- * topology, Γ is also continuous and hence we have the homeomorphism M ∼ = sp M + (M ).
Lemma 3 . 10 .
310If M be a compact Riemannian manifold with boundary, then the space M + (M ) separates points.
we can parallel translate B(x) to a unit 2-blade B(x) any pointx ∈ T γ by parallel translation along γ and then parallel translation in the normal direction. This builds unit 2-blade field B on T γ . Then, on T γ , define the field z ∈ A B (T γ ) using the unit 2-blade field B following eq. (68). Then z(x) = z(y) and z ∈ M + (T γ ). Since T γ is a union of open sets, T γ is open in M , and we can uniformly approximate z by elements of M + (M ). Taking the uniform limit of these approximations yields a function f ∈ M + (M ) satisfying f (x) = f (y).
Specifically, those two ask whether it is true that the algebras ∨M + (M ) and ∨trM + (M ) are isometrically isomorphic. At the moment, we have a partial answer: by the Cauchy integral formula, a monogenic field f ∈ M(M ) is determined by its boundary values therefore the map tr : M(M ) → tr M(M ) is an isomorphism of vector spaces and by the weak maximum principle for elliptic operators, it is also an isometry. This of course applies to the spinor subspace M + (M ). However, it is not clear that the algebras ∨M + (M ) and ∨trM + (M ) are isomorphic. In Clifford analysis, we have the Hardy space H(M ) as the L 2 -completion of tr M(M ) which is studied in both the previously referenced papers[29,16]. Perhaps there is more knowledge about H(M ) that could assist in finding a proof of fact (ii).
f | ∂M . In essence, the Hilbert transform yields boundary values of functions conjugate in the generalized Cauchy-Riemann equations given by ∇f = 0. In fact, if φ k is a k-vector, the Hilbert transform of φ k contains a k − 2-, a k-, and k + 2-vector component. This is part of why we suggest to look beyond pairwise conjugate forms.A reasonable question to ask is: are these two Hilbert transform operators related in some way? Moreover, Sharafutdinov and Shonkwiler extend Belishev and Sharafutdinov's Dirichlet-to-Neumann operator to the complete Dirichlet-to-Neumann operator[8]. If it is not possible to relate Belishev and Sharafutdinov's Hilbert transform using the DN operator on forms to the Hilbert transform in Clifford analysis, is there a related operator defined in terms of the complete DN operator that relates to Clifford analysis?
The maximal ideals of the space of holomorphic functions are exactly the functions that vanish at just a single point. It is not clear that elements in ker δ have this property when the dimension of M exceeds 2. Part of the proof for the 2-dimensional result used by Belishev in [10] follows from [31, Exercise 26.4, pg. 205] which can be proven using sheaves. To that end, it may be useful to think of the space M + (M ) in the context of sheaves.
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| []
|
[
"Electronic vortex structure of Fe-based superconductors: application to LiFeAs",
"Electronic vortex structure of Fe-based superconductors: application to LiFeAs"
]
| [
"B Mencia Uranga \nNiels Bohr Institute\nUniversity of Copenhagen\nUniversitetsparken 5DK-2100CopenhagenDenmark\n",
"Maria N Gastiasoro \nNiels Bohr Institute\nUniversity of Copenhagen\nUniversitetsparken 5DK-2100CopenhagenDenmark\n",
"Brian M Andersen \nNiels Bohr Institute\nUniversity of Copenhagen\nUniversitetsparken 5DK-2100CopenhagenDenmark\n"
]
| [
"Niels Bohr Institute\nUniversity of Copenhagen\nUniversitetsparken 5DK-2100CopenhagenDenmark",
"Niels Bohr Institute\nUniversity of Copenhagen\nUniversitetsparken 5DK-2100CopenhagenDenmark",
"Niels Bohr Institute\nUniversity of Copenhagen\nUniversitetsparken 5DK-2100CopenhagenDenmark"
]
| []
| Detailed tunneling spectroscopy of vortex core states can provide important insight to the momentum structure of the superconducting order parameter. We present a theoretical study of vortex bound states in iron-based superconductors by use of a realistic five-band model relevant to these systems, and superconductivity stabilized by spin-fluctuation generated pairing vertices yielding an s± gap structure. The computed local density of states agrees remarkably well with both the bias dependence of the local conductance and the spatial structure of the low-bias conductance as obtained by scanning tunneling microscopy measurements on LiFeAs [T. Hanaguri et al., Phys. Rev. B 85 214505 (2012)]. | 10.1103/physrevb.93.224503 | [
"https://arxiv.org/pdf/1603.04453v1.pdf"
]
| 118,480,909 | 1603.04453 | 2a2dac786872a72cc0b91b5cccf60d10aaca1218 |
Electronic vortex structure of Fe-based superconductors: application to LiFeAs
B Mencia Uranga
Niels Bohr Institute
University of Copenhagen
Universitetsparken 5DK-2100CopenhagenDenmark
Maria N Gastiasoro
Niels Bohr Institute
University of Copenhagen
Universitetsparken 5DK-2100CopenhagenDenmark
Brian M Andersen
Niels Bohr Institute
University of Copenhagen
Universitetsparken 5DK-2100CopenhagenDenmark
Electronic vortex structure of Fe-based superconductors: application to LiFeAs
(Dated: March 16, 2016)numbers: 7420-z7425Uv7455+v7470Xa
Detailed tunneling spectroscopy of vortex core states can provide important insight to the momentum structure of the superconducting order parameter. We present a theoretical study of vortex bound states in iron-based superconductors by use of a realistic five-band model relevant to these systems, and superconductivity stabilized by spin-fluctuation generated pairing vertices yielding an s± gap structure. The computed local density of states agrees remarkably well with both the bias dependence of the local conductance and the spatial structure of the low-bias conductance as obtained by scanning tunneling microscopy measurements on LiFeAs [T. Hanaguri et al., Phys. Rev. B 85 214505 (2012)].
I. INTRODUCTION
The understanding of the microscopic origin of electron pairing in strongly correlated electron systems remains an ultimate goal in the field of unconventional superconductivity. Generating Cooper pairs from purely repulsive bare Coulomb interactions, as for example in terms of exchange of antiferromagnetic spin excitations, requires distinctive properties of the resulting gap symmetry as given, for example, by a sign-changing superconducting gap in momentum space. Thus, a detailed experimental determination of the superconducting gap structure and its evolution with e.g. temperature and electron filling is of crucial importance in order to unveil the constituents that drive the superconducting instability.
Local perturbations of the superconducting condensate as found, for example, near defect sites or vortex cores, constitute an important means to study the gap symmetry. This is because the low-energy states in the vicinity of such regions are highly dependent on the properties of the gap structure of the bulk superconducting phase. 1,2 In the case of Fe-based superconductors, scanning tunneling spectroscopy (STS) studies of bound, or quasi-bound, states near impurity sites have mapped out a rich series of tunneling spectra that still awaits quantitative theoretical modelling. [3][4][5][6][7][8] Complicating factors in this endeavor include the multi-band electronic structure of these materials and the orbital degrees of freedom of the scattering centers, resulting in substantial parameter dependence of the modelling. [10][11][12] The scattering potential from vortex cores, on the other hand, does not contain orbital complexity/uncertainty, since it is entirely set by the quantum flux lines generated by the external field. In this respect, it may be more straightforward to deduce properties of the host superconducting gap symmetry from the measured tunneling conductance near vortex cores.
In Fe-based superconductors a limited number of experimental studies have measured the detailed sub-gap tunneling conductance near vortex cores. 4,[13][14][15] For example Hanaguri et al. 15 detected and mapped out the voltage and real-space dependence of the sub-gap vortex core states in LiFeAs. 15 This material is particularly suitable for surface-sensitive probes due to its atomically flat nonpolar cleaved surfaces. In addition LiFeAs is superconducting in its stoichiometric composition, i.e. without the complications of dopant disorder, and it exhibits neither a magnetic nor a structural transition. In the bulk homogeneous phase, LiFeAs exhibits a typical two-gap density of states spectrum with the inner (outer) gap edge at 2.9 meV (6.0 meV) 15 composed, presumably, of mainly d xy (d xz /d yz ) orbital character. The latter follows from a comparison of typical DFT bandstructure results and the momentum-resolved gap structure as measured by ARPES. [16][17][18] The main results of the STS study near the vortex cores of LiFeAs in Ref. 15 include a discrete set of resonant in-gap states with a pronounced conductance peak around -0.9 meV, i.e. just below the Fermi level E F , and a smaller and broader conductance peak near -2.3 meV [see Fig. 3(a) and Ref . 15]. The peaks exhibit symmetry partners at positive bias, but significant anisotropy exists in their spectral weight. The conductance peak at -0.9 meV disperses away from E F with increasing distance from the vortex core center and smoothly approaches the inner superconducting gap. The spatial profile of the low-energy core states were found to consist of a fourfold symmetric star shape with high LDOS tails along the nearest As directions, i.e. the 110 directions of the 1-Fe unit cell. At biases beyond the lowest energy peak (at -0.9 meV) the conductance tails split into parallel streaks and become less pronounced [see Fig. 4 and Ref. 15].
Here we study the electronic properties with particular focus on the low-energy sub-gap states in vortex cores in realistic models relevant to Fe-based superconductors. We include all five d orbitals of the Fe sites, and stabilize superconductivity from spin-fluctuation exchange resulting in an s±-wave pairing state. The main findings of this approach is a remarkable resemblance of the theoretically obtained vortex core spectrum to that measured by Hanaguri et al. 15 Based on this agreement and the absence of any tuning parameters of the scattering potential, we conclude that the applied bandstructure and superconducting pairing gap provide a good description of the superconducting phase of LiFeAs.
Several earlier theoretical vortex state studies of Febased superconductors exist in the literature. Some have focussed on the effect of competing magnetism and pairing symmetry dependence of the vortex core electronic structure within simplified two-orbital models. [19][20][21][22][23] Other works have performed a more systematic study of the vortex core bound state spectrum with the number of bands. 24 In a recent publication, we have utilized a fiveband model including multi-orbital Hubbard correlations to investigate vortex-induced stripe magnetism which allowed us to explain recent observed field-enhanced magnetism in Ba(Fe 0.95 Co 0.05 ) 2 As 2 as seen by neutron scattering and µSR experiments. 25 Wang et al. 26 studied the LDOS anisotropy of core states in LiFeAs within a quasi-classical one-band approach, and concluded that the four-fold star shape observed by Hanaguri et al. 15 is a Fermi surface anisotropy effect, and not caused by gap anisotropy. Our calculations presented below within a five-band self-consistent approach support this interpretation of the data in LiFeAs. Finally it should be noted that the experiment by Song et al. 4 in which they found highly anisotropic vortices in FeSe, has stimulated several theoretical studies of vortices in this material. [27][28][29] Before entering the model and result sections, we note that a more quantitative understanding of the vortex cores spectrum in LiFeAs is particularly desirable given the substantial discussion of potential alternative pairing structures of this material. 30 Specifically, the less nested band of this compound 31 and the fact that the largest measured gap appears on the smallest hole pocket around the Z point, has led to a controversy about the pairing origin, and the distribution of signs on the various Fermi pockets. [32][33][34][35][36][37][38] Thus it is currently of interest to include other experimental probes to help resolve the detailed gap structure of LiFeAs in particular, and Fe-based superconductors in general.
II. MODEL
The starting point for the theoretical modelling is the following five-orbital Hamiltonian
H = H 0 + H BCS + H Z ,(1)
where H 0 contains the kinetic energy given by a tightbinding fit to the DFT bandstructure, 11 including hopping integrals of all Fe orbitals to fifth nearest neighbors
H 0 = ij,µν,σ t µν ij e iϕ ij c † iµσ c jνσ − µ 0 iµσ n iµ.σ .(2)
Here, the operators c † iµσ create electrons at the i-th site in orbital µ and spin σ, and µ 0 is the chemical potential used to set the doping δ = n − 6.0. The indices µ and ν run from 1 to 5 corresponding to the Fe orbitals d 3z 2 −r 2 , d yz , d xz , d xy , and d x 2 −y 2 , respectively. The corresponding Fermi surface is shown in Fig. 1(a) in the one Fe unfolded Brillouin zone. It consists of three hole pockets (two smaller Γ-centered circular hole pockets and one M -centered larger and more squarish hole pocket), and two electron pockets at X and Y .
The presence of an external magnetic field is described by standard means by use of the Peierls phases
ϕ ij = −π Φ0 i j A · dr, where Φ 0 = h
2e is the half flux quantum and the integral is a line integral along the straight line joining lattice sites j and i.
The second term in Eq. (1) is given by
H BCS = − i =j,µν [∆ µν ij c † iµ↑ c † jν↓ + H.c.],(3)
with superconducting order parameter defined by ∆ µν ij = αβ Γ βν µα (r ij ) ĉ jβ↓ĉiα↑ . Here Γ βν µα (r ij ) denotes the effective pairing strength between sites (orbitals) i and j (µ, ν, α and β) obtained from the RPA spin-χ RP A s and charge susceptibilities χ RP A c relevant for LiFeAs 12
Γ βν µα (k − k ) = 3 2 U s χ RP A s (k − k )U s + 1 2 U s − 1 2 U c χ RP A c (k − k )U c + 1 2 U c βν µα ,(4)
where U s and U c are 5 × 5 matrices identical to those of Ref. 11. The real-space pairings are then obtained by Γ βν µα (r ij ) = q Γ βν µα (q) exp(iq · (r i − r j )) where we retain all possible orbital combinations up to next-nearest neighbors (NNN). For the present band, the RPA susceptibilities are strongly peaked near (0, ±π) and (±π, 0) favoring an s ± pairing state. A recent theoretical spin fluctuation study of pairing in LiFeAs used a full 3D ARPESderived bandstructure, and also found standard s ± -wave pairing to be the dominant instability. 33 In the current case, the resulting gap structure is shown in Fig. 1(b) where we plot the amplitude of the gap on the various Fermi surface sheets. As seen, the largest gap exists on the smallest inner hole pocket in agreement with experiments. Below, for numerical finite-size reasons we have to use a gap that is larger than the experimental case in order to enhance the spectral resolution at low energies. This, however, causes only minor quantitative changes in the obtained LDOS (for example it makes the inner gap less pronounced).
The last term in Eq. (1), H Z , is the Zeeman term accounting for spin-dependent energy shifts due to the external magnetic field
H Z = h iµ (n iµ↑ − n iµ↓ ).(5)Here, h = − µ B gsB 2
, µ B is the Bohr magneton and g s is electron g-factor.
Performing a standard Bogoliubov transformation applied to Eq. (1) leads to the following multi-band Bogoliubov de-Gennes (BdG) equations 12
jν H µν ijσ ∆ µν ij ∆ µν * ij −H µν * ijσ u n jν v n jν = E n u n iµ v n iµ ,(6)
where
H µν ijσ = t µν ij e iϕ ij + δ ij δ µν [−µ 0 + σh].(7)
We find the stable solutions through iterations of the following self-consistency equations
n iµ↑ = n |u n iµ | 2 f (E n ),(8)n iµ↓ = n |v n iµ | 2 (1−f (E n )),(9)∆ µν ij = αβ Γ βν µα (r ij ) n u n iα v n * jβ f (E n ),(10)
where n denotes summation over all eigenstates n. The self-consistency is unrestricted and the fields are allowed to vary on each site and orbital. Below, the lattice constant a is chosen as the unit of length, and we apply the Landau gauge A(r) = (By, 0), which corresponds to a magnetic field B = B(−ê z ). The magnetic translation operators (MTO), which commute with the Hamiltonian, are given as follows 39
M Rψ (r) = e −i 1 2 χ(r,R)σzψ (r − R),(11)
where σ z is the Pauli matrix,ψ(r) are the wave functions of the quasiparticles with u and v components, χ(r, R) = 2π Φ0 A(R)·r and R = mN xêx +nN yêy with m, n integers and N x ,N y being the dimensions of the magnetic unit cell (MUC). In order to have MTOs that fulfill the composition law M Rm M Rn = M Rm+Rn , it is required that the MUC contains an even number of half flux quantum Φ 0 . The magnetic field is fixed such that the flux going through the MUC is Φ = 2Φ 0 . The fulfilment of the composition law leads to the generalized Bloch theorem, which reads
M Rψk (r) = e −ik·Rψ k (r),(12)
where k= 2πlx Nxê x + 2πly
Nyê y with l x,y = 0, 1, ..., N x,y − 1 are the wave vectors defined in the first Brillouin zone of the vortex lattice andψ k (r) denote eigenstates of the Hamiltonian and the MTO. By use of Eq. (11) and Eq. (12), the eigenfunctions of the Hamiltonian transform under translations as
u n i+Rµ v n i+Rµ = e ik·R e −i 1 2 χ(r,R) u n iµ e i 1 2 χ(r,R) v n iµ ,(13)
where i takes values in the magnetic unit cell and r=i + R. Note that since a minimum of two superconducting flux quanta need to penetrate the MUC, the magnetic field is related to the real-space system size by B ∼ 58500 NxNy T (a = 2.66Å for LiFeAs). For the fiveband model used here, we are restricted numerically to systems of sizes less than (N x , N y ) = (56, 28), which we use in the present calculation, indicating that we have a field of ∼ 37T , whereas a 0.5T field was used in the experiments of Ref. 15. Despite this substantial quantitative discrepancy, we can still reliably study the properties of a single vortex since the neighboring vortices cause only minor quantitative effects. However, in order to make better contact to the experimental conditions by Hanaguri et al., we set B = 0.5T in H z of Eq. (5) in order to avoid unphysically large Zeeman energy splittings. This is only a quantitative issue and does not influence the main points of the theoretical results discussed in the following sections. Figure 2(a) shows the self-consistent superconducting order parameter for Φ = 2Φ 0 . The vortex cores generated by the external field can be clearly seen by the two suppressed regions of ∆(r i ). Here, ∆(r i ) refers to the superconducting order parameter at each site defined by
∆(r i ) ≡ 1 9 µν,j * ∆ µν ij * e iϕ ij * ,(14)
where the index j * includes the set of onsite, nearest neighbor and next-nearest neighbor lattice sites to site i. Figure 2(b-e) show the total and orbitally resolved supercurrents obtained from the expression ?
(∇ ·ĵ) iµ = − ie jνn [t νµ ji e iϕji (u n * jν u n iµ f (E n ) + v n jν v n * iµ f (−E n )) − t µν ij e iϕij (u n * iµ u n jν f (E n ) + v n iµ v n * jν f (−E n ))].(15)
The main contribution to the supercurrents comes from intra-orbital terms of the orbitals that dominate the Fermi surface, i.e. the d yz , d xz , and d xy orbitals, and hence only these contributions are shown in Fig. 2(c-e). The currents from the other two e g orbitals are negligible. Figure 3(a) shows the measured conductance in LiFeAs at the vortex center (red) and away from vortices (blue), reproduced from the publication of Hanaguri et al. 15 . In Fig. 3(b) we show the calculated local density of states (LDOS) ρ(i, ω) given by
ρ(i, ω) = − 1 π Im nµ |u n iµ | 2 w − E n + iη + |v n iµ | 2 w + E n + iη ,(16)
at the vortex center (red) and away from vortices (blue) obtained within the current model. As seen from comparison to Fig. 3(a), we find good agreement between theory and experiment; in both cases there are two clear sub-gap conductance peaks at negative biases with the most pronounced peak just below the Fermi level. These two conductance peaks have particle-hole symmetric partners at positive biases which are, however, strongly suppressed by their associated coherence factors.
To extract information about the orbital content of the sub-gap peaks, we plot the orbitally resolved calculated LDOS in Fig. 3(c). Evidently the main inner peak just below the Fermi level consists of mainly d xy character whereas the outer peak consists of d xz and d yz orbital states. We note that as opposed to the case of impurity bound states, there is no low-energy contribution to the LDOS from the e g orbitals since the vortex states are lowenergy Andreev-like states whereas the impurity bound states can be generated from high-energy states. 11,12 Finally, in Fig. 3(d) we show the spatial evolution of the total LDOS along a line cut through the cortex core. The main in-gap bound state is seen by the bright spot in the center of the plot. When moving away from the core center, the bound state disperses to larger energies and merges with the inner gap edge similar to the experimental finding. 15 We turn now to a discussion of the spatial profile of the conductance at fixed low biases. Figure 4(a) shows the results of the tunneling conductance measurements around a single vortex at low biases inside the fully gapped region. 15 As seen, the conductance displays a four-fold star shape with weight leaking out along the 110 directions. Away from the Fermi level, each of the four tails split up into two sub tails as seen most clearly from Fig. 4(b). This tail-splitting appears to set in earlier (in bias) for positive bias than for negative bias as evident for example from comparison of the panels at ±0.73 mV or ±1.10 mV in Fig. 4(b). Figure 5(a-d) shows a representative set of calculated sub-gap total LDOS patterns around a vortex core. Similar to the experimental results in Fig. 4(a,b), the LDOS exhibit a four-fold star shape with tails along the 110 directions which eventually disappears as the energy becomes comparable to the inner gap edge. With increasing energy, we also find LDOS features that resemble that each tail splits up into two sub tails whereas the more squarish LDOS pattern found experimentally does not seem to be reproduced theoretically. The tail-splitting takes place initially at positive energy as seen by comparing the panels at energies ±0.007 eV.
What is the origin of the star-shaped low-energy LDOS? The spatial profile of core states are known to unveil nodes in the gap, for example, as seen in the case of cuprates. 2,46 That family of materials, however, is known to be prone to competing order which complicates the understanding of the core states due to locally induced spin-and charge density waves. [40][41][42][43][44][45][46][47][48] The gap structure displayed in Fig. 1 5. (a-d) Spatial dependence (28 × 28 sites) of the total LDOS calculated around a single vortex core at selected representative energies inside the gap. (e-h) Same as (a-d) but only the dxy contribution to the LDOS is shown. The star-shape of the LDOS is a property of the dxy orbital, and is evident in the total LDOS only when this orbital contribution dominates. dental) gap nodes but rather a Fermi surface anisotropy with mimima located mainly along the nearest-neighbor Fe-Fe 100 directions. This is certainly true for the d xydominated large hole pocket near the M point, which is also the main orbital character of the inner gap edge in the homogeneous case, and the lowest most pronounced peak in the vortex cores. Thus, the gap structure at the d xy -dominated Fermi surface does not explain the star shape. In Fig. 5(e-h) we show the spatially resolved d xycontribution to the LDOS, revealing that the star shape is a property of the d xy orbital states [As discussed above, the d xz and d yz do not contribute much at this energy, and we have additionally verified that they do not exhibit "star quality"]. From the modelling one may also conclude that the spatially resolved total LDOS exhibits the four-fold star shape only at the energies at which d xy dominates the LDOS. In addition to gap anisotropy, Wang et al. 26 recently studied the role of Fermi surfaceanisotropy on the core states. When applied to LiFeAs, they concluded that the square shape of the large hole pocket near M = (π, π) with significant flat regions along the 100 directions [see also Fig. 1] feed into the spatial structure of the core states, and cause a four-fold star shaped pattern similar to experiment. We have verified that the origin of the star-shaped LDOS obtained in our model is also caused by Fermi surface anisotropy, rather than gap anisotropy. Specifically, it is precisely the squarish form of the d xy -dominated hole pocket as seen from Fig. 1(b) that gives rise to the 110 tails. In Fig. 6(a) we show the Fermi surface obtained by Ikeda et al. 49 , displaying largely the same topology and orbital content as the one used above with the exception of a roughly circular d xy hole pocket at M . From the LDOS shown in Fig. 6(b) one sees directly a significant band dependence of the sub-gap vortex bound states, but in the present case the orbital polarization of the bound states is even more pronounced. Zooming in on the d xydominated bound state at -0.028 eV, we find a perfect rotationally symmetric total and d xy -resolved LDOS as 49 showing the orbital majority with the same color code as in Fig. 1(a). (b) Orbitally resolved LDOS calculated at the center of the vortex core. (c) Spatially resolved total LDOS (28 × 28 sites) at the pronounced sub-gap energy −0.028 eV. (d) Same as (c) but only displaying the dxy contribution to the LDOS. seen from Fig. 6(c) and Fig. 6(d), respectively. Thus, in this case the different directions of the Fermi velocities of the M -centered hole pocket are equally weighted and hence the LDOS star shape has vanished.
(b) exhibits no (acci- (a) (b) (c) (d) (e) y x (f ) (g) (h) FIG.
III. CONCLUSIONS
In summary we have performed a fully self-consistent real-space BdG study of vortex core states in five-orbital models relevant to Fe-based superconductors. The superconducting order was stabilized by spin fluctuationderived pairing vertices generating an s ± -wave gap structure. By application to LiFeAs we find striking agreement with STS measurements by Hanaguri et al. on this compound. 15 In particular the details of the energy dependence and the spatial structure seems in almost quantitative agreement without any tuning parameters. From this fact we conclude that our model provides a reasonable description of LiFeAs, without the necessity to invoke more exotic paring states of this material. In future theoretical studies it would be interesting to compute the vortex core bound states within other pairing states for LiFeAs and compare their vortex core spectrum to experiments.
FIG. 1 .
1(a) Fermi surface with the dominant orbital character indicated by the color notation green (dyz), red (dxz), and blue (dxy). (b) Absolute value of the superconducting gap at the Fermi surface.
FIG. 2 .
2(a) Real-space plot of the amplitude of the superconducting order parameter |∆(r i )| for magnetic flux Φ = 2Φ0 through the unit cell. The cores exhibit the usual suppression of |∆(r i )|. (b-e) Real-space structure of the total (black) and orbitally resolved (green, red, and blue) supercurrents. The currents dyz (green), dxz (red) and dxy (blue) have the arrowheads amplified for visual clarity.
3. (a) Experimental tunneling spectra from Hanaguri et al. 15 in LiFeAs taken at the center of the vortex core (red) and away from vortex core region (blue). (b) Total LDOS calculated at the center of the vortex (red) and away from vortices (blue) to be compared to panel (a). (c) Orbitally resolved DOS calculated at the center of the vortex corresponding to the red curve in panel (b). (d) Total LDOS along a line cut of 28 lattice sites through the vortex core.
FIG. 4 .
4(a) Measured spatial dependence of the tunneling conductance probing the inner sub-gap peak at both negative and positive biases obtained by Hanaguri et al.15 (b) In the second derivative of the conductance one is able to identify a splitting of the tails of the star, and the splitting appears to set in earlier (in bias voltage) for positive biases versus negative biases (compare e.g. the panels at ±0.73 mV.
surface of a band with a circular (π, π) pocket,
IV. ACKNOWLEDGEMENTSWe acknowledge useful discussions with T. Hanaguri
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| []
|
[
"Covariance Estimation for Multivariate Conditionally Gaussian Dynamic Linear Models",
"Covariance Estimation for Multivariate Conditionally Gaussian Dynamic Linear Models"
]
| [
"K Triantafyllopoulos "
]
| []
| []
| In multivariate time series, the estimation of the covariance matrix of the observation innovations plays an important role in forecasting as it enables the computation of the standardized forecast error vectors as well as it enables the computation of confidence bounds of the forecasts. We develop an on-line, non-iterative Bayesian algorithm for estimation and forecasting. It is empirically found that, for a range of simulated time series, the proposed covariance estimator has good performance converging to the true values of the unknown observation covariance matrix. Over a simulated time series, the new method approximates the correct estimates, produced by a non-sequential Monte Carlo simulation procedure, which is used here as the gold standard. The special, but important, vector autoregressive (VAR) and time-varying VAR models are illustrated by considering London metal exchange data consisting of spot prices of aluminium, copper, lead and zinc.Some key words: Multivariate time series, dynamic linear model, Kalman filter, vector autoregressive model, London metal exchange. | 10.1002/for.1039 | [
"https://arxiv.org/pdf/0802.0191v1.pdf"
]
| 12,298,749 | 0802.0191 | 91708c554524bde3a27a2f873e005a96a41ce0a8 |
Covariance Estimation for Multivariate Conditionally Gaussian Dynamic Linear Models
1 Feb 2008 1 March 2006
K Triantafyllopoulos
Covariance Estimation for Multivariate Conditionally Gaussian Dynamic Linear Models
1 Feb 2008 1 March 2006arXiv:0802.0191v1 [stat.ME]
In multivariate time series, the estimation of the covariance matrix of the observation innovations plays an important role in forecasting as it enables the computation of the standardized forecast error vectors as well as it enables the computation of confidence bounds of the forecasts. We develop an on-line, non-iterative Bayesian algorithm for estimation and forecasting. It is empirically found that, for a range of simulated time series, the proposed covariance estimator has good performance converging to the true values of the unknown observation covariance matrix. Over a simulated time series, the new method approximates the correct estimates, produced by a non-sequential Monte Carlo simulation procedure, which is used here as the gold standard. The special, but important, vector autoregressive (VAR) and time-varying VAR models are illustrated by considering London metal exchange data consisting of spot prices of aluminium, copper, lead and zinc.Some key words: Multivariate time series, dynamic linear model, Kalman filter, vector autoregressive model, London metal exchange.
Introduction
Multivariate time series receive considerable attention because a great deal of time series data arrive in vector form. Whittle (1984) and Lütkepohl (1993) discuss VARMA models for vector responses, whilst Harvey (1989, Chapter 8), West and Harrison (1997, Chapter 16) and Durbin and Koopman (2001, Chapter 3) extend this work to state space models for observation vectors. In econometrics most studies of state space models focus on trend estimation, signal extraction and volatility. A review of recent developments of state space models in econometrics can be found in Pollock (2003). Barassi et al. (2005) and Gravelle and Morley (2005) give applications of the Kalman filter to interest rates data and Harvey et al. (1994) use Kalman filter techniques to estimate the volatility of foreign exchange rates using multivariate stochastic volatility (MSV) models. With the exception of multivariate GARCH and MSV models, which focus on the prediction of the volatility, it is usually desirable to use a structural state space model to forecast time series vectors (e.g. foreign exchange rates, monthly sales, interest rates, etc) and to estimate the observation innovation covariance matrix of the underlying time series. For such applications and for short term forecasting the above covariance matrix can be assumed time-invariant, but unknown, and its estimation is the main aim of this paper.
The estimation of the observation covariance matrix plays an important role in forecasting. Firstly we note that, under the general multivariate dynamic linear model (see equation (1) below), the multi-step forecast mean of the response time series vector is a non-linear function of the observation covariance matrix (West and Harrison, 1997, Chapter 16). Secondly, the computation of the standardized forecast error vectors requires a precise estimation of the observation covariance matrix and thus a miss-specification of the observation covariance matrix can lead to false results regarding the evaluation and judgement of the model. Thirdly, the multi-step forecast covariance matrix is a linear function of the observation covariance matrix and the former is of particular interest; the forecast covariance matrix can explain the variability of the forecasts and hence it can enable the computation of confidence bounds for the forecasts. Finally, the precise estimation of the observation covariance matrix gives an accurate estimation of the cross-correlation structure of the several component time series, which is particularly useful, especially for financial time series. For all the above reasons the study of the estimation of the observation covariance matrix is worthwhile and its contribution to forecasting for multivariate time series is paramount.
The problem of the estimation of the observation innovation variance for univariate state space models has been well reported (West and Harrison, 1997, §4.5; Durbin and Koopman, 2001, §2.10), however, for vector time series this problem becomes considerably more complex and the available methodology consists of special cases, approximations and iterative procedures.
Let y t be a p-dimensional observation vector following the Gaussian dynamic linear model (DLM): y t = F ′ θ t + ǫ t and θ t = Gθ t−1 + ω t ,
where θ t is a d-dimensional Markovian state vector, F is a known d × p design matrix and G a known d × d transition matrix. The notation F ′ is used for the transpose matrix of F . The distributions usually adopted for {ǫ t }, {ω t } and θ 0 are the multivariate Gaussian, i.e. ǫ t ∼ N p (0, Σ), ω t ∼ N d (0, Ω) and θ 0 ∼ N d (m 0 , P 0 ), for some known priors m 0 and P 0 . The innovation vectors {ǫ t } and {ω t } are assumed individually and mutually uncorrelated and they are also assumed uncorrelated with the initial state vector θ 0 , i.e. for all t = s: E(ǫ t ǫ ′ s ) = 0, E(ω t ω ′ s ) = 0, and for all t, s > 0: E(ǫ t ω ′ s ) = 0, E(ǫ t θ ′ 0 ) = 0 and E(ω t θ ′ 0 ) = 0, where E(·) denotes expectation. The covariance matrices Σ and Ω are typically unknown and their estimation or specification is a well known problem. The interest is centered on the estimation of Σ, while Ω can be specified a priori (West and Harrison, 1997 Several methods have been proposed, for the estimation of Σ. Harvey (1986) and Quintana and West (1987) independently introduce matrix-variate DLMs, which are matrix-variate linear state space models allowing for covariance estimation. Harvey (1986) proposes a likelihood estimator, while Quintana and West (1987) propose a Bayesian estimation modelling Σ with an inverted Wishart distribution. Harvey (1986)'s model is reported and further developed in Harvey (1989), Fernández and Harvey (1990), Harvey and Koopman (1997) and Moauro and Savio (2005), while Quintana and West (1987)'s model is reported and further developed in Quintana and West (1988), Queen and Smith (1992), West and Harrison (1997), Salvador et al. (2003), Salvador and Gargalo (2004) and . However, both suggestions (Harvey (1989)'s and Quintana and West (1987)'s) are criticized in Barbosa and Harrison (1992) where it is shown that the above models are restrictive in the sense that one can decompose the response vector y t into several scalar time series and model each of these time series individually, using univariate DLMs. Barbosa and Harrison (1992) propose an approximate algorithm for the general DLM (1), but their main assumption seems rather unjustified, since it suggests that for any p × p matrix C it is Σ 1/2 CΣ −1/2 = Σ 1/2 C Σ −1/2 , where Σ is a point estimate of Σ and the notation Σ 1/2 stands for the symmetric square root of Σ (Gupta and Nagar, 1999, p. 7). This assumption holds clearly when Σ 1/2 , C commute and when Σ 1/2 = Σ * , C commute, where (Σ * ) 2 is any particular realization of Σ. However, in general the above assumption is difficult to check since Σ is the unknown covariance matrix subject to estimation. In addition, that assumption seems to be probabilistically quite inappropriate, since it translates that the non-stochastic quantity Σ 1/2 C Σ −1/2 equals the stochastic quantity Σ 1/2 CΣ −1/2 with probability 1. A possible analysis can be obtained in special cases where Σ is diagonal or when the off-diagonal elements of Σ are all common. Triantafyllopoulos and Pikoulas (2002) and Triantafyllopoulos (2006) adopt the model of Harvey (1986) and they provide an improved on-line estimator for Σ based on a standard maximum likelihood technique. The problem is again that the models discussed lack the general formulation of the state space model (1); e.g. one can easily show that all above models are special cases of model (1). Iterative procedures via maximum likelihood and Markov chain Monte Carlo (MCMC) techniques are available, but they tend to be slow, especially as the dimension of the observation vector p increases. Kitagawa and Gersch (1996), Shumway and Stoffer (2000, Chapter 4), Durbin and Koopman (2001, Chapter 7) and Doucet et al. (2001) discuss univariate modelling with iterative methods, but their efficiency in multivariate time series is not yet explored. Barbosa and Harrison (1992) and West and Harrison (1997, §16.2.3) discuss the problem of inefficiency of iterative methods and they point out that the number of parameters to be estimated in Σ is p(p + 1)/2, which rapidly increases with the dimension p of the response vector, e.g. for p = 10 there are 55 distinct parameters in Σ to be estimated. In addition to this Dickey et al. (1986) discuss relevant issues on specifying and assessing the prior distribution of Σ pointing out difficulties in the implementation of iterative procedures.
In this paper we propose a new non-iterative Bayesian procedure for estimating Σ and for forecasting y t . This procedure offers a novel estimator of Σ for the general DLM (1). The proposed estimator is empirically found to converge to the true value of Σ and this estimator approximates well the respective estimators in the special cases of the conjugate univariate and matrix-variate DLMs. A comparison with a non-sequential Monte Carlo simulation shows that the new method produces estimates close to the MCMC. The focus and the benefit employing the new method is on on-line estimation and therefore no attempt has been made to compare the proposed algorithms with sequential iterative procedures. The reason for this is justified by the above discussion and the interested reader should refer to Dickey et al. (1986) and West and Harrison (1997, §16.2.3). The proposed forecasting procedure for model (1) is applied to the important model subclasses of vector autoregressive (VAR) and VAR with time-dependent parameters. These models are illustrated by considering London metal exchange data, consisting of spot prices of aluminium, copper, lead and zinc (Watkins and McAleer, 2004).
We begin by developing the main idea of the paper and giving the proposed algorithm. The performance of this algorithm is illustrated in the following section by considering simulated time series data; a comparison with a Monte Carlo simulation is performed. The proceeding section gives an application to vector autoregressive modelling, which is used to analyze London metal exchange data, in the following section. The appendix details a proof of a theorem in the paper and it describes the MCMC simulation procedure.
Main Results
Denote with y t = (y 1 , y 2 , . . . , y t ) the information set comprising data up to time t, for some positive integer t > 0. Let m t and P t be the posterior mean and covariance matrix of θ t |y t and S t be the posterior expectation of Σ, i.e. E(Σ|y t ) = S t . Let y t (1) = E(y t+1 |y t ) = F ′ Gm t be the one-step forecast mean at time t and Q t+1 = Var(y t+1 |y t ) = F ′ R t+1 F + S t be the one-step forecast covariance matrix at t, where R t+1 = GP t G ′ + Ω. Upon observing y t+1 , we define the one-step forecast error vector as e t+1 = y t+1 − y t (1). The next result (proved in the appendix) gives an approximate property of S t . Theorem 1. Consider the dynamic linear model (1). Let Σ be the covariance matrix of the observation innovation ǫ t and assume that lim t→∞ S t = Σ, where E(Σ|y t ) = S t is the true posterior mean of Σ given y t . Let n 0 be a positive scalar and S 0 = E(Σ) be the prior expectation of Σ. If Σ is bounded, then for large t the following holds approximately
S t = 1 n 0 + t n 0 S 0 + t i=1 S 1/2 i−1 Q −1/2 i e i e ′ i Q −1/2 i S 1/2 i−1 ,(2)
where e i , Q i are defined above and S Conditionally now on Σ = S, for a particular value S, we can apply the Kalman filter to the DLM (1) and obtain the posterior and predictive distributions of θ t |Σ = S, y t and y t+h |Σ = S, y t , for a positive integer h > 0, known as the forecast horizon. Theorem 1 motivates approximating the true posterior mean S t by S = S t , which is produced from application of equation (2), given a particular data set y t = (y 1 , y 2 , . . . , y t ). Thus we obtain the following algorithm:
(b) Posterior distribution at time t: θ t |Σ = S t , y t ∼ N d ( m t , P t ), where e t = y t − y t−1 (1) and m t = G m t−1 + A t e t , P t = G P t−1 G ′ + Ω − A t Q t A ′ t , A t = (G P t−1 G ′ + Ω)F Q −1 t , S t = 1 n 0 + t n 0 S 0 + t i=1 S 1/2 i−1 Q −1/2 i e i e ′ i Q −1/2 i S 1/2 i−1 . (c) h-step forecast distribution at t: y t+h |Σ = S t , y t ∼ N p { y t (h), Q t (h)}, where y t (h) = F ′ G h m t and Q t (h) = F ′ G h P t (G h ) ′ F + h−1 i=0 F ′ G i Ω(G i ) ′ F + S t .
In the special case of matrix-variate DLMs (Harvey, 1986; West and Harrison, 1997, §16.4) the estimator S t approximates the true posterior mean of Σ produced by an application of Bayes' theorem, assuming a prior inverted Wishart distribution for Σ. To see this, note that in the matrix-variate DLM (this model is briefly in page 15, see equation (8)), F is a d-dimensional design vector and Q t = U t S t−1 with U t = F ′ R t F + 1 and so equation (2) can be written recursively as
S t = n −1 t (n t−1 S t−1 + e t e ′ t /U t ) and n t = n t−1 + 1 = n 0 + t.(3)
It is easy to verify that the assumption lim t→∞ S t = Σ is satisfied, since lim t→∞ S t = lim t→∞ E(Σ|y t ) and lim t→∞ Var{vech(Σ)|y t } = 0, where vech(·) denotes the column stacking operator of a lower portion of a symmetric matrix. For p = 1 the matrix-variate DLM is reduced to the conjugate Gaussian/gamma DLM (West and Harrison, 1997, §4.5). It turns out that the estimator S t of equation (2) approximates the analogous estimators of all existing conjugate Gaussian dynamic linear models. It is worth noting that Theorem 1 and Algorithm 1 have been presented for the state space model (1) having time-invariant components F , G and Ω. However, these results apply if some or all of the above components change with time. In addition, if the evolution covariance matrix Ω t is time-dependent, it can be specified via discount factors (West and Harrison, 1997, Chapter 6). This is a useful consideration, because in practice the signal θ t is unlikely to have the same variability over time.
For the application of Algorithm 1 the initial values m 0 , P 0 , n 0 and S 0 must be specified. m 0 can be specified from historical information from the underlying experiment and P 0 can be set as a typically large diagonal matrix, e.g. P 0 = 1000I p , reflecting a low precision (or high uncertainty) on the specification of the moments of θ 0 . The scalar n 0 can be set to n 0 = 1 (in the special case of matrix-variate DLMs, n 0 is the prior degrees of freedom). S 0 is a prior estimate of Σ and requires at least a rough specification. As information is deflated in time series, a miss-specification of S 0 may not affect much the posterior estimate S t , especially in the presence of large data sets. However, in many cases and especially in financial time series, a miss-specification of S 0 can lead to poor estimates of Σ. Here we suggest that a diagonal covariance matrix can be used, where the diagonal elements of S 0 reflect the empirical expectation of the diagonal elements of Σ. This expectation can be obtained by studying historical data and other qualitative pieces of information, which are usually available to practicing experts of the experiment or of the application of interest.
Simulation Studies
Empirical Convergence of S t We have generated 1000 bivariate time series {y it } t=1,2,...,500 i=1,2,...,1000 from several state space models and then we have averaged the 1000 estimates S i,t (produced by each of the 1000 time series) and compared the average S t = 1000 −1 1000 i=1 S i,t with the true value of Σ. Since in practice complicated models are decomposed into simple models comprising local level, polynomial trend and seasonal components (Godolphin and Triantafyllopoulos, 2006), we consider estimation separately in such different component models. We have three modelling situations of interest: situation 1 (bivariate local level models); situation 2 (bivariate linear trend models); and situation 3 (bivariate seasonal models). For each of the above three situations we have generated 1000 bivariate time series, each of length 500, using three different covariance matrices Σ, i.e.
Σ 1 = 2 3 3 5 , Σ 2 =i,0 = I 2 , S(2)
i,0 = 150I 2 and S Table 1 shows the results. There are three blocks of columns, each showing results of the state space model considered, namely local level model (LL or block 1), linear trend model (LT or block 2) and seasonal model (SE or block 3). In each block the first column shows the mean of the average S = 500 −1 500 t=1 S t of all S t . The second column shows the average S 100 at time point t = 100. Likewise the third column shows the respective S 500 averaged over all 1000 series. The rows in Table 1 show the picture of S t over the three different values of Σ, e.g. Σ 1 , Σ 2 and Σ 3 . The average estimate of the correlations is also shown and it is marked in the table by ρ. The results suggest that, generally, the LL model has the best performance as opposed to the LT and the SE model, although we note that σ 12 = 3 (covariance in Σ 1 ) is estimated better from the LT model. It appears that the estimator S t for all models converges to the true values of Σ, but the rate of convergence depends on the underlying state space model (here LL performs faster convergence) and on the prior S 0 . Table 2 shows the averaged (over all 1000 simulated time series) mean vector of squared standardized one-step forecast errors (MSSE (1) ), for each of the three models (LL, LT, SE) and for each of Σ (Σ 1 , Σ 2 , Σ 3 ). For comparison purposes, Table 2 also shows the respective values of the MSSE (2) when Σ i is the true value. The target value of the MSSE (i) is [1 1]. We see that the MSSE (1) approaches the respective MSSE (2) and this demonstrates the accuracy of the estimator S t . We observe that under Σ 3 , the MSSE (1) has values significantly smaller than 1 as compared to the MSSE (2) using the true value of Σ 3 .
Comparison of the Local Level Model with MCMC
We have simulated a single local level model under the observation covariance matrix Σ = Σ 1 and the relevant model components of the local level model of the previous sub-section. We apply Algorithm 1 and we compare it with a state of the art MCMC estimation procedure based on a blocked Gibbs sampler suitable for state space models (Gamerman, 1997, p. 149); the MCMC procedure we use is described in the appendix. The MCMC estimation procedure is an iterative non-sequential MCMC procedure and its role in this section is to provide a means of comparison with the non-iterative procedure of Algorithm 1. MCMC is the gold standard, since it produces (given enough computation) exact computation of S t . But MCMC is impractical; the new proposed method is a quick, practical and easily implemented Table 2: Mean vector of squared standardized one-step forecast errors (MSSE (i) ) of the multivariate dynamic model of Algorithm 1. The index i = 1, 2 refers to when Σ is estimated by the data (i = 1) and when Σ is assumed known (according to the simulations) for comparison purposes (i = 2). The notation LL, LT, SE and Σ 1 , Σ 2 , Σ 3 is the same as in Table 1. Tables 3 and 4 give the results; the former shows the estimates of Σ with both methods (MCMC and Algorithm 1) and the latter shows the performance of the one-step forecast errors for both methods. In Table 4 the one-step forecast error vector e t = [e 1t e 2t ] ′ and the mean vector of squared one-step forecast errors are shown for several values of t under both estimation methods. We observe that the new method (of Algorithm 1) approximates well the MCMC estimates, especially for large values of time t = N . We note that MCMC should not be considered as a better method as compared to the proposal of Algorithm 1, since MCMC is an iterative and in particular in this paper it is a nonsequential estimation procedure. The application of sequential MCMC estimation (Doucet et. al., 2001) often experience several challenges as for example time-constraints, availability for general purpose algorithms, prior-specification, prior-sensitivity, fast monitoring and expert intervention features. The proposal of this paper provides a strong modelling approach allowing for variance estimation in a wide class of conditionally Gaussian dynamic linear models and this section shows that for large time periods its performance is close to Monte Carlo estimation.
MSSE (1) MSSE (2) LL (Σ 1 ) 0.
Application to VAR and TVVAR Time Series Models
The dynamic model (1) is very general and an important subclass of (1) is the popular vector ARMA model. In recent years vector autoregressive (VAR) models have been extensively developed and used, especially for economic time series, as in Doan et al. (1984), Litterman (1986), Karlsson (1993, 1997), Ooms (1994), Johansen (1995), Uhlig (1997), Ni and Sun (2003), Sun and Ni (2004) and Huerta and Prado (2006).
Our discussion in this section includes two important subclasses of model (1), which can be used for a wide-class of stationary and non-stationary time series forecasting. The first is
y t = Φ 1 y t−1 + Φ 2 y t−2 + · · · + Φ ℓ y t−ℓ + ǫ t , ǫ t ∼ N p (0, Σ),(4)
where Φ 1 , Φ 2 , . . . , Φ ℓ are p × p matrices of parameters. In the usual estimation of VAR, stationarity has to be assumed and so the roots of the polynomial (in z)
|I p − Φ 1 z − Φ 2 z 2 − · · · − Φ ℓ z ℓ | = 0
should lie outside the unit circle. In standard theory (4) may not assume a Gaussian distribution for ǫ t , although in practice this is used for operational simplicity. It is also known that for a high order ℓ model (4) approximates multivariate moving average models, which are typically difficult to estimate and this makes the VAR even more attractive in applications. It is also known that for general on-line estimation and forecasting, the covariance matrix Σ either has to be assumed known or it has to be diagonal. This is a major limitation, because it means that either the modeller knows a priori the cross-correlation between the series {y 1t }, {y 2t }, . . . , {y pt }, where y t = [y 1t y 2t · · · y pt ] ′ , or that the p scalar time series are all stochastically uncorrelated, in which case it is more sensible to use several univariate AR models instead. Recently, the need for estimation of Σ as a full covariance matrix (e.g. where Σ has p(p + 1)/2 elements to be estimated) is considered, but the existing estimation procedures include necessarily iterative estimation via importance sampling (Kadiyala and Karlsson, 1997). Ni and Sun (2003) point out that from a frequentist standpoint ordinary least squares and maximum likelihood estimators of (4) are unavailable. These authors state that asymptotic theory estimators may not be applicable for VAR (especially when {y t } is a short-length time series). Ni and Sun (2003), Sun and Ni (2004) and Huerta and Prado (2006) propose Bayesian estimation of the autoregressive parameters Φ i and Σ, based on MCMC. It follows that for model (1) when Σ is unknown, only iterative estimation procedures can be applied. Our proposal for on-line estimation of Σ gives a step forward to the estimation and forecasting of VAR models and it is outlined below. We propose a generalization of the univariate state space representation considered in West and Harrison (1997, §9.4.6). Other state space representations of the VAR are considered in Huerta and Prado (2006), but these representations, usually referred to as canonical representations of the VAR model (Shumway and Stoffer, 2000) are not convenient for the estimation of Σ, because Σ is embedded into the evolution equation of the states θ t . First note that we can rewrite (4) as
y t = ΦX t + ǫ t , where Φ = [Φ 1 Φ 2 · · · Φ ℓ ] and X t = [y ′ t−1 y ′ t−2 · · · y ′ t−ℓ ] ′ and so we can write y t = F ′ t θ + ǫ t = (X ′ t ⊗ I p )vec(Φ) + ǫ t ,(5)
where vec(·) denotes the column stacking operator of a portion of a matrix and ⊗ denotes the Kronecker or tensor product of two matrices. Model (5) can be seen as a regression-type time series model and it can be handled by the general Algorithm 1 for model (1) if we set G = I p , Ω = 0 and if we replace F by the time-varying F t = X t ⊗ I p . Thus we can readily apply Algorithm 1 to estimate Σ and θ or Φ 1 , Φ 2 , . . . , Φ ℓ . Moving to the time-varying vector autoregressive (TVVAR) time series, in recent years there has been a growing literature for TVVAR time series. Kitagawa and Gersch (1996), Dahlhaus (1997), Francq and Gautier (2004) and Anderson and Meerschaert (2005) study parameter estimation based on the asymptotic behaviour of TVVAR and time-varying ARMA models. From a state space standpoint West et al. (1999) propose a state space formulation for a univariate time-varying AR model applied to electroencephalographic data. In this section we extend this state space formulation to a vector of observations and hence we can propose the application of Algorithm 1 in order to estimate the covariance matrix of the error drifts of the TVVAR model.
Consider that the p-vector time series {y t } follows the TVVAR model of known order ℓ defined by y t = Φ 1t y t−1 + Φ 2t y t−2 + · · · + Φ ℓt y t−ℓ + ǫ t , ǫ t ∼ N p (0, Σ),
where Φ 1t , Φ 2t , . . . , Φ ℓt are the time-varying autoregressive parameter matrices. The model can be stationary, locally-stationary or non-stationary depending on the roots of the t polynomials (in z)
|I p − Φ 1t z − Φ 2t z 2 − · · · − Φ ℓt z ℓ | = 0.
Typical considerations include the local stationarity where there are several regimes for which, locally, {y t } is stationary, but globally {y t } is non-stationary. Also the time-dependent parameter matrices Φ it can allow for an improved dynamic fit as opposed to the static parameters of the VAR. In our development we adopt a random walk for the evolution of the parameters Φ it (i = 1, 2, . . . , ℓ), although the modeller might suggest other Markovian stochastic evolution formulae for Φ it . The random walk evolution is the natural consideration when {y t } is assumed locally stationary. Hence we can rewrite model (6) in state-space form as
y t = Φ t X t + ǫ t = F ′ t θ t + ǫ t and θ t = θ t−1 + ω t ,(7)
where
X t = [y ′ t−1 y ′ t−2 · · · y ′ t−ℓ ] ′ , F t = X t ⊗ I p , Φ t = [Φ 1t Φ 2t · · · Φ ℓt ], θ t = vec(Φ t )
and ω t ∼ N p 2 ℓ (0, Ω), for some transition covariance matrix Ω. Model (7) is reduced to (5) when Ω = 0, in which case θ t = θ t−1 = θ. After specifying Ω, we can directly apply Algorithm 1 to the state space model (7) and thus we can obtain an algorithm for the estimation of Σ, for the estimation of θ t or Φ 1t , Φ 2t , . . . , Φ ℓt and for forecasting the series {y t }.
London Metal Exchange Data
In this section we analyze London metal exchange (LME) data consisting of official spot prices (US dollars per tonne of metal). LME is the world's leading non-ferrous metals' market, trading currently highly liquid contracts for metals, such as aluminium, aluminium alloy, copper, lead, nickel, tin and zinc. According to the LME website (http://www.lme.co.uk/) "LME is highly successful with a turnover in excess of US$3,000 billion per annum. It also contributes to the UKs invisible earnings to the sum of more than £250 million in overseas earnings each year." More information about the functions of the LME can be found via its website (see above); the recently growing literature on the econometrics modelling of the LME can be found in the review of Watkins and McAleer (2004). We consider forecasting for four metals exchanged in the LME, namely aluminium, copper, lead and zinc. The data are provided from the LME website for the period of 4 January 2005 to 31 October 2005. After excluding weekends and bank holidays there are N = 210 trading days. We store the data into the 4×1 vector time series {y t } t=1,2,...,210 and y t = [y 1t y 2t y 3t y 4t ] ′ , where y 1t denotes the spot price at time t of aluminium, y 2t denotes the spot price at time t of copper, y 3t denotes the spot price at time t of lead and y 4t denotes the spot price at time t of zinc. The data are plotted in Figure 1.
We propose the VAR and TVVAR models of the previous section; the motivation of this being that from Figure 1 the evolution of the data seems to follow roughly an autoregressive type model. Indeed there is an apparent trend with no seasonality, which can be modelled with a trend model or with a VAR or TVVAR model of the previous section. Here we illustrate the proposal of VAR and TVVAR models, which, according to the previous section, can estimate the covariance matrix of y t , given the state parameters, and thus the correlation structure of {y t } can be studied. Other models for this kind of data have been applied in Triantafyllopoulos (2006) and we can envisage that the models of West and Quintana (1987) can also be applied to the LME data.
First we apply the algorithms of the previous section to several VAR and TVVAR models of different orders in order to find out which model gives the best performance. Performance here is measured via the mean vector of squared standardized one-step forecast errors (MSSE) and the mean vector of absolute percentage one-step forecast errors (MAPE). The first is chosen as a general performance measure taking into account the estimation of the covariance matrix Σ and the second is chosen as a generally reliable percent performance measure. Table 5 shows the results of 10 VAR(i) and TVVAR(i) models (first column) of order i = 1, 2, . . . , 10. The discount factor δ refers to the discounting of the evolution covariance matrix of the state parameters θ t ; δ = 1 refers to a static θ t = θ (VAR model), while δ < 1 refers to a dynamic local level evolution of θ t = θ t−1 + ω t (TVVAR model). Table 5 shows that the performance of the TVVAR is remarkable compared with the performance of VAR, which produces very high MSSE throughout the range of i. Out of the VAR models, the best is the VAR(1), which still produces very large MSSE. This indicates that a moving average (MA) model is unlikely to produce good results at all, as the MSSE of the VAR increases with the order i. Also the approximation of a MA model with a high order VAR model will include a large number of state parameters to be estimated and this will introduce computational problems.
Therefore, our attention is focused on the TVVAR models. From a computational standpoint we note that as the order increases δ can not be too low, because then there are computational difficulties in the calculation of the symmetric square root of Q t , used for the estimation of S t (the estimate of Σ). Lower values of δ work better (Triantafyllopoulos, 2006) 2. δ should not be too low, because then the covariance matrix of θ t will be too large;
3. the MSSE vector should be close to [1 1 1 1] ′ ;
4. the MAPE vector should be as low as possible.
Considering the above criteria we favor the TVVAR(2). Figure 2 shows the estimate of the observation covariance matrix Σ. From the right graph we observe that the estimate of the correlations of y 1t and y jt , given θ t are very high (close to 1) and this means that in forecasting; this provides useful information about the cross-dependence of the four metal prices over time. As mentioned before two competitive models to our TVVAR modelling for the LME data are the matrix-variate DLMs (MV-DLMs) of Quintana and West (1987) and the discount weighted regression (DWR) of Triantafyllopoulos (2006). Next we compare the TVVAR(2) model discussed above with these two modelling approaches. We start by briefly describing the MV-DLM and the DWR. The MV-DLM is defined by
y ′ t = F ′ Θ t + ǫ ′ t and Θ t = GΘ t−1 + ω t ,(8)
where F is a d × 1 design vector, Θ t is a d × p state matrix, G is a d × d transition matrix, ǫ t |Σ ∼ N p (0, Σ) and vec(ω t )|Σ, Ω ∼ N dp (0, Σ ⊗ Ω), where vec(·) denotes the column stacking operator of a lower portion of a matrix and ⊗ denotes the Kronecker product of two matrices. A prior inverted Wishart distribution is assumed for Σ and the resulting posterior distributions as well as further details on the model can be found in Quintana and West (1987) and West and Harrison (1997, Chapter 16) (for more references on this model, see also the Introduction).
In the application of MV-DLMs it is necessary to specify F and G. Following Quintana and West (1987), who consider international exchange rates data, and by consulting the plots of Figure 1 we propose a linear trend model for the LME data. Thus we can set F = 1 0 and G = 1 1 0 1 .
The DWR is defined by
y t = y t−1 + ψ t + ǫ t and ψ t = ψ t−1 + ζ t ,
with ǫ t |Σ ∼ N p (0, Σ) and ζ t ∼ N p (0, Ω t ). This model can be put into state space form as in
y t = [y t−1 I p ] 1 ψ t + ǫ t = F ′ t θ t + ǫ t , θ t = 1 ψ t = 1 ψ t−1 + 0 ζ t = θ t−1 + ω t .
The covariance matrix Ω t is modelled with a discount factor δ and Σ is estimated following Triantafyllopoulos and Pikoulas (2002) and Triantafyllopoulos (2006). Table 6 shows the MSSE and the MAPE of the three models. We see that all models produce reasonable results. For the MSSE the best model is the TVVAR(2) (with the exception of the lead variable where the MV-DLM produces MSSE closer to 1). For the MAPE the best model is the DWR with the TVVAR(2) producing the highest MAPE. Out of the three models, the MV-DLM is limited by its mathematical form, which is constructed to give conjugate analysis (see also the Introduction). The DWR suffers from similar limitations as the MV-DLM, but it provides good results, for linear trend time series without seasonality. The TVVAR model provides a good modelling alternative and considering the numerous applications of VAR time series models in econometrics, it is believed that the TVVAR has a great potential.
In conclusion, the TVVAR model can produce forecasts with good forecast accuracy, while the correlation of the series can be estimated on-line with a fast linear algorithm. A criticism of the model is that its efficiency depends on its order and if high order TVVAR models are required (e.g. as in approximating moving average processes with time-dependent parameters) its efficiency will be similar of that of a vector MA, since the discount factor will have to be close to 1. It will be interesting to know how the order of the TVVAR model is related to the boundness of the eigenvalues of the covariance estimator S t .
Concluding Comments
This paper develops an algorithm for covariance estimation in multivariate conditionally Gaussian dynamic linear models, assuming that the observation covariance matrix is fixed, but unknown. This is a general estimation procedure, which can be applied to any Gaussian linear state space model. The algorithm is empirically found to have good performance providing a covariance estimator which converges to the true value of the observation covariance matrix. The proposed methodology compares well with a non-sequential state of the art MCMC estimation procedure and it is found that the proposed estimates are close to the estimates of the MCMC. The new algorithm is applied (but not limited to) model subclasses of VAR and VAR with time-dependent parameters (TVVAR), which have great application in financial time series. Considering the London metal exchange data, it is found that the TVVAR model has outstanding performance as opposed to the VAR model. It is believed that the development of the TVVAR model is a worthwhile project and the proposed fast, on-line algorithm for the estimation of the observation covariance matrix, is a step forward opening several paths for practical forecasting.
The focus in this paper is on facilitating and advancing non-iterative covariance estimation procedures for vector time series. Such procedures are particularly appealing, because of their simplicity and ease in use. For such wide class of models such us the conditionally Gaussian dynamic linear models, the proposed on-line algorithm enables the computation of the mean vector of standardized errors as well as it enables the computation of the multi-step forecast covariance matrix. Both these computations are valuable considerations in forecasting and they attract interest by academics and practitioners alike.
where A t = n −1/2 t S 1/2 t−1 Q −1/2 t .
Conditional on Σ, we have from an application of the Kalman filter that Cov(e it e jt , e kt e ℓt |Σ, y t−1 ) is bounded, where e t = [e 1t e 2t · · · e pt ] ′ . Since Σ is bounded, S t is also bounded (lim t→∞ S t = Σ), and so all the covariances of e it e jt and e kt e ℓt unconditional on Σ are also bounded. This means that all the elements of Var{vech(e t e ′ t )} are bounded and so Var{vech(e t e ′ t )} is bounded. Now let
X 1 = E(Σ − A t e t e ′ t A ′ t |y t , y t−1 ) = S t − A t e t e ′ t A t and X 2 = E(Σ − A t e t e ′ t A ′ t |y t−1 ) = S t−1 − A t Q t A ′ t .
Then, since lim t→∞ S t = Σ, there exists appropriately a large integer t(L) > 0 such that for every t > t(L) it is E(X 1 − X 2 |y t−1 ) ≈ 0. Also
Var{vech(X 1 − X 2 )|y t−1 } = Var{(A t ⊗ A t )D p vech(e t e ′ t )|y t−1 } = 1 n 2 t E t → 0, with E t = S 1/2 t−1 Q −1/2 t ⊗ S 1/2 t−1 Q −1/2 t D p [Var{vech(e t e ′ t )|y t−1 }]D ′ p Q −1/2 t S 1/2 t−1 ⊗ Q −1/2 t S 1/2 t−1 ,
where D p is the duplication matrix and from the first part of the proof we have that E t is bounded. It follows that for any t > t(L) it is X 1 ≈ X 2 with probability 1 and so we have and by dividing by n t = n 0 + t = n t−1 + 1 we obtain equation (2) as required.
The Gibbs Sampler for Multivariate Conditionally Gaussian DLMs
The following procedure applies to any conditionally Gaussian dynamic linear model in the form of equation (1). For the simulation studies considered in this paper, given data y N = (y 1 , y 2 , . . . , y N ), we are interested in sampling a set of state vectors, θ 1 , θ 2 . . . , θ N and the observation covariance matrix Σ from the full, multivariate posterior distribution of θ 1 , θ 2 , . . . , θ N , Σ|y N . Gibbs sampling involves iterative sampling from the full conditional posterior of each θ t , |θ −t , Σ, y N , for all t = 1, 2, . . . , N , and Σ|θ 1 , θ 2 , . . . , θ N , y N ; in our notation, θ −t means that we are conditioning upon all the components θ 1 , θ 2 , . . . , θ N but θ t . Given the conditionally normal and linear structure of the system, such full conditional distributions are standard, and therefore easily sampled. However, such an implementation of the Gibbs sampler, where each component is updated once at a time, could be very inefficient when applied to the multivariate DLMs discussed in this paper; in fact, the high-correlation of the dynamic system will most likely bring convergence problems. In order to overcome such difficulties, following the early suggestions of Carter and Kohn (1994) and Frühwirth-Schnatter (1994), we have chosen to implement a blocked Gibbs sampler Gamerman (1997, p. 149); within this context, this sampling scheme is better known as the forward filtering, backward sampling algorithm. Following is a concise description of the algorithm used in our studies; for more details, the reader should consult the references above, as well as West and Harrison (1997, Chapter 15).
The first step of the Gibbs sampler involves sampling from the updating distribution of θ N |Σ, y N , which is given by the multivariate normal N d (m M N , P M N ). This is done in the forward filtering phase of the sampler, as follows. Starting at time t = 0 with some given initial values m M 0 , P M 0 and Σ we compute the following quantities at each time t, for t = 1, 2, . . . , N :
(a) the prior mean vector and covariance matrix of θ t |Σ, y t−1 , a t = Gm M t−1 and R M t = GP M t−1 G ′ + Ω.
(b) the mean vector and covariance matrix of the one-step ahead forecast of y t |y t−1 ,
y M t−1 (1) = F ′ a t and Q M t = F ′ R M t F + Σ.
(c) the posterior mean vector and covariance matrix of θ t |y t ,
m M t = a t + A M t e M Σ = N −1 N t=1 ǫ * t (ǫ * t ) ′ .
Finally, with n M 0 being the prior degrees of freedom and S M 0 being the prior estimate of Σ, we sample from the full conditional density of Σ|Θ, y N , which is an inverted Wishart distribution IW p (n M 0 + N + 2p, N Σ N + n M 0 S M 0 ), whose simulation is also standard. This concludes an iteration of the Gibbs sampler.
respectively the symmetric square roots of the matrices S i−1 , Q −1 i based on the spectral decomposition factorization of symmetric positive definite matrices (i = 1, 2, . . . , t).
Algorithm 1 .
1(a) Prior distribution at time t = 0: θ 0 |Σ = S 0 ∼ N d ( m 0 , P 0 ), for some m 0 , P 0 and S 0 .
. . . , 1000). The diagonal choice for the priors S(j) i,0 has been done for: (a) operational simplicity (the user is likely to expect rough values for the diagonal elements of Σ i , rather than for the associated correlations) and (b) judging how the estimation of Σ i is affected by improper priors in the sense of setting the off-diagonal elements of S (j) i,0 to zero, while the true values of Σ i posses high correlations. Throughout the models the remaining settings are n 0 = 1, Ω = I 2 , m 0 = [0 0] ′ and P 0 = 1000I 2 , for all models.
Figure 1 :
1LME data, consisting of aluminium, copper, lead and zinc spot prices (in US dollars per tonne of each metal).
Figure 2 :
2Estimate of the observation covariance matrix Σ = {σ ij } i,j=1,2,3,4 . The left graph shows the estimates of the variances σ ii ; the solid line shows the estimate of σ 11 , the dashed line shows the estimate of σ 22 , the dotted line shows the estimate of σ 33 , the dashed/dotted line shows the estimate of σ 44 . The right graph shows the estimate of the correlations of y 1t |θ t with y jt |θ t (j = 2, 3, 4); the solid line shows the estimate of the correlation of y 1t |θ t with y 2t |θ t , the dashed line shows the estimate of the correlation of y 1t |θ t with y 3t |θ t and the dotted line shows the estimate of the correlation of y 1t |θ t with y 4t |θ t .
proved equation (A-1). Using E(Σ|y t ) = S t , from equation (A-1) we haveE(Σ|y t ) − A t e t e ′ t A ′ t = E(Σ|y t−1 ) − A t E(e t e ′ t |y t−1
, Chapter 6; Durbin and Koopman, 2001, §3.2.2).
Table 1 :
1Performance of the estimator S t (of Algorithm 1) for 1000 simulated bivariate dynamic models generated from a local level model (LL), a linear trend model (LT) and a seasonal model (SE) under three observation covariance matrices Σ 1 , Σ 2 and Σ 3 .Model
LL
LT
SE
Σ = Σ i
S
S 100
S 500
S
S 100
S 500
S
S 100
S 500
σ 11 = 2
1.945
1.938
1.997
2.572
2.721
2.392
2.171
2.207
2.165
σ 12 = 3
2.798
2.770
2.920
2.988
3.029
3.026
2.314
2.186
2.589
σ 22 = 5
4.722
4.685
4.899
4.547
4.489
4.777
4.399
4.283
4.694
ρ = 0.948
0.923
0.919
0.933
0.874
0.867
0.895
0.748
0.711
0.812
σ 11 = 100 100.039 99.931 100.303 98.271 98.157 98.368 98.731 98.806 99.602
σ 12 = 85
83.133 83.028 84.757
79.471 78.969 82.660 79.627 79.427 83.254
σ 22 = 80
80.430 80.277 80.353
78.917 78.865 79.822 79.755 79.886 79.896
ρ = 0.950
0.927
0.927
0.944
0.902
0.897
0.933
0.897
0.894
0.933
σ 11 = 1
1.124
1.135
1.101
1.200
1.234
1.126
1.184
1.202
1.151
σ 12 = 7
6.506
6.457
6.735
5.388
5.177
5.904
5.764
5.623
6.234
σ 22 = 50
49.305 49.375 49.840
48.518 48.579 49.392 48.816 49.038 49.540
ρ = 0.989
0.784
0.862
0.909
0.706
0.668
0.791
0.758
0.732
0.825
Table 3 :
3Bivariate simulated local level dynamic linear model. Showed are: real versus estimated values of Σ = {σ ij } i,j=1,2 and the correlation coefficient ρ. The first column indicates how many observations were used in each estimation.approximation. In this section we compare the new method with the gold standard in order to show how good is the approximation.σ 11
σ 12
σ 22
ρ
N
Real MCMC New Real MCMC New Real MCMC New Real MCMC New
100 2.00
1.68
1.24 3.00
2.35
1.82 5.00
4.12
3.69 0.95
0.90
0.85
150 2.00
1.78
1.34 3.00
2.44
1.91 5.00
3.97
3.72 0.95
0.92
0.86
200 2.00
1.76
1.46 3.00
2.48
2.04 5.00
4.02
3.77 0.95
0.94
0.87
250 2.00
2.04
1.64 3.00
2.89
2.33 5.00
4.50
4.17 0.95
0.95
0.89
300 2.00
1.92
1.65 3.00
2.78
2.39 5.00
4.44
4.31 0.95
0.95
0.90
350 2.00
1.90
1.69 3.00
2.76
2.46 5.00
4.46
4.40 0.95
0.95
0.90
400 2.00
2.01
1.76 3.00
2.90
2.56 5.00
4.59
4.50 0.95
0.96
0.91
450 2.00
2.05
1.82 3.00
2.97
2.66 5.00
4.71
4.65 0.95
0.96
0.92
500 2.00
2.11
1.85 3.00
3.09
2.74 5.00
4.90
4.79 0.95
0.96
0.92
Table 4 :
4Bivariate simulated local level dynamic linear model. Showed are: one-step forecast errors at time t = N and the squared sums of the forecasting errors up to time N . the VAR model of known order ℓ ≥ 1, defined bye 1N
e 2N
N −1 N
t e 2
1t
N −1 N
t e 2
2t
N
MCMC New
MCMC New
MCMC New MCMC New
100
−2.14 −2.22
−2.46 −2.49
4.85
4.84
8.54
8.59
150
−0.14 −0.28
−3.88 −3.92
4.83
4.86
8.21
8.26
200
−1.02 −1.09
−0.12
0.05
5.02
5.04
8.10
8.13
250
−0.24 −0.20
−1.45 −1.47
5.25
5.30
8.72
8.76
300
−1.72 −1.79
−0.61 −0.71
5.01
5.12
8.91
8.95
350
−0.42 −0.46
−1.91 −1.91
5.12
5.12
9.01
9.04
400
−0.42 −0.54
−2.26 −2.29
5.21
5.22
9.09
9.12
450
−3.91 −4.02
−5.42 −5.41
5.28
5.29
9.32
9.33
500
1.48
1.68
−0.93 −0.98
5.24
5.24
9.54
9.54
Table 5 :
5Mean vector of squared standardized one-step forecast errors (MSSE) and mean vector of absolute percentage one-step forecast errors (MAPE) of the multivariate LME time series {y t }. The first column indicates several VAR and TVVAR models.MSSE
MAPE
VAR(1)
6.614 16.782 7.655 18.370
0.033 0.071 0.143 0.057
TVVAR(1): δ = 0.1
2.430
1.764
0.622
1.852
0.059 0.053 0.084 0.076
VAR(2)
19.610 15.966 11.934 10.271
0.081 0.226 0.201 0.101
TVVAR(2): δ = 0.35
1.296
1.743
1.228
1.822
0.065 0.057 0.116 0.095
VAR(3)
11.777 23.715 9.906
9.058
0.585 0.345 0.480 0.246
TVVAR(3): δ = 0.65
2.254
3.149
2.222
2.180
0.074 0.053 0.132 0.108
VAR(4)
39.979 54.892 28.407 19.169
0.159 0.103 0.235 0.161
TVVAR(4): δ = 0.6
1.389
1.802
1.210
1.329
0.101 0.072 0.179 0.147
VAR(5)
18.592 16.605 15.474 12.570
0.203 0.076 0.392 0.248
TVVAR(5): δ = 0.7
1.429
2.269
1.651
1.677
0.114 0.079 0.208 0.171
VAR(6)
24.910 19.085 14.584 17.784
0.206 0.134 0.320 0.197
TVVAR(6): δ = 0.75
1.828
2.705
1.683
1.757
0.132 0.089 0.243 0.197
VAR(7)
21.722 38.054 14.597 14.180
0.330 0.092 0.422 0.490
TVVAR(7): δ = 0.75
1.366
2.044
1.191
1.531
0.148 0.101 0.280 0.223
VAR(8)
28.985 35.867 11.291 16.370
0.515 0.325 0.812 0.563
TVVAR(8): δ = 0.8
2.130
2.900
1.637
1.910
0.168 0.111 0.326 0.249
VAR(9)
40.229 53.798 12.249 19.691
0.393 0.184 0.416 0.411
TVVAR(9): δ = 0.95 14.042 21.011 6.724
8.708
0.207 0.124 0.352 0.284
VAR(10)
46.791 49.869 16.240 23.974
0.611 0.306 0.751 0.694
TVVAR(10): δ = 0.9
4.273
7.541
3.637
5.629
0.205 0.124 0.391 0.296
Table 6 :
6Mean vector of squared standardized one-step forecast errors (MSSE) and mean vector of absolute percentage one-step forecast errors (MAPE) for the LME data and for three multivariate models: TVVAR, MV-DLM and DWR.MSSE
MAPE
TVVAR(2)
1.296 1.743 1.228 1.822
0.065 0.057 0.116 0.095
MV-DLM
1.306 2.436 0.984 1.887
0.019 0.022 0.026 0.025
DWR
2.202 1.610 1.590 1.868
0.013 0.015 0.017 0.017
t and P M t = R M t − A M t Q M t (A M t ) ′ , where A M t = R M t F (Q M t ) −1is the Kalman gain and e M t = y t − y M t−1 (1) is the one-step ahead forecast error vector.An updated vector θ N is thus obtained, and the filtering part of the algorithm is completed. The backwards sampling phase involves sampling from the distribution of θ t |θ t+1 , Σ, y t at all times t = N −1, . . . , 1, 0. Each of such vectors is drawn from a multivariate normalN d (h t , H t ), where h t = m M t + P M t G ′ (R M t+1 ) −1 (θ t+1 − a t+1 ) and H t = P M t {I d − G(R M t+1 ) −1 GP M t ),with I d being the d × d identity matrix. At each time t, we also compute ǫ * t = y t − F ′ θ t . Once the backwards sampling phase is completed, we set
AcknowledgementsI should like to thank M. Aitkin and A. O'Hagan for helpful discussions and suggestions on earlier drafts of the paper. Special thanks are due to G. Montana who helped on the computational part of the paper, in particular regarding the MCMC design and implementation.AppendixProof of Theorem 1Let vech(·) denote the column stacking operator of a lower portion of a symmetric square matrix and let ⊗ denote the Kronecker product of two matrices. First we prove that for large t, it is approximately
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"Finite dimensional simple modules of deformed current Lie algebras",
"Finite dimensional simple modules of deformed current Lie algebras"
]
| [
"Kentaro Wada "
]
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| []
| The deformed current Lie algebra was introduced in [W] to study the representation theory of cyclotomic q-Schur algebras at q = 1. In this paper, we classify finite dimensional simple modules of deformed current Lie algebras. | 10.1016/j.jalgebra.2018.01.006 | [
"https://arxiv.org/pdf/1704.07973v1.pdf"
]
| 55,559,574 | 1704.07973 | 7475f2ee8fb7e6ebfe3092f7d4056dd2c42b6cc2 |
Finite dimensional simple modules of deformed current Lie algebras
26 Apr 2017
Kentaro Wada
Finite dimensional simple modules of deformed current Lie algebras
26 Apr 2017arXiv:1704.07973v1 [math.RT]
The deformed current Lie algebra was introduced in [W] to study the representation theory of cyclotomic q-Schur algebras at q = 1. In this paper, we classify finite dimensional simple modules of deformed current Lie algebras.
§ 0. Introduction 0.1. The deformed current Lie algebra g Q (m) was introduced in [W] to study the representation theory of cyclotomic q-Schur algebras at q = 1. In this paper, we introduce the deformed current Lie algebra sl Q m [x] and gl Q m [x] over C associated with the special linear Lie algebra sl m and general linear Lie algebra gl m respectively. sl Q m [x] (resp. gl Q m [x]) is a deformation of the current Lie algebra sl m [x] = sl m ⊗ C C[x] (resp. gl m [x] = gl m ⊗ C C[x]) with deformation parameters Q = (Q 1 , Q 2 , . . . , Q m−1 ) ∈ C m−1 . Note that sl Q m [x] (resp. gl Q m [x]) is coincide with sl m [x] (resp. gl m [x]) if Q i = 0 for all i = 1, 2, . . . , m − 1. The Lie algebra g Q (m) introduced in [
| β ∈ m−1 i=1 B Q i }, where B Q i = {0} if Q i = 0, C if Q i = 0,
although the 1-dimensional representation of sl m [x] is only the trivial representation (Lemma 6.2). (We remark that L (0,...,0) is the trivial representation of sl Q m [x].) The second difference appears in the evaluation modules. For each γ ∈ C, we can consider the evaluation homomorphism ev γ : U(sl Q m [x]) → U(sl m ) which is a deformation of the evaluation homomorphism for sl m [x] (see the paragraph 1.5 for the definition). Then we can consider the evaluation modules by regarding U(sl m )modules as U(sl Q m [x])-modules through the evaluation homomorphism ev γ . The evaluation homomorphism ev γ is surjective if γ = Q −1 i for all i = 1, 2, . . . , m−1 such that Q i = 0. However, ev γ is not surjective if γ = Q −1 i for some i = 1, 2, . . . , m − 1. Moreover, in general, the evaluation module of a simple U(sl m )-module at γ ∈ C is not simple if γ = Q −1 i for some i = 1, 2, . . . , m − 1 (see Remark 5.10).
0.3. It is a purpose of this paper to classify the finite dimensional simple modules of sl Q m [x] and gl Q m [x]. A classification of the finite dimensional simple modules for the original current Lie algebra is well-known (e.g. [C], [CP]). The classification for sl Q m [x] (resp. gl Q m [x]) is an analogue of the original case. Since sl Q m [x] has the triangular decomposition (Proposition 1.4), we can develop the usual highest weight theory (see §2). In particular, any finite dimensional simple U(sl Q m [x])-module is isomorphic to a highest weight module L(u) of highest weight u ∈ m−1 i=1 t≥0 C (Proposition 2.6). Then it is enough to determine the highest weights such that the corresponding simple highest weight modules are finite dimensional. We obtain a classification of such highest weights as follows. Let C[x] monic be the set of monic polynomials over C with the indeterminate variable x. For each Q ∈ C, put
C[x] Q monic = C[x] monic if Q = 0, {ϕ ∈ C[x] monic | Q −1 is not a root of ϕ} if Q = 0. We define the map m−1 i=1 (C[x] Q i monic × B Q i ) → m−1 i=1 t≥0 C,
(ϕ, β) = ((ϕ i , β i )) 1≤i≤m−1 → u Q (ϕ, β) = (u Q (ϕ, β) i,t ) 1≤i≤m−1,t≥0 , by u Q (ϕ, β) i,t = γ t i,1 + γ t i,2 + · · · + γ t i,n i if Q i = 0, γ t i,1 + γ t i,2 + · · · + γ t i,n i + Q −t i β i if Q i = 0 (0.3.1)
when ϕ i = (x − γ i,1 )(x − γ i,2 ) . . . (x − γ i,n i ) (1 ≤ i ≤ m − 1)
. Then we have the following classification of finite dimensional simple U(sl Q m [x])-modules (Theorem 6.4).
Theorem: {L(u Q (ϕ, β)) | (ϕ, β) ∈ m−1 i=1 (C[x] Q i monic × B Q i )} gives a complete set of isomorphism classes of finite dimensional simple U(sl Q m [x])modules.
We remark that L(u Q (ϕ, β)) is isomorphic to a subquotient of
m−1 j=1 n j k=1 L(ω j ) ev γ j ,k ⊗ L β ,
where {ω j | 1 ≤ j ≤ m − 1} is the set of fundamental weights for sl m , L(ω j ) (1 ≤ j ≤ m − 1) is the simple highest weight U(sl m )-module of highest weight ω j and L(ω j ) ev γ j ,k is the evaluation module of L(ω j ) at γ j,k .
We also see that any finite dimensional simple U(gl Q m [x])-module is isomorphic to a highest weight module L( u) of highest weight u ∈ m j=1 t≥0 C (Proposition 3.3). Note that sl Q m [x] is a Lie subalgebra of gl Q m [x] (Proposition 1.4 (iii)). The difference of representations of gl Q m [x] from one of sl Q m [x] is given by the family
of 1-dimensional U(gl Q m [x])-modules { L h | h ∈ t≥0 C}.
We remark that L h (h ∈ t≥0 C) is isomorphic to the trivial representation L (0,...,0) as a U(sl Q m [x])module when we restrict the action. We obtain the classification of finite dimensional simple U(gl Q m [x])-modules as follows. We define the map m−1 i=1 (C[x] Q i
monic × B Q i ) × t≥0 C → m j=1 t≥0 C, (ϕ, β, h) = ((ϕ i , β i ) 1≤i≤m−1 , (h t ) t≥0 ) → u Q (ϕ, β, h) = ( u Q (ϕ, β, h) j,t ) 1≤j≤m,t≥0 by u Q (ϕ, β, h) j,t = m−1 k=j u Q (ϕ, β) k,t + h t if 1 ≤ j ≤ m − 1 and t ≥ 0, h t if j = m and t ≥ 0,
where u Q (ϕ, β) k,t is determined by (0.3.1). Then we have the following classification of finite dimensional simple U(gl Q m [x])-modules (Theorem 7.4). Theorem: {L( u Q (ϕ, β, h)) | (ϕ, β, h) ∈ m−1 i=1 (C[x] Q i monic × B Q i ) × t≥0 C} gives a complete set of isomorphism classes of finite dimensional simple U(gl Q m [x])-modules. We remark that L( u Q (ϕ, β, h)) is isomorphic to a subquotient of m−1 j=1 n j k=1 L( ω j ) ev γ j ,k ⊗ L β ⊗ L h .
(See §7 for definitions of L( ω j ) ev γ j ,k , L β and L h .) We also remark that x] in this section is different from one of g Q (m) given in [W]. The relation between gl Q m [x] and g Q (m) is given in Lemma 1.7.
L( u Q (ϕ, β, h)) ∼ = L(u Q (ϕ, β)), L( ω j ) ev γ j ,k ∼ = L(ω j ) ev γ j ,k , L β ∼ = L β and L h ∼ = L (0,...,0) as U(sl Q m [x])-Definition 1.1. Put Q = (Q 1 , Q 2 , . . . , Q m−1 ) ∈ C m−1 . We define the Lie algebra sl Q m [x]
over C by the following generators and defining relations:
Generators: X ± i,t , J i,t (1 ≤ i ≤ m − 1, t ≥ 0). Relations: [J i,s , J j,t ] = 0, (L1) [J j,s , X ± i,t ] = ±a ji X ± i,s+t ,(L2)[X + i,t , X − j,s ] = δ ij (J i,s+t − Q i J i,s+t+1 ), (L3) [X ± i,t , X ± j,s ] = 0 if j = i ± 1,(L4)[X + i,t+1 , X + i±1,s ] = [X + i,t , X + i±1,s+1 ], [X − i,t+1 , X − i±1,s ] = [X − i,t , X − i±1,s+1 ],(L5)[X + i,s , [X + i,t , X + i±1,u ]] = [X − i,s , [X − i,t , X − i±1,u ]] = 0, (L6) where we put a ji = 2 if j = i, −1 if j = i ± 1, 0 otherwise.
We also define the Lie algebra gl Q m [x] over C by the following generators and defining relations:
Generators: X ± i,t (1 ≤ i ≤ m − 1, t ≥ 0), I j,t (1 ≤ j ≤ m, t ≥ 0).
Relations:
[I i,s , I j,t ] = 0, (L'1)
[I j,s , X ± i,t ] = ±a ′ ji X ± i,s+t , (L'2) [X + i,t , X − j,s ] = δ ij (J i,s+t − Q i J i,s+t+1 ), where we put J i,t = I i,t − I i+1,t , (L'3)
together with the relations (L4)-(L6) in the above. In the relation (L'2), we
put a ′ ji = 1 if j = i, −1 if j = i + 1, 0 otherwise. 1.2. We call sl Q m [x] (resp. gl Q m [x]
) the deformed current Lie algebra associated with the special linear Lie algebra sl m (resp. the general linear Lie algebra gl m ). If
Q i = 0 for all i = 1, 2, . . . , m − 1, then sl Q m [x] (resp. gl Q m [x]) coincides with the current Lie algebra sl m [x] = sl m ⊗ C C[x] (resp. gl m [x] = gl m ⊗ C C[x]) associated with sl m (resp. gl m ). We can also regard sl Q m [x] (resp. gl Q m [x]) as a filtered deformation of sl m [x] (resp. gl m [x]
) in a similar way as in [W,Proposition 2.13].
1.3. For 1 ≤ i = j ≤ m and t ≥ 0, we define an element E i,j;t ∈ sl Q m [x] (resp. E i,j;t ∈ gl Q n [x]) by E i,j;t = [X + i,0 , [X + i+1,0 , . . . , [X + j−2,0 , X + j−1,t ] . . . ]] if j > i, [X − i−1,0 , [X − i−2,0 , . . . , [X − j+1,0 , X − j,t ] . . . ]] if j < i. In particular, we have E i,i+1;t = X + i,t and E i+1,i;t = X − i,t . Let n + and n − be the Lie subalgebra of sl Q m [x] (also of gl Q m [x]) generated by {X + i,t | 1 ≤ i ≤ m − 1, t ≥ 0} and {X − i,t | 1 ≤ i ≤ m − 1, t ≥ 0}
respectively. Let n 0 (resp. n 0 ) be the Lie subalgebra of sl Q m [x] (resp. gl Q m [x]) generated by
{J i,t | 1 ≤ i ≤ m − 1, t ≥ 0} (resp. {I i,t | 1 ≤ i ≤ m, t ≥ 0}).
By the relation (L1) (resp. (L'1)), we see that n 0 (resp. n 0 ) is a commutative Lie
subalgebra of sl Q m [x] (resp gl Q m [x]).
Proposition 1.4.
(i) {E i,j;t | 1 ≤ i = j ≤ m, t ≥ 0} ∪ {J i,t | 1 ≤ i ≤ m − 1, t ≥ 0} gives a basis of sl Q m [x]. (ii) {E i,j;t | 1 ≤ i = j ≤ m, t ≥ 0} ∪ {I j,t | 1 ≤ j ≤ m, t ≥ 0} gives a basis of gl Q m [x]. (iii)
There exists an injective homomorphism of Lie algebras
Υ : sl Q m [x] → gl Q m [x] such that X ± i,t → X ± i,t , and J i,t → I i,t − I i+1,t .
(iv) We have the triangular decomposition sl Q m [x] = n − ⊕ n 0 ⊕ n + and gl Q m [x] = n − ⊕ n 0 ⊕ n − (as vector spaces).
In particular,
{E i,j;t | 1 ≤ i < j ≤ m, t ≥ 0} (resp. {E i,j;t | 1 ≤ j < i ≤ m, t ≥ 0})
gives a basis of n + (resp. n − ), and
{J i,t | 1 ≤ i ≤ m − 1, t ≥ 0} (resp. {I j,t | 1 ≤ j ≤ m, t ≥ 0})
gives a basis of n 0 (resp. n 0 ).
Proof. (i) and (ii) are proven in a similar way as in the proof of [W,Proposition 2.6]. By checking the defining relations, we see that Υ is well-defined. We also see that Υ is injective by investigating the basis given in (i) and (ii) under the homomorphism Υ. Then we have (iii). (iv) folloes from (i) and (ii).
1.5. Evaluation homomorphisms and evaluation modules. The general linear Lie algebra gl m is a Lie algebra over C generated by e i , f i (1 ≤ i ≤ m − 1) and K j (1 ≤ j ≤ m) together with the following defining relations:
[K i , K j ] = 0, [K j , e i ] = a ′ ji e i , [K j , f i ] = −a ′ ji f i , [e i , f j ] = δ ij H i , where H i = K i − K i+1 , [e i , e j ] = [f i , f j ] = 0 if j = i ± 1, [e i , [e i , e i±1 ]] = [f i , [f i , f i±1 ]] = 0.
The special linear Lie algebra sl m is a Lie subalgebra of gl m generated by
e i , f i , H i (1 ≤ i ≤ m − 1).
For each γ ∈ C, by checking the defining relations, we have the homomorphisms of algebras (evaluation homomorphism)
ev γ : U(sl Q m [x]) → U(sl m ) by X + i,t → (1 − Q i γ)γ t e i , X − i,t → γ t f i , J i,t → γ t H i and ev γ : U(gl Q m [x]) → U(gl m ) by X + i,t → (1 − Q i γ)γ t e i , X − i,t → γ t f i , I j,t → γ t K j . Clearly, the homomorphism ev γ (resp. ev γ ) is surjective if γ = Q −1 i for all i = 1, . . . , m − 1 such that Q i = 0. For a U(sl m )-module M (resp. a U(gl m )-module M), we can regard M as a U(sl Q m [x])-module (resp. a U(gl Q m [x]
)-module) through the evaluation homomorphism ev γ (resp. ev γ ). We call it the evaluation module, and denote it by M evγ (resp. M evγ ). For (i, k) ∈ Γ (m) and j ∈ Z such that 1 ≤ ζ((i, k)) + j ≤ m, put (i + j, k) = ζ −1 (ζ((i, k)) + j). For (i, k) ∈ Γ ′ (m) and (j, l) ∈ Γ (m), put a ′ (j,l)(i,k) = a ′ ζ((j,l))ζ((i,k)) . Take Q = ( Q 1 , . . . , Q r−1 ) ∈ C r−1 . Then the Lie algebra g Q (m) in [W,Definition 2.2] is defined by the generators X ± (i,k),t , I (j,l),t ((i, k) ∈ Γ ′ (m), (j, l) ∈ Γ (m), t ≥ 0) together with the following defining relations:
[I (i,k),s , I (j,l),t ] = 0, [I (j,l),s , X ± (i,k),t ] = ±a ′ (j,l)(i,k) X ± (i,k),s+t , [X + (i,k),t , X − (j,l),s ] = δ (i,k)(j,l) J (i,k),s+t if i = m k , − Q k J (m k ,k),s+t + J (m k ,k),s+t+1 if i = m k , [X ± (i,k),t , X ± (j,l),s ] = 0 if (j, l) = (i ± 1, k), [X + (i,k),t+1 , X + (i±1,k),s ] = [X + (i,k),t , X + (i±1,k),s+1 ], [X − (i,k),t+1 , X − (i±1,k),s ] = [X − (i,k),t , X − (i±1,k),s+1 ], [X + (i,k),s , [X + (i,k),t , X + (i±1,k),u ]] = [X − (i,k),s , [X − (i,k),t , X − (i±1,k),u ]] = 0,
where we put J (i,k),t = I (i,k),t − I (i+1,k),t . Then we have the following isomorphism between gl Q m [x] and g Q (m) under the suitable choice of the deformation parameters Q.
Lemma 1.7. Assume that Q i = 0 for all i = 1, 2, . . . , r − 1. We take Q = (Q 1 , Q 2 , . . . , Q m−1 ) ∈ C m−1 as
Q i = Q −1 k if ζ −1 (i) = (m k , k) for some k = 1, 2, . . . , r − 1, 0
otherwise.
Then we have the isomorphism of Lie algebras
Φ : gl Q m [x] → g Q (m) such taht X + i,t → X + ζ −1 (i),t if ζ −1 (i) = (m k , k) for all k = 1, . . . , r − 1, − Q −1 k X + ζ −1 (i),t if ζ −1 (i) = (m k , k) for some k = 1, . . . , r − 1, X − i,t → X − ζ −1 (i),t , I j,t → I ζ −1 (j),t .
Proof. We see the well-definedness of Φ by checking the defining relations. The inverse homomorphism of Φ is given by k)),t , I (j,l),t → I ζ((j,l)),t . § 2. Representations of sl Q m [x] In this section, we give some fundamental results for finite dimensional U(sl Q m [x])modules by using the standard argument.
X + (i,k),t → X + ζ((i,k)),t if i = m k , − Q k X + ζ((i,k)),t if i = m k , X − (i,k),t → X − ζ((i,2.1. Put h = m−1 i=1 CJ i,0 ⊂ sl Q m [x], then h is a commutative Lie subalgebra of sl Q m [x]. (Note that, if Q i = 0 for all i = 1, . . . , m − 1, h is a Cartan subalgebra of sl m .) Let h * be the dual space of h. For each i = 1, 2, . . . , m − 1, we take α i ∈ h * as α i (J j,0 ) = a ji for j = 1, . . . , m − 1. Put Q + = m−1 i=1 Z ≥0 α i ⊂ h * . We define the partial order on h * by λ ≥ µ if λ − µ ∈ Q + for λ, µ ∈ h. 2.2. For U(sl Q m [x])-mdoule M, we consider the decomposition M = λ∈h * M λ , where M λ = {x ∈ M | (h − λ(h)) N · x = 0 for h ∈ h and N ≫ 0}, namely M =
λ∈h * M λ is the decomposition to the generalized simultaneous eigenspaces for the action of h. By the relation (L2), we have
X ± i,t · M λ ⊂ M λ±α i (1 ≤ i ≤ m − 1, t ≥ 0). Thus, if U(sl Q m [x])-module M = 0 is finite dimensional, there exists λ ∈ h * such that M λ = 0 and X + i,t · M λ = 0 for all i = 1, 2, .
. . , m − 1 and t ≥ 0. On the other hand, M λ (λ ∈ h * ) is closed under the action of n 0 by the relation (L1). Thus, we can take a simultaneous eigenvector v ∈ M λ for the action of n 0 . Then we have the following lemma.
(i) X + i,t · v 0 = 0 for all i = 1, . . . , m − 1 and t ≥ 0, (ii) J i,t · v 0 = u i,t v 0 (u i,t ∈ C) for each i = 1, . . . , m − 1 and t ≥ 0. Moreover, if M is simple, we have M = U(sl Q m [x]) · v 0 .
2.4. Highest weight modules. For U(sl Q m [x])-module M, we say that M is a highest weight module if there exists v 0 ∈ M satisfying the following conditions:
(i) M is generated by v 0 as a U(sl Q m [x])-module. (ii) X + i,t · v 0 = 0 for all i = 1, . . . , m − 1 and t ≥ 0. (iii) J i,t · v 0 = u i,t v 0 (u i,t ∈ C) for each i = 1, . . . , m − 1 and t ≥ 0.
In this case, we say that (u i,t ) 1≤i≤m−1,t≥0 ∈ m−1 i=1 t≥0 C is the highest weight of M, and that v 0 is a highest weight vector of M.
Let M be a highest weight U(sl Q m [x])-module with a highest weight u = (u i,t ) 1≤i≤m−1,t≥0 ∈ m−1 i=1 t≥0 C and a highest weight vector v 0 ∈ M. Thanks to the triangular decomposition (Proposition 1.4 (iv)) together with the above conditions, we have M = U(n − ) · v 0 . Let λ u ∈ h * be as λ u (J i,0 ) = u i,0 for i = 1, . . . , m − 1. By M = U(n − ) · v 0 and the relation (L2), we have the weight space decomposition
M = µ∈h * µ≤λu M µ , where M µ = {x ∈ M | h · x = µ(h) · x for h ∈ h},
(2.4.1) and we also have dim C M λu = 1.
Verma modules. For
u = (u i,t ) ∈ m−1 i=1 t≥0 C, let I(u) be the left ideal of U(sl Q m [x]) generated by X + i,t (1 ≤ i ≤ m − 1, t ≥ 0) and J i,t − u i,t (1 ≤ i ≤ m − 1, t ≥ 0). We define the Verma module M(u) = U(sl Q m [x])/I(u)
. Then M(u) is a highest weight module of highest weight u, and any highest weight module of highest weight u is realized as a quotient of the Verma module M(u). By the weight space decomposition (2.4.1), we see that M(u) has the unique maximal proper submodule rad M(u). Put L(u) = M(u)/ rad M(u), then we have the following proposition.
Proposition 2.6. For u = (u i,t ) ∈ m−1 i=1 t≥0 C, a highest weight simple U(sl Q m [x])- module of highest weight u is isomorphic to L(u). Moreover, any finite dimensional simple U(sl Q m [x])-module is isomorphic to L(u) for some u = (u i,t ) ∈ m−1 i=1 t≥0 C. Proof. By Lemma 2.3, a finite dimensional simple U(sl Q m [x]
)-module is a highest weight module. Then we have the proposition by the above arguments. § 3.
Representations of gl Q m [x] For finite dimensional U(gl Q m [x]
)-modules, we can develop a similar argument as in the case of U(sl Q m [x]) discussed in the previous section. In this section, we give only some notation for U(gl Q m [x])-modules.
3.1. Highest weight modules. For U(gl Q m [x])-module M, we say that M is a highest weight module if there exists v 0 ∈ M satisfying the following conditions:
(i) M is generated by v 0 as a U(gl Q m [x])-module. (ii) X + i,t · v 0 = 0 for all i = 1, . . . , m − 1 and t ≥ 0. (iii) I j,t · v 0 = u j,t v 0 ( u j,t ∈ C) for each j = 1, . . . , m and t ≥ 0.
In this case, we say that ( u j,t ) 1≤j≤m,t≥0 ∈ m j=1 t≥0 C is the highest weight of M, and that v 0 is a highest weight vector of M.
Verma modules. For
u = ( u j,t ) ∈ m j=1 t≥0 C, let I( u) be the left ideal of U(gl Q m [x]) generated by X + i,t (1 ≤ i ≤ m − 1, t ≥ 0) and I j,t − u j,t (1 ≤ j ≤ m, t ≥ 0). We define the Verma module M( u) = U(gl Q m [x])/I( u). Then M( u)
is a highest weight module of highest weight u, and any highest weight module of highest weight u is realized as a quotient of the Verma module M( u). M( u) has the unique maximal proper submodule rad M( u). Put L( u) = M( u)/ rad M( u), then we have the following proposition.
Proposition 3.3. For u = ( u j,t ) ∈ m j=1 t≥0 C, a highest weight simple U(gl Q m [x])- module of highest weight u is isomorphic to L( u). Moreover, any finite dimensional simple U(gl Q m [x])-module is isomorphic to L( u) for some u = (u j,t ) ∈ m j=1 t≥0 C. § 4. Rank 1 case ; some relations in U(sl Q 2 [x]) 4.1. Take Q ∈ C, then sl Q 2 [x]
is a Lie algebra over C generated by X ± t and J t (t ∈ Z ≥0 ) together with the following defining relations:
[J s , J t ] = 0, (L1) [J s , X ± t ] = ±2X ± s+t , (L2) [X + t , X − s ] = J s+t − QJ s+t+1 , (L3) [X ± t , X ± s ] = 0. (L4)
(In the rank 1 case, we omit the first index of the generators since it is trivial.) By checking the defining relations, we see that there exists the algebra anti-
automorphism † : U(sl Q 2 [x]) → U(sl Q 2 [x]) such that †(X + t ) = X − t , †(X − t ) = X + t , †(J t ) = J t . (4.1.1) Clearly, † 2 is the identity on U(sl Q 2 [x]). 4.2. For t, b ∈ Z ≥0 , we define an element X +(b) t (resp. X −(b) t ) of U(sl Q 2 [x]) by X ±(b) t = (X ± t ) b b! .
For convenience, we put X
±(b) t = 0 for b ∈ Z <0 . For t, p, h ∈ Z ≥0 , we define an element X +((p);h) t (resp. X −((p);h) t ) of U(sl Q 2 [x]) by X ±((0);h) t = 1, X ±((p);h) t = p w=0 p w (−Q) w X ± t+ph+w for p > 0. (4.2.1) Clearly, we have †(X +((p);h) t ) = X −((p);h) t . For examples, we have X ±((0);h) t = 1, X ±((1);h) t = X ± t+h + (−Q)X ± t+h+1 , X ±((2);h) t = X ± t+2h + 2(−Q)X ± t+2h+1 + (−Q) 2 X ± t+2h+2 , X ±((3);h) t = X ± t+3h + 3(−Q)X ± t+3h+1 + 3(−Q) 2 X ± t+3h+2 + (−Q) 3 X ± t+3h+3 .
For s, p ∈ Z ≥0 , we define an element J
p s of U(sl Q 2 [x])) inductively on p by J 0 s = 1, J p s = 1 p p z=1 (−1) z−1 z w=0 z w (−Q) w J zs+w J p−z s for p > 0. (4.2.2) For examples, we have J 0 s = 1, J 1 s = J s + (−Q)J s+1 , J 2 s = 1 2 J 2 s − J 2s + 2(−Q) J s J s+1 − J 2s+1 + (−Q) 2 J 2 s+1 − J 2s+2 , J 3 s = 1 3 J 3 s − 2J s J 2s + J 3s + 3(−Q) J 2 s J s+1 − J s J 2s+1 − J s+1 J 2s + J 3s+1 + (−Q) 2 3J s J 2 s+1 − 2J s J 2s+2 − 4J s+1 J 2s+1 + 3J 3s+2 + (−Q) 3 J 3 s+1 − 2J s+1 J 2s+2 + J 3s+3 . Lemma 4.3. For s, t, p ∈ Z ≥0 , we have the following relations in U(sl Q 2 [x]). (i) [J p s , X + t ] = p z=1 (−1) z+1 (z + 1)J p−z s X +((z);s) t . (ii) [J p s , X − t ] = − p z=1 (−1) z+1 (z + 1)X −((z);s) t J p−z s .
Proof. (ii) follows from (i) by applying the algebra anti-automorphism † defined in (4.1.1). Then, we prove only (i) by the induction on p.
If p = 0, (i) is clear. If p > 0, by the definition (4.2.2), we have J p s X + t = 1 p p z=1 (−1) z−1 z w=0 z w (−Q) w J zs+w J p−z s X + t .
By the assumption of the induction, we have
J p s X + t = 1 p p z=1 (−1) z−1 z w=0 z w (−Q) w J zs+w × X + t J p−z s + p−z k=1 (−1) k+1 (k + 1)J p−z−k s X +((k);s) t = 1 p p z=1 (−1) z−1 z w=0 z w (−Q) w (X + t J zs+w + 2X + t+zs+w )J p−z s + 1 p p z=1 z w=0 p−z k=1 (−1) z+k z w (k + 1)(−Q) w J zs+w J p−z−k s X +((k);s) t .
Applying the assumption of the induction again, we have
J p s X + t = X + t 1 p p z=1 (−1) z−1 z w=0 z w (−Q) w J zs+w J p−z s + 2 1 p p z=1 (−1) z−1 z w=0 z w (−Q) w J p−z s X + t+zs+w − 2 1 p p z=1 (−1) z−1 z w=0 z w (−Q) w p−z k=1 (−1) k+1 (k + 1)J p−z−k s X +((k);s) t+zs+w + 1 p p z=1 z w=0 p−z k=1 (−1) z+k z w (k + 1)(−Q) w J zs+w J p−z−k s X +((k);s) t . (4.3.1)
By the definition (4.2.2), we have
X + t 1 p p z=1 (−1) z−1 z w=0 z w (−Q) w J zs+w J p−z s = X + t J p s . (4.3.2) By the definition (4.2.1), we have p z=1 (−1) z−1 z w=0 z w (−Q) w J p−z s X + t+zs+w = p z=1 (−1) z−1 J p−z s X +((z);s) t . (4.3.3) Put ( * ) = p z=1 (−1) z−1 z w=0 z w (−Q) w p−z k=1 (−1) k+1 (k + 1)J p−z−k s X +((k);s) t+zs+w .
By the definition (4.2.1), we also have
( * ) = p z=1 (−1) z−1 z w=0 z w (−Q) w p−z k=1 (−1) k+1 (k + 1)J p−z−k s × k l=0 k l (−Q) l X + (t+zs+w)+ks+l = p z=1 p−z k=1 (−1) z+k (k + 1)J p−(z+k) s z w=0 k l=0 z w k l (−Q) w+l X + t+(z+k)s+(w+l) Put z ′ = z + k and w ′ = w + l, we have ( * ) = p z ′ =2 z ′ −1 k=1 (−1) z ′ (k + 1)J p−z ′ s z ′ w ′ =0 min{k,w ′ } l=max{0,w ′ −(z ′ −k)} z ′ − k w ′ − l k l (−Q) w ′ X + t+z ′ s+w ′ .
By the induction on k, we can show that
min{k,w ′ } l=max{0,w ′ −(z ′ −k)} z ′ − k w ′ − l k l = z ′ w ′ . (4.3.4) Then, we have ( * ) = p z ′ =2 (−1) z ′ z ′ −1 k=1 (k + 1) J p−z ′ s z ′ w ′ =0 z ′ w ′ (−Q) w ′ X + t+z ′ s+w ′ ,
and by the definition of (4.2.1), we have
( * ) = p z ′ =2 (−1) z ′ (z ′ − 1)(z ′ + 2) 2 J p−z ′ s X +((z ′ );s) t . (4.3.5) By the definition (4.2.2), we have p z=1 z w=0 p−z k=1 (−1) z+k z w (k + 1)(−Q) w J zs+w J p−z−k s X +((k);s) t = p−1 k=1 p−k z=1 z w=0 (−1) z+k z w (k + 1)(−Q) w J zs+w J p−z−k s X +((k);s) t = p−1 k=1 (−1) k+1 (k + 1)(p − k) 1 p − k p−k z=1 (−1) z−1 z w=0 z w (−Q) w J zs+w J (p−k)−z s X +((k);s) t = p−1 k=1 (−1) k+1 (k + 1)(p − k)J p−k s X +((k);s) t .
(4.3.6) Combining (4.3.1) with (4.3.2), (4.3.3), (4.3.5) and (4.3.6), we have
J p s X + t = X + t J p s + 1 p p z=1 (−1) z+1 2 + (z − 1)(z + 2) + (z + 1)(p − z) J p−z s X +((z);s) t = X + t J p s + p z=1 (−1) z+1 (z + 1)J p−z s X +((z);s) t . Lemma 4.4. For s, t, h ∈ Z ≥0 and p ∈ Z >0 , we have [X + t , X −((p);h) s ] = p w=0 p w (−Q) w J 1 s+t+ph+w .
Proof. By the definitions (4.2.1), (4.2.2) and the defining relation (L3), we have
X + t X −((p);h) s = p w=0 p w (−Q) w X + t X − s+ph+w = p w=0 p w (−Q) w X − s+ph+w X + t + J s+t+ph+w + (−Q)J s+t+ph+w+1 = X −((p);h) s X + t + p w=0 p w (−Q) w J 1 s+t+ph+w .
Lemma 4.5. For s, t, h ∈ Z ≥0 and p ∈ Z >0 , we have the following relations.
(i) [J 1 s , X +((p);h) t ] = 2X +((p+1);h) s+t−h . (ii) [J 1 s , X −((p);h) t ] = −2X −((p+1);h) s+t−h .
Proof. (ii) follows from (i) by applying the algebra anti-automorphism † defined in (4.1.1). Then, we prove (i). By the definition (4.2.1), we have
J 1 s X +((p);h) t = p w=0 p w (−Q) w J 1 s X + t+ph+w .
Applying Lemma 4.3 (i), we have
J 1 s X +((p);h) t = p w=0 p w (−Q) w X + t+ph+w J 1 s + 2X +((1);s) t+ph+w
Then, by the definition (4.2.1) again, we have
J 1 s X +((p);h) t = X +((p);h) t J 1 s + 2 p w=0 p w (−Q) w X + s+t+ph+w + (−Q)X + s+t+ph+w+1 .
On the other hand, we have
p w=0 p w (−Q) w X + s+t+ph+w + (−Q)X + s+t+ph+w+1 = X + s+t+ph + p w=1 p w + p w − 1 (−Q) w X + s+t+ph+w + (−Q) p+1 X + s+t+ph+p+1 = p+1 w=0 p + 1 w (−Q) w X + s+t−h+(p+1)h+w = X +(p+1);h s+t−h .
Thus, we have (i).
Lemma 4.6. For s, t, c ∈ Z ≥0 , we have
[X + t , X −(c) s ] = X −(c−1) s J 1 s+t − X −(c−2) s X −((1);s+t) s .
Proof. We prove the lemma by the induction on c. If c = 0, it is clear. If c = 1, it is the defining relation (L3). If c > 1, by the assumption of the induction, we have
X + t X −(c) s = 1 c X + t X −(c−1) s X − s = 1 c X −(c−1) s X + t + X −(c−2) s J 1 s+t − X −(c−3) s X −((1);s+t) s X − s .
Then, by the defining relations (L3), (L4) and Lemma 4.3 (ii), we have
X + t X −(c) s = 1 c X −(c−1) s X − s X + t + J 1 s+t + X −(c−2) s X − s J 1 s+t − 2X −((1);s+t) s − X −(c−3) s X − s X −((1);s+t) s = 1 c cX −(c) s X + t + X −(c−1) s J 1 s+t + (c − 1)X −(c−1) s J 1 s+t − 2X −(c−2) s X −((1);s+t) s − (c − 2)X −(c−2) s X −((1);s+t) s = X −(c) s X + t + X −(c−1) s J 1 s+t − X −(c−2) s X −((1);s+t) s . 4.7. A partition λ = (λ 1 , λ 2 , . . . )
is a non-increasing sequence of non-negative integers which has only finitely many non-zero terms. The size of a partition λ is the sum of all terms of λ, and we denote it by |λ|. Namely, we have |λ| = i≥1 λ i . If |λ| = n, we say that λ is a partition of n, and we denote it by λ ⊢ n. The length of λ is the maximal i such that λ i = 0, and we denote the length of λ by ℓ(λ). For a partition λ = (λ 1 , λ 2 , . . . ), let m j (λ) (j ∈ Z >0 ) be the multiplicity of j in λ. Then, for a partition λ and t, h ∈ Z ≥0 , we define an element X
+(λ;h) t (resp. X −(λ;h) t ) of U(sl Q 2 [x]) by X ±(λ;h) t = j≥1 (X ±((j);h) t ) m j (λ) m j (λ)! , (4.7.1)
where we note the defining relation (L4). Clearly, we have †(X
+(λ;h) t ) = X −(λ;h) t . For examples, we have X ±((0);h) t = 1, X ±((1);h) t = X ±((1);h) t , X ±((2);h) t = X ±((2);h) t , X ±((1,1);h) t = (X ±((1);h) t ) 2 2! , X ±((3);h) t = X ±((3);h) t , X ±((2,1);h) t = X ±((2);h) t X ±((1);h) t , X ±((1,1,1);h) t = (X ±(1);h) t ) 3 3! , X ±((3,3,2,2,2,1,1);h) t = (X ±((3);h) t ) 2 2! (X ±((2);h) t ) 3 3! (X +((1);h) t ) 2 2! .
For t, h, k, b, p ∈ Z ≥0 , we define an element X
+(b;p|k;h) t (resp. X −(b;p|k;h) t ) of U(sl Q 2 [x]) by X ±(b;p|k;h) t = λ⊢k X ±(λ;h) t X ±(b−p−ℓ(λ)) t .
(4.7.2) Note the defining relation (L4), we see that †(X
+(b;p|k;h) t ) = X −(b;p|k;h) t . For examples, we have X ±(b;p|0;h) t = X ±(b−p) t , X ±(b;p|1;h) t = X ±((1);h) t X ±(b−p−1) t , X ±(b;p|2;h) t = X ±((2);h) t X ±(b−p−1) t + X ±((1,1);h) t X ±(b−p−2) t , X ±(b;p|3;h) t = X ±((3);h) t X ±(b−p−1) t + X ±((2,1);h) t X ±(b−p−2) t + X ±((1,1,1);h) t X ±(b−p−3) t .
For the element X ±(b;p|k;h) t ∈ U(sl Q 2 [x]), we prepare the following technical formulas.
Lemma 4.8. For t, h, k, b, p ∈ Z ≥0 , we have the following equations for the element X
±(b;p|k;h) t of U(sl Q 2 [x]). (i) If b − p < 0, we have X ±(b;p|k;h) t = 0. (ii) If k = 0, we have X ±(b;p|0;h) t = X ±(b−p) t . If k = 1, we have X ±(b;p|1;h) t = X ±((1);h) t X ±(b−p−1) t . (iii) If p = b, we have X ±(b;b|k;h) t = 1 if k = 0, 0 if k = 0. (iv) If b, p > 0, we have X ±(b;p|k;h) t = X ±(b−1;p−1|k;h) t . (v) If b, k > 0, we have X ±(b;p|k;h) t = 1 k k z=1 zX ±((z);h) t X ±(b−1;p|k−z;h) t . (vi) If b > 0, we have (b − p + k)X ±(b;p|k;h) t = X ± t X ±(b−1;p|k;h) t + k z=1 (z + 1)X ±((z);h) t X ±(b−1;p|k−z;h) t .
Proof. (i), (ii), (iii) and (iv) are clear from definitions. We prove (v). Note that z≥1 zm z (λ) = k for a partition λ of k. Then, by the definition (4.7.2), we have
X ±(b;p|k;h) t = λ⊢k X ±(λ;h) t X ±(b−p−ℓ(λ)) t = 1 k λ⊢k z≥1 zm z (λ) X ±(λ;h) t X ±(b−p−ℓ(λ)) t .
On the other hand, by the definition (4.7.1), we have
X ±(λ;h) t = j≥1 (X ±((j);h) t ) m j (λ) m j (λ)! = 1 m z (λ) X ±((z);h) t (X ±((z);h) t ) mz(λ)−1 (m z (λ) − 1)! j≥1 j =z (X ±((j);h) t ) m j (λ) m j (λ)! for each z such that m z (λ) = 0. Thus, we have X ±(b;p|k;h) t = 1 k λ⊢k z≥1 mz (λ) =0 zX ±((z);h) t (X ±((z);h) t ) mz(λ)−1 (m z (λ) − 1)! j≥1 j =z (X ±((j);h) t ) m j (λ) m j (λ)! X ±(b−p−ℓ(λ)) t = 1 k k z=1 zX ±((z);h) t λ⊢k mz (λ) =0 (X ±((z);h) t ) mz(λ)−1 (m z (λ) − 1)! j≥1 j =z (X ±((j);h) t ) m j (λ) m j (λ)! X ±(b−p−ℓ(λ)) t = 1 k k z=1 zX ±((z);h) t µ⊢k−z j≥1 (X ±((j);h) t ) m j (µ) m j (µ)! X ±(b−p−(ℓ(µ)+1)) t = 1 k k z=1 zX ±((z);h) t µ⊢k−z X ±(µ;h) t X ±((b−1)−p−ℓ(µ)) t = 1 k k z=1 zX ±((z);h) t X ±(b−1;p|k−z;h) t .
We prove (vi). By the definition (4.7.2), we have
(b − p + k)X ±(b;p|k;h) t = kX ±(b;p|k;h) t + λ⊢k ℓ(λ)X ±(λ;h) t X ±(b−p−ℓ(λ)) t + λ⊢k (b − p − ℓ(λ))X ±(λ;h) t X ±(b−p−ℓ(λ)) t . Note that ℓ(λ) = z≥1 m z (λ), (b − p − ℓ(λ))X ±(b−p−ℓ(λ)) t = X ± t X ±(b−p−ℓ(λ)−1) t and the defining relation (L4), we have (b − p + k)X ±(b;p|k;h) t = kX ±(b;p|k;h) t + λ⊢k z≥1 m z (λ) X ±(λ;h) t X ±(b−p−ℓ(λ)) t + X ± t λ⊢k X ±(λ;h) t X ±(b−1−p−ℓ(λ)) t .
In a similar argument as in the proof of (v), we have
(b − p + k)X ±(b;p|k;h) t = kX ±(b;p|k;h) t + k z=1 X ±((z);h) t X ±(b−1;p|k−z;h) t + X ± t λ⊢k X ±(λ;h) t X ±((b−1)−p−ℓ(λ)) t .
Then, by (v) and the definition (4.7.2), we have
(b − p + k)X ±(b;p|k;h) t = k z=1 zX ±((z);h) t X ±(b−1;p|k−z;h) t + k z=1 X ±((z);h) t X ±(b−1;p|k−z;h) t + X ± t X ±(b−1;p|k;h) t = X ± t X ±(b−1;p|k;h) t + k z=1 (z + 1)X ±((z);h) t X ±(b−1;p|k−z;h) t .
Lemma 4.9. For s, t, c, p, k ∈ Z ≥0 , we have
[X + t , X −(c;p|k;s+t) s ] = k z=0 k−z w=0 k − z w (−Q) w X −(c;p+1|z;s+t) s J 1 (k−z+1)(s+t)+w − (k + 1)X −(c;p+1|k+1;s+t) s .
(4.9.1)
Proof. If c = 0, the equation (4.9.1) follows from Lemma 4.8 (i) and (iii). Then, we prove (4.9.1) by the induction on k in the case where c > 0. If k = 0, we see that (4.9.1) is just the formula in Lemma 4.6 by Lemma 4.8 (ii). If k > 0, by Lemma 4.8 (v) and the defining relation (L4), we have
X + t X −(c;p|k;s+t) s = 1 k k z=1 zX + t X −(c−1;p|k−z;s+t) s X −((z);s+t) s .
Applying the assumption of the induction, we have
X + t X −(c;p|k;s+t) s = 1 k k z=1 z X −(c−1;p|k−z;s+t) s X + t + k−z y=0 X −(c−1;p+1|y;s+t) s k−z−y w=0 k − z − y w (−Q) w J 1 (k−z−y+1)(s+t)+w − (k − z + 1)X −(c−1;p+1|k−z+1;s+t) s X −((z);s+t) s .
Applying Lemma 4.4 and Lemma 4.5 (ii), we have Put z ′ = k − z in ( * 2) and apply Lemma 4.8 (iv), we have
X + t X −(c;p|k;s+t) s = 1 k k z=1 zX −(c−1;p|k−z;s+t) s X −((z);s+t) s X + t + z w=0 z w (−Q) w J 1 (z+1)(s+t)+w + 1 k k z=1 k−z y=0 zX −(c−1;p+1|y;s+t) s k−z−y w=0 k − z − y w (−Q) w × X −((z);s+t) s J 1 (k−z−y+1)(s+t)+w − 2X −((z+1);s+t) s+(k−z−y)(s+t)+w − 1 k k z=1 z(k − z + 1)X −(c−1;p+1|k−z+1;s+t) s X −((z);s+t) s . Put ( * 1) = 1 k k z=1 zX −(c−1;p|k−z;s+t) s X −((z);s+t) s X + t , ( * 2) = 1 k k z=1 zX −(c−1;p|k−z;s+t) s z w=0 z w (−Q) w J 1 (z+1)(s+t)+w ( * 3) = 1 k k z=1 k−z y=0 zX −(c−1;p+1|y;s+t) s k−z−y w=0 k − z − y w (−Q) w X −((z);s+t) s J 1 (k−z−y+1)(s+t)+w ( * 4) = 1 k k z=1 k−z y=0 zX −(c−1;p+1|y;s+t) s k−z−y w=0 k − z − y w (−Q) w X −((z+1);s+t) s+(k−z−y)(s+t)+w , ( * 5) = 1 k k z=1 z(k − z + 1)X −(c−1;p+1|k−z+1;s+t)( * 2) = 1 k k−1 z ′ =0 (k − z ′ )X −(c;p+1|z ′ ;s+t) s k−z ′ w=0 k − z ′ w (−Q) w J 1 (k−z ′ +1)(s+t)+w . (4.9.4) Put h = z + y in ( * 3) , we have ( * 3) = 1 k k h=1 h z=1 zX −(c−1;p+1|h−z;s+t) s X −((z);s+t) s k−h w=0 k − h w (−Q) w J 1 (k−h+1)(s+t)+w .
Applying Lemma 4.8 (v) together with (L4), we have Put h = k − y + 1, we have
( * 3) = 1 k k h=1 hX −(c;p+1|h;s+t) s k−h w=0 k − h w (−Q) w J 1 (k−h+1)(s+t)+w .( * 4) = 1 k k+1 h=2 h−1 z=1 zX −(c−1;p+1|k−h+1;s+t) s h−z−1 w=0 h − z − 1 w (−Q) w X −((z+1);s+t) s+(h−z−1)(s+t)+w . Put (♯) = h−z−1 w=0 h − z − 1 w (−Q) w X −((z+1);s+t) s+(h−z−1)(s+t)+w .
By (4.2.1), we have
(♯) = h−z−1 w=0 h − z − 1 w (−Q) w z+1 y=0 z + 1 y (−Q) y X − s+h(s+t)+w+y .
Put y ′ = w + y, we have where the last equation follows from Lemma 4.8 (v). By (4.9.2), (4.9.3), (4.9.6) and (4.9.7), we have
(♯) = h y ′ =0 min{h−(z+1),y ′ } w=max{0,y ′ −(z+1)} h − (z + 1) w z + 1 y ′ − w (−Q) y ′ X − s+h(s+t)+y ′ . Note that min{h−(z+1),y ′ } w=max{0,y ′ −(z+1)} h − (z + 1) w z + 1 y ′ − w = h y ′ by (4.3.4), we have (♯) = h y ′ =0 h y ′ (−Q) y ′ X − s+h(s+t)+y ′ = X −((h);s+t)X + t X −(c;p|k;s+t) s = X −(c;p|k;s+t) s X + t + k z=0 k−z w=0 k − z w (−Q) w X −(c;p+1|z;s+t) s J 1 (k−z+1)(s+t)+w − (k + 1)X −(c;p+1|k+1;s+t) s . Proposition 4.10. For s, t, b, c ∈ Z ≥0 , we have [X +(b) t , X −(c) s ] = min{b,c} p=1 p k=0 p−k l=0 (−1) k+l X −(c;p|k;s+t) s J p−(k+l) s+t X +(b;p|l;s+t) t .
(4.10.1) Proof. We prove (4.10.1) by the induction on b. If b = 1, (4.10.1) follows from Lemma 4.6 together with Lemma 4.8.
If b > 1, we have
X +(b) t X −(c) s = 1 b X + t X (b−1) t X −(c) s = 1 b X + t X −(c) s X +(b−1) t + min{b−1,c} p=1 p k=0 p−k l=0 (−1) k+l X −(c;p|k;s+t) s J p−(k+l) s+t X +(b−1;p|l;s+t) t
by the assumption of the induction. Applying Lemma 4.6 and Lemma 4.9, we have
X +(b) t X −(c) s = 1 b X −(c) s X + t + X −(c−1) s J 1 s+t − X −(c−2) s X −((1);s+t) s X +(b−1) t + min{b−1,c} p=1 p k=0 p−k l=0 (−1) k+l × X −(c;p|k;s+t) s X + t + k z=0 k−z w=0 k − z w (−Q) w X −(c;p+1|z;s+t) s J 1 (k−z+1)(s+t)+w − (k + 1)X −(c;p+1|k+1;s+t) s J p−(k+l) s+t X +(b−1;p|l;s+t) t .
On the other hand, by Lemma 4.3, we have
X −(c;p|k;s+t) s X + t J p−(k+l) s+t X +(b−1;p|l;s+t) t = X −(c;p|k;s+t) s J p−(k+l) s+t X + t − p−(k+l) z=1 (−1) z+1 (z + 1)J p−(k+l)−z s+t X +((z);s+t) t X +(b−1;p|l;s+t) t . Put ( * 1) = bX −(c) s X +(b) t + X −(c−1) s J 1 s+t X +(b−1) t − X −(c−2)× X + t X +(b−1;p|l;s+t) t + l z=1 (z + 1)X +((z);s+t) t X +(b−1;p|l−z;s+t) t ,
where we note that 0 z=1 X +((z);s+t) t X +(b−1;p|l−z;s+t) t = 0. Applying Lemma 4.8 (vi), we have
( * 2) + ( * 3) = min{b,c} p=1 p k=0 p−k l=0 (−1) k+l (b − p + l)X −(c;p|k;s+t) s J p−(k+l) s+t X +(b;p|l;s+t) t = (b − 1)X −(c;1|0;s+t) s J 1 s+t X +(b;1|0;s+t) t − bX −(c;1|0;s+t) s J 0 s+t X +(b;1|1;s+t) t − (b − 1)X −(c;1|1;s+t) s J 0 s+t X +(b;1|0;s+t) t + min{b,c} p=2 p k=0 p−k l=0 (−1) k+l (b − p + l)X −(c;p|k;s+t) s J p−(k+l) s+t X +(b;p|l;s+t) t .
(4.10.4) Put p ′ = p + 1 in ( * 4), we have
( * 4) = min{b,c} p ′ =2 p ′ −1 k=0 p ′ −k−1 l=0 k z=0 k−z w=0 (−1) k+l k − z w (−Q) w × X −(c;p ′ |z;s+t) s J 1 (k−z+1)(s+t)+w J p ′ −(k+l)−1 s+t X +(b−1;p ′ −1|l;s+t) t ,
where we note that X
+(b−1;p ′ −1|l;s+t) t = 0 if p ′ = b+1, and X −(c;p ′ |z;s+t) s = 0 if p ′ = c+1 bys × p−l−1 k=z k−z w=0 (−1) k+l k − z w (−Q) w J 1 (k−z+1)(s+t)+w J p−(k+l)−1 s+t X +(b−1;p−1|l;s+t) t . Put k ′ = k − z + 1, we have p−l−1 k=z k−z w=0 (−1) k+l k − z w (−Q) w J 1 (k−z+1)(s+t)+w J p−(k+l)−1 s+t = p−l−z k ′ =1 k ′ −1 w=0 (−1) k ′ +z+l−1 k ′ − 1 w (−Q) w J 1 k ′ (s+t)+w J p−k ′ −z−l s+t . Since J 1 k ′ (s+t)+w = J k ′ (s+t)+w + (−Q)J k ′ (s+t)+w+1 , we see that k ′ −1 w=0 k ′ − 1 w (−Q) w J 1 k ′ (s+t)+w = k ′ w=0 k ′ w (−Q) w J k ′ (s+t)+w .
Thus we have
p−l−1 k=z k−z w=0 (−1) k+l k − z w (−Q) w J 1 (k−z+1)(s+t)+w J p−(k+l)−1 s+t = (−1) z+l p−l−z k ′ =1 (−1) k ′ −1 k ′ w=0 k ′ w (−Q) w J k ′ (s+t)+w J p−k ′ −z−l s+t = (−1) z+l (p − l − z)J p−l−z s+t ,
where the last equation follows from (4.2.2). Then we have
( * 4) = min{b,c} p=2 p−1 z=0 p−z−1 l=0 (p − l − z)(−1) z+l X −(c;p|z;s+t) s J p−l−z s+t X +(b−1;p−1|l;s+t) t = min{b,c} p=2 p z=0 p−z l=0 (p − l − z)(−1) z+l X −(c;p|z;s+t) s J p−l−z s+t X +(b−1;p−1|l;s+t) t .
Applying Lemma 4.8 (iv), we have
( * 4) = min{b,c} p=2 p z=0 p−z l=0 (p − l − z)(−1) z+l X −(c;p|z;s+t) s J p−l−z s+t X +(b;p|l;s+t) t .
(4.10.5) Put p ′ = p + 1 in ( * 5), we have
( * 5) = min{b,c} p ′ =2 p ′ −1 k=0 p ′ −k−1 l=0 (−1) k+l+1 (k + 1)X −(c;p ′ |k+1;s+t) s J p ′ −k−l−1 s+t X +(b−1;p ′ −1|l;s+t) t ,
where we note that X
+(b−1;p ′ −1|l;s+t) t = 0 if p ′ = b + 1, and X −(c;p ′ |k+1;s+t) s = 0 if p ′ = c + 1 by Lemma 4.8 (i). Put k ′ = k + 1, we have ( * 5) = min{b,c} p ′ =2 p ′ k ′ =1 p ′ −k ′ l=0 (−1) k ′ +l k ′ X −(c;p ′ |k ′ ;s+t) s J p ′ −k ′ −l s+t X +(b−1;p ′ −1|l;s+t) t = min{b,c}X +(b) t X −(c) s = X −(c) s X +(b) t + min{b,c} p=1 p k=0 p−k l=0 (−1) k+l X −(c;p|k;s+t) s J p−(k+l) s+t X +(b;p|l;s+t) t . § 5. Rank 1 case ; finite dimensional simple modules of U(sl Q 2 [x])
In this section, we classify the finite dimensional simple U(sl
Q 2 [x])-modules. 5.1. 1-dimensional representations. First, we consider 1-dimensional represen- tations of sl Q 2 [x]. Let L = Cv be a 1-dimensional U(sl Q 2 [x])-module with a basis {v}, then J t (t ≥ 0) acts on v as a scalar multiplication. If X + t · v = 0 (resp. X − t · v = 0), then X + t · v (resp. X − t · v)
is an eigenvector for the action of J 0 whose eigenvalue is different from one of v by the defining relation (L2). This is a contradiction since L is 1-dimensional. Thus, we have X ± t · v = 0 for t ≥ 0. Moreover, by the defining relation (L3), we have
(J t − QJ t+1 ) · v = (X + t X − 0 − X − 0 X + t ) · v = 0. This implies that J t · v = 0 for t ≥ 0 if Q = 0, and that J t · v = Q −t J 0 · v for t > 0 if Q = 0.
We define the set B Q by
B Q = {0} if Q = 0, C if Q = 0.
For each β ∈ B Q , we can define the 1-dimensional U(sl
Q 2 [x])-module L β = Cv 0 such that X ± t · v 0 = 0, J t · v 0 = 0 if Q = 0, Q −t βv 0 if Q = 0 (t ∈ Z ≥0 )
by checking the defining relations of sl Q 2 [x]. Note that L 0 is the trivial representation. Now we obtain the following lemma.
Lemma 5.2. Any 1-dimensional U(sl Q 2 [x])-module is isomorphic to L β for some β ∈ B Q .
5.3.
Recall from §2, a finite dimensional simple U(sl Q 2 [x])-module is isomorphic to a simple highest weight module L(u) for some highest weight u = (u t ) ∈ t≥0 C (Proposition 2.6), where we omit the first index for the highest weight. Then, in order to classify the finite dimensional simple U(sl Q 2 [x])-module, it is enough to classify the highest weight u such that L(u) is finite dimensional.
In order to obtain a necessary condition for u such that L(u) is finite dimensional, we prepare the following lemma.
Lemma 5.4. Let M be a finite dimensional U(sl Q 2 [x])-module. Take an element v ∈ M satisfying X + t · v = 0, J t · v = u t v (t ∈ Z ≥0 ), X −(n) 0 · v = 0 and X −(n+1) 0 · v = 0
for some u t ∈ C (t ∈ Z ≥0 ) and n ∈ Z ≥0 . (In fact, a such element exists by Lemma 2.3.) Then, for s, t ∈ Z ≥0 , we have
n w=0 n w (−Q) w J 1 t+ns+w · v = n−1 k=0 (−1) n−k+1 k w=0 k w (−Q) w J 1 t+ks+w J n−k s · v.
Proof. By the assumption X −(n+1) 0
· v = 0 and Proposition 4.10, we have By the definition, we have
0 = X +(n) s X −(n+1) 0 · v = X −(X −(n+1;n|k;s) 0 = λ⊢k X −(λ;s) 0 X −(1−ℓ(λ)) 0 = X − 0 if k = 0, X −((k);s) 0 if k = 0. Thus, we have 0 = X − 0 J n s · v + n k=1 (−1) k X −((k);s) 0 J n−k s · v.
By multiplying X + t from left to this equation, we have
0 = X + t X − 0 J n s · v + n k=1 (−1) k X + t X −((k);s) 0 J n−k s · v = n k=0 (−1) k k w=0 k w (−Q) w J 1 t+ks+w J n−k s · v,
where we use Lemma 4.4 and the fact X + t J n−k s ·v = 0. This implies the Lemma.
This Lemma implies the following proposition which gives a necessary condition for u such that L(u) is finite dimensional.
Proposition 5.5. Let M be a finite dimensional U(sl Q 2 [x])-module. Take an ele- ment v ∈ M satisfying X + t · v = 0, J t · v = u t v (t ∈ Z ≥0 ), X −(n) 0 · v = 0 and X −(n+1) 0 · v = 0
for some u t ∈ C (t ∈ Z ≥0 ) and n ∈ Z ≥0 .
(i) If Q = 0, we have u 0 = n, and there exist γ 1 , γ 2 , . . . , γ n ∈ C such that
u t = p t (γ 1 , γ 2 , . . . , γ n ) (t > 0),
where p t (γ 1 , . . . , γ n ) = γ t 1 + γ t 2 + · · · + γ t n . (ii) If Q = 0, there exist β, γ 1 , γ 2 , . . . , γ n ∈ C such that
u 0 = n + β and u t = p t (γ 1 , γ 2 , . . . , γ n ) + Q −t β (t > 0),
where p t (γ 1 , . . . , γ n ) = γ t 1 + γ t 2 + · · · + γ t n .
Proof. (i). Assume that Q = 0. Then, sl 0 2 [x] coincides with the current Lie algebra sl 2 [x] of sl 2 . Moreover, the Lie subalgebra of sl 2 [x] generated by X ± 0 and J 0 is isomorphic to sl 2 . Thus, by the representation theory of sl 2 , we have u 0 = n.
For u 1 , . . . , u n , there exist γ 1 , . . . , γ n ∈ C such that u k = p k (γ 1 , . . . , γ n ) for k = 1, . . . , n (5.5.1) by Lemma A.2.
By the definition, we have
J k 1 = 1 k k z=1 (−1) z−1 J z J k−z 1 (5.5.2)
since we assume Q = 0. By the induction on k together with (5.5.1), (5.5.2) and (A.1.1), we can show that J k 1 · v = e k (γ 1 , . . . , γ n )v for k = 1, . . . , n,
(5.5.3)
where e k (γ 1 , . . . , γ n ) = 1≤i 1 <i 2 <···<i k ≤n γ i 1 γ i 2 . . . γ i k .
By the induction on t, we prove that
u t = p t (γ 1 , . . . , γ n ) (t > 0). (5.5.4)
If t ≤ n, (5.5.4) follows from (5.5.1). If t > n, by Lemma 5.4 in the case where s = 1, we have
u t v = J (t−n)+n · v = n−1 k=0 (−1) n−k+1 J (t−n)+k J n−k 1 · v.
By the assumption of the induction together with (5.5.3) and (A.1.2), we have u t v = n−1 k=0 (−1) n−k+1 p t−n+k (γ 1 , . . . , γ n )e n−k (γ 1 , . . . , γ n )v = p t (γ 1 , . . . , γ n )v.
(ii). Assume that Q = 0. For u 0 , u 1 , . . . , u n , there exist β, γ 1 , . . . , γ n ∈ C such that u 0 = n + β and u k = p k (γ 1 , . . . , γ n ) + Q −k β for k = 1, . . . , n (5.5.5) by Lemma A.2.
By the induction on k, we prove that J k 0 · v = e k (θ 1 , θ 2 , . . . , θ n )v for k = 1, . . . , n, (5.5.6) where θ i = 1 − Qγ i (1 ≤ i ≤ n) and e k (θ 1 , . . . , θ n ) = 1≤i 1 <i 2 <···<i k ≤n θ i 1 θ i 2 . . . θ i k .
In the case where k = 1, we have J 1 0 · v = (J 0 + (−Q)J 1 ) · v. Then we have J 1 0 · v = e 1 (θ 1 , . . . , θ n )v by (5.5.5).
In the case where 1 < k ≤ n, by the definition, we have
J k 0 · v = 1 k k z=1 (−1) z−1 z w=0 z w (−Q) w J w J k−z 0 · v.
since J 1 (t−n−1)+w = J (t−n−1)+w + (−Q)J (t−n−1)+w+1 . Then, by (5.5.11) and the assumption of the induction, we have
n w=0 n w (−Q) w J 1 (t−n−1)+w · v = (−Q) n+1 J t · v + n w=0 n + 1 w (−Q) w p (t−n−1)+w (γ 1 , . . . , γ n ) + Q −((t−n−1)+w) β v (5.5.12) and k w=0 k w (−Q) w J 1 (t−n−1)+w · v = k+1 w=0 k + 1 w (−Q) w p (t−n−1)+w (γ 1 , . . . , γ n ) + Q −((t−n−1)+w) β v (5.5.13)
for k = 0, 1, . . . , n − 1. Moreover, by the direct calculations, we have k+1 w=0 k + 1 w (−Q) w p (t−n−1)+w (γ 1 , . . . , γ n ) + Q −((t−n−1)+w) β = p (γ) k+1 (θ 1 , . . . , θ n ) (5.5.14)
for k ≥ 0, where p 2 + · · · + γ t−n−1 n θ k+1 n . Then, by (5.5.10), (5.5.12), (5.5.13) and (5.5.14), we have
(−Q) n+1 J t · v − (−Q) n+1 p t (γ 1 , . . . , γ n ) + Q −t β v + p (γ)
n+1 (θ 1 , . . . , θ n ) = n−1 k=0 (−1) n−k+1 e n−k (θ 1 , . . . , θ n )p (γ) k+1 (θ 1 , . . . , θ n ).
Applying (A.3.2) to the right-hand side, we have
J t · v = p t (γ 1 , . . . , γ n ) + Q −t β v.
5.6. By Lemma 5.2 and Proposition 5.5, we see that the highest weight u = (u t ) t≥0 of a simple highest weight U(sl
u 0 = n if Q = 0 n + β if Q = 0,
u t = p t (γ 1 , γ 2 , . . . , γ n ) if Q = 0, p t (γ 1 , γ 2 , . . . , γ n ) + Q −t β if Q = 0 (5.6.1) for some n ∈ Z ≥0 and β, γ 1 , γ 2 , . . . , γ n ∈ C if L(u) is finite dimensional. Let C[x] be the polynomial ring over C with the indeterminate variable x, and let C[x] monic be the subset of C[x] consisting of monic polynomials. We define the set C[x] Q monic by
C[x] Q monic = C[x] monic if Q = 0, {ϕ ∈ C[x] monic | Q −1 is not a root of ϕ} if Q = 0. Recall that B Q = {0} if Q = 0, C if Q = 0.
We define the map
C[x] Q monic × B Q → t≥0 C, (ϕ, β) → u Q (ϕ, β) = (u Q (ϕ, β) t ) t≥0 (5.6.2) by u Q (ϕ, β) t = deg ϕ + β if t = 0, p t (γ 1 , γ 2 , . . . , γ n )
if t > 0 and Q = 0, p t (γ 1 , γ 2 , . . . , γ n ) + Q −t β if t > 0 and Q = 0, when ϕ = (x − γ 1 )(x − γ 2 ) . . . (x − γ n ). We see that the map (5.6.2) is injective, and it gives a bijection between C[x] Q monic × B Q and the set of highest weight u = (u t ) ≥0 satisfying (5.6.1), where we note that p t (γ 1 , . . . , γ n , Q −1 , . . . , Q −1 k ) + Q −t β = p t (γ 1 , . . . , γ n ) + Q −t (β + k).
Then we have the following corollary of Lemma 5.2 and Proposition 5.5.
Corollary 5.7. Any finite dimensional simple U(sl Q 2 [x])-module is isomorphic to L(u Q (ϕ, β)) for some (ϕ, β) ∈ C[x] Q monic × B Q . Moreover, L(u Q (ϕ, β)) ∼ = L(u Q (ϕ ′ , β ′ )) if (ϕ, β) = (ϕ ′ , β ′ ).
5.8. Recall the evaluation modules from the paragraph 1.5. Let L(2) be the twodimensional simple U(sl 2 )-module, and v 0 ∈ L(2) be a highest weight vector. We consider the evaluation module L(2) evγ at γ ∈ C, then we see that ϕ,β) . Then (5.8.1) and (5.8.2) imply that N ′ (ϕ,β) is a highest weight module of highest weight u Q (ϕ, β), and N ′ (ϕ,β) / rad N ′ (ϕ,β) is isomorphic to the simple highest weight module L(u Q (ϕ, β)). From the construction, L(u Q (ϕ, β)) ∼ = N ′ (ϕ,β) / rad N ′ (ϕ,β) is finite dimensional for each (ϕ, β) ∈ C[x] Q monic × B Q . Combining with Corollary 5.7, we have the following classification of finite dimensional simple U(sl
X + t · v 0 = 0, J t · v 0 = γ t v 0 (t ≥ 0) in L(2) ev γ . For (ϕ = (x − γ 1 )(x − γ 2 ) . . . (x − γ n ), β) ∈ C[x] Q monic × B Q , we consider the U(sl Q 2 [x])-module N (ϕ,β) = L(2) ev γ 1 ⊗ L(2) ev γ 2 ⊗ · · · ⊗ L(2) ev γn ⊗ L β , where L β is the 1-dimensional U(sl Q 2 [x])-module given in the paragraph 5.1. Let v (k) 0 ∈ L(2) evγ k (1 ≤ k ≤ n) be a highest weight vector, and L β = Cw 0 . Put v (ϕ,β) = v (1) 0 ⊗ v (2) 0 ⊗ · · · ⊗ v (n) 0 ⊗ w 0 . Then, for t ≥ 0, we have X + t · v (ϕ,β) = 0 (5.8.1) and J t · v (ϕ,β) = (n + β)v (ϕ,β) if t = 0, p t (γ 1 , γ 2 , . . . , γ n )v (ϕ,β) if t > 0 and Q = 0, (p t (γ 1 , γ 2 , . . . , γ n ) + Q −t β)v (ϕ,β) if t > 0 and Q = 0. (5.8.2) Let N ′ (ϕ,β) be the U(sl Q 2 [x])-submodule of N (ϕ,β) generated by v (L(u Q (ϕ, β)) ∼ = L(u Q (ϕ ′ , β ′ )) ⇔ (ϕ, β) = (ϕ ′ , β ′ ) for (ϕ, β), (ϕ ′ , β ′ ) ∈ C[x] Q monic × B Q . Moreover, {L(u Q (ϕ, β)) | (ϕ, β) ∈ C[x] Q monic × B Q }
gives a complete set of isomorphism classes of finite dimensional simple U(sl Remark 5.10. If Q = 0, the evaluation module L(2) ev Q −1 at Q −1 is not simple. Recall that L(2) is the two dimensional simple U(sl 2 )-module with a highest weight vector v 0 . Put v 1 = f · v 0 . Then we see that U(sl
= (u i,t ) ∈ m−1 i=1 t≥0 C.
Thus, it is enough to classify the highest weight u such that L(u) is finite dimensional.
6.1. 1-dimensional representations. First, we consider the 1-dimensional representations of sl Q m [x]. For each i = 1, 2, . . . , m−1, by checking the defining relations, we have the homomorphism of algebras
ι i : U(sl Q i 2 [x]) → U(sl Q m [x]) by X ± t → X ± i,t , J t → J i,t (t ≥ 0). (6.1.1) Let L = Cv be a 1-dimensional U(sl Q m [x])-module. For each i = 1, 2, . . . , m − 1, when we regard L as a U(sl Q i 2 [x])-module through the homomorphism ι i , we see that L is isomorphic to L β i for some β i ∈ B Q i by Lemma 5.2. Thus, we have X ± i,t · v = 0, J i,t · v = 0 if Q i = 0, Q −t i β i v if Q i = 0 (1 ≤ i ≤ m − 1, t ≥ 0) (6.1.2) for some β = (β i ) 1≤i≤m−1 ∈ m−1 i=1 B Q i .
On the other hand, by checking the defining relations, we can define the 1dimensional U(sl Q m [x])-module L β = Cv by (6.1.2) for each β = (β i ) ∈ m−1 i=1 B Q i . Now we proved the following lemma.
Lemma 6.2. Any 1-dimensional U(sl Q m )-module is isomorphic to L β for some β ∈ m−1 i=1 B Q i . 6.3. For u = (u i,t ) ∈ m−1 i=1 t≥0 C, let v 0 be a highest weight vector of the simple highest weight U(sl Q m [x])-module L(u). When we regard L(u) as a U(sl Q i 2 [x]
)module through the homomorphism ι i in (6.1.1) for each i = 1, . . . , m−1, we see that the U(sl
Q i 2 [x])-submodule of L(u) generated by v 0 is a highest weight U(sl Q i 2 [x])- module of highest weight u i = (u i,t ) t≥0 ∈ t≥0 C with the highest weight vector v 0 . Then, if L(u) is finite dimensional, we see that u i = u Q i (ϕ i , β i ) for some (ϕ i , β i ) ∈ C[x]
Q i monic × B Q i by Theorem 5.9 (or Corollary 5.7).
For (ϕ, β) = ((ϕ i , β i )) 1≤i≤m−1 ∈ m−1 i=1 C[x] Q i monic × B Q i , we define u Q (ϕ, β) = (u Q (ϕ, β) i,t ) 1≤i≤m−1, t≥0 ∈ m−1 i=1 t≥0 C by u Q (ϕ, β) i,t = deg ϕ i + β i if t = 0, p t (γ i,1 , γ i,2 , . . . , γ i,n i ) if t > 0 and Q i = 0, p t (γ i,1 , γ i,2 , . . . , γ i,n i ) + Q −t i β i if t > 0 and Q i = 0, (6.3.1) when ϕ i = (x − γ i,1 )(x − γ i,2 ) . . . (x − γ i,n i ) (1 ≤ i ≤ m − 1). Then we have that (u Q (ϕ, β) i,t ) t≥0 = u Q i (ϕ i , β i )
for each i = 1, 2, . . . , m − 1. From the definition, we see that
u Q (ϕ, β) = u Q (ϕ ′ , β ′ ) ⇔ (ϕ, β) = (ϕ ′ , β ′ ) for (ϕ, β), (ϕ ′ , β ′ ) ∈ m−1 i=1 C[x] Q i monic × B Q i . By the above argument, any fi- nite dimensional simple U(sl Q m [x])-module is isomorphic to L(u Q (ϕ, β)) for some (ϕ, β) ∈ m−1 i=1 C[x] Q i monic × B Q i . On the other hand, for each (ϕ, β) ∈ m−1 i=1 C[x] Q i monic × B Q i , we can construct a finite dimensional highest weight U(sl Q m [x]
)-module of highest weight u Q (ϕ, β) as follows.
Let ω j (1 ≤ j ≤ m − 1) be the fundamental weight of sl m , and L(ω j ) be the simple highest weight U(sl m )-module of highest weight ω j . Let v 0 ∈ L(ω j ) be a highest weight vector, then we have e i · v 0 = 0 and H i · v 0 = δ ij v 0 (1 ≤ i ≤ m − 1) by the definition. Recall that L(ω j ) ev γ is the evaluation module of L(ω j ) at γ ∈ C. From the definition, we see that
X + i,t · v 0 = 0, J i,t · v 0 = δ ij γ t v 0 (1 ≤ i ≤ m − 1, t ≥ 0) (6.3.2) in L(ω j ) evγ . For (ϕ, β) = ((ϕ i , β i )) 1≤i≤m−1 ∈ m−1 i=1 C[x] Q i monic × B Q i , we consider the U(sl Q m [x])-module N (ϕ,β) = m−1 j=1 n j k=1 L(ω j ) evγ j,k ⊗ L β ,
where n j and γ j,k (1 ≤ k ≤ n j ) are determined by ϕ j = (x − γ j,1 )(x − γ j,2 ) . . . (x − γ j,n j ) for each j = 1, 2, . . . , m − 1, and β = (
β i ) 1≤i≤m−1 . Let v (j,k) 0 ∈ L(ω j ) ev γ j,k (1 ≤ j ≤ m − 1, 1 ≤ k ≤ n j ) be a highest weight vector, and L β = Cw 0 . Put v (ϕ,β) = (⊗ m−1 j=1 ⊗ n j k=1 v (j,k) 0
) ⊗ w 0 ∈ N (ϕ,β) , then we have
X + i,t · v (ϕ,β) = 0, J i,t · v (ϕ,β) = u Q (ϕ, β) i,t v (ϕ,β) (1 ≤ i ≤ m − 1, t ≥ 0) (6.3.3) by (6.3.2). Let N ′ (ϕ,β) be the U(sl Q m [x]
)-submodule of N (ϕ,β) generated by v (ϕ,β) . Then (6.3.3) implies that N ′ (ϕ,β) is a finite dimensional highest weight module of highest weight u Q (ϕ, β). Then we obtain the following classification of finite dimensional simple U(sl Q m [x])-modules. β)) of highest weight u Q (ϕ, β) is finite dimensional, and we have that
Theorem 6.4. For (ϕ, β) ∈ m−1 i=1 C[x] Q i monic × B Q i , the highest weight simple U(sl Q m [x])-module L(u Q (ϕ,L(u Q (ϕ, β)) ∼ = L(u Q (ϕ ′ , β ′ )) ⇔ (ϕ, β) = (ϕ ′ , β ′ ) for (ϕ, β), (ϕ ′ , β ′ ) ∈ m−1 i=1 C[x] Q i monic × B Q i Moreover, {L(u Q (ϕ, β)) | (ϕ, β) ∈ m−1 i=1 C[x] Q i monic × B Q i }7.1. 1-dimensional representations. For β = (β i ) 1≤i≤m−1 ∈ m−1 i=1 B Q i , by checking the defining relations, we can define the 1-dimensional U(gl Q m [x])-module L β = Cv by X ± i,t · v = 0, J i,t · v = 0 if Q i = 0, Q −t i β i v if Q i = 0 (1 ≤ i ≤ m − 1, t ≥ 0), I j,t · v = m−1 k=j J k,t · v (1 ≤ j ≤ m − 1, t ≥ 0), I m,t · v = 0 (t ≥ 0).
Note that J j,t = I j,t − I j+1,t in U(gl Q m [x]), we see that L β ∼ = L β as U(sl Q m [x])modules when we restrict the action on L β to U(sl Q m [x]) through the injective homomorphism Υ in the proposition 1.4 (iii).
For h = (h t ) t≥0 ∈ t≥0 C, we can also define the 1-dimensional
U(gl Q m [x])- module L h = Cv by X ± i,t · v = 0, I j,t · v = h t v (1 ≤ i ≤ m − 1, 1 ≤ j ≤ m, t ≥ 0).
We see that L h ∼ = L 0 as U(sl Q m [x])-modules when we restrict the action on L h to
U(sl Q m [x]) where 0 = (0) 1≤i≤m−1 ∈ m−1 i=1 B Q i (i.e.
L 0 is the trivial representation). Then we have the following classification of 1-dimensional U(gl Q m )-modules.
Lemma 7.2. Any 1-dimensional U(gl Q m [x])-module is isomorphic to L β ⊗ L h for some β ∈ m−1 i=1 B Q i and h ∈ t≥0 C. We have that L β ⊗ L h ∼ = L β ′ ⊗ L h ′ ⇔ (β, h) = (β ′ , h ′ ). Moreover, we see that L β ⊗ L h ∼ = L β as U(sl Q m [x])-modules when we restrict the action on L β ⊗ L h to U(sl Q m [x]).
Proof. Let L = Cv be a 1-dimensional U(gl Q m )-module. By restricting the action on L to U(sl Q m ) through the injective homomorphism Υ in the proposition 1.4 (iii), we have
X ± i,t · v = 0, J i,t · v = (I i,t − I i+1,t ) · v = 0 if Q i = 0, Q −t i β i v if Q i = 0 (1 ≤ i ≤ m − 1, t ≥ 0) (7.2.1)
for some β = (β i ) ∈ m−1 i=1 B Q i by Lemma 6.2. On the other hand, for t ∈ Z ≥0 , there exists h t ∈ C such that I m,t · v = h t v (7.2.2) since dim L = 1. Then (7.2.1) and (7.2.2) imply taht
I j,t · v = m−1 k=j J k,t + h t · v (1 ≤ j ≤ m − 1, t ≥ 0), I m,t · v = h t v (t ≥ 0).
Then we see that L ∼ = L β ⊗ L h . The remaining statements are clear.
7.3.
For u = ( u j,t ) ∈ m j=1 t≥0 C, let v 0 be a highest weight vector of the simple highest weight U(gl Q m [x])-module L( u). By restricting the action on L( u) to U(sl Q m [x]), Theorem 6.4 implies that u i,t − u i+1,t = u Q (ϕ, β) i,t (1 ≤ i ≤ m − 1, t ≥ 0) (7.3.1) for some (ϕ, β) ∈ m−1 i=1 (C[x] Q i monic × B Q i ) if L( u) is finite dimensional. For t ∈ Z ≥0 , let h t ∈ C be such that u m,t = h t . (7.3.2) By (7.3.1) and (7.3.2), we have From the definition, we see that
u j,t = m−1 k=j u Q (ϕ, β) k,t + h t (1 ≤ j ≤ m − 1, t ≥ 0), u m,t = h t (t ≥ 0) for some (ϕ, β) ∈ m−1 i=1 (C[x] Q i monic × B Q i ) and h = (h t ) ∈ t≥0 C if L( u) is finite dimensional. For (ϕ, β, h) = ((ϕ i , β i ) 1≤i≤m−1 , (h t ) t≥0 ) ∈ m−1 i=1 (C[x] Q i monic × B Q i ) × t≥0 C, we define u Q (ϕ, β, h) = ( u Q (ϕ, β, h) j,t ) ∈u Q (ϕ, β, h) = u Q (ϕ ′ , β ′ , h ′ ) ⇔ (ϕ, β, h) = (ϕ ′ , β ′ , h ′ )
for (ϕ, β, h), (ϕ ′ , β ′ , h ′ ) ∈ m−1 i=1 (C[x] Q i monic ×B Q i )× t≥0 C. By the above argument, any finite dimensional simple U(gl Q m [x])-module is isomorphic to L( u Q (ϕ, β, h)) for some (ϕ, β, h)
∈ m−1 i=1 (C[x] Q i monic × B Q i ) × t≥0 C.
On the other hand, for each (ϕ, β, h) ∈ m−1 i=1 (C[x] Q i monic ×B Q i )× t≥0 C, we can construct a finite dimensional highest weight U(gl Q m [x])-module of highest weight u Q (ϕ, β, h) as follows.
Let P = m i=1 Zε i be the weight lattice of gl m . Put ω l = ε 1 + ε 2 + · · · + ε l for l = 1, 2, . . . , m−1. Let L( ω l ) be the simple highest weight U(gl m )-module of highest weight ω l , and v 0 ∈ L( ω l ) be a highest weight vector. Then, we have e i · v 0 = 0 (1 ≤ i ≤ m − 1) and K j · v 0 = v 0 if 1 ≤ j ≤ l, 0 if l < j ≤ m.
Recall that L( ω l ) ev γ is the evaluation module of L( ω l ) at γ ∈ C. From the definition, we see that
X + i,t · v 0 = 0 (1 ≤ i ≤ m − 1, t ≥ 0), I j,t · v 0 = γ t v 0 if 1 ≤ j ≤ l, 0 if l < j ≤ m (t ≥ 0) (7.3.3)
in L( ω l ) evγ . (We remark that L( ω l ) evγ ∼ = L(ω l ) evγ as U(sl Q m [x])-modules when we restrict the action on L( ω l ) ev γ to U(sl Q m [x]).)
For (ϕ, β, h) = ((ϕ i , β i ) 1≤i≤m−1 , (h t ) t≥0 ) ∈ m−1 i=1 (C[x] Q i monic × B Q i ) × t≥0 C, we consider the U(gl Q m [x])-module N (ϕ,β,h) = m−1 l=1 n l k=1 L( ω l ) evγ l,k ⊗ L β ⊗ L h ,
where n l and γ l,k (1 ≤ k ≤ n l ) are determined by ϕ l = (x−γ l,1 )(x−γ l,2 ) . . . (x−γ l,n l ) for each l = 1, 2, . . . , m − 1, and we put β = (β i ) 1≤i≤m−1 and h = (h t ) t≥0 . Let v (l,k) 0 ∈ L( ω l ) evγ l,k (1 ≤ l ≤ m − 1, 1 ≤ k ≤ n l ) be a highest weight vector, L β = Cw 0 and L h = Cz 0 . Put v (ϕ,β,h) = (⊗ m−1 l=1 ⊗ n l k=1 v (l,k) 0 ) ⊗ w 0 ⊗ z 0 ∈ N (ϕ,β) , then we have X + i,t · v (ϕ,β,h) = 0, I j,t · v (ϕ,β,h) = u Q (ϕ, β, h) j,t v (ϕ,β,h) (7.3.4) for 1 ≤ i ≤ m − 1, 1 ≤ j ≤ m and t ≥ 0 by (7.3.3). Let N ′ (ϕ,β,h) be the U(gl Q m [x])submodule of N (ϕ,β,h) generated by v (ϕ,β,h) . Then (7.3.4) implies that N ′ (ϕ,β,h) is a finite dimensional highest weight module of highest weight u Q (ϕ, β, h). Now we obtain the following classification of finite dimensional simple U(gl Q m [x])-modules.
Theorem 7.4. For (ϕ, β, h) ∈ m−1 i=1
C[x]
Q i monic ×B Q i × t≥0 C, the highest weight simple U(gl Q m [x])-module L( u Q (ϕ, β, h)) of highest weight u Q (ϕ, β, h) is finite dimensional, and we have that
L( u Q (ϕ, β, h)) ∼ = L( u Q (ϕ ′ , β ′ , h ′ )) ⇔ (ϕ, β, h) = (ϕ ′ , β ′ , h ′ ) for (ϕ, β, h), (ϕ ′ , β ′ , h ′ ) ∈ m−1 i=1 C[x] Q i monic × B Q i × t≥0 C. Moreover, {L( u Q (ϕ, β, h)) | (ϕ, β, h) ∈ m−1 i=1 C[x] Q i monic × B Q i × t≥0 C}
gives a complete set of isomorphism classes of finite dimensional simple U(gl Q m [x])modules.
We also have the following corollary.
We also put e (b) 0 = 1. Note that e (b) k (x 1 , . . . , x n ) = 0 if k > n. Put 1 = (1, 1, . . . , 1), then we have e (1) k (x 1 , . . . , x n ) = ke k (x 1 , . . . , x n ) and p (1) k (x 1 , . . . , x n ) = p k (x 1 , . . . , x n ). We consider the generating functions E(t), E (b) (t) and P (b) (t) by Then, we have
E(t) =E(t) = n i=1 (1 + x i t), P (b) (t) = n i=1 b i x i 1 + x i t and P (b) (t)E(t) = n i=1 b i x i n j=1,j =i (1 + x i t) = E (b) (t).
This implies that, for k ≥ 0,
1. 6 .
6In the rest of this section, we give a relation with the Lie algebra g Q (m) introduced in [W, Definition 2.2]. Let m = (m 1 , . . . , m r ) be an r-tuple of positive integers such that r k=1 m k = m. Put Γ (m) = {(i, k) | 1 ≤ i ≤ m k , 1 ≤ k ≤ r} and Γ ′ (m) = Γ (m) \ {(m r , r)}. Then we have the bijective map ζ : Γ (m) → {1, 2, . . . , m} such that (i,
Lemma 2 . 3 .
23For a finite dimensional U(sl Q m [x])-module M = 0, there exists v 0 ∈ M (v 0 = 0) satisfying the following conditions:
t X −(c;p|k;s+t) s = ( * 1) + ( * 2) + ( * 3) − 2( * 4) − ( * 5).
(
−1) k+l X −(c;p|k;s+t)
−1) k+l X −(c;p|k;s+t)
Q2
[x])-module L(u) has the form
.
For (ϕ, β) ∈ C[x] Q monic × B Q , the highest weight simple U(sl Q 2 [x])module L(u Q (ϕ,β)) of highest weight u Q (ϕ, β) is finite dimensional, and we have that
Q2
[x]) · v 1 = Cv 1 is a proper U(sl Q 2 [x])-submodule of L(2) ev Q −1 . Moreover, we have L(2) ev Q −1 /Cv 1 ∼ = L 1 and Cv 1 ∼ = L −1 as U(sl Q 2 [x])-modules. § 6. Finite dimensional simple U(sl Q m [x])-modules In this section, we classify the finite dimensional simple U(sl Q m [x])-modules. By Proposition 2.6, any finite dimensional simple U(sl Q m [x])-module is isomorphic to the simple highest weight module L(u) of highest weight u
(ϕ, β) k,t + h t if 1 ≤ j ≤ m − 1 and t ≥ 0, h t if j = m and t ≥ 0.
k≥0 e k (x 1 , . . . , x n )t k ∈ Z[x 1 , . . . , x n ][b 1 , . . . , b n ][[t]], x 1 , . . . , x n )t k ∈ Z[x 1 , . . . , x n ][b 1 , . . . , b n ][[t]], x 1 , . . . , x n )t k ∈ Z[x 1 , . . . , x n ][b 1 , . . . , b n ][[t]].
x 1 , . . . , x n )e k−z (x 1 , . . . , x n ). (A.3.1)In the case where k = n, x 1 , . . . , x n )e n−z (x 1 , . . . , x n ) x 1 , . . . , x n )e n−z (x 1 , . . . , x n ) = p(b) n+1 (x 1 , . . . , x n ). (A.3.2)
W] is isomorphic to gl Q m [x] under a suitable choice of deformation parameters Q (Lemma 1.7). 0.2. The differences of the representation theory of sl Q m [x] from one of sl m [x] appear in the following two points. The deformed current Lie algebra sl Q m [x] has a family of 1-dimensional representations {L β
modules when we restrict the action. Acknowledgements: This work was supported by JSPS KAKENHI Grant Number JP16K17565. § 1. Deformed current Lie algebras sl Q m [x] and gl Q m [x] In this section, we give a definition of deformed current Lie algebras sl Q m [x] and gl Q m [x], and also give some basic facts. The definition of gl Q m [
gives a complete set of isomorphism classes of finite dimensional simple U(sl Q m [x])modules. § 7. Finite dimensional simple U(gl Q m [x])-modules In this section, we classify the finite dimensional simple U(gl Q m [x])-modules. By Proposition 1.4 (iii), sl Q m [x] is a Lie subalgebra of gl Q m [x]. The difference of representations of gl Q m [x] from one of sl Q m [x] is given by the family of 1-dimensional U(gl Q m [x])-modules { L h | h ∈ t≥0 C}. We remark that L h (h ∈ t≥0 C) is isomorphic to the trivial representation as a U(sl Q m [x])-module when we restrict the action.
(t−n−1)+w · v = n−1 k=0 (−1) n−k+1 e n−k (θ 1 , . . . , θ n ) k w=0 k w (−Q) w J 1 (t−n−1)+w · v.(5.5.10)On the other hand, for k ≥ 0, we havek w=0 k w (−Q) w J 1 (t−n−1)+w = k+1 w=0k + 1 w (−Q) w J (t−n−1)+w (5.5.11)
Applying the assumption of the induction to the right-hand side, we havewhere we note that J 0 0 = e 0 (θ 1 , . . . , θ n ) = 1. On the other hand, by (5.5.5), we have(5.5.8)where we note that z w=0 z w (−1) w = 0. By (5.5.7) and (5.5.8) together with (A.1.1), we have (5.5.6). By the induction on t, we prove thatIf t ≤ n, (5.5.9) follows from (5.5.5). If t > n, by Lemma 5.4 in the case where s = 0, we haveBy (5.5.6) , we have). Proof. We prove that L( u Q (ϕ, β, h)) is also simple when we restrict the action to U(sl Q m [x]). Then the isomorphism follows from the definitions of u Q (ϕ, β, h) and u Q (ϕ, β).Let v 0 ∈ L( u Q (ϕ, β, h)) be a highest weight vector as the U(gl Q m [x])-module. Then we haveby the triangular decomposition in Proposition 1.4 (iv). This implies that). This is a contradiction.Appendix A. Some combinatorics A.1. Let Z[x 1 , . . . , x n ] be the ring of polynomials in independent variables x 1 , . . . , x n over Z. For k ∈ Z >0 , putNamely, p k (x 1 , . . . , x n ) is the power sum symmetric polynomial of degree k, and e k (x 1 , . . . , x n ) is the elementary symmetric polynomial of degree k. We also put e 0 (x 1 , . . . , x n ) = 1. Then, for k > 0, we havewhere we note that e s−z (x 1 , . . . , x n ) = 0 if z < s − n. Put w = z − s + n, we have n−1 w=0 (−1) n−w+1 p s−n+w (x 1 , . . . , x n )e n−w (x 1 , . . . , x n ) = p s (x 1 , . . . , x n ) (A.1.2) for s > n.Lemma A.2. For n ∈ Z >0 and u 1 , u 2 , . . . , u n ∈ C, the simultaneous equationshas a solution in C.Proof. We prove the lemma by the induction on n. In the case where n = 1, it is clear. If n > 1, the equations (A.2.1) are equivalent to the equations p 1 (x 1 , x 2 , . . . , x n−1 ) = u 1 − x n , p 2 (x 1 , x 2 , . . . , x n−1 ) = u 2 − x 2 n , . . .where we note that e n (x 1 , x 2 , . . . , x n−1 ) = 0. On the other hand, we can write e n−i (x 1 , x 2 , . . . , x n−1 ) = λ⊢n−i α λ p λ (x 1 , x 2 , . . . , x n−1 ) for some α λ ∈ C, where p λ (x 1 , x 2 , . . . , x n−1 ) = ℓ(λ) j=1 p λ j (x 1 , x 2 , . . . , x n−1 ) for λ = (λ 1 , λ 2 , . . . ) ⊢ n − i. Thus we have, x 2, . . . , x n−1 )p λ (x 1 , x 2 , . . . , x n−1 ).(Note that {p µ (x 1 , x 2 , . . . , x n−1 ) | µ ⊢ k} is not linearly independent if k ≥ n. For an example, we have p (3) (x 1 , x 2 ) = 3 2 p (2,1) (x 1 , x 2 ) − 1 2 p (1,1,1) (x 1 , x 2 ).) Then the equations (A.2.2) are equivalent to the equationsLet β n be a solution of the equation ( * 1) for the variable x n . By the assumption of the induction, the simultaneous equations p 1 (x 1 , x 2 , . . . , x n−1 ) = u 1 − β n , p 2 (x 1 , x 2 , . . . , x n−1 ) = u 2 − β 2 n , . . . p n−1 (x 1 , x 2 , . . . , x n−1 ) = u n−1 − β n−1 n for variables x 1 , x 2 , . . . , x n−1 has a solution. We denote it by (x 1 , x 2 , . . . , x n−1 ) = (β 1 , β 2 , . . . , β n−1 ). Then (x 1 , x 2 , . . . , x n ) = (β 1 , β 2 , . . . , β n ) gives a solution of (A.2.3).A.3. We consider some modifications of the formulas (A.1.1) and (A.1.2) as follows. Let b = (b 1 , . . . , b n ) be n independent variables, and we consider the ring of polynomials Z[x 1 , . . . , x n ][b 1 , . . . , b n ]. For k ∈ Z >0 , put e (b) k (x 1 , . . . , x n ) = 1≤i 1 <i 2 <···<i k ≤n k (x 1 , . . . , x n ) = b 1 x k 1 + b 2 x k 2 + · · · + b n x k n ∈ Z[x 1 , . . . , x n ][b 1 , . . . , b n ].
Integrable representations of affine Lie algebras. V Chari, Invent. Math. 85V. Chari, Integrable representations of affine Lie algebras, Invent. Math. 85 (1986), 317- 335.
New unitary representations of loop groups. V Chari, A Pressley, Math. Ann. 275V. Chari and A. Pressley, New unitary representations of loop groups, Math. Ann. 275 (1986), 87-104.
I G Macdonald, Symmetric Functions and Hall Polynomials. Oxford Univ. Press2nd editionI.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford Univ. Press, 1995.
New Realization of Cyclotomic q-Schur Algebras. K Wada, Publ. RIMS Kyoto Univ. 52K. Wada, New Realization of Cyclotomic q-Schur Algebras, Publ. RIMS Kyoto Univ. 52 (2016), 497-555.
Asahi 3-1-1, Matsumoto 390-8621. Department of Mathematics, Faculty of Science, Shinshu UniversityJapan E-mail address: [email protected] of Mathematics, Faculty of Science, Shinshu University, Asahi 3- 1-1, Matsumoto 390-8621, Japan E-mail address: [email protected]
| []
|
[
"Moduli stabilization with open and closed string fluxes",
"Moduli stabilization with open and closed string fluxes"
]
| [
"Ignatios Antoniadis \nDepartment of Physics\nCERN -Theory Division\nCH-1211Geneva 23Switzerland\n\nOn leave from CPHT (UMR du CNRS 7644), Ecole Polytechnique\nF-91128Palaiseau Cedex\n\nIntroduction\n\n",
"Alok Kumar \nInstitute of Physics\n751 005BhubaneswarIndia\n",
"‡ ",
"Tristan Maillard \nDepartment of Physics\nCERN -Theory Division\nCH-1211Geneva 23Switzerland\n\nInstitut für Theoretische Physik\nETH Hönggerberg\nCH-8093ZürichSwitzerland\n\nIntroduction\n\n",
"Tristan Maillard@cern ",
"Ch "
]
| [
"Department of Physics\nCERN -Theory Division\nCH-1211Geneva 23Switzerland",
"On leave from CPHT (UMR du CNRS 7644), Ecole Polytechnique\nF-91128Palaiseau Cedex",
"Introduction\n",
"Institute of Physics\n751 005BhubaneswarIndia",
"Department of Physics\nCERN -Theory Division\nCH-1211Geneva 23Switzerland",
"Institut für Theoretische Physik\nETH Hönggerberg\nCH-8093ZürichSwitzerland",
"Introduction\n"
]
| []
| We study the stabilization of all closed string moduli in the T 6 /Z 2 orientifold, using constant internal magnetic fields and 3-form fluxes that preserve N = 1 supersymmetry in four dimensions. We first analyze the stabilization of Kähler class and complex structure moduli by turning on magnetic fluxes on different sets of D9 branes that wrap the internal space T 6 /Z 2 . We present explicit consistent string constructions, satisfying in particular tadpole cancellation, where the radii can take arbitrarily large values by tuning the winding numbers appropriately. We then show that the dilaton-axion modulus can also be fixed by turning on closed string constant 3-form fluxes, consistently with the supersymmetry preserved by the magnetic fields, providing at the same time perturbative values for the string coupling. Finally, several models are presented combining open string magnetic fields that fix part of Kähler class and complex structure moduli, with closed string 3-form fluxes that stabilize the remaining ones together with the dilaton. * On leave from CPHT (UMR du CNRS 7644), Ecole Polytechnique, F-91128 Palaiseau Cedex. † [email protected] ‡ [email protected] § [email protected] consistently. One may think that the results of this work can be obtained simply by a Tduality from the toroidal case with O9 planes analyzed in[3]. This is indeed true only for invertible magnetic field matrices. On the contrary, if the magnetic flux has a zero eigenvalue, in the T-dual theory it becomes infinite and the analysis does not go through. Thus, the study of moduli stabilization in the T 6 /Z 2 orientifold case is non-trivial and cannot be obtained by a T-duality from the toroidal analysis of ref.[3].Actually, we are interested to find an explicit solution, where the toroidal geometry of T 6 is fixed to a factorized form, T 6 ≡ (T 2 ) 3 . Thus, one needs in particular to set the off-diagonal components of the complex structure to zero, implying the presence of magnetic fluxes with 1 For partial Kähler moduli stabilization, see also[4,5]. | null | [
"https://arxiv.org/pdf/hep-th/0505260v2.pdf"
]
| 8,556,461 | hep-th/0505260 | 9700f7a33c5c0f49842c2f369c42dcf8b988daa4 |
Moduli stabilization with open and closed string fluxes
Jul 2005
Ignatios Antoniadis
Department of Physics
CERN -Theory Division
CH-1211Geneva 23Switzerland
On leave from CPHT (UMR du CNRS 7644), Ecole Polytechnique
F-91128Palaiseau Cedex
Introduction
Alok Kumar
Institute of Physics
751 005BhubaneswarIndia
‡
Tristan Maillard
Department of Physics
CERN -Theory Division
CH-1211Geneva 23Switzerland
Institut für Theoretische Physik
ETH Hönggerberg
CH-8093ZürichSwitzerland
Introduction
Tristan Maillard@cern
Ch
Moduli stabilization with open and closed string fluxes
Jul 2005arXiv:hep-th/0505260v2 7
We study the stabilization of all closed string moduli in the T 6 /Z 2 orientifold, using constant internal magnetic fields and 3-form fluxes that preserve N = 1 supersymmetry in four dimensions. We first analyze the stabilization of Kähler class and complex structure moduli by turning on magnetic fluxes on different sets of D9 branes that wrap the internal space T 6 /Z 2 . We present explicit consistent string constructions, satisfying in particular tadpole cancellation, where the radii can take arbitrarily large values by tuning the winding numbers appropriately. We then show that the dilaton-axion modulus can also be fixed by turning on closed string constant 3-form fluxes, consistently with the supersymmetry preserved by the magnetic fields, providing at the same time perturbative values for the string coupling. Finally, several models are presented combining open string magnetic fields that fix part of Kähler class and complex structure moduli, with closed string 3-form fluxes that stabilize the remaining ones together with the dilaton. * On leave from CPHT (UMR du CNRS 7644), Ecole Polytechnique, F-91128 Palaiseau Cedex. † [email protected] ‡ [email protected] § [email protected] consistently. One may think that the results of this work can be obtained simply by a Tduality from the toroidal case with O9 planes analyzed in[3]. This is indeed true only for invertible magnetic field matrices. On the contrary, if the magnetic flux has a zero eigenvalue, in the T-dual theory it becomes infinite and the analysis does not go through. Thus, the study of moduli stabilization in the T 6 /Z 2 orientifold case is non-trivial and cannot be obtained by a T-duality from the toroidal analysis of ref.[3].Actually, we are interested to find an explicit solution, where the toroidal geometry of T 6 is fixed to a factorized form, T 6 ≡ (T 2 ) 3 . Thus, one needs in particular to set the off-diagonal components of the complex structure to zero, implying the presence of magnetic fluxes with 1 For partial Kähler moduli stabilization, see also[4,5].
Introduction
String theory is known to possess a large number of vacua which contain the basic structure of grand unified theories, and in particular of the Standard Model. However, one of the major stumbling blocks in making further progress along these lines has been the lack of a guiding principle for choosing the true ground state of the theory, thus implying the loss of predictivity. In particular, string vacua depend in general on continuous parameters, characterizing for instance the size and shape of the compactification manifold, that correspond to vacuum expectation values (VEVs) of the so-called moduli fields. These are perturbative flat directions of the scalar potential, at least as long as supersymmetry remains unbroken. It is therefore of great interest that during the last few years there has been a considerable success in fixing the string ground states, by invoking principles similar to the spontaneous symmetry breaking mechanism, now in the context of string theory. In particular, it has been realized that closed,
as well as open, string background fluxes can be turned on, fixing the VEVs of the moduli fields and therefore providing the possibility for choosing a ground state as a local isolated minimum of the scalar potential of the theory. This line of approach allows string theory to play directly a role in particle unification, predicting the strength of interactions and the mass spectrum. In particular, the string coupling becomes a calculable dynamical parameter that fixes the value of the fine structure constant and determines the Newtonian coupling in terms of the string length.
On one hand, moduli stabilization using closed string 3-form fluxes has been discussed in a great detail in the literature [1,2]. N = 1 space-time supersymmetry and various consistency requirements imply that the 3-form fluxes must satisfy the following conditions formulated on the complexified flux defined as G = F − φH, where F and H are the R-R (Ramond) and
NS-NS (Neveu-Schwarz) 3-forms, respectively, and φ is the axion-dilaton modulus: (1) The only non-vanishing components of G are of the type (2, 1), pointing along two holomorphic and one anti-holomorphic directions, implying that its (1,2), (3,0) and (0, 3) components are zero and (2) G is primitive, requiring J ∧ G = 0 with J being the Kähler form. This approach has been applied to orientifolds of both toroidal models as well as of Calabi-Yau compactifications. However, a drawback of the method is that the Kähler class moduli remain undetermined due to the absence of an harmonic (1, 0) form on Calabi-Yau spaces, implying that the constraint J ∧ G = 0 is trivially satisfied. In the toroidal orientifold case, it turns out that one is able to stabilize the Kähler class moduli only partially, but in particular the overall volume remains always unfixed.
On the other hand, in [3] two of the present authors have shown that both complex structure and Kähler class moduli can be stabilized in the type I string theory compactified down to four dimensions. 1 This is achieved by turning on magnetic fluxes which couple to various D9 branes, that wrap on T 6 , through a boundary term in the open string world-sheet action. The latter modifies the open string Hamiltonian and its spectrum, and puts constraints on the closed string background fields due to their couplings to the open string action. More precisely, supersymmetry conditions in the presence of branes with magnetic fluxes, together with conditions which define a meaningful world-volume theory, put restrictions on the values of the moduli and fix them to specific constant values. This also breaks the original N = 4 supersymmetry of the compactified type I theory to an N = 1 supersymmetric gauge theory with a number of chiral multiplets. A detailed analysis of the final spectrum, as well as other related issues have been discussed in [6].
In the simplest case, the above model has only O9 orientifold planes and several stacks of magnetized D9 branes. The main ingredients for moduli stabilization are then: (1) the introduction of "oblique" magnetic fields, needed to fix the off-diagonal components of the metric, that correspond to mutually non-commuting matrices similar to non-abelian orbifolds; (2) the property that magnetized D9 branes lead to negative 5-brane tensions; and (3) the non-linear part of Dirac-Born-Infeld (DBI) action which is needed to fix the overall volume. Actually, the first two ingredients are also necessary for satisfying the 5-brane tadpole cancellation without adding D5 branes or O5 planes, while the last two properties are only valid in four-dimensional compactifications (and not in higher dimensions).
In this paper we construct N = 1 supersymmetric models with stabilized moduli in T 6 /Z 2 orientifold compactifications of type IIB theory, following the earlier work in [3]. In the simplest case, our models have only O3 orientifold planes and several stacks of magnetized D9 branes that behave as D3 branes. The induced 7-brane tadpoles cancel without the addition of extra D7 branes or O7 planes. We write down the relevant supersymmetry requirements, and demand that the world-volume theory should be well defined. We then analyze these conditions for several situations, to examine what magnetic fluxes can be turned on along the D9 branes non-zero off-diagonal components, mixing the T 2 's as in [3]. However, unlike that case, the consistency conditions now imply that one has to simultaneously turn on certain diagonal (in T 2 's) fluxes as well. Concerning the branes with purely diagonal form of magnetic flux along the three T 2 's, we find that it is allowed to have a zero flux along one T 2 and two negative fluxes along the remaining two tori. In such situtations, it is also possible to turn on off-diagonal components of magnetic fluxes in the directions orthogonal to the T 2 with zero flux. In fact, we make use of such purely diagonal fluxes, as well as the ones with off-diagonal components, since they all provide conditions on moduli without contributing to the 3-brane tadpoles.
The restrictions we find in this paper on the possible allowed fluxes, turn out to be more restrictive than in [3]. Nevertheless, we have been able to use them for the purpose of stabilizing all the complex structure and Kähler class moduli. In fact, additional restrictions on the string construction emerge from the requirement of 7-brane and 3-brane tadpole cancellations. As mentioned above, the contribution of 7-brane tadpoles depends on the D9 brane winding around the corresponding transverse 2-cycles. It turns out that the 7-brane tadpole contribution within a stack of branes can take positive or negative values along the various 2-cycles. On the other hand, the 3-brane tadpole contribution within a stack of branes is not affected by the windings, and is restricted to be positive. We then keep them to their minimum positive value in order to have the possibility of introducing closed string 3-form fluxes as well, so as to finally fix the only remaining closed string moduli field, namely the axion-dilaton modulus. Indeed, we are able to find consistent models within this framework where the string coupling is fixed to perturbative values. By tuning appropriately the magnetic fluxes, we also find an infinite but discrete series of solutions with stabilized moduli, where some radii can take arbitrarily large values and the dilaton can be fixed at arbitrarily weak coupling. We finally present models where part of the Kähler and complex structure moduli are stabilized using the closed string 3-form flux and the other part by open string magnetic fluxes. In these cases, we are able to obtain even smaller values for the string coupling.
In this work, we do not address the issue of open string moduli stabilization. In particular, we study only vacua where gauge symmetries are unbroken. If one allows the possibility of gauge symmetry breaking, other vacua should exist where Kähler moduli mix with open string D-term flat directions and thus only one linear combination is fixed by the presence of the corresponding magnetic field [7]. In principle, the remaining directions can be also fixed by adding more magnetic fields but such an analysis goes beyond the scope of the present paper.
The rest of the paper is organized as follows. In Section 2, we write down the consistency conditions for magnetic fluxes on D9 branes in T 6 /Z 2 orientifold models, leaving unbroken N = 1 supersymmetry in four dimensions. Supersymmetry conditions are analyzed in subsection 2.1, while tadpole cancellation and positivity requirements are discussed in subsection 2.3. We also describe the general mechanism for moduli stabilization in subsections 2.2 and 2.4. The notations are the same as in ref. [3] but for self-consistency, in Appendix A, we present briefly the torus T 6 parametrization. In Section 3, we review the supersymmetry and consistency conditions of closed string 3-form fluxes and discuss the effects of turning on a non-trivial NS-NS B-field background. In Section 4, we give in advance the various brane stacks and choices for the magnetic fluxes that will be used in the examples of string constructions of the following sections. In Section 5, we present an explicit model in detail (called model-A), using twelve magnetized D9 branes, contributing the lowest possible value to the 3-brane tadpole, q 3,R = 6. We show that our choice of magnetic fields satisfies all consistency requirements, leading to a N = 1 supersymmetric vacuum where all complex structure and Kähler class moduli are fixed and the metric becomes diagonal in the internal coordinates. In the subsection 5.6, we also show that the dilaton-axion modulus is also fixed by turning on appropriate 3-form fluxes at weak values of the string coupling. Therefore, all closed string moduli get fixed. Finally, in the last subsection 5.7, we present a possible alternative based on a minimal number of nine magnetized D9 branes, leading to an infinite (but discrete) family of solutions with the same values of all geometric moduli, but with different spectrum and couplings. 2 In Section 6, we show how the above solution can be 'rescaled' to generate large values for (some of) the internal radii [8]. In Section 7, we repeat the analysis for another example (called model-B), which uses 15 magnetized D9 branes contributing 12 units of 3-brane charge, q 3,R = 12.
Many technical details of the model, such as the choice of fluxes and windings, as well as the tadpole cancellation conditions are given in Appendix B. In Section 8, we present a different model (called model-C), where closed string 3-form fluxes are also used to stabilize part of the geometric moduli, besides the dilaton. In this way, the number of magnetized branes and their contribution to the 3-brane tadpole is lower than before. Finally, Section 9 contains a brief summary of our results.
2 Magnetic fluxes and supersymmetric vacua 2.1 Condition for N = 1 supersymmetry
The presence of a constant internal magnetic field generically breaks supersymmetry by shifting the masses of the four dimensional bosons and fermions [9]. However, for suitable choice of the fluxes and moduli, a four dimensional supersymmetry can be recovered [10]. Written in the complex basis (A.4) of Appendix A where the field strength F splits in purely (anti-) holomorphic (F (0,2) ), F (2,0) and mixed F (1,1) parts, the condition for N = 1 supersymmetry in four dimensions can be written as [11]:
(iJ + F ) 3 = e iθ |g 6 + F | V 6 |g 6 | (2.1) F (2,0) = 0 , (2.2)
where V 6 is the volume form of T 6 and g 6 is its metric. Eq. (2.1) can be put in the form:
tan θ (J ∧ J ∧ F − F ∧ F ∧ F ) = J ∧ J ∧ J − J ∧ F ∧ F , (2.3)
where the wedge product A N is defined with an implicit normalization factor 1/N!. Note that only the (1, 1)-part of F contributes in this formula. Formally, (2.3) can be also written as
Im e −iθ Φ = 0 , (2.4) with Φ = (iJ + F ) ∧ (iJ + F ) ∧ (iJ + F ) . (2.5)
The constant phase θ selects which supersymmetry the magnetized brane preserves. In the case of type I string theory, the supercharges preserved by the magnetic background field is consistent with the presence of the orientifold plane O9 for the choice of θ = − π 2 . Consider on the other hand the orientifold compactification T 6 /Z 2 , where the Z 2 orientifold projection is given by ΩR(−) F L . This is a composition of the world-sheet parity Ω with the parity R on the torus T 6 : z i → −z i and the spacetime left handed fermionic number (−1) F L . The orientifold projection has 64 fixed points on T 6 , giving rise to 64 O3-planes. Each of them carries negative tension and charge and preserves a common supersymmetry with the magnetized D9 branes for the special choice of phase θ = 0 [4]. The supersymmetry condition (2.4) reduces then to the formula
J ∧ J ∧ J − J ∧ F ∧ F = 0. (2.6)
The supersymmetry condition (2.6) can also be understood in a type IIA T-dual representation in terms of the angles between different stacks of D6 branes. To illustrate this fact, let us consider a coordinate basis u k , k = 1, . . . , 6, on the torus where the metric is the identity, g kl = δ kl , and the magnetic flux is block-diagonal F = iσ 2 ⊗ (f 1 , f 2 , f 3 ). We denote the radii of the coordinates as R k . The fluxes are then quantized as
qf i = m i n i R 2i−1 R 2i , i = 1, 2, 3, (2.7)
where q is the quantum of the electric charge, m i are the first Chern numbers and n i are the winding numbers of the D9 brane around the cycles [u 2i u 2i−1 ]. The boundary conditions of the open string coordinates in this magnetic background deform the pure Neumann conditions to
∂ σ u 2i−1 − qf i ∂ τ u 2i = 0 at σ = 0 qf i ∂ τ u 2i−1 + ∂ σ u 2i = 0 at σ = π i = 1, 2, 3 (2.8)
where σ and τ are the usual world-sheet coordinates. Upon three T-dualities along the directions
u 2i , R 2i → 1/R 2i ,(2.9)
the boundary conditions are modified as
∂ σ (u 2i−1 − qf i u 2i ) = 0 at σ = 0 ∂ τ (qf i u 2i−1 + u 2i ) = 0 at σ = π i = 1, 2, 3. (2.10)
The T-dualities also map the quantized D9 brane fluxes to the D6 brane angles
qf i = m i n i R 2i−1 R 2i →f i = m i R 2i n i R 2i−1 = tan φ i . (2.11)
In fact, the first Chern numbers m i are mapped into the winding numbers of the D6 brane along the coordinates u 2i while n i become the winding numbers along the directions u 2i−1 . Furthermore, as the three T-dualities map the O3 planes into O6 planes sitting along the u 2i axis, the new boundary conditions (2.10) correspond then to a D6 brane wrapped on a 3-cycle defined by the angles with respect to the u 2i−1 axis given by tan φ i =f i (Figure 1). In these new variables, the supersymmetry condition (2.6) reads
i φ i = 3π 2 mod2π. (2.12)
The sum over the angles defined with respect to the vertical axis where the O6 plane sits is then zero, as argued above.
Moduli stabilization
From now on, we will focus our attention to the orientifold compactification T 6 /Z 2 with θ O3 = 0. Following our analysis of eqs. (2.2) and (2.3), we have seen that a single magnetized D9 brane stack preserves N = 1 supersymmetry in four dimensions for a restricted closed string moduli space. As we will see below, if we introduce several magnetic fluxes in the world-volume of different stacks of D9 branes, it will be possible to fix completely all closed string moduli but the dilaton.
As in [3], eqs. (2.2) and (2.6) can be interpreted as conditions which fix the moduli in terms of the magnetic fluxes. More specifically, we consider K stacks of N a D9 branes, with
R 2i u 2i R 2i−1 u 2i−1 φ i O−plane i i
(m ,n ) = (2,1) Figure 1: T-duality along the vertical axis u 2i . The D-brane dual to a magnetized brane forms an angle π 2 − φ with respect to the vertical axis. The orientifold plane, T-dual to an O3-plane, sits along the vertical axis. a = 1, · · · , K. We introduce on each stack a background magnetic field with constant field strength F a on the corresponding world-volume and endpoint charge q a . The magnetic fields are separately quantized, following the Dirac condition [12] q a F a kl = 2π · m a kl n a kl ≡ 2π · p a kl , p a kl ∈ Q , a = 1, · · · , K . (2.13) Written in the complex coordinates (A.4), the field strength is decomposed in a purely holomorphic and mixed part. The supersymmetry conditions for each stack ask then for a vanishing purely holomorphic field strength:
F (2,0) = 0 → τ T p a xx τ − τ T p a xy − p a yx τ + p a yy = 0, (2.14)
where the matrices (p a xx ) ij , (p a xy ) ij and (p a yy ) ij enter in the quantized field strength (2.13) in the directions (x i , x j ), (x i , y j ) and (y i , y j ), respectively, where τ is the complex structure matrix. 3 The second condition (2.6) restricts the Kähler moduli to satisfy
J ∧ J ∧ J − J ∧ F a ∧ F a = 0 , a = 1, · · · , K . (2.15)
We have used the fact that the phases θ a 's of all stacks have to be the same in order for each stack to preserve the same supersymmetry: θ a = 0, ∀a = 1, . . . , K. Furthermore, when the condition (2.14) is fulfilled, the expression for the magnetized field strength F , denoted F in the complex basis (A.10), reduces to the matrix:
F a = 0 Y a Y a † 0 ; Y a = 1 2 (2π) 2 α ′ Imτ −1 T p a yx − τ T p a xx . (2.16)
This splits in the real and imaginary parts:
ReY a = (2π) 2 α ′ 2 Imτ −1 T p a yx − Reτ T p a xx ,(2.17)ImY a = − (2π) 2 α ′ 2 p a xx . (2.18)
Inspection of eqs. (2.14) and (2.15) shows that for each stack of magnetized D9 branes, we have up to three complex conditions for the moduli of the complex structure, depending on the directions in which the fluxes are switched on, whereas only one real condition can be set on the Kähler moduli. Therefore, in order to fix the Kähler moduli, we must add more stacks of branes compared to the ones needed to fix the same number of complex structure moduli and at least nine in order to fix them all. 4
Consistency conditions
The presence of constant internal magnetic field strength induces lower dimensional charges and tensions. In a consistent compactification, these have to be cancelled by the contribution of lower dimensional objects (branes or orientifold planes) or other kinds of fluxes (such as 3-form fluxes). In the case of a T 6 /Z 2 compactification where the supersymmetry conditions (2.2) and
(2.6) are satisfied, the Dirac-Born-Infeld (DBI) and Wess-Zumino (WZ) action of magnetized D9 branes read:
V DBI = −T 9 K a=1 N a M a 10 |g + F a | = −T 9 K a=1 N a M 4 |g 4 | M a 6 Re [e −iθa (iJ + F a ) 3 ] = −T 9 K a=1 N a M 4 |g 4 | M a 6 F a ∧ F a ∧ F a − J ∧ J ∧ F a V W Z = µ 9 K a=1 N a M a 10 Ce F a = µ 9 K a=1 N a M a 10 C 4 ∧ F a ∧ F a ∧ F a + C 8 ∧ F a (2.19)
where T 9 and µ 9 are the D9 brane tension and R-R charge, respectively, while the integral over the internal manifold M a 6 takes into account the winding numbers n a kl of the different branes. The terms involving the R-R potentials C 10 and C 6 terms do not appear in the WZ action as they are projected out by the orientifold projection.
Consider now the real basis ω r of H 2 (T 6 ), with r = 1, · · · , h 2 , in which the quantization condition (2.13) for the magnetic fluxes reads:
1 2π q a F a r = m a r n a r = p a r .
(2.20)
We now define the quantity
K rst = T 6 ω r ∧ ω s ∧ ω t (2.21)
which is a sign, following the orientation choice given in (A.1). The 3-brane R-R charge, q 3,R , coming from the first term of the last line of (2.19), reads
q 3,R = K a=1 N a r,s,t K rst m a r m a s m a t . (2.22)
Since we start with a T 6 /Z 2 orientifold with O3 planes carrying −16 units of R-R charge, the R-R tadpole cancellation condition implies
q 3,R = 16 . (2.23)
The second set of conditions comes from the induced 7-brane R-R charges, emerging from the second term of eqs. (2.19). For each 2-cycle C (2) t of the torus T 6 , there is a localized 7-brane charge, given by q t 7,R :
q t 7,R = K a=1 N a r,s K rst n a r n a s m a t =: K a=1 N a q a t . (2.24)
In the T 6 /Z 2 compactification, 7-dimensional orientifold planes are absent and the total 7-brane tadpole contribution must thus vanish for any 2-cycle t:
q t 7,R = 0 , ∀t = 1, · · · , h 2 . (2.25)
As a result, we will impose the R-R tadpole cancellation conditions (2.23) and (2.25): q 3,R = 16
and q t 7,R = 0, together with the supersymmetry constraints (2.2) or equivalently (2.14), and (2.15).
Furthermore, even if magnetized antibranes may preserve the same supersymmetry as the orientifold T 6 /Z 2 , satisfying a different condition than (2.15) [13], here we will consider only a setup without antibranes. In the T-dual picture of D6 branes at angles presented in the previous section, the O6 plane is located along the axis u 2i . Then, from Figure 2, the image of a brane with quantum numbers (m i , n i ), i = 1, 2, 3, under the orientifold projection is a brane with quantum numbers (m i , −n i ). Moreover, an antibraneD6 is obtained by a rotation by an angle π from the corresponding D6 brane in an odd number of cycles [u 2i−1 u 2i ], corresponding to a brane with winding numbers (−m i , −n i ). Therefore, the absence of antibranes is expressed as a condition on the winding numbers along the u 2i axis, or equivalently on the first Chern numbers:
K rst m a r m a s m a t > 0 ∀a = 1, . . . K.
(2.26)
The limiting case where one of the first Chern numbers m i vanishes, along the coordinates u 2i corresponds to the situation where the brane is horizontal in one of the 2-cycles [u 2i−1 u 2i ]. Switching the sign of the winding number n i corresponds then to switch a brane into an antibrane. The condition for the absence of antibranes in this case then reads:
K rst m a r m a s n a t > 0 ∀a = 1, . . . K . (2.27)
Next, a condition of positivity for the real part of Φ a defined in eq. (2.5) has to be satisfied for each a, as it corresponds to the generalized world-volume element of each separate brane stack:
Re(e −iθa Φ a ) > 0 , ∀ a = 1, · · · , K , (2.28)
with Φ a = (iJ + F a ) ∧ (iJ + F a ) ∧ (iJ + F a ) . (2.29)
For θ a = 0, it reduces to the condition:
F a ∧ F a ∧ F a − J ∧ J ∧ F a > 0 . (2.30)
Let us consider two cases which will arise in the examples of the following sections.
• When only the diagonal Kähler form elements J i =: J iī are non-zero and all off-diagonal fluxes vanish F ij = 0, the positivity condition (2.30), together with the supersymmetry condition (2.15), reads:
−J 1 J 2 J 3 (H 2 + H 3 ) 1 + H 2 1 > 0, (2.31)
where we use the notation F i = F iī and H i = F i /J i . As all Kähler moduli J i are volumes, they are positive and the above condition becomes
H 2 + H 3 < 0 . (2.32)
• In the last case we will consider, there are also non-diagonal fluxes, like for example F 12 , together with a diagonal one F 3 . Eq. (2.30) then reads −F 3 |F 12 | 2 1 + J 1 J 2 |F 12 | 2 > 0, (2.33) implying that the diagonal component F 3 has to be negative.
Finally, we compute the intersection number I ab between the stacks a and b, which gives the number of chiral fermions. As it has been shown in [6], I ab in the presence of a general magnetic flux can be written as
I ab = − W a W b (2π) 3 T 6 c 3 (L a ⊗ L b ) = W a W b rst K rst (m a r n b r − m b r n a r )(m a s n b s − m b s n a s )(m a t n b t − m b t n a t ) n a r n a s n a t n b r n b s n b t ,(2.34)
where W a/b is the winding number of the stack a/b around the whole T 6 , L a/b corresponds to the U(1) line bundle associated to the magnetic flux and c 3 (L a ⊗ L b ) is the third Chern class.
The intersection number I ab in (2.34) is associated to the degeneracy of the Landau levels and therefore has to be integer. An obvious solution of this requirement is to ask for the winding numbers n a r of each stack a to satisfy 5 W a = n a r n a s n a t , ∀r, s, t with K rst = 0.
(2.35)
Since I ab depends only on the product of W a and W b , the above restriction is valid up to a sign ambiguity. For each brane a, there is a unique winding number W a around the whole torus T 6 which is given up to a sign by the product of winding numbers of orthogonal 2-cycles. It corresponds to the geometrical picture where the fundamental cycles of the torus are six 1-cycles and the winding numbers n r around the fifteen different 2-cycles are not independent, but given in terms of products of winding numbers around 1-cycles. Note that in this case, the 7-brane charge q a t defined in (2.24) reduces to q a t = K rst n r n s m t without a sum over the indices r, s.
R-R Moduli
We have seen above that under strong constraints on the magnetic fluxes, it is in principle possible to find N = 1 supersymmetric vacua in four dimensions with stabilized metric moduli. In sections 4-5 and 7, we will give explicit examples where this is indeed achieved. Here, we want to address the question of the remaining moduli. In the orientifold compactification T 6 /Z 2 , apart from the metric and dilaton moduli, the four dimensional spectrum contains massless 2-forms, which arise in the R-R sector. They correspond to the internal components of the R-R 4-form C (4) which survived the Z 2 -orientifold action defined in Section 2.1. They are decomposed in elements of three different cohomology classes H 1,1 (T 6 ), H 2,0 (T 6 ) and H 0,2 (T 6 ):
(C (4) ) µνij , (C (4) ) µνij , (C (4) ) µνīj , i, j = 1, . . . , 3 (2.36) where the indices µ, ν refer to four dimensional spacetime : µ, ν = 0, . . . , 3. The first nine 2-forms (C (4) ) µνij are dual to pseudo-scalars in four dimensions; they actually form linear multiplets with the Kähler moduli J ij . When the latter are fixed in the presence of magnetized fluxes, they give rise to Stückelberg couplings that provide masses to some U(1) gauge fields. This can be seen explicitly from the Wess Zumino action (2.19) in ten dimensions: Consider the gauge potential
A M = (A µ , A i ) of a magnetized U(1) with A k = − 1 2 F kl u l . Its spacetime field strength F (2) = dA then couples to the 2-form B ij
(2) = (C (4) ) µνij as:
M 10 C 4 ∧ F ∧ F ∧ F → q ij M 4 B ij (2) ∧ F (2) , (2.37)
where the couplings q ij are functions of the internal magnetic fluxes. As a result, some combination of the nine R-R 2-forms (C (4) ) µνij is absorbed in the U(1) gauge field which becomes massive.
The situation with the last six massless 2-forms in (2.36) is different. They are harmonic (2, 0) and (0, 2) forms on the internal torus and therefore elements of the cohomologies H 2,0 (T 6 ) and H 0,2 (T 6 ). By contraction with the holomorphic 3-form Ω of T 6 , we can construct from (C (4) ) µνij and (C (4) ) µνīj harmonic (2, 1) and (1, 2)-forms on the torus:
B µνijl ∼ Ωk ij (C (4) ) µνkl B µνījl ∼Ω k ij (C (4) ) µνkl . (2.38)
To each harmonic (2, 0) and (0, 2) form, we can then associate a harmonic (2, 1) and (1, 2)-form, associated to the complex structure moduli. Thus, the nine elements of the complex structure τ ij correspond to six purely (anti-) holomorphic metric moduli and three (anti-) holomorphic R-R moduli. As shown in [6], the stabilization of the latter via the condition (2.2) can be understood by a potential generated through their mixing with the NS-NS moduli.
Closed string fluxes
As argued in section 2.2, all geometric moduli can be stabilized by turning on internal magnetic background fields. Moreover, the introduction of nine stacks of magnetized D9 branes can fix all complex structure and Kähler class moduli. Furthermore, the R-R moduli complexifying the Kähler class are absorbed into the longitudinal degrees of freedom of the U(1) gauge fields, which become massive. The remaining unfixed moduli correspond to the (complex) dilatonaxion field.
Dilaton stabilization
A possible stabilization mechanism for the dilaton is by turning on R-R and NS-NS 3-form closed string fluxes, that for generic Calabi-Yau compactifications can fix also the complex structure [1]. As we are going to combine the two mechanisms, in this section we review briefly the main properties of 3-form fluxes. Let H (3) and F (3) be the field strengths of the NS-NS 2-form B (2) and of the R-R 2-form C (2) , respectively, (2) , subject as usual to the Dirac quantization condition in the compact space. In the basis (α a , β b ) chosen in (A.2) of Appendix A, H (3) and F (3) can be written as
H (3) = dB (2) F (3) = dC1 (2π) 2 α ′ H (3) = h 2,1 a=0 (h a 1 α a + h a 2 β a ) 1 (2π) 2 α ′ F (3) = h 2,1 a=0 (f a 1 α a + f a 2 β a ) ,(3.1)
where h a 1 , h a 2 , f a 1 and f a 2 are integers. Using the complex dilaton modulus, one can then form
the 3-form G (3) G (3) = F (3) − φH (3) , φ = C (0) + ig −1 s , (3.2)
where g s is the string coupling. The 3-form background fields preserve then a common supersymmetry with the Z 2 -orientifold projection of T 6 /Z 2 if the following conditions are fulfilled: G (3) has to be a primitive (2, 1) form [14]:
G (3) ∧ J = 0 , G (3) ∈ H 2,1 . (3.3)
Actually, the second of the conditions above corresponds to finding a minimum of the GVW superpotential [15] W =
T 6 G (3) ∧ Ω, (3.4)
which then has to be covariantly constant with respect to all moduli, D I W = 0, or equivalently:
W = 0 , ∂ φ W = 0 , ∂ τ ij W = 0, (3.5)
where φ is defined in (3.2). Note that all primitive (2, 1)-forms are imaginary self dual (ISD),
⋆ 6 G 2,1 = iG 2,1 ,
where the star map ⋆ 6 is the usual Hodge map on the torus. Let us analyze further the supersymmetry conditions (3.5). For given flux quanta (3.1), they can be understood as conditions on the dilaton and complex structure moduli. More precisely, using the symplectic structure (A.3), the superpotential (3.4) reads
W = 1 (2π) 2 α ′ T 6 G (3) ∧ Ω = −(f 0 1 − φh 0 1 )detτ + (f 0 2 − φh 0 2 ) + (f ij 1 − φh ij 1 )(cofτ ) ij + (f ij 2 − φh ij 2 )τ ij .
(3.6) We can now express the three supersymmetry conditions (3.5) explicitly in the form :
0 = −(f 0 1 − φh 0 1 )detτ + (f 0 2 − φh 0 2 ) + (f ij 1 − φh ij 1 )(cofτ ) ij + (f ij 2 − φh ij 2 )τ ij (3.7) 0 = h 0 1 detτ − h 0 2 − h ij 1 (cofτ ) ij − h ij 2 τ ij (3.8) 0 = −(f 0 1 − φh 0 1 )(cofτ ) kl + (f kl 2 − φh kl 2 ) + (f ij 1 − φh ij 1 )ǫ ikm ǫ jln τ mn ,(3.9)
where cofτ = (detτ )τ −1,T . These are eleven conditions on the complex structure, parametrized by the nine elements τ ij and the (complex) dilaton field φ. It is then in principle possible to fix all complex structure and dilaton moduli in terms of adequate quanta [1]. Let us now examine the primitivity condition G (2,1) ∧J = 0. We could naively think that this can be interpreted, for given fluxes, as conditions on the Kähler moduli. However, this condition is trivially satisfied in the case of generic Calabi-Yau compactifications, because there are no harmonic (3, 2) forms on these manifolds. Therefore, this condition can only become partially non-trivial on Kähler moduli for compactification manifolds with more symmetries, such as the torus. There exist however alternative possibilities to fix the metric moduli. As shown in section 2, the presence of internal magnetic fluxes leads to conditions on both the Kähler class (2.15) and complex structure moduli (2.14). For generic Calabi-Yau spaces one can fix only the former, while for toroidal compactifications it is possible to fix all metric moduli by a suitable choice of stacks of magnetized D9 branes. An explicit example will be shown in section 4. On the other hand, the dilaton modulus remains unfixed, but can be stabilized using 3-form closed string fluxes. In fact, for fixed complex structure, the conditions (3.7), (3.8) and (3.9) constrain exclusively the dilaton. Moreover, as the Kähler form is fixed by the presence of magnetic fields, the primitivity condition G (2,1) ∧ J = 0 restricts the possible fluxes G (2,1) we can switch on. Finally, the value of the string coupling we can obtain in this way is strongly constrained by the tadpole conditions. The latter can be read off from the topological coupling of the 3-form fluxes with the R-R 4-form C (4) potential in the effective action of the ten-dimensional type IIB supergravity:
S CS = 1 4i(2π) 7 α ′4 M 10 C (4) ∧ G (3) ∧Ḡ (3) Imφ = −µ 3 1 2 1 (2π) 4 α ′ 2 M 10 C (4) ∧ H (3) ∧ F (3) , (3.10) where we defined the R-R charge µ 3 in terms of α ′ as µ 3 = (2π) −3 α ′ −2 .
The coupling to C (4) of the magnetized D9 branes is given in (2.19), while the coupling of the O3 orientifold plane reads
S O3 = µ 3 Q O3 M 4 C (4) , (3.11)
where the charge Q O3 of O3 planes has been defined in section 2.3. Therefore, the integrated Bianchi identity for the modified R-R 5-form field strength F (5) reads
− 1 2 1 (2π) 4 α ′2 T 6 H (3) ∧ F (3) + q 3,R + Q O3 = 0, (3.12)
where the factor 1 2 comes from the fact that the volume of the orientifold T 6 /Z 2 is half the volume of the torus T 6 . 6 It follows from the ISD condition, that the contribution to (3.12) coming from the 3-form flux is always positive :
N 3 =: − 1 2 1 (2π) 4 α ′2 T 6 H (3) ∧ F (3) = 1 2g s T 6 H (3) ∧ ⋆ 6 H (3) > 0. (3.13)
The second source for 3-brane charges in (3.12) comes from the internal magnetic fluxes. As shown in section 2.3, each stack of magnetized D9 branes with magnetic fluxes switched on in three orthogonal directions of T 6 contributes positively to the 3-brane charge (2.22). Finally, the 3-brane tadpole could also receive contributions from ordinary D3 branes. All together, the tadpole condition (2.23) is now modified as where Q O3 = −16. As the first three terms in the l.h.s. of equation (3.14) contribute positively, the possible values of N 3 as well of q 3,R are bounded. This restricts strongly the possible values of the string coupling g s . Since the tadpole condition (3.14) asks for N 3 to remain of order one, the only possibility for the string coupling to be fixed at a small value is to get a large contribution from the integral
N 3 + q 3,R + N D3 + Q O3 = 0,(3.T 6 H (3) ∧ ⋆ 6 H (3)
. This depends on the quanta h a 1 , h a 2 of (3.1) and on the Hodge star operator. The latter only depends on the complex structure [17]. It is therefore in principle possible to fix the string coupling g s at small value and to keep the contribution N 3 at fixed value by stabilizing the integral T 6 H (3) ∧⋆ 6 H (3) at large value with the help of either internal magnetic fields or 3-form fluxes. This will be discussed in more details in section 6.
Quantized NS-NS B field
Further restrictions on fluxes arise from the quantization condition in the orientifold T 6 /Z 2 , as compared to the torus. As explained in [1], the quanta of NS-NS and R-R 3-form fluxes have to be even along any 3-cycle of T 6 /Z 2 . This remains valid in the presence of magnetic fluxes, as well. However, the situation changes if one introduces a non-trivial NS-NS B field in some of the 2-cycles of the torus. Consider for instance the case where the B field is switched on only in one 2-cycle, say [x 3 y 3 ]: B x 3 y 3 = α ′ (2π) 2 b, where b = 0 or 1/2. Its consequences are: • A change in the spectrum of the open string sector [18]. The first Chern number m a x 3 y 3 of the magnetic fluxes of all stacks a = 1, · · · , K gets shifted tom x 3 y 3 = m a x 3 y 3 + bn a x 3 y 3 .
• A modification of the configuration of O3-planes. In the orientifold compactification T 6 /Z 2 , there are 64 fixed points where the different O3 planes sit. All of them have negative tension and charge. However, for b = 1/2, 16 of the 64 O3 planes become of the type O3 + , which have positive tension and charges [19]. The remaining 48 are of the usual type O3 ≡ O3 − . The 3-brane tadpole condition is therefore modified. As in our conventions each O3 orientifold plane carries 1/4 unit of (negative) charge, the total charge contribution of the different orientifold planes for b = 1/2 is not anymore −16 but −8:
N O3 + − N O3 − = 1 4 (16 − 48) = −8. (3.15)
The tadpole condition (3.14) is then modified to
N 3 +q 3,R + N D3 = 8 ,q 3,R = K a=1 N a r,s,t K rstm a r m a s m a t . (3.16)
In the modified 3-brane chargeq 3,R induced by the magnetic fields, it is implicitly assumed that the only shifted Chern numbersm a r correspond to the 2-cycle carrying the B-field; in our example, it ism a x 3 y 3 .
• A modification of the quantization condition for the NS-NS 3-form fluxes H (3) [1]. Consider first a NS-NS 3-form switched on in a 3-cycle γ of the torus T 6 . If γ crosses an odd number of orientifold planes of the type O3 + , the corresponding quanta h γ have to be odd integers, while when the crossing number is even, h γ has to be even. Let us consider now the case where b = 1/2. As depicted in Figure 4, the sixteen O3 + planes are located at one of the four fixed points of the third torus [x 3 y 3 ]. We can easily see that the only 3-cycles of T 6 , whose crossing number with the O3 + planes is odd are the following ones: The constraints on the complex structure matrix τ ij are derived from eq. (2.14). We notice that, in order to have the off-diagonal components of the complex structure moduli vanishing, one needs to turn on certain off-diagonal components of magnetic fluxes on the D9 branes.
[x i y i x 3 ] , [x i y i y 3 ] , i = 1, 2 .
They are characterized by rational numbers p a 's, defined as the ratios of the quantum numbers m and n given in eq. (2.13). These fluxes turn out to be of the type p a x i y j , p a x i x j and p a y i y j , with i = j. However, we will find out later that off-diagonal fluxes of these types have to be necessarily accompanied by certain diagonal fluxes of the type p a x i y i (i = 1, 2, 3), as well. Taking these restrictions into account, the following non-zero fluxes are turned on along the branes in stack-1:
1. [p 1 x 1 y 2 , p 1 x 2 y 1 , p 1 x 1 y 1 , p 1 x 3 y 3 ] = 0, (4.1)
with the remaining components of the flux being set to zero, by choosing the corresponding Chern numbers m = 0 in eq. (2.13). The windings n r can be zero along some of the 2-cycles C r (2) , even if the corresponding magnetic flux vanishes. However, since the magnetized branes are D9's, they have to cover the whole internal space T 6 /Z 2 . This means that the effective winding number K rst n a r n a s n a t around (the 6-cycle of) T 6 /Z 2 has to be non-zero. Similarly to (4.1), we choose for stack-2:
2. [p 2 x 2 y 3 , p 2 x 3 y 2 , p 2 x 2 y 2 , p 2 x 1 y 1 ] = 0 (4.2)
and for stack-3:
3. [p 3 x 3 y 1 , p 3 x 1 y 3 , p 3 x 3 y 3 , p 3 x 2 y 2 ] = 0 . (4.3)
As we will see below, the supersymmetry condition (2.14) on the stacks of branes 1-3, with fluxes turned on according to eqs. (4.1)-(4.3), imply that all off-diagonal components of τ ij (i = j) are set to zero. Moreover, these conditions fix the ratios of the diagonal components τ ii in terms of the ratios of p a x i y j in the different brane stacks. We will also show that some magnetized branes will play a role in setting three independent combinations of the off-diagonal components of the Kähler class moduli J ij to zero. In order to get similar conditions on the remaining off-diagonal components of the Kähler moduli, we introduce three more brane stacks with the following non-vanishing flux components:
4. [p 4 x 1 x 2 , p 4 y 1 y 2 , p 4 x 1 y 1 , p 4 x 3 y 3 ] = 0, (4.4) 5. [p 5 x 2 x 3 , p 5 y 2 y 3 , p 5 x 2 y 2 , p 5 x 1 y 1 ] = 0, (4.5) and 6. [p 6 x 3 x 1 , p 6 y 3 y 1 , p 6 x 3 y 3 , p 6 x 2 y 2 ] = 0. (4.6)
Studying various possibilities of string constructions incorporating moduli stabilization, we will also introduce in some cases six more copies of brane stacks, called stack-1 ′ -stack-6 ′ .
These branes have the same diagonal fluxes (and with brane multiplicities N a ′ = N a ) as their unprimed counterparts, but off-diagonal components with opposite sign:
8. [p 8 x 1 y 1 = 0, p 8 x 2 y 2 = 0, p 8 x 3 y 3 = 0], (4.14) 9. [p 9 x 1 y 1 = 0, p 9 x 2 y 2 = 0. p 9 x 3 y 3 = 0]. (4.15)
Another possibility to satisfy the consistency conditions mentioned in section 2.3 is to introduce some stacks of branes with off-diagonal fluxes which do not contribute to the 3-brane tadpole:
10. [p 10
x 1 y 1 , p 10
x 3 y 3 , p 10 x 1 y 3 , p 10 x 3 y 1 ] = 0, 10 ′ . [p 10 ′ x 1 y 1 , p 10 ′ x 3 y 3 , p 10 ′ x 1 x 3 , p 10 ′ y 3 y 1 ] = 0, (4.16) 11. [p 11 x 1 y 1 , p 11 x 2 y 2 , p 11 x 1 y 2 , p 11 x 2 y 1 ] = 0, 11 ′ . [p 11 ′ x 1 y 1 , p 11 ′ x 2 y 2 , p 11 ′ x 1 x 2 , p 11 ′ y 2 y 1 ] = 0, (4.17) 12. [p 12 x 2 y 2 , p 12 x 3 y 3 , p 12 x 2 y 3 , p 12 x 3 y 2 ] = 0, 12 ′ . [p 12 ′ x 2 y 2 , p 12 ′ x 3 y 3 , p 12 ′ x 2 x 3 , p 12 ′ y 3 y 2 ] = 0. (4.18)
Of course, one has also the possibility of introducing branes with non-zero fluxes along all diagonal elements. Such branes are, however, not used in the examples we present below, for simplicity and for minimizing the 3-brane tadpole contribution.
We are now ready to examine the moduli stabilization when different combinations of branes, mentioned above, are used.
Explicit construction: Model-A with q 3,R = 6
In this section, we analyze the conditions (2.14), (2.15), (2.30), (2.25), (2.23),(2.26) and (2.27) in more detail and present explicit examples when the twelve brane stacks 1-6, 10-12 and 10 ′ -12 ′ are used. We first discuss complex structure moduli stabilization, and next, in subsections 5.2-5.3, we show the stabilization of the Kähler class moduli, as well. These branes together contribute q 3,R = 6 to the 3-brane tadpoles; tadpole cancellation will be discussed in subsection 5.5.
Stabilization of complex structure moduli
We show below that all complex structure moduli are stabilized using only the stacks of branes 1-6, with magnetic fluxes given in eqs. (4.1)-(4.6). In fact the situation remains similar to the (T-dual) case studied in [3], and we only give the final conditions following from the vanishing of the F (2,0) components (c.f. (2.2)), as given in (2.14).
First, the brane stacks 1-3 restrict the off-diagonal elements of the complex structure matrix by a set of six linear equations for the six variables, τ 12 , τ 23 , τ 31 , τ 21 , τ 32 , τ 13 :
0 p 1 x 2 y 1 −p 1 x 3 y 3 −p 2 x 1 y 1 0 p 2 x 3 y 2 p 3 x 1 y 3 −p 3 x 2 y 2 0 τ 12 τ 23 τ 31 = −p 1 x 1 y 1 τ 13 −p 2 x 2 y 2 τ 21 −p 3 x 3 y 3 τ 32 , (5.1) −p 1 x 1 y 2 0 p 1 x 3 y 3 0 p 3 x 2 y 2 −p 3 x 3 y 1 p 2 x 1 y 1 −p 2 x 2 y 3 0 τ 13 τ 21 τ 32 = 0. (5.2)
As we will see later on, for the specific values of magnetic fluxes that we turn on along the branes, the matrix appearing in eq. (5.2) turns out to be singular and implies the equality:
τ 13 = τ 21 = τ 32 . (5.3)
Moreover, the matrix appearing in the l.h.s. of eq. (5.1) is also singular, and using the result (5.3), one obtains:
τ 13 = 0 ; τ 12 = τ 23 = τ 31 . (5.4)
Finally, using the constraint (2.14) for one of the branes 4, 5 or 6, one obtains that all offdiagonal components of the complex-structure vanish:
τ 12 = τ 13 = τ 21 = τ 23 = τ 31 = τ 32 = 0 . (5.5)
The brane stacks 1-6 also restrict the diagonal elements of the matrix τ , and they satisfy the following conditions:
τ 11 τ 22 = p 1 x 2 y 1 p 1 x 1 y 2 ≡ K 1 , τ 22 τ 33 = p 2 x 3 y 2 p 2 x 2 y 3 ≡ K 2 , τ 33 τ 11 = p 3 x 1 y 3 p 3 x 3 y 1 ≡ K 3 ,(5.6)
and
τ 11 τ 22 = − p 4 y 1 y 2 p 4 x 1 x 2 ≡ −K 4 , τ 22 τ 33 = − p 5 y 2 y 3 p 5 x 2 x 3 ≡ −K 5 , τ 33 τ 11 = − p 6 y 3 y 1 p 6 x 3 x 1 ≡ −K 6 .(5.7)
Following [3], we use K 1 , K 3 and K 4 as independent parameters. Consistency between eqs. (5.6) and (5.7) then implies:
K 2 = 1 K 1 K 3 , K 5 = K 3 K 4 , K 6 = K 1 K 3 K 4 . (5.8)
Since we will look for solutions where τ ii are all purely imaginary, this further imposes a positivity condition on K i 's:
K i > 0, for i = 1, .., 6. (5.9)
The solutions for the diagonal elements τ ii (i = 1, 2, 3) are then given by:
τ 11 = i K 1 K 4 , τ 22 = i K 4 K 1 , τ 33 = i K 1 K 4 K 3 . (5.10)
We have therefore determined the complex structure moduli completely, given by the equations (5.5) and (5.10). Using this form of the complex structure, it can also be easily verified that the stacks of branes 7-9, having fluxes only along diagonals x i y i , do not impose any further constraints on it. We go on now to the stabilization of the Kähler class moduli.
Stabilization of Kähler class moduli: constraints on fluxes
In this subsection we derive the constraints on magnetic fluxes, for the stack of branes 1-6, 10-12 and 10 ′ -12 ′ , defined in section 4, in order to obtain the stabilization of the Kähler class moduli. For this purpose, we analyze the supersymmetry condition (2.15) for these stacks. As the complex structure has been stabilized to the diagonal form τ ij = iδ ij , the flux content given earlier in eqs. We now analyze the supersymmetry condition (2.15) and find that it puts several restrictions on the fluxes that are turned on. Expressing eq. (2.15) in components, we obtain for brane-1: 20) where in writing the last term we have also made use of the condition F 1 21 = F 1 12 given in eq. (5.11). Similarly, we have for brane-2 and brane-3:
(J ∧ J ∧ J) 112233 = J 22 F 1 11 F 1 33 − J 33 F 1 12 F 1 21 − (J 12 + J 21 )F 1 33 F 1 21 ,(5.(J ∧ J ∧ J) 112233 = J 33 F 2 22 F 2 11 − J 11 F 2 23 F 2 32 − (J 23 + J 32 )F 2 11 F 2 32 , (5.21) (J ∧ J ∧ J) 112233 = J 11 F 3 33 F 3 22 − J 22 F 3 31 F 3 13 − (J 31 + J 13 )F 3 22 F 3 13 . (5.22)
For branes 4-6, on the other hand, we get: The fluxes along different branes are constrained in order to satisfy the supersymmetry conditions (5.20)-(5.28). As we are interested in Kähler moduli solutions with vanishing offdiagonal components J ij = 0, and using the positivity of the volume form J ∧ J ∧ J, the above fluxes are restricted as: Moreover, for branes 10-12 and 10 ′ -12 ′ one has to impose:
(J ∧ J ∧ J) 112233 = J 22 F 4 11 F 4 33 − J 33 F 4 12 F 4 21 − (J 12 − J 21 )F 4 33 F 4 21 , (5.23) (J ∧ J ∧ J) 112233 = J 33 F 5 22 F 5 11 − J 11 F 5 23 F 5 32 − (J 23 − J 32 )F 5 11 F 5 32 ,(5.J 22 F 1 11 F 1 33 > J 33 FF 10 33 F 10 11 − |F 10 13 | 2 > 0, F 11 11 F 11 22 − |F 10 12 | 2 > 0, F 12 33 F 12 22 − |F 12 23 | 2 > 0, F 10 ′ 33 F 10 ′ 11 − |F 10 ′ 13 | 2 > 0, F 11 ′ 11 F 11 ′ 22 − |F 11 ′ 12 | 2 > 0, F 12 ′ 33 F 12 ′ 22 − |F 12 ′ 23 | 2 > 0. (5.30)
We have therefore given a set of conditions to be used for solving the supersymmetry equations (5.20)-(5.28). We postpone the discussion on their solutions for the next subsection and examine now the additional constraints imposed on fluxes from the positivity requirement (2.30). For stacks 1-6, this condition reduces to the form (2.33). The diagonal fluxes are then restricted to the domain where
F 1 x 3 y 3 < 0, F 2 x 1 y 1 < 0, F 3 x 2 y 2 < 0, F 4 x 3 y 3 < 0, F 5 x 1 y 1 < 0, F 6 x 2 y 2 < 0. (5.31)
When combined with conditions (5.29), this further implies that the remaining diagonal fluxes in branes 1-6 have to be negative, as well:
F 1 x 1 y 1 < 0, F 2 x 2 y 2 < 0, F 3 x 3 y 3 < 0, F 4 x 1 y 1 < 0, F 5 x 2 y 2 < 0, F 6 x 3 y 3 < 0. (5.32)
Finally, since the stacks 10-12 and 10 ′ -12 ′ satisfy F a ∧ F a ∧ F a = 0, they do not contribute to the 3-brane charge, and the positivity conditions follow from eq. (2.32). Combined with eq. (5.30), we get that the magnetic fluxes for these branes must also be negative:
F 10 x 3 y 3 < 0, F 10 x 1 y 1 < 0, F 10 ′ x 3 y 3 < 0, F 10 ′ x 1 y 1 < 0, F 11 x 1 y 1 < 0, F 11 x 2 y 2 < 0, F 11 ′ x 1 y 1 < 0, F 11 ′ x 2 y 2 < 0, F 12 x 2 y 2 < 0, F 12 x 3 y 3 < 0, F 12 ′ x 2 y 2 < 0, F 12 ′ x 3 y 3 < 0. (5.33)
Explicit solutions: fluxes and moduli
We now present an explicit solution for the fluxes along all stacks of branes satisfying the restrictions given in equations (5.8), (5.9), (5.29), (5.30) and (5.31)-(5.33). These fluxes are defined in terms of the first Chern numbers m i 's and winding numbers n i 's, introduced earlier in eq. (2.13), along the various 2-cycles of T 6 /Z 2 . We choose for branes 1-3:
(m 1 x 1 y 2 , n 1 x 1 y 2 ), (m 1 x 2 y 1 , n 1 x 2 y 1 ) (m 1 x 1 y 1 , n 1 x 1 y 1 ) (m 1 x 2 y 2 , n 1 x 2 y 2 ) (m 1 x 3 y 3 , n 1 x 3 y 3 ) = (m 2 x 2 y 3 , n 2 x 2 y 3 ), (m 2 x 3 y 2 , n 2 x 3 y 2 ) (m 2 x 2 y 2 , n 2 x 2 y 2 ) (m 2 x 3 y 3 , n 2 x 3 y 3 ) (m 2 x 1 y 1 , n 2 x 1 y 1 ) = (m 3 x 3 y 1 , n 3 x 3 y 1 ), (m 3 x 1 y 3 , n 3 x 1 y 3 ) (m 3 x 3 y 3 , n 3 x 3 y 3 ) (m 3 x 1 y 1 , n 3 x 1 y 1 ) (m 3 x 2 y 2 , n 3 x 2 y 2 ) = (−1, 1), (1, −1) (2, −1) (0, −1) (1, −1) ,(5.34)
Similarly, for branes 4-6 we choose:
(m 4 x 1 x 2 , n 4
x 1 x 2 ), (m 4 y 1 y 2 , n 4 y 1 y 2 ) (m 4
x 1 y 1 , n 4 x 1 y 1 ) (m 4
x 2 y 2 , n 4
x 2 y 2 ) (m 4 x 3 y 3 , n 4 x 3 y 3 ) = (m 5 x 2 x 3 , n 5 x 2 x 3 ), (m 5 y 2 y 3 , n 5 y 2 y 3 ) (m 5 x 2 y 2 , n 5 x 2 y 2 ) (m 5 x 3 y 3 , n 5 x 3 y 3 ) (m 5 x 1 y 1 , n 5 x 1 y 1 ) = (m 6
x 3 x 1 , n 6
x 3 x 1 ), (m 6 y 3 y 1 , n 6 y 3 y 1 ) (m 6 x 3 y 3 , n 6 x 3 y 3 ) (m 6
x 1 y 1 , n 6 x 1 y 1 ) (m 6
x 2 y 2 , n 6
x 2 y 2 ) = (−1, 1), (1, −1) (2, −1) (0, −1) (1, −1) . (5.35)
For branes 10-12, the values of the fluxes are given by:
(m 10 x 1 y 3 , n 10 x 1 y 3 ), (m 10 x 3 y 1 , n 10 x 3 y 1 ) (m 10
x 1 y 1 , n 10 x 1 y 1 ) (m 10
x 2 y 2 , n 10 x 2 y 2 ) (m 10 x 3 y 3 , n 10
x 3 y 3 ) = (m 11
x 2 y 1 , n 11 x 2 y 1 ), (m 11 x 1 y 2 , n 11 x 1 y 2 ) (m 11
x 2 y 2 , n 11 x 2 y 2 ) (m 11
x 3 y 3 , n 11 x 3 y 3 ) (m 11
x 1 y 1 , n 11
x 1 y 1 ) = (m 12
x 3 y 2 , n 12 x 3 y 2 ), (m 12 x 2 y 3 , n 12 x 2 y 3 ) (m 12 x 3 y 3 , n 12 x 3 y 3 ) (m 12
x 1 y 1 , n 12 x 1 y 1 ) (m 12
x 2 y 2 , n 12
x 2 y 2 ) = (1, −1), (−1, 1) (−1, 1) (0, 1) (−2, 1) . (5.36)
Finally, the flux quanta for the stacks 10 ′ -12 ′ are the same as for the stacks 10-12 and are given by:
(m 10 ′ x 1 x 3 , n 10 ′ x 1 x 3 ), (m 10 ′ y 1 y 3 , n 10 ′ y 1 y 3 ) (m 10 ′ x 1 y 1 , n 10 ′ x 1 y 1 ) (m 10 ′ x 2 y 2 , n 10 ′ x 2 y 2 ) (m 10 ′ x 3 y 3 , n 10 ′ x 3 y 3 ) = (m 11 ′ x 2 x 1 , n 11 ′ x 2 x 1 ), (m 11 ′ y 1 y 1 , n 11 y 2 y 1 ) (m 11 ′ x 2 y 2 , n 11 ′ x 2 y 2 ) (m 11 ′ x 3 y 3 , n 11 ′ x 3 y 3 ) (m 11 ′ x 1 y 1 , n 11 ′ x 1 y 1 ) = (m 12 ′ x 3 x 2 , n 12 ′ x 3 x 2 ), (m 12 ′ y 3 y 2 , n 12 ′ y 3 y 2 ) (m 12 ′ x 3 y 3 , n 12 x 3 y 3 ) (m 12 ′ x 1 y 1 , n 12 x 1 y 1 ) (m 12 ′ x 2 y 2 , n 12 x 2 y 2 ) = (1, −1), (−1, 1) (−1, 1) (0, 1) (−2, 1) . (5.37)
Using the above values of m and n, the non-zero fluxes defined in eq. (2.13) and used in complex structure moduli stabilization of section 5.1, for branes 1-6 read:
p 1 x 1 y 2 p 1 x 2 y 1 p 1 x 1 y 1 p 1 x 3 y 3 = p 2 x 2 y 3 p 2 x 3 y 2 p 2 x 2 y 2 p 2 x 1 y 1 = p 3 x 3 y 1 p 3 x 1 y 3 p 3 x 3 y 3 p 3 x 2 y 2 = −1 −1 −2 −1 , (5.38) p 4 x 1 x 2 p 4 y 1 y 2 p 4 x 1 y 1 p 4 x 3 y 3 = p 5 x 2 x 3 p 5 y 2 y 3 p 5 x 2 y 2 p 5 x 1 y 1 = p 6 x 3 x 1 p 6 y 3 y 1 p 6 x 3 y 3 p 6 x 2 y 2 = −1 −1 −2 −1 . (5.39)
Similarly, for branes 10-12 and 10 ′ -12 ′ , the non-zero fluxes are given by:
p 10
x 1 y 3 p 10
x 3 y 1 p 10
x 1 y 1 p 10 x 3 y 3 = p 11
x 1 y 2 p 11
x 2 y 1 p 11
x 2 y 2 p 11
x 1 y 1 = p 12
x 2 y 3 p 12
x 3 y 2 p 12
x 3 y 3 p 12 x 2 y 2 = p 10 ′ x 1 x 3 p 10 ′ y 1 y 3 p 10 ′ x 1 y 1 p 10 ′ x 3 y 3 = p 11 ′ x 1 x 2 p 11 ′ y 1 y 2 p 11 ′ x 2 y 2 p 11 ′ x 1 y 1 = p 12 ′ x 2 x 3 p 12 ′ y 2 y 3 p 12 ′ x 3 y 3 p 12 ′ x 2 y 2 = −1 −1 −1 −2 . (5.40)
One can now verify that the above magnetic fluxes (5.38)-(5.40) satisfy the conditions (5.8) and the parameters K i defined in eqs. (5.6) and (5.7) read:
K 1 = K 2 = K 3 = K 4 = K 5 = K 6 = 1,(5. 0 −1 1 1 0 −1 −1 1 0 ,(5.43)
while the matrix appearing in eq. (5.2) We will now obtain the values of the Kähler class moduli by solving the supersymmetry conditions. We will show that all off-diagonal components vanish, while the diagonal ones (in real coordinates) are J x i y i = (2π) 2 α ′ (for i = 1, 2, 3), defining a meaningful solution of the supersymmetry conditions (5.20)-(5.28).
1 0 −1 0 −1 1 −1 1 0
Solving the supersymmetry conditions to fix the Kähler form
Here, we analyze the supersymmetry conditions (5.20)-(5.28), which consist of nine independent non-linear equations for nine variables. The reason is that the three equations in the r.h.s. of
J 11 = J 22 = J 33 = (2π) 2 α ′ 2 , (5.47)
or in terms of real coordinates: and for the branes 10 ′ -12 ′ :
J x 1 y 1 = J x 2 y 2 = J x 3 y 3 = (2π) 2 α ′ .F 1 12 = F 1 21 = F 2 23 = F 2 32 = F 3 31 = F 3 13 = − 1 2 4π 2 α ′ ,2F 10 ′ 11 = F 10 ′ 33 = 2F 11 ′ 22 = F 11 ′ 11 = 2F 12 ′ 33 = F 12 ′ 22 = −4π 2 α ′ , 2F 10 ′ 13 = 2F 10 ′ 31 = 2F 11 ′ 21 = F 11 ′ 12 = 2F 12 ′ 32 = F 12 ′ 23 = 4π 2 iα ′ . (5.53)
Tadpole cancellations
We now analyze the tadpole cancellation conditions, written in equations (2.25), (2.23) and (3.14) for model-A, specified by the quantum numbers (m, n) of eqs. (5.34)-(5.37). We start with the analysis of the 7-brane R-R tadpoles (2.25). The expressions for the tadpole contributions q a t from the a-th brane, localized at the 2-cycle C (2) t , are given in eq. (2.24). For example, brane-1 has a potential contribution in the following 2-cycles:
q 1 [x 1 y 2 ] = −n 1 x 3 y 3 n 1 x 2 y 1 m 1 x 1 y 2 , q 1 [x 2 y 1 ] = −n 1 x 3 y 3 n 1 x 1 y 2 m 1 x 2 y 1 , (5.54) q 1 [x 1 y 1 ] = n 1 x 2 y 2 n 1 x 3 y 3 m 1 x 1 y 1 , q 1 [x 2 y 2 ] = n 1 x 1 y 1 n 1 x 3 y 3 m 1 x 2 y 2 , q 1 [x 3 y 3 ] = n 1 x 1 y 1 n 1 x 2 y 2 m 1 x 3 y 3 . (5.55)
By inserting the values of m's and n's from eq. (5.34), we obtain for brane-1:
q 1 [x 1 y 2 ] = 1, q 1 [x 2 y 1 ] = 1, q 1 [x 1 y 1 ] = 2, q 1 [x 2 y 2 ] = 0, q 1 [x 3 y 3 ] = 1 ,(5.56)
and similarly for brane-2:
q 2 [x 2 y 3 ] = 1, q 2 [x 3 y 2 ] = 1 q 2 [x 2 y 2 ] = 2, q 2 [x 3 y 3 ] = 0, q 2 [x 1 y 1 ] = 1 ,(5.
57) and brane-3:
q 3 [x 3 y 1 ] = 1, q 3 [x 1 y 3 ] = 1 q 3 [x 3 y 3 ] = 2, q 3 [x 1 y 1 ] = 0, q 3 [x 2 y 2 ] = 1 . (5.58)
The computation is similar for brane-4 to brane-6 with fluxes given in eqs. (5.35). The result for the 7-brane charges is:
q 4 [x 1 x 2 ] = 1, q 4 [y 1 y 2 ] = 1, q 4 [x 1 y 1 ] = 2, q 4 [x 2 y 2 ] = 0, q 4 [x 3 y 3 ] = 1 , (5.59) q 5 [x 2 x 3 ] = 1, q 5 [y 2 y 3 ] = 1. q 5 [x 2 y 2 ] = 2, q 5 [x 3 y 3 ] = 0, q 5 [x 1 y 1 ] = 1 , (5.60) q 6 [x 3 x 1 ] = 1, q 6 [y 3 y 1 ] = 1, q 6 [x 3 y 3 ] = 2, q 6 [x 1 y 1 ] = 0, q 6 [x 2 y 2 ] = 1 . (5.61)
Assuming that each stack contains only one brane N a = 1 ∀a, and adding the above contributions to the 7-brane tadpoles from branes 1-6, we obtain a non-vanishing result for the Similarly, the contributions of the branes 10 ′ -12 ′ read:
diagonal 2-cycles [x 1 y 1 ], [x 2 y 2 ], [x 3 y 3 ]: 6 a=1 N a q a t = 6 , t = [x 1 y 1 ], [x 2 y 2 ], [x 3 y 3 ] .q 11 [x 2 y 2 ] = n 11 x 3 y 3 n 11 x 1 y 1 m 11 x 2 y 2 = −1 , q 11 [x 1 y 1 ] = n 11 x 2 y 2 n 11 x 3 y 3 m 11 x 1 y 1 = −2, q 11 [x 2 y 1 ] = n 11 x 3 y 3 n 11 x 1 y 2 m 11 x 2 y 1 = −1 , q 11 [x 1 y 2 ] = n 11 x 2 y 2 n 11 x 2 y 1 m 11 x 1 y 2 = −1, (5.65) q 12 [x 3 y 3 ] = n 12 x 1 y 1 n 12 x 2 y 2 m 12 x 3 y 3 = −1 , q 12 [x 2 y 2 ] = n 12 x 3 y 3 n 12 x 1 y 1 m 12 x 2 y 2 = −2,q 10 ′ [x 1 y 1 ] = −1 , q 10 ′ [x 3 y 3 ] = −2 q 10 ′ [x 1 x 3 ] = −1 , q 10 ′ [y 1 y 3 ] = −1 , (5.67) q 11 ′ [x 2 y 2 ] = −1 , q 11 ′ [x 1 y 1 ] = −2, q 11 ′ [x 1 x 2 ] = −1 , q 11 ′ [y 1 y 2 ] = −1 , (5.68) q 12 ′ [x 3 y 3 ] = −1 , q 12 ′ [x 2 y 2 ] = −2, q 12 ′ [x 2 x 3 ] = −1 , q 12 ′ [y 2 y 3 ] = −1. (5.69)
Adding the results of eqs. (5.64)-(5.69), we obtain the (non-vanishing) contributions of branes 10-12 and 10 ′ -12 ′ to the 7-brane tadpoles:
q t 7,R = 12 a=10 N a q a t + 12 ′ a=10 ′ N a q a t = −6 , t = [x 1 y 1 ], [x 2 y 2 ], [x 3 y 3 ] . (5.70)
For off-diagonal 2-cycles one now has: Let us now discuss the 3-brane tadpole cancellation. It can be directly verified, using (2.23), the brane multiplicities N a = 1 for each of the twelve stacks, and the quantum numbers m specified in eqs. (5.34)-(5.37) that each of the branes 1-6 contributes q a 3,R = 1 (a = 1, .., 6) to the 3-brane tadpole, whereas for branes 10-12 and 10 ′ -12 ′ one obtains a vanishing contribution q a 3,R = 0 (a = 10-12, 10 ′ -12 ′ ). The total 3-brane tadpole contribution is therefore equal to 12 a=1 q a 3,R = 6. One possibility to cancel the 3-brane tadpole, namely to satisfy eq. (2.23), is to either take multiple copies of various branes 1-6, or/and add ordinary D3 branes to the system. On the other hand, one can also cancel the 3-brane tadpoles by turning on closed string 3-form fluxes, as discussed in section 3. In this way, one also has the advantage that the remaining closed string moduli, corresponding to the axion and dilaton, can be stabilized as well, specifying the string coupling uniquely.
12 a=10 N a q a t + 12 ′ a=10 ′ N a q a t = −1 , t = [x i y j ] , [x i x j ] , [y i y j ] , i = j.
Stabilization of the axion-dilaton moduli
As shown above, the model presented in section 5.3 is a consistent supersymmetric four dimensional perturbative vacuum with all closed string moduli but the dilaton fixed. In particular, the metric moduli, represented by the complex structure and the Kähler class are stabilized at the values Since the complex structure has been fixed to the purely imaginary diagonal form (5.72), the supersymmetry conditions (3.7) and (3.8) are trivially satisfied whereas eq. (3.9) fixes the dilaton in terms of the complex structure element τ 33 = i:
J ij = 4π 2 α ′ 2 δ ij , τ ij = iδ ij .(f 12 2 − φh 12 2 = τ 33 (f 21 1 − φh 21 1 ) , f 21 2 − φh 21 2 = τ 33 (f 12 1 − φh 12 1 ). (5.75)
The two equations can not be independent and give rise to a constraint on the allowed fluxes. Actually, these conditions are equivalent to the requirement that the G (3) flux must be of the type (2, 1). Indeed, in the complex coordinates (A.4), the only non-vanishing components of G (2,1) are G 113 and G 223 : if we assume that the complex structure modulus τ 33 has no real part, as we found in model-A. However, we keep track of the dependence of g s on the complex structure in order to examine (in the next section) the possible stabilization of the string coupling at small values.
−2iG (2,1) = (f 21 1 − φh 21 1 )dz 1 ∧ dz1 ∧ dz 3 − (f 12 1 − φh 12 1 )dz 2 ∧ dz2 ∧ dz 3 .
To simplify the discussion, let us further reduce the number of flux components to the case where only four of them are different than zero: As the flux quanta (5.79) have to be even in the absence of a B-field, the minimal value we can get for the string coupling is given by f 12 1 = −4 and h 12 2 = 2, which (for τ 33 = i) corresponds to g s = 1/2.
In fact, a smaller value of g s can be obtained in the presence of a non-trivial NS-NS B field. Consider for instance the case where a B x 3 y 3 = 0 is introduced in the third torus. As explained in section 3.2, the presence of b = 1/2 induces a different quantization of the flux quanta (3.18), which can now be odd integers. Moreover, the 3-brane tadpole condition (3.14) is modified to (3.16). Since the first Chern number m x 3 y 3 is shifted by b = 1/2, its minimal value is m x 3 y 3 = 1/2. It is therefore in principle possible to find a model similar to Model-A, where the six stacks of branes contribute half of the previous 3-brane chargeq 3,R = 3 instead of 6. The tadpole condition (3.16) then reads Since the quantum of the R-R flux f 12 1 has to be even, the minimal value for the string coupling g s is given by the choice of fluxes h 12 2 = 1 and f 12 1 = −4, with N D3 = 1. It follows from eq. (5.80), that the string coupling is then stabilized to the value g s = 1/4.
Another possibility: Model-A ′
In this section, we present another supersymmetric solution with total 3-brane tadpole contribution q 3,R = 6 to the 3-brane charge, vanishing 7-brane charges and complex structure and Kähler moduli fixed at the same values as before:
τ ij = iδ ij , J ij = (2π) 2 α ′ 2 δ ij . (5.86)
However, instead of introducing 12 stacks of branes, we only introduce nine, namely the stacks 1 to 9 given in eqns. (4.1)-(4.6) and (4.13)-(4.15), relaxing the "geometric" constraint (2.35) but keeping all intersection numbers to integer values. As non-trivial flux configurations, we choose for branes 1-3:
(m 1 x 1 y 2 , n 1 x 1 y 2 ), (m 1 x 2 y 1 , n 1 x 2 y 1 ) (m 1 x 2 y 3 , n 1 x 2 y 3 ), (m 1 x 3 y 2 , n 1 x 3 y 2 ) (m 1 x 1 y 3 , n 1 x 1 y 3 ), (m 1 x 3 y 1 , n 1 x 3 y 1 ) (m 1 x 1 y 1 , n 1 x 1 y 1 ) (m 1 x 2 y 2 , n 1 x 2 y 2 ) (m 1 x 3 y 3 , n 1 x 3 y 3 ) = (m 2 x 2 y 3 , n 2 x 2 y 3 ), (m 2 x 3 y 2 , n 2 x 3 y 2 ) (m 2 x 3 y 1 , n 2 x 3 y 1 ), (m 2 x 1 y 3 , n 2 x 1 y 3 ) (m 2 x 2 y 1 , n 2 x 2 y 1 ), (m 2 x 1 y 2 , n 2 x 1 y 2 ) (m 2 x 2 y 2 , n 2 x 2 y 2 ) (m 2 x 3 y 3 , n 2 x 3 y 3 ) (m 2 x 1 y 1 , n 2 x 1 y 1 ) = (m 3 x 3 y 1 , n 3 x 3 y 1 ), (m 3 x 1 y 3 , n 3 x 1 y 3 ) (m 3 x 1 y 2 , n 3 x 1 y 2 ), (m 3 x 2 y 1 , n 3 x 2 y 1 ) (m 3 x 3 y 2 , n 3 x 3 y 2 ), (m 3 x 2 y 3 , n 3 x 2 y 3 ) (m 3 x 3 y 3 , n 3 x 3 y 3 ) (m 3 x 1 y 1 , n 3 x 1 y 1 ) (m 3 x 2 y 2 , n 3 x 2 y 2 ) = (−1, 1), (1, −1) (0, 1), (0, −1) (0, 1), (0, 1) (2, −1) (0, l) (1, −1) , (5.87)
where the integer l in the last column specifies the numerical value of the windings along the indicated 2-cycles. For the time being, l is left arbitrary. As we will see later on, this parameter does not affect any of the previous discussions on moduli stabilization, as the flux along this particular 2-cycle is zero. Similarly, for branes 4-6 we choose:
(m 4 x 1 x 2 , n 4 x 1 x 2 ), (m 4 y 1 y 2 , n 4 y 1 y 2 ) (m 4 x 2 x 3 , n 4 x 2 x 3 ), (m 4 y 2 y 3 , n 4 y 2 y 3 ) (m 4 x 3 x 1 , n 4 x 3 x 1 ), (m 4 y 3 y 1 , n 4 y 3 y 1 ) (m 4 x 3 y 1 , n 4 x 3 y 1 ), (m 4 x 2 y 3 , n 4 x 2 y 3 ) (m 4 x 1 y 1 , n 4 x 1 y 1 ) (m 4 x 2 y 2 , n 4 x 2 y 2 ) (m 4 x 3 y 3 , n 4 x 3 y 3 ) = (m 5 x 2 x 3 , n 5 x 2 x 3 ), (m 5 y 2 y 3 , n 5 y 2 y 3 ) (m 5
x 3 x 1 , n 5
x 3 x 1 ), (m 5 y 3 y 1 , n 5 y 3 y 1 ) (m 5
x 1 x 2 , n 5
x 1 x 2 ), (m 5 y 1 y 2 , n 5 y 1 y 2 ) (m 5
x 1 y 2 , n 5 x 1 y 2 ), (m 5 x 3 y 1 , n 5 x 3 y 1 ) (m 5
x 2 y 2 , n 5 x 2 y 2 ) (m 5
x 3 y 3 , n 5 x 3 y 3 ) (m 5
x 1 y 1 , n 5
x 1 y 1 ) = (m 6
x 3 x 1 , n 6
x 3 x 1 ), (m 6 y 3 y 1 , n 6 y 3 y 1 ) (m 6
x 1 x 2 , n 6
x 1 x 2 ), (m 6 y 1 y 2 , n 6 y 1 y 2 ) (m 6
x 2 x 3 , n 6 x 2 x 3 ), (m 6 y 2 y 3 , n 6 y 2 y 3 ) (m 6
x 2 y 3 , n 6 x 2 y 3 ), (m 6 x 1 y 2 , n 6 x 1 y 2 ) (m 6
x 3 y 3 , n 6 x 3 y 3 ) (m 6
x 1 y 1 , n 6 x 1 y 1 ) (m 6
x 2 y 2 , n 6
x 2 y 2 ) = (−1, 1), (1, −1) (0, 1), (0, −1) (0, 1), (0, 1) (0, 1), (0, 1) (2, −1) (0, l) (1, −1) . (5.88)
Finally, for branes 7-9, the values of the fluxes are given by:
(m 7 x 1 y 1 , n 7 x 1 y 1 ) (m 7 x 2 y 2 , n 7 x 2 y 2 ) (m 7 x 3 y 3 , n 7 x 3 y 3 ) = (m 8 x 2 y 2 , n 8 x 2 y 2 ) (m 8 x 3 y 3 , n 8 x 3 y 3 ) (m 8 x 1 y 1 , n 8 x 1 y 1 ) = (m 9 x 3 y 3 , n 9 x 3 y 3 ) (m 9
x 1 y 1 , n 9 x 1 y 1 ) (m 9
x 2 y 2 , n 9 Thus, all off-diagonal contributions vanish for each brane separately.
x 2 y 2 ) = (−1, 1) (0, 3(−l + 1)) (−1, 1)) .
This model therefore satisfies all consistency conditions listed in section 2.3. Its special feature is the presence of an additional parameter l that represents the winding number of some 2-cycles where the branes have vanishing first Chern number. As a consequence, it does affect neither the magnetic fluxes F r = mr nr nor the supersymmetry conditions (2.1)-(2.2) and therefore does not change the values of the fixed moduli. However, since the tadpoles q t and the overall winding number W a of the brane stacks are sensitive to l, the different vacua will have different couplings and spectra. Thus, the presence of this parameter implies the existence of an infinite family of vacua with identical values for the geometrical moduli but with different couplings and spectra. It is therefore important to check further the consistency of this model by computing for instance its partition function.
Large dimensions
Here, we examine the possibility to stabilize the transverse to the D3 branes volume modulus at large values. In the T-dual case presented in [3], for a supersymmetric vacuum compatible with the presence of O9-planes, it is possible to obtain for instance two large radii longitudinal to the magnetized D9 branes. This was achieved by an appropriate rescaling of the magnetic fluxes m r , which is compatible with all tadpole cancellation conditions. On the other hand, the winding numbers n r can not be rescaled, because they are constrained by the 9-brane tadpole condition. Similarly, by a uniform rescaling of all magnetic fluxes, one could obtain a family of solutions with all six radii large.
The situation in our case is similar. The vacuum presented in the section 5.3 corresponds to the case of three orthogonal tori T 2 × T 2 × T 2 with radii R 1 i and R 2 i . The Kähler form and complex structure (5.72) correspond to the areas and ratios :
J x i y i = 4π 2 R 1 i R 2 i , τ ii = i R 2 i R 1 i , i = 1, 2, 3. (6.1)
Unlike the T-dual case, now the 3-brane tadpole condition (2.23) restricts strongly the possible rescaling of the first Chern numbers m r , but it does not constrain the winding numbers. There exists therefore a set of different families of an infinite but discrete number of vacua, starting for instance from those found in the previous section:
• All radii are rescaled uniformly at values lower than the string length √ α ′ . Thus, Kähler moduli are rescaled whereas the complex structure remains at the original value:
J r = (2π) 2 Λ −1 α ′ , τ ij = iδ ij . (6.2)
This is achieved be a rescaling of all winding numbersn a r = Λn a r , resulting into a decrease of all magnetic fluxesF a r = Λ −1 F a r . Indeed, the supersymmetry condition (2.15) is then satisfied by rescaled Kähler moduliĴ r = Λ −1 J r , ∀r. On the other hand, the complex structure moduli in eq. (5.10) are given by ratios of fluxes. Therefore, a general rescaling of the latter does not affect the complex structure. As a result, the radii of the different tori T 2 i remain equal even after the rescaling:R 2 i =R 1 i . This rescaling is also compatible with the 7-brane tadpoles. In fact, in the setup presented in section 5.5, the 7-brane charges induced by the stacks 1 to 6 are cancelled by the contributions of the stacks 7 to 9. As all 7-brane charges (2.24) are quadratic in the winding numbers, the tadpole conditions (2.25) are left invariant after the rescaling. It is therefore possible to obtain arbitrary small radiiR 2 i = 1 √ Λ R 2 i by a general rescaling of all winding numbers. It follows that from the explicit example of a supersymmetric vacuum with fixed moduli (5.72), there exists an infinity of discrete supersymmetric vacua with the same complex structure τ ij = iδ ij and arbitrary small volume moduliĴ x i y i , i = 1, 2, 3. It is easy to see that in the T-dual version, this corresponds actually to large "longitudinal" dimensions, along the world volume of the branes.
• A single radius is smaller than the string length √ α ′ , say R 1 3 , whereas the others remain of order of the string length. This corresponds to the case where the Kähler class moduli J x 1 y 1 and J x 2 y 2 remain fixed, as well as τ 11 and τ 22 , whereas the area J x 3 y 3 of the third T 2 is small and its radii ratio τ 33 is big: • Two (or three) of the complex structure moduli are big and two (or three) of the T 2 's areas become smaller than the string scale. For instance,
τ 11 = τ 22 = i , J x 1 y 1 = J x 2 y 2 = (2π) 2 α ′ ;τ 33 = iΛ ,Ĵ x 3 y 3 = (2π) 2 Λ −1 α ′ .τ 11 = i , J x 1 y 1 = (2π) 2 α ′ ; τ 22 = τ 33 = iΛ ,Ĵ x 2 y 2 =Ĵ x 3 y 3 = (2π) 2 Λ −1 α ′ . (6.4)
In this example, the radii R 1 3 and R 1 2 are fixed to a value smaller than the string length, keeping the other ones of order √ α ′ . This can be achieved by the rescaling of all winding numbers involving the directions x 3 and x 2 , namelŷ n a x 2 x 3 = Λ 2 n a x 2 x 3 ,n a x j y i = Λn a x j y i ,n a x j x 1 = Λn a x j x 1 for i = 1, 2, 3, j = 2, 3 . (6.5)
• The areas can be fixed at small values while keeping the radii ratios fixed. For instance, we can rescale one area, say of the last T 2 :
τ ij = iδ ij , J x 1 y 1 = J x 2 y 2 = (2π) 2 α ′ , J x 3 y 3 = (2π) 2 Λ −2 α ′ . (6.6)
Here, the radii R 1 3 and R 2 3 are increased by the rescaling of all winding numbers which involves the directions x 3 and y 3 , namelŷ n a x 3 y 3 = Λ 2 n a x 3 y 3 ,n a x i y 3 = Λn a x i y 3 ,n a x 3 y i = Λn a x 3 y i ,n a x 3 x i = Λn a x 3 x i ,n a y 3 y i = Λn a y 3 y i , (6.7)
for i = 1, 2. The same method can be used in order to fix more than two areas at values much lower than the string scale α ′ .
7 Model-B with q 3 = 12
We now present another consistent model for the stabilization of Kähler and complex structure moduli using open and closed string fluxes. In this example, as seen by comparing eqs. (4.1)-(4.6) with (4.7)-(4.12), certain components of fluxes (of the type p a x i y j , p a x i x j , p a y i y j , i = j) in branes-a and a ′ (a = 1, .., 6) are equal in magnitude and opposite in sign. Their contributions to the 7-brane tadpoles are also equal and opposite, and such tadpoles cancel between pairs of brane-a and brane-a ′ (a = 1, .., 6). Thus, one is left with non-zero contributions to the 7brane tadpoles from branes 1-6 (and 1 ′ -6 ′ ) only along the diagonal directions [x 1 y 1 ], [x 2 y 2 ] and [x 3 y 3 ], which then cancel with the opposite contributions from branes 7-9. To show the tadpole cancellation explicitly and to find out the resulting stabilized values of the complex structure and Kähler class moduli, we choose the (m, n) quantum numbers along various branes as given in Appendix B. These values give the same magnetic fluxes for the branes 1-6 as in eqs. (5.38)-(5.39). On the other hand, the magnetic fluxes for branes 1 ′ -3 ′ are given by:
p ′1 x 1 y 2 p ′1 x 2 y 1 p ′1 x 1 y 1 p 1 x 3 y 3 = p ′2 x 2 y 3 p ′2 x 3 y 2 p ′2 x 2 y 2 p ′2 x 1 y 1 = p ′3 x 3 y 1 p ′3 x 1 y 3 p ′3 x 3 y 3 p ′3 x 2 y 2 = 1 1 −2 −1 . (7.1)
Similarly, for branes 4 ′ -6 ′ the non-zero fluxes read:
p ′4 x 1 x 2 p ′4 y 1 y 2 p ′4 x 1 y 1 p ′4 x 3 y 3 = p ′5 x 2 x 3 p ′5 y 2 y 3 p ′5 x 2 y 2 p ′5 x 1 y 1 = p ′6 x 3 x 1 p ′6 y 3 y 1 p ′6 x 3 y 3 p ′6 x 2 y 2 = 1 1 −2 −1 . (7.2)
We can now discuss the stabilization of the complex structure and Kähler class moduli for model-B, specified by branes 1-6, 1 ′ -6 ′ and 7-9. These branes alone stabilize the moduli in the present case, as well, to the same values:
τ ij = 0, (i = j), τ 11 = τ 22 = τ 33 = i. (7.3) J ij = 0, (i = j), J x 1 y 1 = J x 2 y 2 = J x 3 y 3 = (2π) 2 α ′ . (7.4)
However, one now has the additional branes 1 ′ -6 ′ and we must therefore make sure that their presence maintains the moduli stabilization values (7.3), (7.4). To see that this is indeed the case, we first notice that the values of the complex structure given from eqs. (2.14) and (7.3) remain invariant if one changes the sign of all the magnetic flux components of the type, p a x i y j , p a
x i x j , p a y i y j , i = j, while keeping the diagonal fluxes p a x i y i unchanged. Since this is precisely the change induced in branes 1 ′ -6 ′ , we conclude that model-B gives still the same solution for the complex structure moduli as in eq. (7.3). Next, we note that the supersymmetry conditions, written for branes 1-9 in eqs. (5.20)-(5.28), are respected by the branes 1 ′ -6 ′ as well. More precisely, the r.h.s. of eqs. (5.20)-(5.25), as well as eqs. (5.29), written for branes 1 ′ -6 ′ , are identical with those for branes 1-6. Similarly, eqs. (5.31) and (5.32), imposing the positivity condition (2.30), remain also identical for branes 1 ′ -6 ′ as for branes 1-6. Finally eq. (5.46), used in determining the explicit value of the diagonal components of the Kähler moduli, also remains intact when one replaces the branes 1-6 by 1 ′ -6 ′ . We therefore have the solution of the Kähler moduli for model-B as in eq. (7.4).
To show the cancellation of the 7-brane and 3-brane tadpoles in this model, we first note that the general expression for the 7-brane tadpole contribution remains the same as in section 5.5 for model-A. In Appendix B, we give the tadpole contributions from every brane and show the 7-brane tadpole cancellations. The 3-brane tadpole cancellation in this model is also similar to the one discussed in section 5.5. Each of the branes 1-6 and 1 ′ -6 ′ contributes q a 3,R = 1 to the 3-brane tadpole, whereas this contribution is zero for branes 7-9. One therefore obtains the total 3-brane tadpole:
6 a=1 q a 3,R + 6 a ′ =1 q a ′ 3,R + 9 a=7
q a 3,R = 12, (7.5) if only a single brane of each stack is used, N a = N a ′ = 1. To satisfy the condition (2.23), one can for instance add four space filling D3 branes to the system, or alternatively consider multiple copies of some of the D9 branes 1-6 and 1 ′ -6 ′ . Moreover, in this model , it is also possible to introduce some R-R and NS-NS 3-form fluxes in order to fix the dilaton. Let us assume for instance the same configuration of quanta as in section 5.6. The value for the string coupling in terms of the 3-form quanta is still given by eq. (5.80), but the tadpole condition (5.83) changes because of the higher contribution (7.5) due to the additional magnetized branes. The condition (3.14) now reads −h 12 2 f 12 1 = 4, (7.6) and the minimal value for the string coupling is then given by f 12 1 = −2 and h 12 2 = 2, which corresponds to g s = 1.
Moduli stabilization using open and closed string fluxes
In the previous sections, we have shown in several examples that both complex structure and Kähler class moduli stabilization can be achieved in string theory involving wrapped D9 branes, using magnetic fluxes that are turned on along the compactified directions. In this section, we present models where some of the complex structure and Kähler class moduli are fixed using the 3-form fluxes that were introduced in sections 5.6 and 7 to stabilize the axion-dilaton field. To this end, we make use of the primitivity condition (3.3) and the superpotential variation eqs. (3.7)-(3.9) to put several constraints on the geometric moduli. The remaining ones are then fixed by the magnetic fluxes along the branes, as in sections 5 and 7.
Model-C with q 3 = 4 and 3-form fluxes
As an explicit example, we present a model (called model-C), in which the 3-form fluxes involve four non-vanishing parameters of eq. (5.79). The conditions imposed by this flux on the complex structure and dilaton moduli are given in (3.7)-(3.9). Eqs. (3.9) give rise to five conditions on the nine complex structure matrix elements: The above relations (8.1), (8.2) and (8.3) assure that the 3-form flux G (3) is of the type G (2,1) . If we anticipate the fact that the magnetic fluxes fix the remaining off diagonal complex structure component to zero, τ 12 = 0, the 3-form flux reads: In addition, we use the three stacks of branes with only diagonal fluxes, 7-9 given in eqs. (4.13)-(4.15). However, the quantum numbers (m, n) for these branes are now different from the ones in eq. (5.89) as will be specified later in eqs. (8.8) and (8.9). We have already seen in section 5.1 that branes 1-3 fix the ratios of the diagonal components of the complex structure, according to (5.6). Branes 1-4 then completely determine all diagonal components τ 11 , τ 22 and τ 33 . The presence of these magnetized D9 branes also fixes the remaining off-diagonal component of τ ij to zero. We have thus stabilized all complex structure moduli, using the corresponding 3-form fluxes (5.79) and branes 1-4, to the value τ ij = iδ ij , as in eq. (5.42). Now, to stabilize the remaining Kähler class moduli, we use branes 7-8, as well as branes 1-4. After we have shown the Kähler and complex structure moduli stabilization for model-C, we can discuss the tadpole cancellation conditions. In fact, brane-9 is needed only for tadpole cancellation and does not in any way disturb the moduli stabilization obtained above. The 7-brane tadpole contributions from branes 1-4 is then given by (using l = −1): To cancel these tadpoles using branes 7-9, we modify the values of the corresponding quantum numbers (m, n) compared to the ones of eqs. (5.89) to:
−2iG (2,1) = f 21 1 Imτ 11 dz 1 ∧ dz1 ∧ dz 3 − f 12 1 Imτ 22 dz 2 ∧ dz2 ∧ dz 3 .
(m 7 x 1 y 1 , n 7 x 1 y 1 ) (m 7
x 2 y 2 , n 7 x 2 y 2 ) (m 7
x 3 y 3 , n 7 (m 9 x 3 y 3 , n 9 x 3 y 3 ) (m 9
x 1 y 1 , n 9 x 1 y 1 ) (m 9
x 2 y 2 , n 9
x 2 y 2 ) = (−1, 1) (0, 2) (−1, 1) (8.9)
We then obtain the following 7-brane tadpole contributions from these branes: On the other hand, the total 3-brane tadpole in this model (from single copies of branes 1-4, and 7-9) is equal to q R 3 = 4 which, after adding the 3-form flux contribution should satisfy eq. (2.23). Using eq. (3.14), we get:
Conclusion
In this work, we presented several consistent string models based on T 6 /Z 2 orientifolds of type IIB theory, having N = 1 supersymmetry in four dimensions and stabilized complex structure and Kähler class moduli using open string magnetic fluxes. We have also shown that the dilaton-axion modulus can be stabilized by turning on closed string 3-form fluxes consistently with the leftover supersymmetry and the fixed values of the geometric moduli in the presence of the magnetic fields. By tuning the fluxes appropriately, we found an infinite but discrete series of vacua where some radii are fixed at arbitrarily large values, while the dilaton can be stabilized at arbitrarily weak values for the string coupling.
An advantage of fixing moduli using internal magnetic fields is that the method has an exact string description and the spectrum, as well as the effective interactions, are calculable in terms of modified boundary conditions for the world-sheet fields. The method has also a direct application to string model building based on intersecting branes, while it can in principle be generalized to include open string moduli breaking gauge symmetries. Finally, we have presented examples where some of the complex structure and Kähler class moduli are stabilized by the magnetic fluxes whereas the remaining ones, as well as the axion-dilaton, are stabilized using the 3-form fluxes. Among interesting open problems is to study nonsupersymmetric vacua with stabilized moduli and count consistent solutions in this corner of the string landscape.
choose then the orientation 8 T 6 dx 1 ∧ dy 1 ∧ dx 2 ∧ dy 2 ∧ dx 3 ∧ dy 3 = 1 (A.1) and define the basis of the cohomology H 3 (T 6 , Z)
α 0 = dx 1 ∧ dx 2 ∧ dx 3 α ij = 1 2 ǫ ilm dx l ∧ dx m ∧ dy j (A.2) β ij = − 1 2 ǫ jlm dy l ∧ dy m ∧ dx i β 0 = dy 1 ∧ dy 2 ∧ dy 3 ,
forming a symplectic structure on T 6 :
T 6 α a ∧ β b = −δ b a , for a, b = 1, · · · , h 3 /2 , (A.3)
with h 3 = 20, the dimension of the cohomology H 3 (T 6 , Z).
We can also choose complex coordinates
z i = x i + τ ij y j , (A.4)
where τ ij is a complex 3 × 3 matrix parametrizing the complex structure. In this basis, the cohomology H 3 (T 6 , Z) decomposes in four different cohomologies corresponding to the purely holomorphic parts and those with mixed indices: The purely holomorphic cohomology H 3,0 is one-dimensional and is formed by the holomorphic three-form Ω for which we choose the normalization
Ω = dz 1 ∧ dz 2 ∧ dz 3 . (A.6)
In terms of the real basis (A.2), this can be written as
Ω = α 0 + τ ij α ij − cofτ ij β ij + detτ β 0 , (A.7)
where cofτ ij is given by cofτ = (detτ ) τ −1,T . We can then define the periods of the holomorphic 3-form to be τ a = As the Kähler form is a real form, its elements satisfy the reality condition J † ij = J jī . Therefore J depends only on nine real parameters.
B Quantum numbers (m, n) in Model-B
In this appendix we give some more details on model-B presented in section 7.
[(m 1
x 1 y 2 , n 1 x 1 y 2 ), (m 1 x 2 y 1 , n 1 x 2 y 1 ), (m 1 x 1 y 1 , n 1 x 1 y 1 ), (m 1 x 3 y 3 , n 1 [(m ′ 1 x 1 y 2 , n ′ 1 x 1 y 2 ), (m ′ 1 x 2 y 1 , n ′ 1 x 2 y 1 ), (m ′ 1 x 1 y 1 , n ′ 1 x 1 y 1 ), (m ′ 1 x 3 y 3 , n ′ 1 The 3-brane tadpole cancellation in this model is discussed in the text.
Figure 2 :
2Example of a brane at angle with its orientifold image and antibrane. The O-plane is situated on the vertical axis.
Figure 3 :
3O3 plane configuration in case of discrete torsion in the direction [x 3 y 3 ].
to 3 cycles wrapping a 'diagonal' 2-cycle [x i y i ] as well as one of the 1-cycles x 3 or y 3 . They are located at one fixed point of the last 2-torus. As a result, the following quanta (3.1) of H (3) can be odd: section we present the different stacks of magnetized D9 branes we need in order to satisfy the supersymmetry conditions (2.14), (2.15), the positivity requirements (2.26), (2.27) and (2.30), and the tadpole cancellations (2.25) and(2.23). For the shake of simplicity, our aim is to stabilize the moduli to a geometry of a factorized torus T 6 as T 2 × T 2 × T 2 . This implies
Figure 4 :
4Example of 3-cycle with an odd crossing number of O3 + 's. The cycle [x 1 y 1 x 3 ] crosses a single O3 + . in particular that the off-diagonal components of the complex structure, defined in terms of the real coordinates x i , y i (i = 1, 2, 3) through eq. (A.4), should vanish.
solve both eqs.(5.8) and(5.9). Moreover, the diagonal components of the complex structure moduli are fixed using eq. (5.10) to:τ 11 = τ 22 = τ 33 = i.(5.42) Since all diagonal components of the fluxes (5.38)-(5.40) are negative, they also obviously satisfy the conditions (5.31)-(5.33). The conditions (5.29) and (5.30) are also satisfied, as will be shown in the following subsection 5.4. We have therefore shown that the explicit choice for the fluxes presented in eqs. (5.38)-(5.40) satisfy the consistency requirements imposed earlier. Obviously, this choice is not unique. For instance, it is possible to modify them in a way that the products appearing in the supersymmetry conditions (5.20)-(5.28) involving also the Kähler class moduli remain invariant. Before ending this section, we also give the matrices entering in eqs. (5.1) and (5.2), for the values of fluxes (5.38)-(5.40). The 3 × 3 matrix appearing in the l.h.s. of eq. (5.1) reads:
and implies the relation (5.3): τ 21 = τ 32 = τ 13 . Using this equality in the r.h.s. of eq. (5.1) with the result (5.43), one finds: τ 12 = τ 23 = τ 31 and τ 21 = τ 32 = τ 13 = 0. Finally, using brane 4 (or brane 5-6) one obtains the result (5.5) that all off-diagonal components of the complex structure τ ij are zero.
( 5 .
526)-(5.28), related to the stacks 10-10 ′ , 11-11 ′ and 12-12 ′ , are trivially satisfied because of our choice of fluxes. Even if the system could in principle be solved exactly, we only present here the solution where the off-diagonal components of the Kähler form vanish, J ij = 0 for i = j. This solution is consistent with eqs. (5.26)-(5.28), arising from the brane stacks 10-12 and 10 ′ -12 ′ with the choice of fluxes given in (5.40), for a restricted Kähler class moduli space where J 11 = J 22 = J 33 . (5.45) Moreover, the brane stacks 1-6 restrict further the Kähler moduli to
that the conditions (5.29) and (5.30) are satisfied, we rewrite the fluxes in the complex coordinates (A.10), using (2.16) and eqs. (5.38)-(5.40):
then easy to see that the conditions (5.29) are satisfied, using the result J 11 = J 22 = J 33 . Similarly, the conditions (5.30) are satisfied using the following expressions for the magnetic fluxes along the branes 10-12 in complex coordinates:
other hand, for each of the twelve off-diagonal 2-cycles: [x i y j ],[x i x j ],[y i y j ] for i = j we have: 6 a=1 N a q a t = 1 , t = [x i y j ] , [x i x j ] , [y i y j ] , i = j. (5.63) The 7-brane tadpole contributions for the branes 10-12 with fluxes and quantum numbers (m, n) given in eq. (5.36) are also non-vanishing and read: q 10 [x 1 y 1 ] = n 10 x 2 y 2 n 10 x 3 y 3 m 10 x 1 y 1 = −1 , q 10 [x 3 y 3 ] = n 10 x 1 y 1 n 10 x 2 y 2 m 10 x 3 y 3 = −2 , q 10 [x 1 y 3 ] = −n 10 x 2 y 2 n 10 x 3 y 1 m 10 x 1 y 3 = −1 , q 10 [x 3 y 1 ] = −n 10 x 2 y 2 n 10 x 1 y 3 m 10 x 3 y 1 = −1 , (5.64)
q 12 [x 3 y 2 ] = −n 12 x 1 y 1 n 12 x 2 y 3 m 12 x 3 y 2 = −1 , q 12 [x 2 y 3 ] = −n 12 x 1 y 1 n 12 x 3 y 2 m 12 x 2 y 3 = −1. (5.66)
. (5.62), (5.63) and (5.70), (5.71), we conclude that the total 7-brane R-R tadpoles vanish for all 2-cycles, when the contributions of all the branes are added.
for the values (5.72) of the Kähler form , the primitivity condition G (3) computation shows that under the restriction on the fluxes coming from eqs. (5.75) and from the primitivity condition (5.77), the string coupling is given by:
this restriction is only possible in the absence of a B-field, as explained in section 3.2. Indeed, in the presence of a B-field (b = 1/2), the flux components h 12 1 and h 21 1 have to be odd and can therefore not be set to zero. With the reduced number of non-vanishing elements (5.79), the string coupling (5.78) is then fixed to the to compute the value of the dilaton, we first have to analyze the 3-form contribution to the 3-brane tadpole(3.13). From the symplectic structure (A.3) and the restriction (
the only non-vanishing components of the 3-form fluxes are still the ones of eq. (5.79), the tadpole condition (5.
easy to see that this configuration of fluxes satisfy the consistency conditions imposing the absence of antibranes (2.26) or (2.27), the positivity condition (2.30) and the supersymmetry conditions (2.1) and (2.2) for the values of the moduli given in (5.86). However, the condition (2.35) is obviouly not satisfied for generic values of the parameter l; it is satisfied only for the values l = ±1. Despite this fact, the intersection numbers (2.34) are integers for any pair of stacks presented in (5.87)-(5.89). Furthermore, all tadpole conditions are satisfied: • The first six stacks give rise to a 3-brane charge q 3,R = 6 for the simple case in which each stack is composed of a single brane. • The 7-brane charges induced in the diagonal directions t = [x 1 y 1 ], [x 2 y 2 ], [x 3 y 3 ] from the 6 first stacks are canceled by the choice of fluxes in the last three stacks.
• 7 -
7brane charges along the off-diagonal directions, t = [x i y j ], [x i x j ], [y i y j ] where i = j, can be in principle induced only by the stacks 1-6, since branes 7-9 have only diagonal fluxes. However, we have chosen the winding numbers in (5.87) and (5.88) in such a way, so that the effective winding around the 4-cycle perpendicular to each 2-cycle C
be achieved by a rescaling of the windings of model-A which involves the direction x 3 , 7 namelyn a x 3 y i = Λn a x 3 y i andn a x 3 x i = Λn a x 3 x i , for i = 1, 2, 3 and for all stacks of branes a = 1, . . . , 8. Indeed, from eqs. (5.6) and (5.7), we notice that the complex structure moduli τ 11 and τ 22 are not rescaled, in contrast to τ 33 which gets rescaled asτ 33 = Λτ 33 . Furthermore, the solutions to the supersymmetry conditions (5.20)-(5.28) remain valid for a rescaled areaĴ x 3 y 3 = Λ −1 J x 3 y 3 . By a similar argument as in the previous case, it can be finally checked that even with the rescaled winding numbers, the tadpole cancellation conditions (2.23) and (2.25) are still satisfied.This family of discrete vacua provides a new interesting feature: It allows the rescaling of the string coupling (5.80) g s = −Λ the tadpole condition (5.83). This does not come from a rescaling of the 3-form quanta h 21 2 and f 12 1 and therefore its tadpole contribution to (5.83) remains invariant (of order unity). Note however that upon T-duality where the small dimension becomes large longitudinal, the string coupling becomes again of order one.
τ 13 = τ 23 = τ 31 = τ 32 = 0 , f 12 1 τ 21 + f 21 1 τ 12 = 0. (8.1) Condition (3.7) is then trivially satisfied, while eq. (3.8) restricts the flux parameters by f 12 1 h 12 2 = f 21 1 h 21 2 . (8.2) Finally, the condition (3.9) relates the axion-dilaton field φ to the yet undetermined complex structure element τ 33 φh 12 2 = −τ 33 f 21 1 . (8.3)
turn on to the restriction on the Kähler form coming from the primitivity condition G (2,1) ∧ J = 0 which is a (3, 2)-form. As there exists three of them on T 6 /Z 2 , this condition could give rise to a maximum of three complex conditions on the Kähler form. In our case, the choice of fluxes made in eq. (5.79) restricts the Kähler moduli space to J 13 = J 23 = 0 in a supersymmetric vacuum, the presence of the closed string fluxes (5.79) restricts the metric moduli space. There are five complex structure and three Kähler class moduli which are fixed. They correspond to a factorized geometry of the form T 4 × T 2 , where the complex structure τ 11 , τ 22 and τ 12 of the T 4 and τ 33 of the T 2 remains unfixed. In the same way, the Kähler moduli J 11 , J 22 and J 12 of the T 4 , as well as the area J 33 of the T 2 are not stabilized by the closed string moduli. They correspond to four complex parameters for the complex structure and four real ones for the Kähler class.In order to fix the remaining moduli, we switch on internal magnetic fields, using branes 1-4 presented of section 4, with fluxes given in eqs. (4.1)-(4.4). In this example, we also use the quantum numbers (m, n) for the branes 1-4, given in eqs. (5.87)-(5.88).
The corresponding supersymmetry conditions (5.26), (5.27) and (5.20)-(5.23) has as solution: J 11 = J 22 = J 33 , J 12 = J 21 = 0. (8.6) Furthermore, the actual value for the diagonal Kähler components is the same as in eqs. (5.47).
N 3 =
3D3 = 0, corresponding to the case when no space-filling D3 branes are introduced, we get for the 3-form flux: −the axion-dilaton modulus is stabilized by the 3-form fluxes at a value given in eq. (5.80). To obtain a weak coupling string theory solution, we choose the maximum possible value for the dilaton modulus, by choosing h 12 2 = 2, f 12 1 = −6, implying the value for the string coupling g s = 1 3 . (8.13)
H 3 (
3T 6 ) = H 3,0 (T 6 ) ⊕ H 2,1 (T 6 ) ⊕ H 1,2 (T 6 ) ⊕ H 0,3 (T 6 ). (A.5)
the period F b can be written as the derivative of a prepotential F :F b = ∂ τ b F .Similarly, the cohomology H 2 (T 6 , Z) decomposes also in three cohomologiesH 2 (T 6 ) = H 2,0 (T 6 ) ⊕ H 1,1 (T 6 ) ⊕ H 0,2 (T 6 ). (A.9)We choose the basis e ij of H 1,1 to be of the forme ij = idz i ∧ dzj. (A.10)The Kähler form can therefore by parametrized as J = J ij e ij . (A.11)
x 1 x 2 , n 4 x 1 x 2 ), (m 4 y 1 y 2 , n 4 y 1 y 2 ), (m 4 x 1 y 1 , n 4 x 1 y 1 ), (m 4 x 3 y 3 , n 4 x 3 y 3
22221133−1, 1), (1, −1), (2, −1), (1, −1)] (B.2)
x 3 y 3 NN a q a [x 3 y 3 ]
33)1 x 2 ] = −1, q 4 ′ [y 1 y 2 ] = −1 , q 4 ′ [x 1 y 1 ] = 0, q 4 ′ [x 2 y 2 ] = 0 , q 4 ′ [x 3 y 3 ] = 1 , (B.15) q 5 ′ [x 2 x 3 ] = −1, q 5 ′ [y 2 y 3 ] = −1 , q 5 ′ [x 2 y 2 ] = 0, q 5 ′ [x 3 y 3 ] = 0 , q 5 ′ [x 1 y 1 ] = 1 , (B.16) q 6 ′ [x 3 x 1 ] = −1, q 6 ′ [y 3 y 1 ] = −1 , q 6 ′ [x 3 y 3 ] = 0, q 6 ′ [x 1 y 1 ] = 0 , q 6 ′ [x 2 y 2 ] = 1 . (B.17)Adding the contributions from branes 1-6 and 1 ′ -6 ′ , we obtain non-zero values only for tadpoles corresponding to the three diagonal directions (x 1 y 1 ), (x 2 y 2 ), (x 3 y 3 ). a ′ q a ′ [x 3 y 3 ] = 4It can then be verified that the above tadpole contributions are cancelled by those of branes 7-9 for the choice of quantum numbers (m, n) given in eq. (B.5). Indeed, = −4.
This model however satisfies weaker constraints and further work is needed to establish its consistency.
See parametrization in Appendix A.
As mentioned in the introduction, the above counting of conditions holds for vacua with unbroken gauge symmetries, without open string moduli switched on.
We thank R. Blumenhagen for useful communications on this point.
Note that it does not come from the factor 1 2 in (3.10) which is compensated by the magnetic coupling to C (4) ; see[16] for more details.
′ . [−p 1 x 1 y 2 , −p 1 x 2 y 1 , p 1 x 1 y 1 , p 1 x 3 y 3 ] = 0, (4.7) 2 ′ . [−p 2 x 2 y 3 , −p 2 x 3 y 2 , p 2 x 2 y 2 , p 2 x 1 y 1 ] = 0, (4.8) 3 ′ . [−p 3 x 3 y 1 , −p 3 x 1 y 3 , p 3 x 3 y 3 , p 3 x 2 y 2 ] = 0, (4.9) 4 ′ . [−p 4 x 1 x 2 , −p 4 y 1 y 2 , p 4 x 1 y 1 , p 4 x 3 y 3 ] = 0, (4.10) 5 ′ . [−p 5 x 2 x 3 , −p 5 y 2 y 3 , p 5 x 2 y 2 , p 5 x 1 y 1 ] = 0, (4.11) 6 ′ . [−p 6 x 3 x 1 , −p 6 y 3 y 1 , p 6 x 3 y 3 , p 6 x 2 y 2 ] = 0. (4.12)The stacks 1-6 (or alternatively stacks 1 ′ -6 ′ ), when used with some other branes with diagonal fluxes along [x i y i ] (called stacks 7-9), give six independent conditions on the Kähler moduli J ij , (i = j) and force them to vanish. In our examples, we choose the stacks 7-9 having only two non-zero diagonal components of magnetic fluxes. The magnetic fields along these branes are required to satisfy the consistency conditions mentioned in section 2.3 and are sufficient to fix all diagonal components of the Kähler moduli J, as well. More precisely, the fluxes in stacks 7-9 read: 7. [p 7x 1 y 1 = 0, p 7 x 2 y 2 = 0, p 7 x 3 y 3 = 0],(4.13)
Note that the coordinate x 3 in (6.1) has periodicity x 3 ≡ x 3 + 2πR 1 3 .
This is the orientation of[21], which is different from the one of[1].
AcknowledgmentsWe would like to thank for useful discussions Massimo Bianchi, Ralph Blumenhagen, Juan Cascales, Hans Jockers, Elisa Trevigne and Angel Uranga. AK thanks the CERN Theory Division for warm hospitality during the course of this work. TM thanks the Swiss Army for kind hospitality, where part of this work has been done. This work was supported in part by the European Commission under the RTN contract MRTN-CT-2004-503369, and in part by the INTAS contract 03-51-6346.A NotationsA.1 Parametrization of T 6Consider a six-dimensional torus T 6 having six coordinates u k , k = 1, . . . , 6 with periodicity normalized to unity x i = x i + 1, y i = y i + 1[21]. Writing the coordinates u k as x i , y i , we x 2 y 3 , n ′ 2B.1 Tadpole cancellation in model-BThe 7-brane R-R tadpole contributions, using the (m, n) quantum numbers of eq. (B.1) for branes 1-3, are given as:Similarly, for branes 1 ′ -3 ′ the expressions are:
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. G W Moore, arXiv:hep-th/9807087G. W. Moore, arXiv:hep-th/9807087.
| []
|
[
"Finite-temperature behavior of an impurity in the spin-1/2 XXZ chain",
"Finite-temperature behavior of an impurity in the spin-1/2 XXZ chain"
]
| [
"Ryoko Yahagi [email protected] ",
"Jun Sato \nResearch Center for Advanced Science and Technology\nUniversity of Tokyo\n4-6-1 Komaba, Meguro-ku153-8904TokyoJapan\n",
"Tetsuo Deguchi ",
"\nDepartment of Physics\nGraduate School of Humanities and Sciences\nOchanomizu University\n2-1-1 Ohtsuka, Bunkyo-ku112-8610TokyoJapan\n"
]
| [
"Research Center for Advanced Science and Technology\nUniversity of Tokyo\n4-6-1 Komaba, Meguro-ku153-8904TokyoJapan",
"Department of Physics\nGraduate School of Humanities and Sciences\nOchanomizu University\n2-1-1 Ohtsuka, Bunkyo-ku112-8610TokyoJapan"
]
| []
| We study the finite-temperature behavior of the integrable spin-1/2 XXZ periodic chain with an impurity by the algebraic and thermal Bethe ansatz methods. Through the graphs of the impurity entropy versus temperature we show that the impurity spin effectively vanishes through the many-body effect in the bulk chain if pseudo-decoupling of the impurity spin occurs in which the exchange interactions between the impurity and other spins are very small, while in low temperature it couples strongly to them such as the Kondo effect. Thus, we observe not only the crossover from the high-to the low-temperature regime but also another one from the N -site to the (N − 1)-site chain with effective freezing of the impurity spin. We evaluate the impurity local magnetization at zero temperature analytically, and show how easily the impurity spin becomes saturated even with an infinitesimally small magnetic field when it is pseudo-decoupled. Furthermore we show universality that the Wilson ratio at low temperature is independent of the impurity parameter if its absolute value is small, and the universality class is described by the XXZ anisotropy in terms of the dressed charge. * As we shall see in §2 the term H N −1 (0) gives the Hamiltonian of the spin-1/2 periodic XXZ chain with no impurity. Applying a small magnetic field, h, on the whole system of M down spins we have M < N/2. Here the Hamiltonian is given by H ′ = H N (x) − 2hS z , where S z denotes the eigenvalue of the z-component of the total spin operator. We also assume that the XXZ spin chain, H N −1 (0), has the anti-ferromagnetic interactions. Then, the ground-state energy of H N −1 (0) with M down spins decreases as the number of down spins, M, increases, and any down spin on the impurity site is attracted to the the (N − 1)-site chain due to the anti-ferromagnetic interactions. Here we remark that the energy reduction is of the order of 1/N. We also remark that the magnetic energy term, −2hS z , in the total Hamiltonian H ′ does not change whether the impurity spin has an up spin or down spin in the sector of M down spins. Moreover, we give a conjecture that the number of such eigenstates that have an up spin at the impurity site and have the | 10.1088/1742-5468/2014/11/p11020 | [
"https://arxiv.org/pdf/1407.4187v3.pdf"
]
| 118,626,726 | 1407.4187 | f52b567bed506da85c715e2d2c403c4ae417c899 |
Finite-temperature behavior of an impurity in the spin-1/2 XXZ chain
30 Aug 2014 September 2, 2014
Ryoko Yahagi [email protected]
Jun Sato
Research Center for Advanced Science and Technology
University of Tokyo
4-6-1 Komaba, Meguro-ku153-8904TokyoJapan
Tetsuo Deguchi
Department of Physics
Graduate School of Humanities and Sciences
Ochanomizu University
2-1-1 Ohtsuka, Bunkyo-ku112-8610TokyoJapan
Finite-temperature behavior of an impurity in the spin-1/2 XXZ chain
30 Aug 2014 September 2, 20141
We study the finite-temperature behavior of the integrable spin-1/2 XXZ periodic chain with an impurity by the algebraic and thermal Bethe ansatz methods. Through the graphs of the impurity entropy versus temperature we show that the impurity spin effectively vanishes through the many-body effect in the bulk chain if pseudo-decoupling of the impurity spin occurs in which the exchange interactions between the impurity and other spins are very small, while in low temperature it couples strongly to them such as the Kondo effect. Thus, we observe not only the crossover from the high-to the low-temperature regime but also another one from the N -site to the (N − 1)-site chain with effective freezing of the impurity spin. We evaluate the impurity local magnetization at zero temperature analytically, and show how easily the impurity spin becomes saturated even with an infinitesimally small magnetic field when it is pseudo-decoupled. Furthermore we show universality that the Wilson ratio at low temperature is independent of the impurity parameter if its absolute value is small, and the universality class is described by the XXZ anisotropy in terms of the dressed charge. * As we shall see in §2 the term H N −1 (0) gives the Hamiltonian of the spin-1/2 periodic XXZ chain with no impurity. Applying a small magnetic field, h, on the whole system of M down spins we have M < N/2. Here the Hamiltonian is given by H ′ = H N (x) − 2hS z , where S z denotes the eigenvalue of the z-component of the total spin operator. We also assume that the XXZ spin chain, H N −1 (0), has the anti-ferromagnetic interactions. Then, the ground-state energy of H N −1 (0) with M down spins decreases as the number of down spins, M, increases, and any down spin on the impurity site is attracted to the the (N − 1)-site chain due to the anti-ferromagnetic interactions. Here we remark that the energy reduction is of the order of 1/N. We also remark that the magnetic energy term, −2hS z , in the total Hamiltonian H ′ does not change whether the impurity spin has an up spin or down spin in the sector of M down spins. Moreover, we give a conjecture that the number of such eigenstates that have an up spin at the impurity site and have the
Introduction
Several decades have passed since the Kondo effect was discovered experimentally in the 1930s [1] and was first explained theoretically in the 1960s [2]. However, the Kondo problem and related subjects are still quite attractive in both theoretical and experimental studies [3,4,5]. Quantum impurity systems show universal critical behavior at low temperature, which is characterized by the ratio of the impurity magnetic susceptibility χ imp to the impurity specific heat c imp with temperature T [6] r = π 2 3
χ imp c imp /T . (1.1)
We call it the Wilson ratio. It was exactly shown by the Bethe ansatz that it is given by 2 for the Kondo model [7,8].
The effect of an impurity embedded in a one-dimensional interacting quantum system or in a Tomonaga-Luttinger liquid [9] has been one of the most attractive topics during the 1990s and also in the last decade. It has been investigated in a variety of systems by different methods such as renormalization group techniques [10,11,12], conformal field theories (CFT) [13,14], numerical techniques with CFT [15,16], and the Betheansatz method [17,18,19,20,21,22,23,24,25,26]. Recently, it is studied by functional renormalization group [27] and also experimentally in a quasi-one-dimensional conductor [28]. However, it is still rare that the finite-temperature thermodynamic behavior is explicitly and exactly shown by a theoretical method for a large but finite lattice system without making any approximation or assumption.
In this paper we show explicitly the finite-temperature behavior of an integrable model of the spin-1/2 XXZ periodic chain with a spin-1/2 impurity. We remark that the spin part of the Kondo model is quite similar to the Heisenberg spin chain, i.e. the XXX model. It is thus interesting to study the integrable spin-chains with the spin-1/2 impurity which are solved by the Bethe ansatz. In the XXZ impurity model we investigate exactly the effect of an impurity embedded in the interacting quantum spin chain similarly as the Kondo model. Schlottmann studied the integrable spin-S XXZ spin chain with one spin-S ′ impurity [23] in association with the multi-channel Kondo effect [18,19,25]. The spin-1/2 impurity in the open spin-1/2 XXX chain was studied by Frahm and Zvyagin [21] and it was shown that the Kondo-like temperature exists, while the periodic XXZ chain with an impurity was studied by Eckle et al. [20]. However, the analytic expressions of the impurity magnetization and the impurity specific heat at low temperature in Ref. [23] are not accurate enough to evaluate the Wilson ratio correctly. They are not consistent with the numerical estimates evaluated by solving the Bethe ansatz equations, in particular, when the impurity parameter is small. Moreover, the crossover temperature has not been evaluated numerically, while its analytic expression [23] is not appropriate. The thermodynamic behavior of the spin-1/2 impurities in the spin-1/2 XXZ chain was shown in Ref. [24] for several distributions of the impurity parameters, and in the homogeneous case the results are consistent with ours. In the present paper we show how the thermodynamic behavior of the single spin-1/2 impurity depends on the impurity parameter. Although it looks simpler than that of many impurities, it is still nontrivial and interesting from the viewpoint of universality and the crossover from the N-site to the (N − 1)-site chain.
We consider the integrable model of the spin-1/2 XXZ periodic chain with a spin-1/2 impurity, i.e., the case of S = S ′ = 1 2 in the XXZ impurity model of Ref. [23], and investigate the finite-temperature behavior of the integrable XXZ impurity model by numerically solving the truncated integral equations of the thermal Bethe ansatz [29,30,31,32]. We evaluate the specific heat and entropy numerically by solving them. Then, by plotting graphs of the impurity specific heat and entropy versus temperature we show that the impurity-spin degree of freedom effectively vanishes through pseudo-decoupling from other spins, while in low temperature it couples strongly to them such as the Kondo effect. Here, if the exchange coupling between the impurity spin and other spins is very small, we call it pseudo-decoupling. It occurs in the present model when the absolute value of the impurity parameter is very large. We derive the impurity susceptibility at zero temperature analytically by the Wiener-Hopf method, and give an analytic expression for the impurity specific heat at low temperature. We also evaluate the impurity susceptibility numerically by solving the Bethe ansatz equations, and confirm that the analytic expression is valid when the absolute value of the impurity parameter is small. We then show that the Wilson ratio at low temperature is independent of the impurity parameter and the universality class is described by the XXZ anisotropy through the dressed charge when the absolute value of the impurity parameter is small.
We evaluate the local magnetization at the impurity site under a given small magnetic field at zero temperature and derive an analytic expression by the algebraic Bethe ansatz method. It is consistent with the impurity susceptibility derived through the Bethe ansatz equations. We thus confirm that the identification of the impurity contribution to the susceptibility is valid. Moreover, we show how easily the impurity spin becomes saturated even with an infinitesimally small magnetic field if pseudo-decoupling occurs.
Let us denote the Hamiltonian of the XXZ spin chain with the spin-1/2 impurity consisting of N sites by H N (x), which depends on the impurity parameter, x. If the absolute value of the impurity parameter, |x|, becomes large, the exchange coupling of the impurity spin to neighboring spins becomes small, and hence the Hamiltonian of the N-site chain is decomposed into that of the (N − 1)-site chain, H N −1 (x), and that of the impurity site, H imp , as follows.
H N (x) = H N −1 (0) + H imp + O(e −|x| )
for |x| ≫ 1 .
(1.2) energy eigenvalues which are close to the ground-state energy under the magnetic field h within the order of 1/N is larger than that of a down spin, if M < N/2. Here we remark that the number of such eigenstates that have energies close to the ground state energy should increase if the number of down spins in the bulk chain increases when M < N/2, and also that in the sector of M down spins the number of down spins in the bulk part is given by M if the impurity spin is up, while it is given by M − 1 if the impurity spin is down. We thus suggest that the impurity spin tends to be given only by an up spin and all the M down-spins on the whole chain tend to be located on the (N − 1)-site chain if the absolute value of the impurity parameter is very large, that is, if pseudo-decoupling of the impurity spin occurs. We thus have the crossover from the N-site chain to the (N − 1)-site chain with no flipping impurity spin, as a consequence of the many-body effect in the XXZ spin chain. We also call it the effective freezing of the impurity spin if the impurity spin does not flip when it is pseudo-decoupled from other spins.
In the XXX case of the integrable impurity model the Wilson ratio has been evaluated numerically and systematically by applying the quantum transfer-matrix method [22], in which it is necessary to consider logarithmic corrections for the magnetization in low temperature due to the SU(2) symmetry [33]. In the XXZ impurity model, however, it is not necessary to consider logarithmic corrections, and hence it is easier to derive the susceptibility at a very low temperature.
The contents of the present paper consist of the following. In section 2, we introduce the spin-1/2 XXZ chain with a spin-1/2 impurity and give an explicit expression of the Hamiltonian in terms of the spin operators. We show that the impurity spin operator becomes separate from the other spin operators, i.e., pseudo-decoupling occurs, if we send the absolute value of the impurity parameter to infinity. In section 3 we derive an analytic solution to the ground state under a weak magnetic field by the Wiener-Hopf method and show an analytic expression of the impurity susceptibility at zero temperature when the absolute value of the impurity parameter is small. We evaluate the local magnetization at the impurity site by the algebraic Bethe ansatz analytically. We also confirm the analytical results with numerical plots. We also evaluate numerically the impurity susceptibility at zero temperature by solving the Bethe-ansatz equations. In section 4 we evaluate the impurity specific heat by numerically solving the truncated integral equations of the thermal Bethe ansatz and show the finite-temperature behavior of the XXZ impurity model, explicitly: The impurity spin tends to vanish if the absolute value of the impurity parameter is large, which is clearly seen in high temperature particularly, while it couples strongly to them in low temperature, that is, the Kondo effect appears. We also give an analytic expression of the impurity specific heat at low temperature for small absolute values of the impurity parameter. We numerically determine the crossover temperature from the high-to low-temperature regimes as a function of the impurity parameter and give a good fitting formula to the data. In the last section, we derive analytically the universal value of the Wilson ratio at low temperature for small absolute values of the impurity parameter, and express it in terms of the dressed charge of the XXZ spin chain. We also evaluate the Wilson ratio numerically, and confirm that universality is described by the dressed charge [34] of the XXZ spin chain in the case of small absolute values of the impurity parameter.
Integrable model of an impurity 2.1 An inhomogeneous transfer matrix of the XXZ spin chain
We now introduce one of the integrable XXZ models with a spin-1/2 impurity through the algebraic Bethe ansatz. Let V 0 , V 1 , . . . , V N be two-dimensional vector spaces over complex numbers C. Here we denote by N the number of sites on the XXZ spin chain. We call V 0 the auxiliary space and the tensor product V 1 ⊗ · · · ⊗ V N the quantum space, where V j corresponds to the jth site of the XXZ spin chain for j = 1, 2, . . . , N. On the tensor product V 0 ⊗ V n we define the R-matrix by
R 0n (λ) = 1 b(λ) c(λ) c(λ) b(λ) 1 [0,n]
.
(2.1)
Here the suffix [0, n] means that the matrix is defined for the tensor product V 0 ⊗ V n . We define the transfer matrix of the spin-1/2 XXZ spin chain by the trace of the Nth product of the R matrices R 0j (λ − ξ j ) for j = 1, 2, . . . , N, over the auxiliary space 0:
τ 1···N (λ|ξ 1 , · · · , ξ N ) = tr 0 R 0N (λ − ξ N ) · · · R 01 (λ − ξ 1 ), (2.2)
where ξ j (j = 1, 2, . . . , N) are arbitrary. We call them the inhomogeneity parameters. The functions b(λ) and c(λ) are given by
b(λ) = ϕ(λ) ϕ(λ + iζ) , c(λ) = ϕ(iζ) ϕ(λ + iζ) , (2.3)
where ϕ(λ) = sinh λ and ∆ = cos ζ with 0 < ζ ≤ π in the gapless regime of the XXZ spin chain. We consider only the gapless regime throughout the paper,.
Definition of the XXZ chain with an impurity: H XXZ (x)
We now define the Hamiltonian of the impurity model, H XXZ (x). We assume that only the 1st site has a different value for the inhomogeneity parameter ξ 1 :
ξ 1 = iζ 2 − x , for x ∈ R ,(2.4)
while the inhomogeneous parameters of the other sites ξ j for j = 2, 3, . . . , N, have the same value iζ/2. Here we assume that the impurity parameter x is real. We define the Hamiltonian of the impurity model, H XXZ (x), by the logarithmic derivative of the transfer matrix of the spin-1/2 XXZ chain as follws.
H XXZ (x) = ϕ(iζ) 2 d dλ log τ 1···N λ ξ 1 , iζ 2 , · · · , iζ 2 λ→ iζ 2 .
(2.5)
By putting x = 0 (i.e., x → 0), the Hamiltonian H XXZ (x) reduces to the standard homogeneous spin-1/2 XXZ Hamiltonian, which has no impurity. It is clear that the Hamiltonian (2.5) is integrable. It is a consequence of the construction in terms of the algebraic Bethe ansatz. Let M be the number of down spins. The Bethe-ansatz equations (BAE) for the impurity Hamiltonian (2.5) are given by
(N − 1)θ 1 (z l ) + θ 1 (z l + x) = 2πI l + M j =l θ 2 (z l − z j ), for l = 1, . . . , M,(2.6)
where I l are called the Bethe quantum numbers, and they are integers or half-integers according to the following rule:
I l ≡ N − M + 1 2 (mod 1), for l = 1, 2, . . . , M. (2.7)
Here the functions θ n (z) are given by
θ n (z) = i log − ϕ(z + inζ 2 ) ϕ(z − inζ 2 )
.
(2.8)
We remark that in the special case, i.e. for the XXX model with x = 1, the BAEs (2.6) are similar to the Kondo model. By solving the Bethe-ansatz equations (2.6), we can evaluate the energy of the Bethe ansatz eigenvector with M down spins. Let z ℓ be a solution of the Bethe ansatz equations (2.6). The energy for the eigenstate with Bethe roots z ℓ is expressed in terms of the set of Bethe roots z i as
E = − M l=1 sin 2 ζ cosh 2z l − cos ζ . (2.9)
We remark that the energy depends on the impurity parameter x, since the solutions z j contain the information of x, though it does not appear explicitly.
Expression of H XXZ (x) in terms of local spin operators
We now present an explicit expression of the XXZ Hamiltonian with a spin-1/2 impurity, H XXZ (x), defined by eq. (2.5) in terms of the local spin operators. The expression is useful for confirming analytic results. Making use of the explicit expression of H XXZ (x) we can perform the exact diagonalization of the Hamiltonian (2.5), and compare numerical results with analytic or numerical results which are obtained by solving the Bethe ansatz equations (2.6).
Through a direct but straightforward calculation we obtain the following compact expression of H XXZ (x) in terms of local spin operators S ± j and S z j .
H XXZ (x) = N −1 n=2 1 2 (S + n S − n+1 + S − n S + n+1 ) + ∆ S z n S z n+1 − 1 4 + c + c − ϕ ′ (x) 2 (S + N S − 1 + S − N S + 1 ) + ∆ S z N S z 1 − 1 4 + c + c − ϕ ′ (x) 2 (S + 1 S − 2 + S − 1 S + 2 ) + ∆S z 1 S z 2 + b + b − ∆ 2 (S + N S − 2 + S − N S + 2 ) + ∆S z N S z 2 − ∆ 4 + b + c − ϕ ′ (x)(S + N S − 2 − S − N S + 2 )S z 1 −∆(S + N S − 1 − S − N S + 1 )S z 2 −∆(S + 1 S − 2 − S − 1 S + 2 )S z N , (2.10)
where the symbols b ± and c ± are given by
b ± = b(±x) = ϕ(x) ϕ(x ± iζ) , c ± = c(±x) = ± ϕ(iζ) ϕ(x ± iζ)
.
(2.11)
Here, the spin operators S ± j and S z j are expressed in terms of the Pauli matrices as follows.
S ± j = 1 2 σ x j ± i 1 2 σ y j , S z j = 1 2 σ z j , for j = 1, 2, . . . , N . (2.12)
Some details of the derivation of (2.10) are given in Appendix A. We remark that the XXZ Hamiltonian with the spin-1/2 impurity, H XXZ (x), given in eq. (2.10) is Hermitian when the impurity parameter x is real.
In the cases of N ≦ 10 we have confirmed explicitly that the lowest energy level of the XXZ Hamiltonian with the spin-1/2 impurity (2.10) is completely consistent with the eigenvalue evaluated through eq. (2.9) from the ground-state solution of the Bethe ansatz equations. We therefore assume that the quantum numbers I ℓ of the ground state do not change under the existence of the impurity. They are given by
I ℓ = j − N − M − 1 2 for j = 1, 2, . . . , M . (2.13)
We can confirm this assumption by directly diagonalizing the XXZ Hamiltonian with an impurity for a larger number of sites. We should remark that the integrable model of an impurity (2.5) corresponds to the spin-1 2 case of the integrable XXZ model with the spin-1/2 impurity studied by Schlottmann [23]. Here we recall that the spin part of the Kondo model [5] is similar to the spin-1/2 XXX chain.
Pseudo-decoupling of the impurity spin for large |x|
If one takes the limit x → ∞, the Hamiltonian (2.10) reduces to
H XXZ (∞) = N −1 n=2 1 2 (S + n S − n+1 + S − n S + n+1 ) + ∆ S z n S z n+1 − 1 4 + ∆ 2 S + N S − 2 + S − N S + 2 + ∆ S z N S z 2 − 1 4 +2 √ 1 − ∆ 2 (S x N S y 2 − S y N S x 2 ) S z 1 .
(2.14)
The impurity spin, i.e., the spin operators defined on the 1st site such as S ± 1 and S z 1 , appears only in the last term of the reduced Hamiltonian in eq. (2.14). The exchange coupling of the impurity spin to the neighboring spins vanishes in H XXZ (∞). In terms of eq. (1.2) the bulk part H N −1 (0) and the impurity part H imp (0) corresponds to the first two lines and the third line of the right-hand side of (2.14), respectively. If one takes the XXX limit (∆ → 1), the terms associated with the impurity site vanish. The Hamiltonian now reduces to the homogeneous spin-1/2 XXX Hamiltonian defined on the N − 1 sites from site 2 to site N:
H XXX (∞) = N n=2 S x n S x n+1 + S y n S y n+1 + S z n S z n+1 − 1 4 . (2.15)
However, even if the impurity spin is pseudo-decoupled from other spins, it is not independent of other spins due to the many-body effect in the bulk chain. Here we recall that if the exchange interactions of the impurity spin with other spins are very small but nonzero, we call it pseudo-decoupling of the impurity spin.
Under a magnetic field, h > 0, the number of down spins, M, becomes smaller than that of the half-filling case, i.e., M < N/2. Let us assume that when the impurity spin is pseudo-decoupled from other spins, we can effectively define the energy of the bulk part H N −1 (0) for the eigenstates with M down spins in the bulk chain and denote it by E bulk N −1 (M). We then suggest that any down spin on the impurity site will be attracted to the bulk chain since the bulk energy decreases as the number of down spins in the bulk increases: E bulk N −1 (M) < E bulk N −1 (M − 1) due to the anti-ferromagnetic interactions. Here the energy reduction is of the order of 1/N.
We give a conjecture that in a sector of M down spins in the whole chain, the number of such eigenstates that have an up spin on the impurity site and have the energy eigenvalues close to the ground-state energy under the magnetic field h within the order of 1/N, is much larger than that of a down spin on the impurity site, if M ≤ N/2. For instance, in the sector of M = 2, the number of such eigenstates that have an up spin on the impurity site is given by N C 2 = N(N − 1)/2 while the number of such eigenstates that have a down spin on the impurity site is given by N C 1 = N. The former is (N − 1)/2 times larger than the latter. Here, for simplicity, we have assumed that all the energy eigenvalues are almost equal to the ground-state energy under the magnetic field h within the order of 1/N. In general, we assume that the number of such eigenstates that have energies close to the ground state energy within the order of 1/N should increase if the number of down spins in the bulk chain increases when M < N/2. We remark that in the sector of M down spins the number of down spins in the bulk part is given by M if the impurity spin is up, while it is given by M − 1 if the impurity spin is down.
Thus, from the viewpoint of the energy reduction and the increase in the number of eigenstates with an up spin on the impurity site, we suggest that the impurity spin tends to be given by an up spin and all the M down-spins on the whole chain tend to be located on the (N − 1)-site chain for any number M of down spins even in finite temperatures if pseudo-decoupling occurs under a magnetic field.
We thus have the crossover from the N-site chain to the (N − 1)-site chain with no flipping impurity spin, as a consequence of the many-body effect in the XXZ spin chain. We recall that we call it the effective freezing of the impurity spin if the impurity spin does not flip when it is pseudo-decoupled from other spins.
We show effective freezing of the impurity spin for large absolute values of x in the graphs of the impurity specific heat in §4.
Magnetic susceptibility due to impurity
The Bethe anzatz equations with an impurity
We shall calculate the magnetic susceptibility by adding a small magnetic field h to the Hamiltonian defined by (2.5):
H ′ = H XXZ (x) − 2h N n=1 S z n . (3.1)
We recall that the Hamiltonian H XXZ (x) is expressed explicitly in terms of spin operators in (2.10). Let us assume that the Bethe quantum numbers I ℓ of the ground state do not change under the existence of the impurity. By taking the thermodynamic limit of the BAE (2.6) through the Euler-Maclaurin formula, we derive the following inregral equation:
ρ(z, x) + B −B a 2 (z − z ′ )ρ(z ′ , x)dz ′ = 1 N {(N − 1)a 1 (z) + a 1 (z + x)} . (3.2)
Here parameter B denotes the Fermi point determined by the magnetic field h, and ρ(z, x) the density of the Bethe roots in the ground-state solution of the BAE (2.6), where functions a n (z) for n = 1, 2, . . . , are defined by
a n (z) = θ ′ n (z) 2π = − i 2π ϕ(inζ) ϕ(z + inζ 2 )ϕ(z − inζ 2 ) . (3.3)
We remark that we have the infinite Fermi point, B = ∞, for h = 0. We define the Fourier transform of a given function f (z) bỹ
f (k) = ∞ −∞ f (z)e ikz dz. (3.4)
Let us denote by ρ 0 (z, x) the root density for the ground-state solution of the BAE with no magnetic field for (2.5). We now derive the analytic expression of ρ 0 (z, x). By taking the Fourier transform of the following integral equation
ρ 0 (z, x) + ∞ −∞ a 2 (z − z ′ )ρ 0 (z ′ , x)dz ′ = 1 N {(N − 1)a 1 (z) + a 1 (z + x)} ,(3.5)
we haveρ
0 (k, x) = (N − 1 + e −ikx )ã 1 (k) N(1 +ã 2 (k)) . (3.6)
Hereã 1 (k) andã 2 (k) are given bỹ
a 1 (k) = sinh k 2 (π − ζ) sinh kπ 2 ,ã 2 (k) = sinh k 2 (π − 2ζ) sinh kπ 2 . (3.7)
Taking the inverse transform of (3.6), one finds
ρ 0 (z, x) = 1 2Nζ (N − 1)sech πz ζ + sech π(z + x) ζ . (3.8)
In the case of x = 0, it reduces to the bulk density, σ 0 (z), which is given by
σ 0 (z) = 1 2ζ sech πz ζ . (3.9)
Hereafter we shall sometimes abbreviate ρ(z, x) and ρ 0 (z, x) as ρ(z) and ρ 0 (z), respec- Figure 1: Root densities ρ 0 (z, x) and σ 0 (z) of the solution of the Bethe-ansatz equations for N = 50 and ∆ = 0.6. ρ 0 (z, x) (x = 3) with an impurity (red curve); σ 0 (z) (x = 0) with no impurity (blue curve). Note that the red curve has a small peak at z = −x = −3.
tively.
Let us evaluate the largest value of the impurity parameter x for which thermodynamic expressions are valid. We shall compare several numerical data obtained for the system of a finite number of N with such analytic expressions that are derived in the thermodynamic limit. We approximately estimate it by setting σ 0 (x) > 1/N. If the root density is smaller than 1/N, then the continuous approximation to the discrete distribution of the Bethe roots should be not good. From Fig. 1 we have x = 1.2 for N = 50.
Let us introduce a function R(z) by
∞ −∞ (a 2 (z − z ′ )R(z − z ′′ )) dz = a 2 (z ′′ − z ′ ) − R(z ′′ − z ′ ) (3.10)
or equivalently byã
2 (k)R(k) =ã 2 (k) −R(k). (3.11)
Multiplying R(z − z ′′ ), the integral equation (3.2) and (3.5) are merged into the following:
ρ(z) = ρ 0 (z) + |z ′ |>B R(z − z ′ )ρ(z ′ )dz ′ . (3.12)
In the Wiener-Hopf method the integral equation (3.12) plays a fundamental role. We denote by s z (x) the magnetization per site under magnetic field h. We shall give the relation between the Fermi point B and the magnetic field h by the Wiener-Hopf method in §3.2. Let us recall that the energy eigenvalue of the Hamiltonian H XXZ (x) is given by
E N = −π sin ζ 1 N M j=1 a 1 (z j ) . (3.13)
We consider the difference of the ground-state energy per site under magnetic field h from that of no magnetic field, and we denote it by e(x). Then, physical quantities such as s z (x) and e(x) are written in terms of the inetgrals of the root density ρ(z) as follows.
s z (x) = 1 2 − |z|<B ρ(z)dz = π 2π − 2ζ |z|>B ρ(z)dz, (3.14) e(x) = −π sin ζ |z|<B a 1 (z)ρ(z)dz − −π sin ζ ∞ −∞ a 1 (z)ρ 0 (z)dz = π sin ζ |z|>B σ 0 (z)ρ(z)dz. (3.15)
Hereafter, we shall often abbreviate the magnetization per site and the energy per site simply as the magnetization and the energy, respectively.
Wiener-Hopf integral equation with an impurity
Let us define function y(z, x) by
y(z, x) = ρ(z + B, x). (3.16)
By making use of the invariance of eq. (3.2) under the replacement of z and x by −z and −x, respectively, we show the following relation:
y(z, −x) = ρ(−z − B, x). (3.17)
We define functionsy ± (k, x) by
y ± (k, x) = ∞ −∞ H(±z)y(z, x)e ikz dz. (3.18)
Here H(z) is the Heaviside step function: H(z) = 1 for z > 0 and H(z) = 0 for z < 0. In terms of functions y(z, ±x) the integral equation (3.51) is now expressed as follows.
y(z, x) = ρ 0 (z + B) + ∞ 0 R(z − z ′ )y(z ′ , x)dz ′ + ∞ 0 R(z + z ′ + 2B)y(z ′ , −x)dz ′ , (3.19) y(z, −x) = ρ 0 (−z − B) + ∞ 0 R(z − z ′ )y(z ′ , −x)dz ′ + ∞ 0 R(z + z ′ + 2B)y(z ′ , x)dz ′ . (3.20)
Let us now assume that the Fermi point B is very large. The magnetization and the energy are expressed in terms of functionsy ± (k, x) as follows.
s z (x) = π 2π − 2ζ {y + (0, x) +y + (0, −x)} , (3.21) e(x) ≃ π sin ζ ζ e − πB ζ y + ( iπ ζ , x) +y + ( iπ ζ , −x) . (3.22)
Here we recall that eq. (3.22) is valid only for large values of B: exp(−πB/ζ) ≪ 1. It is known that integral equations such as eqs. (3.19) and (3.20) can be solved by the Wiener-Hopf method (see [5], [36], [32]). In the following analysis we neglect the terms with R(z +z ′ +2B) in eqs. (3.20) since the corrections from them are negligible for h ≪ 1. Expressing k n = πi(2n + 1)/ζ for n ∈ Z, we thus havȇ
y + (k, x) = i Nζ G + ζk 2 ∞ n=0 (−1) n G − − ζkn 2 e iknB k + k n N − 1 + e −iknx ,(3.23)
where functions G ± (k) are given by
G + (iπz) = 2π(1 − 1 γ ) (γ − 1) γ−1 γ γ z Γ(γz + 1) Γ( 1 2 + z)Γ((γ − 1)z + 1) ,(3.24)
with γ = π/ζ, and by the following relation:
G − (z) = G + (−z) . (3.25)
Then we have
s z (x) = π 2π − 2ζ 2i Nζ G + (0) ∞ n=0 (−1) n G − − ζkn 2 e iknB k n (N − 1 + cosh ik n x) , (3.26) e(x) = π sin ζ ζ e − πB ζ 2i Nζ G + iπ 2 × ∞ n=0 (−1) n G − − ζkn 2 e iknB iπ ζ + k n (N − 1 + cosh ik n x) . (3.27)
However, the Fermi point B remains unknown, yet. It is determined as a function of h by the following condition:
∂ ∂B (e(x) − 2hs z (x)) = 0. (3.28)
We therefore have
exp − πB ζ = G + (0)ζh (π − ζ) sin ζG + iπ 2 .
(3.29)
The relation between the Fermi point B and the magnetic field h is shown explicitly in . The cut-off B goes to infinity in the limit of h → 0. However, it dramatically decreases to values such as 2 or 3, which are not very large, when h becomes nonzero such as h = 10 −3 . By comparing Figure 2 with the profile of the root density ρ 0 (z) shown in Fig. 1, we suggest that the Wiener-Hopf method is valid only when the impurity parameter x is small enough such as |x| < 1 so that the small peak at z = −x in the root density ρ 0 (z) is located in the interval between the Fermi point z = −B and the origin z = 0.
Impurity susceptibility at zero temperature
Recall that the symbol s z (x) denotes the magnetization per site in the XXZ spin chain which has the spin-1/2 impurity with impurity parameter x. Through the Wiener-Hopf method the magnetization per site s z (x) is given by
s z (x) ≃ iπ Nζ(π − ζ) G + (0) G − − iπ 2 iπ ζ G + (0)hζ (π − ζ) sin ζG + iπ 2 N − 1 + cosh πx ζ = 2ζh Nπ(π − ζ) sin ζ N − 1 + cosh πx ζ . (3.30)
Here, the magnetic field h is very small, so that only linear terms of h may appear. Let us define the magnetic susceptibility per site, χ(x), by
χ(x) = ∂ 2s z (x) ∂h h=0 . (3.31)
Consequently, we have
χ(x) = 4ζ Nπ(π − ζ) sin ζ N − 1 + cosh πx ζ . (3.32)
We now extract the impurity susceptibility χ imp induced by the spin-1/2 impurity of parameter x. We first define the impurity magnetization s z imp (x) by
Ns z (x) = (N − 1)s z (0) + s z imp (x) . (3.33)
Here, Ns z (x) gives the total magnetization of the XXZ spin chain with the impurity parameter x. We also express it by S z tot = Ns z (x). We define the impurity susceptibility χ imp (x) by
χ imp = ∂ 2s z imp (x) ∂h h=0 . (3.34)
The impurity susceptibility is therefore given by
χ imp (x) = 4ζ π(π − ζ) sin ζ cosh πx ζ . (3.35)
3.4 Local magnetization at the impurity site
Derivation through the algebraic Bethe ansatz
Let us now derive the local magnetization at the impurity site, s z 1 = 1 2 σ z 1 g,h , under a small magnetic field h at zero temperature. We take the expectation value of the local operator σ z 1 for the ground state under the magnetic field h. Here we remark that if the magnetic field is zero, the local magnetization at the impurity site vanishes.
We now define the monodromy matrix of the spin-1/2 XXZ spin chain by the Nth product of the R matrices R 0j (λ − ξ j ) for j = 1, 2, . . . , N:
R 0, 1···N (λ|ξ 1 , · · · , ξ N ) = R 0N (λ − ξ N ) · · · R 01 (λ − ξ 1 ),(3.36)
where ξ j (j = 1, 2, . . . , N) are the inhomogeneity parameters given by ξ 1 = iζ/2 − x and ξ 2 = ξ 3 = · · · = ξ N = iζ/2. We define the operator-valued matrix elements of the monodromy matrix by
A(λ) B(λ) C(λ) D(λ) = R 0, 1···N (λ|ξ 1 , · · · , ξ N ) . (3.37)
Let us denote the ground state under a small magnetic field h by |Ψ g,h . We express the ground-state solution of the Bethe-ansatz equations with M down spins under a magnetic field h as λ b for b = 1, 2, . . . , M. We have
|Ψ g,h = B(λ 1 ) · · · B(λ M )|0 (3.38)
where |0 denotes the vacuum state, which has no down spin. The Hermitian conjugate vector is given by
Ψ g,h | = (|Ψ g,h ) † = (−1) M 0|C(λ 1 ) · · · C(λ M ) . (3.39)
We therefore define the impurity local magnetization s z 1 by
1 2 σ z 1 g,h = 1 2 Ψ g,h | σ z 1 |Ψ g,h / Ψ g,h |Ψ g,h = 1 2 0|C(λ 1 ) · · · C(λ M ) σ z 1 B(λ 1 ) · · · B(λ M )|0 0|C(λ 1 ) · · · C(λ M )B(λ 1 ) · · · B(λ M )|0 . (3.40)
We now evaluate it through the expectation value of the local operator e 22 1 as follows.
1 2 σ z 1 g,h = 1 2 − e 22 1 g,h (3.41)
where e a ,b 1 denote the two-by-two matrices with only nonzero matrix element 1 at the entry of (a, b) for a, b = 1, 2. Hereafter, we call σ z 1 g,h the local magnetization at the impurity site, not s z 1 = σ z 1 g,h /2, for simplicity. We introduce the Gaudin matrix Φ ′ a,b ({λ α }) with real parameters z k defined by ξ k = iζ/2 + z k for k = 1, 2, . . . , N, as follows.
Φ ′ a,b = 2πiN ρ(λ a ) δ a,b + 1 N a 2 (λ a − λ b ) (3.42)
where function ρ(λ) is defined by
ρ(λ) = 1 N N k=1 a 1 (λ − z k ) − 1 N M b=1 a 2 (λ − λ b ) . (3.43)
Putting the ground-state solution under a magnetic field, {λ a }, with z 1 = x and z 2 = · · · = z N = 0, we have
ρ(λ) = 1 N ((N − 1)a 1 (λ) + a 1 (λ + x)) − 1 N M b=1 a 2 (λ − λ b ) = 1 N ((N − 1)a 1 (λ) + a 1 (λ + x)) − B −B a 2 (λ − λ ′ )dλ ′ + O(1/N 2 ) = ρ(λ) + O(1/N 2 ) .(3.e 22 1 = D(ξ 1 )(A(ξ 1 ) + D(ξ 1 )) −1 (3.45)
and Slavnov's scalar product formula we can show
e 22 1 g,h = M a=1 det M (Φ ′ ({λ β }) −1 Ψ ′ ({λ β } \ {λ a } ∪ {ξ 1 }) = 1 N M a=1 1 ρ(λ a ) ρ I (λ a , x) + O(1/N 2 ) = B −B ρ I (λ, x)dλ + O(1/N 2 ) . (3.46)
Here ρ I (λ, x) denotes the solution of the integral equation
ρ I (z, x) + B −B a 2 (z − z ′ )ρ I (z ′ , x)dz ′ = a 1 (z + x) ,(3.M matrix Ψ ′ ({λ β } \ {λ a } ∪ {ξ 1 }) are given by Ψ ′ b,c = 2πi a 1 (λ a + x) if b = a Φ ′ b,c otherwise . (3.48)
Analytic expression of the impurity local magnetization for small |x|
Let us consider the case when the absolute value of the impurity parameter x is small enough with respect to a given value of magnetic field h. We assume that it is much smaller than the Fermi point B: |x| ≪ B, where B is related to the magnetic field h by (3.29). In this case we solve eq. (3.47) by the Wiener-Hopf method. We define ρ I 0 (z, x) by the solution of the integral equation (3.47) with B = ∞. Explicitly, we have ρ I 0 (z, x) = σ 0 (z + x). We define function y I (z, x) by By solving the integral equation
y I (z, x) = ρ I (z + B, x).ρ I (z, x) = ρ I 0 (z, x) + |z ′ |>B R(z − z ′ )ρ I (z ′ , x)dz ′ (3.51)
we evaluate e 22 1 g,h through the following:
B −B ρ I (z, x)dz = 1 2 − π 2π − 2ζ |z|>B ρ I (z, x)dz = 1 2 − π 2π − 2ζ y I + (0, x) +y I + (0, −x) . (3.52)
We thus obtain the analytic expression of the impurity local magnetization
σ z 1 g,h = 4ζ π(π − ζ) sin ζ cosh πx ζ h + O(1/N 2 ). (3.53)
Here we recall that the magnetic field h is related to the Fermi point B by (3.29). The analytic expression (3.53) of the impurity local magnetization is consistent with the impurity susceptibility (3.35). Therefore, it follows that the identification of the impurity contribution to the magnetic susceptibility in terms of the Bethe ansatz equations is valid. Here we remark that expression (3.53) is valid only when the absolute value of impurity parameter x is small enough such as |x| < B so that we can apply the Wiener-Hopf method to solve the integral equation (3.47).
Pseudo-decoupled case
Let us now consider the case when the absolute value of the impurity parameter x is large. For instance, we assume that it is larger than the logarithm of the system size N: |x| > ln N. It follows that we have exp(−x) < 1/N, and the energy gap due to the exchange interaction between the impurity site and the neighboring sites is smaller than the bulk excitation energy gaps, which are of the order of 1/N, and hence can be neglected with respect to them. Here we remark that the energy gap in the bulk chain consisting of the (N − 1) sites, which corresponds to the bulk part H N −1 (0) in eq. (1.2), is given by the order of 1/N, since the bulk spectrum is gapless in the large-N limit. Then, the mixing of such eigenstates with a up spin on the impurity site and those with a down spin there does not occur very much. Consequently, the impurity local magnetization should be approximately given by the largest value 1 for such an eigenstate that has almost an up spin on the impurity site, while it is given by −1 for such an eigenstate that has a down spin on the impurity site. Under a small magnetic field, the former states are preferred since the number of down spins are less than in the half-filling case (i.e., M < N/2), and hence down spins tend to be located on the bulk part due to anti-ferromagnetic interactions.
If the absolute value of the impurity parameter x is very large such as x ≫ ln N and x ≫ B, we can show that the impurity local magnetization, σ z 1 g,h , approaches the largest value 1 exponentially with respect to the impurity parameter x. The right hand side of the linear integral equation (3.47), which gives the source term to it, is proportional to exp(−2x) if impurity parameter x is very large such as x ≫ B. Therefore, we have e Let us show that the impurity local magnetization becomes easily saturated even with an infinitesimally small magnetic field, if pseudo-decoupling occurs. In fact, even if the absolute value of the impurity parameter x is very large such as x ≫ ln N, we can take a very small magnetic field h such that the value of the Fermi point B is much larger than |x|. Then, we can apply the Wiener-Hopf method to the integral equation (3.47), and we obtain the expression (3.53) of the impurity local magnetization for a very small magnetic field h with a large value of x.
We now estimate the small magnetic field h for which the expression
1/N > exp(−x) > exp(−B) , (h) 1/p 0 ∝ exp(−B) ,(3.55)
where p 0 = π/ζ. For instance, p 0 = 3 for ζ = π/3. We suggest that the magnetic field h which triggers the impurity local magnetization is given by h < 1/N 3 , which is very small. We thus expect that if we add a small magnetic field h such as h ∼ 1/N 3 , then the local magnetization at the impurity site readily approaches the largest value 1. 1.
Numerical estimates of the impurity local magnetization
x Σ 1 z Figure 3: Local magnetization at the impurity site, σ z 1 g,h , at T = 0 and ζ = π/3 versus impurity parameter x for different values of magnetic field (purple, blue, green, orange, and red, in increasing order) for N = 10. The purple line corresponds to the case of the zero magnetic field.
Let us now confirm the analytic results of the impurity local magnetization by numerical data. We plot the numerical estimates of the local magnetization at the impurity site, σ z 1 g,h , against the impurity parameter x in the case of N = 10 in Fig. 3. The estimates are evaluated by numerically diagonalizing the Hamiltonian H XXZ (x) with N = 10.
For small values of the impurity parameter x, the impurity local magnetization, σ z 1 g,h , is proportional to the magnetic field h, and it increases with respect to x similarly as cosh πx/ζ. However, for large values of the impurity parameter x, it approaches the largest value 1.0. Moreover, under an infinitesimally small magnetic field h, the impurity spin takes very quickly a positive large value less than 1.0 . We also observe that it vanishes under zero magnetic field for any values of the impurity parameter x.
Numerical estimates of the impurity susceptibility
We now evaluate numerically the magnetic susceptibility by calculating the energy eigenvalues through solutions of the Bethe ansatz equations. For a given number M satisfying M ≤ N/2 we solve the BAE (2.6) for the ground state with M down spins, and evaluate the ground state energy E(M) through the solution by making use of eq. (2.9). We also evaluate the energy E(M − 1) for the ground state with M − 1 down spins. Then, we determine the magnetic field h(M) by the following relation:
2h(M) = E(M − 1) − E(M) .
(3.56)
Here we recall that the total magnetization S z tot is given by S z tot = (N − 2M)/2 in the case of M down spins. We define the total susceptibility χ tot (M) under magnetic field h(M) by
χ tot (M) = (2S z tot (M − 1) − 2S z tot (M))/(h(M − 1) − h(M)) . (3.57)
We can evaluate the susceptibility at h = 0, i.e. χ tot (M) for M = N/2, by taking the difference through eq. (3.57). It is close to the definition (3.31) of the susceptibility by the derivative of 2S z tot with respect to magnetic field h. The susceptibility per site χ(x) is given by χ(x) = χ tot /N. We then evaluate the impurity susceptibility χ imp (x) from the total susceptibility χ tot by the following relation: χ tot = (N − 1)χ(0) + χ imp (x). 1 In Figs. 4 and 5 the numerical estimates of the impurity susceptibility χ imp (x) at h = 0 are plotted against impurity parameter x. The graph is downward convex in the small range of x such as |x| < 1 as shown in Fig. 4. When the absolute value of x is small, the impurity susceptibility χ imp (x) increases with respect to the absolute value of impurity parameter x, and we consider that it is due to the Kondo effect. When |x| is large, however, χ imp (x) has double peaks around at x = ±2 as shown in Fig. 5. Furthermore, the impurity susceptibility χ imp (x) almost vanishes when the absolute value of x becomes very large such as x = ±10.
Let us explain the reason why the impurity susceptibility χ imp (x) vanishes for large values of |x|. The estimates of the impurity susceptibility χ imp (x) for |x| > 2 plotted in Fig. 5 correspond to the impurity local magnetization σ z 1 g,h with large absolute values 1 Numerically we have evaluated the susceptibility χ tot at h = 0 and then the impurity susceptibility χ imp (x) at h = 0 as follows. We make a graph of magnetization 2S Z = N − 2M versus h(M ), and interpolate the data points of 2S z versus h(M ) as a polynomial of h of some degree. We employ the coefficient of the h-linear term as the susceptibility χ tot at h = 0. Here we remark that the total susceptibility χ tot is O(N ), i.e. of the order of the system size N . We repeat the evaluation of χ tot for different values of the system size N , and plot the total susceptibility divided by N , χ tot /N , against the inverse of the system size, 1/N . Then, we interpolate the data points of χ tot /N versus 1/N as a polynomial of y = 1/N of some degree and employ the coefficient of the y 1 term as the shift ∆χ = χ imp (x) − χ(0). By adding the susceptibility per site with no impurity, χ(0), to it, we obtain χ imp (x). of impurity parameter x as shown in (3.54). They are also shown in Fig. 3. We suggest that the coefficient of the h-linear term in the total susceptibility χ tot gives the derivative of the impurity local magnetization at a nonzero value of magnetic field h: h > 0, which decreases exponentially with respect to impurity parameter x. The graph of the analytic formula of the impurity susceptibility (3.35) is shown together with the numerical estimates of the impurity susceptibility χ imp in Fig.4. When the absolute value of impurity parameter x is small such as |x| < 1, the analytic formula and numerical estimates are completely consistent as shown in Fig. 4. The numerical estimates (blue dots) are located exactly on the theoretical curve (red line). However, they become different as the absolute value of the impurity parameter x increases, for instance, |x| is lager than 2 or 3, as shown in Fig. 5.
By applying the Wiener-Hopf method formulated in Ref. [36] we can show that the xdependence of the susceptibility per site χ(x) changes if we add higher-order terms with respect to the expansion parameter exp(−πB/ζ). However, it seems that the same xdependence as in eq. (3.35) is reproduced for the impurity susceptibility at zero magnetic field since higher-order terms as in Ref. [36] do not remain when h is small.
In Ref. [23], it is stated that the impurity susceptibility χ imp should depend on the parameter x such as exp (π|x|/ζ) in terms of the notation of the present paper, which is different from cosh (πx/ζ). However, the exponential dependence is not exact as an analytic solution in the Wiener-Hopf method. Furthermore, it is not consistent with the numerical estimates of the impurity susceptibility χ imp even for small x (see also Fig. 4).
Let us now express physical quantities in terms of ρ j (z) and ρ h j (z). The energy of this system is described by
e(x, T ) = κ 3 j=1 ∞ −∞ a j (z)ρ j (z)dz , (4.7)
where κ is given by κ = −π sin ζ, (4.8) and the entropy is expressed as
s = 3 j=1 ∞ −∞ ρ j (z) log 1 + ρ h j (z) ρ j (z) + ρ h j (z) log 1 + ρ j (z) ρ h j (z) dz. (4.9)
Let us take the free energy f = e − T s (T : temperature) with respect to ρ j under the conditions (4.2). Then, it is clear that the thermal equilibrium condition δf = 0 is equivalent to
log η j = κ T a j + 3 k=1 sign(q k )T jk * log(1 + η −1 j ), (4.10)
where functions η j are defined by
η j = ρ h j ρ j . (4.11)
The impurity parameter x does not remain after by taking the variation with respect to ρ j and ρ h j under the constraint of BAE (4.2), and hence the above equations are the same as those of the homogeneous spin-1/2 XXZ spin chain. By putting them back to the free energy, its final form is given by the following:
f = − T N 3 j=1 sign(q j ) ∞ −∞ {(N − 1)a j (z) + a j (z + x)} log(1 + η j (z) −1 )dz. (4.12)
Equations (4.10) are solved numerically [31]. As we said before, j = 2 and j = 3 indicate the same equation. Only two of the three eta-functions are independent and the equations can be reformulated as follows.
h 1 = log 1 + exp − κ T b 1 − 2b 2 * h 1 − 2b 1 * h 2 h 2 = log 1 + exp − κ T b 2 − b 1 * h 1 − 2b 2 * h 2 , (4.13)
where h j and the Fourier transforms of b j are given by (4.16)
h j (z) = log(1 + η −1 j (z)), (4.14) b 1 (ω) =ã 1 1 −ã 2 ,b 2 (ω) =ã 2 1 −ã 2 .
Here, the expressions of functions b j are specific to ζ = π 3 . In terms of these functions, the free energy is written by
f = − T N ∞ −∞ α 1 h 1 + α 2 2h 2 + κ T b 2 + b 1 * h 1 + 2b 2 * h 2 dz, (4.17) α j (z) = (N − 1)a j (z) + a j (z + x). (4.18)
The specific heat is calculated by the same approach. Let us set the following quantities:
u j = T 2 ∂h j ∂T , v j := ∂u j ∂T . (4.19)
Then, the specific heat (c(x,
T ) = ∂e ∂T , e = −T 2 ∂ ∂T f T ) becomes c(x, T ) = 1 N ∞ −∞ {α 1 v 1 + α 2 (2v 2 + b 1 * v 1 + 2b 2 * v 2 )} dz. (4.20)
During the procedure, we also have its energy
e = 1 N ∞ −∞ {α 1 u 1 + α 2 (2u 2 − κb 2 + b 1 * u 1 + 2b 2 * u 2 )} dz. (4.21)
The equations for u j are given by
u 1 = 1 − e −h 1 [−κb 1 − 2b 2 * u 1 − 2b 1 * u 2 ] u 2 = 1 − e −h 2 [−κb 2 − b 1 * u 1 − 2b 2 * u 2 ] ,(4.22)
and those for v j by
v 1 = 1 T 2 u 1 2 e h 1 −1 + 1 − e −h 1 [−2b 2 * v 1 − 2b 1 * v 2 ] v 2 = 1 T 2 u 2 2 e h 2 −1 + 1 − e −h 2 [−b 1 * v 1 − 2b 2 * v 2 ] .
Specific heat with an impurity near zero temperature
The low temperature behavior of the specific heat is important when we consider the Wilson ratio defined by eq. (1.1). We shall suggest in §5 that it is universal at low temperature if |x| is small.
The specific heat per site c(x, T ) at low temperature T = 0.001 is plotted against the impurity parameter x in Fig. 6. At x = 0 the graph is convex and almost flat, i.e. the value of the specific heat c(x, T ) does not change very much near x = 0. However, it has two peaks at x = 2 ∼ 3 and x = −3 ∼ −2. If the absolute value of x is larger than the peak positions x ≈ ±3, the specific heat of the total system Nc(x, T ) reduces to the value of (N − 1)c(0, T ), i.e. equal to the specific heat of the remaining N − 1 sites with no impurity. We shall give the reason why it reduces to that of the N − 1 sites, shortly (see also Appendix B).
In Fig. 6 the spin degree of freedom on the impurity site effectively vanishes when the absolute value of parameter x is large such as |x| > 4. We suggest that it is a consequence of pseudo-decoupling of the impurity spin and the many-body effects of the XXZ spin chain. Here we recall the arguments with eq. (1.2).
In Fig. 7 the graph of the specific heat per site c(x, T ) at low temperature with T = 0.001 against impurity parameter x is drawn in the neighborhood of x = 0. Here the scale of parameter x is magnified by twenty times from that of Fig. 6. We observe in Fig. 7 that the numerical estimates of the specific heat c(x, T ) in the interval [−0.3, 0.3] near the origin of x = 0 are fitted as a function of x completely to the following analytic expression:
c(x, T ) = 2ζT 3N sin ζ N − 1 + cosh πx ζ . (4.24)
Here we have no fitting parameter. We remark that the x-dependence of the specific heat c(x, T ) is exactly the same as that of the magnetic susceptibility at zero temperature. The specific heat (4.24) reduces to the low-temperature result of the homogeneous XXZ spin chain (see e.g. [30]), when we put x = 0 into it. Let us define the shift of the specific heat per site with impurity x at a low temperature T from that of the homogeneous chain, ∆c(x, T ), by ∆c(x, T ) = c(x, T ) − c(0, T ).
(4.25)
It follows from eq. (4.24) that we have
∆c(x, T ) = 2ζT 3N sin ζ cosh πx ζ − 1 . (4.26)
We suggest that expressions of (4.24) and (4.26) for the low-temperature specific heat per site and the difference from its bulk value, respectively, should be valid for any value of ζ in the massless regime, although we have evaluated them only for the case of ζ = π 3 .
Impurity specific heat at finite temperatures
We now investigate the temperature dependence of the specific heat and the entropy. We evaluate the entropy by s = (e − f )/T from the energy e and the free energy f . We shall show that the impurity spin has two effects in finite temperatures: the Kondo effect in low temperature and the freezing effect through pseudo-decoupling of the impurity spin in high temperature.
We denote the specific heat on the impurity site with parameter x at temperature T by c imp (x, T ). We call it the impurity specific heat. We define the impurity specific heat c imp (x, T ) by the following equation:
Nc(x, T ) = (N − 1)c(0, T ) + c imp (x, T ) .
(4.27)
Here we remark that the specific heat of the total system C tot (x, T ) is given by Nc(x, T ): C tot (x, T ) = Nc(x, T ). If the absolute value of parameter x is small and the temperature T is low enough, the analytical expression of c imp (x, T ) is derived from that of the specific heat per site c(x, T ) (4.24) as follows.
c imp (x, T ) = 2ζT 3 sin ζ cosh πx ζ .
(4.28)
We now introduce the impurity entropy s imp (x, T ). Similarly as the impurity specific heat c imp (x, T ), we define it by
Ns(x, T ) = (N − 1)s(0, T ) + s imp (x, T ) .
(4.29)
Here we remark that the entropy of the total system, S tot (x, T ), is given by the following: S tot (x, T ) = Ns(x, T ). Let us show numerical results. The numerical estimates of the impurity specific heat c imp (x, T )and the impurity entropy s imp (x, T ) are plotted in Figs.8 and 9, respectively. We have several important observations: (i) The impurity specific heat c imp (x, T ) has a peak approximately at T = 0.4 in Fig.8 for small absolute values of parameter x such as |x| = 0.3; (ii) The impurity specific heat c imp (x, T ) grows linearly with respect to the temperature T near zero temperature, and the gradient of the linear specific heat becomes larger as the impurity parameter x increases; (iii) The peak position of the impurity specific heat c imp (x, T ) shifts to lower temperatures as the impurity parameter x increases; (iv) The peak value of the impurity specific heat c imp (x, T ) decreases as the peak position shifts to lower temperatures.
We now argue that the Kondo effect leads to the peak in the graph of the impurity specific heat c imp (x, T ) versus temperature T shown in Fig. 8. We recall that the Bethe ansatz equations (2.6) in the XXX limit with x = 1 are quite similar to those of the Kondo model [5]. We therefore conclude that the peak in Fig. 8 is due to the Kondo effect. Furthermore, it seems that the increase of the gradient of the impurity specific heat c imp (x, T ) at low temperature is consistent with the analytic expression (4.28).
In high temperatures such as T = 1.0, the impurity entropy s imp (x, T ) approaches the value ln 2 ≈ 0.69 for x satisfying 0 < x ≤ 3.0, as shown in Fig.9. It is standard that the impurity spin becomes thermally independent of the other spins in the chain in high temperature when the absolute value of impurity parameter x is small such as |x| ≤ 3.0. It is simply due to thermal fluctuations. However, the impurity entropy s imp (x, T ) in high temperature decreases as the absolute value of impurity parameter x increases when it is large such as |x| = 10, as shown in Fig. 9.
Let us consider the decrease of the impurity entropy s imp (x, T ) in the high-temperature regime when the absolute values of parameter x is large. We suggest that it is due to the effective freezing of the impurity spin, which is a result of pseudo-decoupling of the impurity spin and the many-body effect in the XXZ spin chain.
By comparing the specific heat shift ∆c(x, T ) with the bulk specific heat c(0, T ) we can show explicitly that freezing of the impurity spin occurs when the absolute value of impurity parameter x is large. We show it in Appendix B through graphs of the specific heat shift ∆c(x, T ) versus temperature T for x = 20.
Crossover temperature
Let us now determine the crossover temperature between the two regimes with respect to temperature: the low-temperature regime where the Kondo effect is dominant and the high-temperature regime where pseudo-decoupling or freezing of the impurity spin is dominant. We have the whole picture of the finite-temperature behavior of the XXZ impurity model if we combine the two effects: the Kondo effect in low temperature and pseudo-decoupling of the impurity spin in high temperature.
The numerical estimates of the specific heat shift ∆c(x, T ) are plotted against temperature T in Fig.10. Each dotted curve shows the graph of the data with a fixed value of the impurity parameter x. We suggest from the graphs of Fig. 10 that we can determine the crossover temperature systematically. In fact, the values of the specific heat shift ∆c(x, T ) are not always positive. It can be negative in high temperature, while it can be positive at low temperature for small absolute values of parameter x. We consider that the Kondo efffect occurs if the specific heat shift ∆c(x, T ) is positive; the freezing of the impurity spin occurs if the specific heat shift ∆c(x, T ) is negative. The graphs of the specific heat shift ∆c(x, T ) versus temperature T for much smaller values of x are shown in Fig. 11. Even if the absolute value of x is small, the specific heat shift ∆c(x, T ) may be negative in high temperatures. When the absolute value of parameter x is small, a peak due to the Kondo effect appears at low temperature in Figs.10 and 11. We define the crossover temperature by the temperature at which the specific heat shift ∆c(x, T ) vanishes. If the shift ∆c(x, T ) is negative, we consider that the freezing effect through pseudo-decoupling of the impurity spin is more dominant than the Kondo effect. If the shift ∆c(x, T ) is positive, we consider that the Kondo effect is more dominant than the freezing effect through pseudo-decoupling of the impurity spin.
We plot the estimates of the crossover temperature T X as a function of the impurity parameter x in Fig. 12. Furthermore, we find that the following formula gives a good fitting curve to the numerical estimates of the crossover temperature:
T X = a cosh bx (4.30)
where the best estimates of parameters a and b are given by a = 0.269 and b = 1.160, respectively. It is suggested from the impurity susceptibility (3.35) that the Kondo temperature T K is given by T K ∝ 1/ cosh(πx/ζ). However, the numerical estimate of parameter b of crossover temperature T X in (4.30) is given by almost half the value of π/ζ = 3. Thus, the functional form of the crossover temperature T X is the same with the Kondo temperature T K , while the estimate of the parameter b is different. 5 Wilson ratio 5.1 Universality at low temperature for small |x| Let us recall that we have defined the Wilson ratio by eq. (1.1), which is given by the ratio of the impurity magnetic susceptibility χ imp (x) to the impurity specific heat c imp (x, T ) with some normalization factor. We also recall that the impurity susceptibility is given by eq. (3.35) as
χ imp (x) = 4ζ π(π − ζ) sin ζ cosh πx ζ .
We have shown in §4 that when the absolute value of the impurity parameter x is small, the impurity specific heat c imp (x, T ) is given by eq. (4.28) for low temperature. Thus, by making use of (3.35) and (4.28), we evaluate the Wilson ratio (1.1) in the XXZ impurity model for small x at low temperature T by
r = π 2 3 χ imp c imp /T = 2π π − ζ . (5.1)
We observe that the Wilson ratio r does not depend on the impurity parameter x. Therefore, the Hamiltonians of the XXZ impurity model with the same value of ζ but different values of x are classified in the same universality class. Furthermore, the Wilson ratio (5.1) is expressed in terms of the dressed charge Z as
r = 4Z 2 . (5.2)
Here the dressed charge Z denotes the dressed charge function Z(λ) evaluated at the Fermi point: Z = Z(B). The dressed charge Z(λ) is defined by the solution of the integral equation [34]:
Z(λ) + B −B a 2 (λ − µ)Z(µ)dµ = 1 . (5.3)
It is shown in Ref. [34] that by applying the Wiener-Hopf method to eq. (5.3) for very small magnetic field h we have
Z(B) = π 2(π − ζ)
.
(5.4)
Therefore, we obtain relation (5.2). The Wilson ratio of the XXZ impurity model (2.10) is expressed in terms of the dressed charge of the XXZ spin chain. From the CFT viewpoint we give a physical argument for the Wilson ratio expressed with the dressed charge (5.2) at low temperature for small |x| in Appendix C.
Numerical estimates of the Wilson ratio
The numerical estimates of the Wilson ratio are plotted in Fig. 13 against impurity parameter x. Here we have evaluated the Wilson ratio numerically by making use of the estimates of the impurity susceptibility χ imp (x) of Fig. 5 in §3 and those of the impurity specific heat c imp (x, T ) at low temperature with T = 10 −3 in §4.
For small absolute values of impurity parameter x such as |x| < 1 the Wilson ratio is almost constant with respect to x and is approximately given by 3.0. We suggest that the low-temperature impurity effect with small |x| should be due to the Kondo effect and the Wilson ratio is consistent with eq. (5.1). The universality class is described by the XXZ coupling through the dressed charge as shown in eq. (5.2). Here we recall that the estimated largest value of x for which thermodynamic expressions should be valid is given by x = 1.2 from Fig. 1. It is also consistent with the range of impurity parameter x where the estimate of the Wilson ratio is given by the universal value.
For large absolute values of impurity parameter x, however, the Wilson ratio is not constant with respect to x and not given by 3.0 even for ζ = π/3, as shown in Fig. 13. We suggest that when the absolute value of impurity parameter x is large, the effect of the spin-1/2 impurity is not only given by the Kondo effect but also by other effects such as freezing of the impurity spin as discussed in §3 and §4. The Wilson ratio is not given by a simple value for large |x|, since the different impurity effects appear simultaneously.
Acknowledgment
The authors would like to thank F.H.L. Essler and K. Sakai for helpful comments. In particular, we thank K. Sakai for useful comments on the Wiener-Hopf method and the specific heat of the XXZ spin chain. We also thank S. Okuda for helpful discussion. The present study is partially supported by Grant-in-Aid for Scientific Research No. 24540396.
A Expression of H XXZ (x) with spin operators
We now derive expression (2.10) of the XXZ Hamiltonian with a spin-1/2 impurity which is defined by the logarithmic derivative of the transfer matrix as shown in eq. (2.5).
Here we express the Hamiltonian explicitly in terms of local spin operators S ± j and S z j . First, we remark that the R-matrix at zero rapidity R i,j (0) gives the permutation operator P i,j acting on the tensor product space V j ⊗V j : R i,j (0) = P i,j . We consider the two factors of the logarithmic derivative (2.5), separately.
H XXZ (x) = ϕ(iζ) 2 τ 1···N iζ 2 ξ 1 , iζ 2 , · · · , iζ 2 −1 × d dλ τ 1···N λ ξ 1 , iζ 2 , · · · , iζ 2 λ→ iζ 2 . (A.1)
For the first factor of (A.1) we have
τ 1···N iζ 2 ξ 1 , iζ 2 , · · · , iζ 2 = Π 2···N R N,1 (x), (A.2)
where Π 2···N denotes the cyclic permutation operator, which is also expressed as the product of permutation operators: Π abc···d = P ad · · · P ac P ab . For the second factor of (A.1) we have
ϕ(iζ) 2 d dλ τ 1···N λ ξ 1 , iζ 2 , · · · , iζ 2 λ→ iζ 2 = N n=2
tr 0 [P 0N · · · h 0n · · · P 02 R 01 (x)] + tr 0 [P 0N · · · P 02 h 01 (x)], (A.3) where symbols h i,j (λ) denote the derivatives of the R-matrices with respect to rapidity λ
h i,j (λ) = ϕ(iζ) 2 d dλ R i,j (λ). (A.4)
For eq. (A.3) we consider three cases in the following:
• For 3 ≤ n ≤ N we have tr 0 [P 0N · · · h 0n · · · P 02 R 01 (x)] = Π 2···n−1 R n−1,1 (x)Π n−1,n+1,···N h n−1,n . (A.5)
Applying the inverse of (A.2) to (A.5), we have R −1 N,1 (x) Π 2···N −1 Π 2···n−1 R n−1,1 (x)Π n−1,n+1,···N h n−1,n =ȟ n−1,n . (A.6)
Here, we used the abbreviationǍ i,j := P i,j A i,j .
• For n = 2 we have tr 0 [P 0N · · · P 03 h 02 R 01 (x)] = Π 3···N h N,2 R N,1 (x). (A.7)
Then, by applying the inverse of (A.2), we have The first term of (A.11) is given by
R −1 N,1 (x) Π 2···N −1 Π 3···N h N,2 R N,1 (x) =Ř −1 N,1 (x)ȟ 12ŘNR −1 N,1 (x)ȟ N,1 (x) = c + c − 2 0 −∆ ϕ ′ (x) ϕ ′ (x) −∆ 0 [N,1] = c + c − ϕ ′ (x) 2 (S + N S − 1 + S − N S + 1 ) + ∆ S z N S z 1 − 1 4 . (A.13)
Here we recall the notation given in eq. (2.11). The second term of (A.11) is given by [1] .
R −1 N,1 (x)ȟ 12ŘN,1 (x) = 1 2 ( 1 2 + S z N ) + c − ( 1 2 − S z N ) b − S − N b − S + N c − ( 1 2 + S z N ) + ( 1 2 − S z N ) [1] × −∆( 1 2 − S z 2 ) S − 2 S + 2 −∆( 1 2 + S z 2 ) [1] × ( 1 2 + S z N ) + c + ( 1 2 − S z N ) b + S − N b + S + N c + ( 1 2 + S z N ) + ( 1 2 − S z N )
(A.14)
By a straightforward calculation, we derive the rest of (2.10).
B Freezing through pseudo-decoupling for large x
We now show numerically that the freezing effect occurs through pseudo-decoupling. Let us consider the case of large parameter x. The specific heat shift ∆c converges to a function of temperature T as impurity parameter x becomes very large.
In the case of x = 20 the graph of ∆c versus T is shown in Fig. B.1. The graph is precisely given by the same graph as − 1 N c(0, T ) versus T . It is shown in Fig. B.3 that ∆c(x, T ) and −c(0, T )/N coincide completely as functions of T . Here we recall that c(0, T ) denotes the specific heat of the homogeneous XXZ chain with no impurity, while N is the total number of lattice sites.
Thus, the impurity spin is completely frozen up when parameter x is very large such as x = 20.
C A possible CFT argument for the Wilson ratio
Let us explain the Wilson ratio (5.2) expressed in terms of the dressed charge through a physical argument based on the possible CFT. We assume that the low-lying excitation spectrum of the XXZ spin chain with the spin-1/2 impurity is described by a CFT.
We first derive the ratio of the bulk susceptibility χ s to the bulk specific heat C. The magnetic susceptibility χ bulk s of the homogeneous XXZ spin chain with no impurity is given by [34] where v bulk s is the group velocity of spin excitation at the Fermi point: v s = dǫ/dk(B)/2πσ(B) with ǫ(k) being the dressed energy. For the homogeneous XXZ spin chain with no impurity the specific heat C at low temperature is given by
C bulk = π 3v bulk s T. (C.2)
For the bulk susceptibility and the bulk specific heat we thus have π 2 3
χ bulk s C bulk /T = 4Z 2 . (C.3)
We now extend the calculation for the homogeneous chain [34] to the XXZ chain with the spin-1/2 impurity. We thus have
χ bulk s + 1 N χ imp s = 4Z 2 π(v bulk s + v imp s /N) ≈ 4Z 2 πv bulk s 1 − 1 N v imp s v bulk s (C.4)
where we have assumed the contribution of the impurity in the Fermi velocity, v imp s /N. It follows from the assumption that the low-lying excitation spectrum of the XXZ spin chain with the spin-1/2 impurity is given by the CFT of central charge c = 1 that the specific heat is given by
C bulk + 1 N C imp = π T 3(v bulk s + v imp s /N) ≈ π T 3v bulk s 1 − 1 N v imp s v bulk s . (C.5)
Thus, the Wilson ratio for the impurity susceptibility χ imp s and the impurity specific heat C imp is given by We have thus obtained the Wilson ratio given by eq. (5.1).
For integrable impurity models such as the Anderson and the Kondo models the boundary CFT is applied to characterize the impurity susceptibilities of the models [37]. However, it seems that it is nontrivial to show that some conformal field theory describes the low-lying excitation spectrum of the XXZ spin chain with the spin-1/2 impurity.
We remark that the above argument for the Wilson ratio in eq. (C.6) does not depend on the ratio of the impurity group velocity to the bulk one v imp s /v bulk s . However, it is nontrivial to evaluate the impurity contributions χ imp s and C imp so that they are consistent with numerical estimates, as we have shown in §3 and §4 of the present paper.
Figure 2 :
2Fermi point B as a function of the magnetic field h.
Fig. 2 .
2The Fermi point B plays the role of the cut-off in the integral of eq. (3.2)
47) and the matrix elements of the M ×
(3. 49 )
49Since the equation (3.47) does not change by replacing z and x by −z and −x, respectively, we have y I (z, −x) = ρ I(−z − B, x). We now define functionsy I ± (k, ±z)y I (z, x)e ikz dz.(3.50)
22 1 ∼
1exp(−2x) for large values of x, which leads to the following: 1 − σ z 1 g,h ∼ exp(−2x) for x ≫ B and x ≫ ln N . (3.54)
( 3 .
347) is valid in the case of pseudo-decoupling. If we determine the magnetic field by 2h(M) = E(M − 1) − E(M) in the N-site chain with M − 1 down spins, then h(M) is given by of the order of 1/N. However, the magnetic field is much smaller than O(1/N) if pseudo-decoupling occurs. Let us now assume the following conditions:
Figure 4 :
4Impurity susceptibility χ imp (x) versus impurity x with small x for ζ = π 3 .Extracted from the data for N = 30 ∼ 100 interpolated with N = ∞. Red curve: analytic formula (3.35) Blue dots: numerical estimates.
Figure 5 :
5Impurity susceptibility χ imp (x) versus x in a wide range of x for ζ = π 3 . Extracted from the data for N = 30 ∼ 100 interpolated with N = ∞.
Figure 6 :
6Specific heat c(x, T ) versus impurity x at T = 10 −3 . for ζ = π 3 and N = 100.
Figure 7 :
7Specific heat c(x, T ) versus x in a narrow range of x = 0 at T = 10 −3 for ζ = π 3 , N = 100. Red curve is given by eq. (4.24).
Figure 8 :
8Impurity specific heat c imp (x, T ) versus temperature T for ζ =
Figure 9 :
9Impurity entropy s imp (x, T ) versus temperature T ζ = π 3 , x =1.
Figure 10 :
10Specific heat shift ∆c(x, T ) versus temperature T for ζ = π 3 , N = 100: x =0.3 (red), 0.5 (orange), 0.7 (yellow), 1.0 (green),1.5 (blue), 2.0 (purple), 3.0 (gray), 10.0 (brown).
Figure 11 :
11Specific heat shift ∆c with small values of impurity x for ζ = π 3 and N = 100. x =0.3 (red), 0.2 (orange), 0.1 (yellow), 0.05 (green), 0.01 (blue).
Figure 12 :
12Crossover temperature T X versus impurity parameter x. Estimates of T X for each x (blue dots). Fitting curve: T X = a/cosh bx with a = 0.269, b = 1.160 (red curve).
Figure 13 :
13Wilson ratio r versus impurity parameter x for ζ = π 3 through numerical estimates of χ imp (x) and c imp (x, T ).
•
For n = 1 we have tr 0 [P 0N · · · P 02 h 01 (x)] = Π 2···N h N,1 (x) Π 2···N −1 Π 2···N h N,1 (x) =Ř −1 N,1 (x)ȟ N,1 (x). (A.10)Taking the sum of these terms, we haveH XXZ (x) =Ř −1 N,1 (x)ȟ N,1 +Ř −1 N,1 (x)ȟ 1,2ŘN,1 express expression (A.11) in the form of matrices. The third term of (A.11) is exactly the same matrix which appears in the XXZ Hamiltonian. Actually, we have
Figure B. 1 :
1Specific heat shift ∆c(x, T ) for ζ = π 3 , N = 100, x = 20. (∆c(x, T ) = c(x, T ) − c(0, T )) Figure B.2: Bulk specific heat: c(0, T ) for ζ = π 3 , N = 100.
Figure B. 3 :
3Coincidence of ∆c(x, T ) and −c(0, T )/N for x = 20: Red dots: Fig. B.1 Blue dots: − 1 N × Fig. B.2 .
4 Specific heat due to impurity4.1 Thermal Bethe ansatzWe now evaluate the specific heat by solving the truncated integral equations of the thermal Bethe ansatz[29,30,31,32]. We show explicitly how the specific heat depends on the impurity parameter x. We restrict the anisotropy parameter ζ as ζ = π 3 throughout this section.Let us consider the string solutions for the BAE at ζ = π 3 . It is known that there are only three different types of solutions z j (j = 1, 2, 3) for ζ = π 3 . Explicitly, we havewhere each z R j gives the real part or the center of the string solution z j . We remark that sinh(z ± ζ 2 p 0 i) = ±i cosh z. In terms of string solutions the BAE is now expressed as follows.Here, unknown functions ρ j (z) and ρ h j (z) denote the particle and hole densities, respectively, with respect to the real part z R j of the string solutions z j , and the symbol f * g denotes the convolution of given functions f and g. The functions a j (z) for j = 1, 2, 3 are given by a j (z) = 1 2π 2 sin ζq j cosh 2z + cos ζq j ,where q j are given byThe matrix T is defined by(4.5)Here we remark that a 1 and a 2 are the same functions, as defined by(3.3). By making use of the identity a 2 = −a 3 (4.6)we reduce the number of equations of (4.2) from 3 to 2.
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| []
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[
"Orbifolds and Exact Solutions of Strongly-Coupled Matrix Models",
"Orbifolds and Exact Solutions of Strongly-Coupled Matrix Models"
]
| [
"Clay Córdova \nSchool of Natural Sciences\nInstitute for Advanced Study\n08540PrincetonNJUSA\n",
"Ben Heidenreich be-mail:[email protected]:[email protected] \nPerimeter Institute for Theoretical Physics\nN2L 2Y5WaterlooOntarioCanada\n",
"Alexandr Popolitov \nInstitute for Theoretical and Experimental Physics\n117218MoscowRussia\n\nInstitute for Information Transmission Problems\n127994MoscowRussia\n\nKorteweg-de Vries Institute for Mathematics\nUniversity of Amsterdam\nP.O. Box 942481090 GEAmsterdamThe Netherlands\n",
"Shamil Shakirov de-mail:[email protected] \nInstitute for Information Transmission Problems\n127994MoscowRussia\n\nSociety of Fellows\nHarvard University\n20138CambridgeMAUSA\n"
]
| [
"School of Natural Sciences\nInstitute for Advanced Study\n08540PrincetonNJUSA",
"Perimeter Institute for Theoretical Physics\nN2L 2Y5WaterlooOntarioCanada",
"Institute for Theoretical and Experimental Physics\n117218MoscowRussia",
"Institute for Information Transmission Problems\n127994MoscowRussia",
"Korteweg-de Vries Institute for Mathematics\nUniversity of Amsterdam\nP.O. Box 942481090 GEAmsterdamThe Netherlands",
"Institute for Information Transmission Problems\n127994MoscowRussia",
"Society of Fellows\nHarvard University\n20138CambridgeMAUSA"
]
| []
| We find an exact solution to strongly-coupled matrix models with a single-trace monomial potential. Our solution yields closed form expressions for the partition function as well as averages of Schur functions. The results are fully factorized into a product of terms linear in the rank of the matrix and the parameters of the model. We extend our formulas to include both logarthmic and finite-difference deformations, thereby generalizing the celebrated Selberg and Kadell integrals. We conjecture a formula for correlators of two Schur functions in these models, and explain how our results follow from a general orbifold-like procedure that can be applied to any one-matrix model with a single-trace potential. | 10.1007/s00220-017-3072-x | [
"https://arxiv.org/pdf/1611.03142v1.pdf"
]
| 119,708,190 | 1611.03142 | edd08146ef8b6e8cbbacb6d8769d6062de37cb50 |
Orbifolds and Exact Solutions of Strongly-Coupled Matrix Models
November 11, 2018 10 Nov 2016
Clay Córdova
School of Natural Sciences
Institute for Advanced Study
08540PrincetonNJUSA
Ben Heidenreich be-mail:[email protected]:[email protected]
Perimeter Institute for Theoretical Physics
N2L 2Y5WaterlooOntarioCanada
Alexandr Popolitov
Institute for Theoretical and Experimental Physics
117218MoscowRussia
Institute for Information Transmission Problems
127994MoscowRussia
Korteweg-de Vries Institute for Mathematics
University of Amsterdam
P.O. Box 942481090 GEAmsterdamThe Netherlands
Shamil Shakirov de-mail:[email protected]
Institute for Information Transmission Problems
127994MoscowRussia
Society of Fellows
Harvard University
20138CambridgeMAUSA
Orbifolds and Exact Solutions of Strongly-Coupled Matrix Models
November 11, 2018 10 Nov 2016
We find an exact solution to strongly-coupled matrix models with a single-trace monomial potential. Our solution yields closed form expressions for the partition function as well as averages of Schur functions. The results are fully factorized into a product of terms linear in the rank of the matrix and the parameters of the model. We extend our formulas to include both logarthmic and finite-difference deformations, thereby generalizing the celebrated Selberg and Kadell integrals. We conjecture a formula for correlators of two Schur functions in these models, and explain how our results follow from a general orbifold-like procedure that can be applied to any one-matrix model with a single-trace potential.
Introduction
In this paper we study a class of matrix models, those with single-trace potential of monomial form S[X] = Tr(X r ) , (1.1) and generalizations thereof. When r = 2 the model is quadratic and free, but for r > 2 the models we study are interacting and can be viewed as the infinite coupling limit of more familiar Gaussian plus interaction potentials.
We demonstrate that any such monomial matrix model is exactly solvable and provide a completely factorized form of the correlators. The solution depends on a non-perturbative choice of contour of integration in the space of matrices, which introduces an additional hidden integral parameter 0 ≤ a < r into the model. A taste of the type of formulas that we provide is the following expression for the partition function 2) where N is the rank of the matrix and δ r,a (N ) = 0, ±1, depending on N , see (3.21). In §3 we provide a similar formula for the expectation value of Schur polynomial insertions s λ (X) where λ is a Young diagram of at most N rows:
Z (r,a) N = δ r,a (N ) (2π) N N −1 i=0 Γ i r + 1 Γ i − a r + a r + 1 ,(1.
s λ (X) = δ r (λ) r |λ|/r x∈λ N + c λ (x) r,0 N + c λ (x) r,a h λ (x) r,0 .
(1.3)
Here c λ and h λ are respectively the contents and hook length of the box x ∈ λ, δ r (λ) = 0, ±1 depending on λ (see Appendix A), and n r,a = n n ≡ a mod r 1 otherwise .
(1.4)
These formulas-and their logarithmic and q-deformed (finite difference calculus) analogs that we obtain in §4.2 and §4.3-are similar to the celebrated Selberg and Kadell integrals [1][2][3][4], and reduce to them when r = 1. To illustrate their simple and explicit nature, we show an example of applying (1.3) in Figure 1.
In §4 we show that our results are related to a general orbifold-like structure in matrix models, where the partition function and Schur polynomial averages for the potential W (X r ) factor into r-fold products of those for the potential W (X). This factorization is related to a combinatoric identity involving the Vandermonde, and occurs in an natural basis of complex integration contours. (For other integration contours the partition function where the 3-signature δ 3 (λ) = −1 can be found using one of several methods discussed in Appendix A. and insertions can be expressed as a sum over factorized components.)
We emphasize that these results go far beyond the observation that one-matrix models with single-trace potentials are solvable, e.g., by the method of orthogonal polynomials. Generically, even after determining the moments of the eigenvalue potential, computing the partition function (with or without insertions) requires the evaluation of an N × N determinant, by diagonalization (as in the method of orthogonal polynomials) or otherwise. By contrast, in the examples we study these determinants can be evaluated in closed form for arbitrary N . One illustration of the simplicity of our results is the fact that correlation functions of N -independent operators are rational functions of N (once discrete data such as a and (N mod r) are fixed). This suggests that these models fall into a class which is some matrix-model analog of exactly solvable quantum field theories, such as integrable systems, rational CFTs, and Liouville theory. (However, we know no precise definition of such a class.)
Our results have a variety of possible applications. As suggested above, the simple potential S[X] = Tr(X 2 ) + λTr(X r ) , (1.5) can be viewed as a toy model of variety of interacting quantum systems, where λ controls the strength of the interactions. When λ is small perturbation theory can be applied, but the resulting perturbative expansion is not convergent, and advanced techniques are required to make sense of it (see, e.g., [5] for a recent discussion). On the other hand, when the coupling λ is large, it is more appropriate to view the Gaussian potential Tr(X 2 ) as a perturbation around the monomial model that we solve exactly. As we demonstrate in §2.4, the resulting strong coupling expansion converges for any λ = 0, and provides a different perspective on the interacting system.
Apart from their use as toy models, our monomial matrix models may also find applications in calculations of certain observables in ordinary quantum field theory. For instance the case r = 1 (the Selberg integral) is related to the integral expression for the superconformal index of a class of four-dimensional N = 1 gauge theories [6,7]. 5 The fact that the integral can be evaluated in closed form is a reflection of s-confinement [10][11][12][13]: this theory has an alternative infrared description in terms of free fields. The r > 1 models we study here may be related to the superconformal index of the same class of s-confining gauge theories on a Lens space [14][15][16]. We comment further on this in §5.3.
Related matrix models (r = 3) have also appeared in the study of five-dimensional gauge theory on the five-sphere [17]. In that case the quadratic piece of the model controls the Yang-Mills term and the cubic descends from the Chern-Simons interaction. The pure monomial model describes the strongly-coupled pure Chern-Simons theory that arises at infinite Yang-Mills coupling.
Finally, another possible avenue of application for our results is described in §5.2 concerns a generalization of the AGT conjecture [18,19]. In [20] the partition function of four-dimensional gauge theory on an orbifold C 2 /Z r was studied resulting in matrix models similar to those considered here. It would be interesting to determine the precise connection, and to understand if our results may be used to compute three-point functions in para-Liouville theory (see, e.g., [21][22][23][24][25]), thus generalizing [26]. 6 We leave this as a potential direction for future research.
Matrix Model Review
In this section we review general properties of random matrix models which are pertinent to our results. For more comprehensive reviews, see, e.g., [27,28]. We focus on the relation between perturbative and non-perturbative approaches, the role of the integration contour in the latter and the role of reflection positivity.
A matrix model is an average over random matrices, of the schematic form:
Z = 1 Vol G DX e −S[X] ,(2.1)
where S[X] is an action that depends on one or more matrices X, DX is some measure for integration over these matrices, and G is a gauge group, i.e. a symmetry of the action whose singlet sector defines the set of observables. Observables are defined by inserting X-dependent operators into the partition function:
Z[O] ≡ 1 Vol G DX O[X] e −S[X] , (2.2)
whereas normalized expectation values are obtained by dividing by the partition function
O ≡ Z[O] Z . (2.3)
In the case where G is non-trivial, only G-invariant operators are permissible.
There is an obvious analogy between matrix models and quantum field theories; in essence, a matrix model is a quantum field theory in zero dimensions. We pursue this analogy in more detail below, as it will provide an interesting context for our main results.
As an example, Hermitian one-matrix models are defined by the measure
DX = N i=1 dX ii N i<j dX ij dX * ij (2.4)
where X is a Hermitian N × N matrix and the independent real components are integrated from −∞ to ∞. The measure is invariant under U (N ) transformations X → U * XU , (2.5) where U * denotes the Hermitian conjugate of U , so it is natural to take G = U (N ).
Henceforward, we focus on such U (N ) Hermitian matrix models with single-trace potentials of the form S[X] = Tr W (X), where W is any analytic function. Expanding about a critical point of W , we obtain
S[X] = 1 2 Tr X 2 + ∞ p=3 λ p Tr X p . (2.6)
A Gaussian matrix model has S[X] = 1 2 Tr X 2 , and is free in the sense that Tr X 2 = i,j X ij X ji = i,j |X ij | 2 , so that the integral factors into one-dimensional integrals. Models with λ p = 0 for p > 2 are interacting, in that the integral no longer factors in this way. Two examples of interacting models that we will use frequently are the cubic model, S[X] = 1 2 Tr X 2 + λ 3 Tr X 3 , and the quartic model S[X] = 1 2 Tr X 2 + λ 4 Tr X 4 . Interacting theories can be studied perturbatively by splitting S = 1 2 Tr X 2 + S int and expanding e −S int in a formal power series in the coupling constants.
The loop equations
An alternative approach to these matrix models is to systematically exploit integration by parts identities (a.k.a. Ward identities). The resulting formulas are known as loop equations (see e.g. [27]), and we review them below.
Let M [X] = M 0 (Tr X i ) + M 1 (Tr X i )X + M 2 (Tr X i )X 2 + . .
. be a polynomial function of X and its traces, and introduce the matrix differential operator
(∂ X ) i,j = ∂ ∂(X) j,i . (2.7)
Now consider the total derivative:
0 = 1 Vol G DX Tr ∂ X (M [X]e −S[X] ) ,(2.8)
where we assume that the corresponding boundary term vanishes. We then obtain:
0 = Z[Tr(∂ X M ) − Tr(M ∂ X )S] . (2.9)
In particular, taking M = OX k+1 for k −1 for some gauge invariant operator O = O(Tr X i ) and using ∂ X X n+1 = n i=0 (Tr X i )X n−i and ∂ X Tr X n+1 = X n , we find:
0 = Z O k i=0 Tr X i Tr X k−i + Tr(X k+1 ∂ X )O − O Tr(X k+1 ∂ X )S ,(2.10)
for k −1.
These are the loop equations for the one-matrix model (2.1). Besides the action S[X], the only information about the matrix integral that (2.10) encodes is the vanishing of the boundary term (2.8). Nonetheless, the loop equations are a powerful tool for solving the model. For instance, the Gaussian matrix model, S = 1 2 Tr X 2 , has the loop equations
O Tr X k+2 = k i=0 O Tr X i Tr X k−i + Tr(X k+1 ∂ X )O . (2.11)
Suppose that O is a polynomial in X, and split it into monomials. Notice that the total power of X on the left-hand side (LHS) of the equation is two greater than on the righthand side (RHS). We can compute any polynomial correlator by iteratively replacing terms of the form O Tr X k+2 , k −1, with the RHS of (2.11). Since every non-trivial gaugeinvariant monomial in X takes this form, and the overall power of X is strictly decreasing, this procedure reduces every polynomial correlator to 1 = 1, allowing all such correlators to be computed.
Thus, the loop equations provide a nearly-complete solution to the Gaussian matrix model, fixing everything but the partition function itself (which requires a direct evaluation of the integrals). The loop equations also provide a perturbative solution to interacting models. The action (2.6) gives the loop equations:
O Tr X k+2 = k i=0 O Tr X i Tr X k−i + Tr(X k+1 ∂ X )O − p>2 pλ p O Tr X k+p . (2.12)
Because of the last term, the largest power of X on the RHS is now larger than on the LHS, and we cannot solve the model exactly using the same method as before. However, if we truncate at some fixed order λ kp p in perturbation theory with p k p finite, then the power of X increases in at most p k p steps by at most p (p − 2)k p in total, and the same iterative procedure as for the free theory terminates in a finite number of steps.
Thus, the loop equations fix the correlators of both the free theory and perturbatively in interacting theories.
Contour dependence
It is interesting to ask whether the loop equations likewise solve interacting theories nonperturbatively. As an example, we consider the cubic matrix model S = 1 2 Tr X 2 + λ 3 Tr X 3 . We rewrite the loop equations as
3λ 3 O Tr X k+3 = k i=0 O Tr X i Tr X k−i + Tr(X k+1 ∂ X )O − O Tr X k+2 ,(2.13)
so that the LHS has a power of X one greater than the RHS. We can then iteratively apply these equations to simplify any polynomial insertion, just as for the Gaussian theory. As before, this procedure terminates in a finite number of steps due to the strictly decreasing maximum power of X. However, unlike in the Gaussian case, not every monomial insertion is of the form O Tr X k+3 for k −1, hence the best we can do is to express every correlator in terms of the unknown averages (Tr X) p for p > 0, where (Tr X) 0 = 1. This is not the full story, because for finite-dimensional N × N matrices there are operator equations 7 relating Tr X P , P > N , to polynomials in Tr X p , p N . These trace relations exist because gauge-invariant polynomials in X only depend on the N eigenvalues of X, so there are only N independent gauge-invariant operators. For instance,
Tr X N +1 = (−1) N +1 N ! (Tr X) N +1 + . . . ,(2.14)
where the omitted terms involve at least one factor of Tr X p , 2 p N . Applying the loop equations to (2.14), we can express (Tr X) N +1 in terms of (Tr X) p , p N . Multiplying (2.14) by (Tr X) k and applying the loop equations once more, we find that in general (Tr X) P for P > N can be expressed in terms of the N unknowns (Tr X) p for 1 p N .
Although it is not yet clear, one can show that there are no further relations between the unknowns (Tr X) p , 1 p N . Thus, non-perturbatively the loop equations do not completely solve the cubic matrix model. Instead, the correlators depend on N additional parameters which were invisible in perturbation theory.
To determine the nature of these parameters, we consider the case N = 1, corresponding to the integral:
Z = 1 2π ∞ −∞ dx e − 1 2 x 2 −λx 3 . (2.15)
However, we notice an immediate problem: this integral diverges unless Re λ = 0! This problem, arising from the factor of e −(Re λ)x 3 at either x → ∞ or x → −∞, is invisible in perturbation theory, but makes the Hermitian matrix model ill-defined non-perturbatively (unless λ is imaginary).
Recall that the loop equations (hence also perturbation theory) are insensitive to the choice of integration contour. A natural way to define a non-perturbative completion of the model is to choose a different integration contour C such that the integral (2.15) is finite. To avoid introducing boundary terms into the loop equations, the integrand must vanish on ∂C. This occurs asymptotically at |x| → ∞ with | arg(λx 3 )| < π 2 (or with | arg(λx 3 )| = π 2 and | arg(x 2 )| < π 2 ). There are three such regions, centered on arg(λ 1/3 x) = 0, ± 2π 3 , so there are two linearly-independent closed contours C 1 , C 2 connecting these regions, see Figure 2. For a general linear combination C = c 1 C 1 +c 2 C 2 , one can check that the integrals I 0 ≡ dx e − 1 2 x 2 −λx 3 and I 1 ≡ dx xe − 1 2 x 2 −λx 3 depend on different linear combinations of c 1,2 , hence a choice of I 0,1 is equivalent to a choice of contour. Since Z = I 0 /2π and Tr X = I 1 /I 0 , the information not specified by the loop equations is precisely the choice of integration contour.
Thus, we conclude that the choice of integration contour C is a non-perturbative ambiguity in the N = 1 cubic matrix model, and this ambiguity precisely accounts for the missing information in the loop equations. The need to introduce contours in field space to discuss the theory non-perturbatively To extend this analysis to N > 1, it is convenient to re-express the random matrix X
Z = 1 N ! N i=1 dx i 2π i<j (x i − x j ) 2 e − i W (x i ) , (2.16) where i<j (x i − x j ) 2 -the square of the Vandermonde determinant det x j−1 i
-is the gauge-fixing determinant, and we specialize to a single-trace potential S[X] = Tr W (X) for simplicity. The residual gauge-symmetry is S N U (1) N , with volume (2π) N N !, where U (1) N acts trivially and S N permutes the eigenvalues. Gauge invariant operators are (sufficiently regular) symmetric functions of the x i , the simplest class of which are symmetric polynomials, corresponding to the vector space generated by multitrace operators in the original matrix model.
The permissible integration contours for the cubic model W (x) = 1 2 x 2 +λx 3 are the same as those for N = 1 described above, except that the contour for each eigenvalue can be chosen separately. For S N invariant insertions, only the number of eigenvalues integrated along each contour will affect the answer, hence there is a basis C N 1 ,N 2 of integration contours with N 1 eigenvalues integrated along C 1 and N 2 eigenvalues integrated along C 2 . A general contour takes the form: the choice of integration contour. As before, this is a non-perturbative ambiguity (invisible in perturbation theory).
C = N i=0 c i,N −i C i,N −i .
Thus, the cubic eigenvalue model is sensitive to integration contour at the non-perturbative level, with N +1 independent possible contours. Except in the special case where λ is purely imaginary, the real axis is not a possible choice of integration contour.
The quartic model W (x) = 1 2 x 2 + λx 4 is an interesting counterpoint. In this case, the integral along the real axis converges for Re λ 0, hence there is a "canonical" choice of integration contour. Nonetheless, other integration contours are available -a total of (N +1)(N +2) 2
are linearly independent -with the same perturbation series and loop equations. In this sense, non-perturbative ambiguities persist, and the canonical resolution of these ambiguities is just one of many possibilities.
Reflection positivity
Nonetheless, the real axis is a distinguished contour, because only for this contour is the eigenvalue integral equivalent to an integral over Hermitian matrices (X * = X), with the measure (2.4). In principle, for other contours the eigenvalue integral can be written as a matrix integral of the holomorphic form DX = where the first equation specifies that X is normal and the f i are N real functions of Tr X, . . . , Tr X N and their conjugates which specify the eigenvalue contour implicitly. For instance, the contour described by: 9
[X, X * ] = 0 , Im(Tr X 2k ) = 0 , (k = 1, . . . , N ) ,(2.20)
is relevant to the quartic model. Equivalently, this contour consists of normal matrices X whose squares are Hermitian.
Despite the fact that X = X * on a general contour, the eigenvalue model -and the In particular, [1] = 0 and Z[Tr X 3 ] = 0, the latter implying that Z is independent of λ. This is a contradiction, because Z[1] = 0 at λ = 0 is incompatible with an analysis of the Gaussian. 9 The constraints on Tr X 2k , k = 1, . . . , N can in principle be rewritten as constraints on Tr X, . . . , Tr X N . related matrix integral -retains some of the formal properties of a Hermitian matrix model. To illustrate this, we define an antilinear involution † on the operator algbera by X † ≡ X (noting that in general X † = X * ). The action of † on an arbitrary operator is specified by antilinearity together with (Tr X p ) † = (Tr X p ). Provided that we choose a real potential and a contour satisfying C = C * , this implies the formal property
Z[O † ] = Z[O] * ,(2.21)
for any operator O.
To distinguish between eigenvalue models with these formal reality properties and an actual integral over Hermitian matrices, we note that the latter satisfies reflection positivity:
Z[O † O] > 0 for any operator O = 0. (2.22)
By contrast, the cubic model is in general not reflection-positive. Consider the N = 1 model, for example. Reflection positivity requires that A † i A j is a positive-definite matrix for any set of linearly-independent operators {A i }. Choosing the operators {1, Tr X, Tr X 2 } and applying the loop equations, we find that the matrix A † i A j has at least one non-positive eigenvalue unless |λ| < 2 −1/2 3 −7/4 0.1. The constraint on λ becomes successively tighter as we consider larger operators (Tr X p for p > 2) and indeed the non-polynomial operator:
O = (Tr X)e 1 4 Tr X 2 (2.23)
satisfies O † O = 0, independent of λ = 0. Thus, the N = 1 model is not reflection-positive, regardless of the choice of integration contour. 10 We expect the same to be true for N > 1.
The cubic matrix model is therefore analogous to a non-unitary QFT. The quartic matrix model, by contrast, is manifestly reflection-positive when the integration contour is chosen to be the real line. A similar analysis to above shows that other contours are not reflection-positive, with increasingly large operators required to violate reflection-positivity as the integration contour approaches the real line. Thus, while the quartic integrated along the real line is analogous to a unitary QFT, other integration contours behave like non-unitary QFTs.
The weak-and strong-coupling expansions
The existence of non-perturbative ambiguities is closely related to the divergence of perturbation theory, which typically defines only an asymptotic series near an essential singularity on the Riemann sphere. As an example, consider the N = 1 quartic model, integrated along the real axis:
Z = 1 2π ∞ −∞ dx e − 1 2 x 2 −λx 4 . (2.24)
Expanding the partition function in powers of λ, we obtain formally:
Z = 1 2π ∞ p=0 (−1) p p! λ p ∞ −∞ dx x 4p e − 1 2 x 2 = 1 √ 2π ∞ p=0 (−1) p (4p − 1)!! p! λ p (2.25) where n!! ≡ n(n − 2)(n − 4) . . .. Since (4p − 1)!! p! = 2 2p Γ 2p + 1 2 Γ(p + 1)Γ 1 2 ∼ p p , (p 1) , (2.26)
we conclude that the radius of convergence of the formal perturbation series (2.25) is zero. Similar divergences appear with insertions and in normalized correlators.
Heuristically, perturbation theory diverges because for |x| 1/ |λ| the quartic coupling dominates the integral (2.24), hence a perturbative expansion in λ is not justified. In particular, the integral diverges for Re λ < 0, whereas it converges for Re λ > 0, implying that λ = 0 is an essential singularity in the holomorphic function Z(λ). This is similar to how contour dependence appears non-perturbatively. Since the quartic coupling dominates for |x| 1/ |λ|, there are additional integration contours where the quadratic term e − 1 2 x 2 diverges but the quartic term keeps the integral finite -such as the imaginary axis -and the choice of integration contour generates a non-perturbative ambiguity. 11
Thus, these twin problems of perturbation theory -divergence and insensitivity to non-perturbative physics -are both linked to the dominance of interactions at large field values. A novel approach would be to instead expand the exponential in the quadratic coupling, keeping the interaction term fixed. To do so, we rescale x → x/λ 1/4 to obtain the model
Z = 1 2π ∞ −∞ dx e − 1 2g 2 x 2 −x 4 , (2.27)
up to a normalizing factor for Z, where g = √ λ. If we now expand the exponential about the strong-coupling limit, g → ∞, we obtain
Z = 1 2π ∞ p=0 (−1) p 2 p p! g −2p ∞ −∞ dx x 2p e −x 4 = 1 4π ∞ p=0 (−1) p Γ p 2 + 1
Using Stirling's approximation, we conclude that
Γ p 2 + 1 4 2 p p! ∼ p −p/2 , (2.29)
hence the perturbation series converges! The sum can be performed explicitly,
Z = 1 4π ∞ p=0 (−1) p Γ p 2 + 1 4 2 p p! g −2p = 1 32π 2 g 2 e 1 32g 4 K 1 4 1 32g 4 , (2.30)
where K ν (z) is the modified Bessel function of the second kind, a result which can be verified by direct integration of (2.27).
Provided that a solution in the strong coupling limit, g → ∞, is available, the strongcoupling expansion g 1 has much better properties than the weak-coupling expansion g 1. In particular, the value of g = 0 does not affect the convergence of the partition function, so we expect that Z(1/g 2 ) is analytic at 1/g 2 = 0, and the expansion in g 1 should converge. For the same reason, there are no analogs of non-perturbative ambiguities. Indeed, the pure quartic model, g = ∞, depends on the same set of contours as at any intermediate coupling g = 0, so the ambiguities that went unresolved perturbatively at weak coupling are fixed at strong coupling, even before perturbing! In the next section, we explore the feasibility of solving the cubic or quartic model in the strong coupling limit g → ∞ for arbitrary finite N , which would enable a solution for any g via the strong-coupling expansion described above.
Exact Solutions at Strong Coupling
Motivated by the above considerations, we analyze one-matrix models in the limit where an r-point coupling blows up, λ r → ∞. After a field redefinition, these correspond to monomial matrix models S[X] = Tr X r . We begin by considering the quartic with a real (reflectionpositive) integration contour, before generalizing to other contours and potentials.
The real-line quartic
We consider the pure quartic model:
Z N = 1 N ! ∞ −∞ N i=1 dx i 2π e −x 4 i i>j (x i − x j ) 2 . (3.1)
We can rewrite the partition function as a determinant using a standard trick; note that
i>j (x i − x j ) = det V ij = det x j−1 i , (3.2) where V ij = x j−1 i
is the Vandermonde matrix. Fixing the S N permutation symmetry of the partition function, we obtain
Z N = ∞ −∞ N i=1 dx i 2π x i−1 i e −x 4 i × (det x k−1 j ) = det N ×N Z 1 [x i+j−2 ] , (3.3) where det N ×N denotes the determinant of the upper-left N × N block, i, j = 1, . . . , N .
Evaluating the integral directly, we find
Z 1 [x p ] = ∞ −∞ dx 2π x p e −x 4 = 1 4π Γ p+1 4 p ∈ 2Z 0 p ∈ 2Z + 1 . (3.4)
Thus,
Z N = 1 (4π) N det N ×N 1 + (−1) i+j 2 · Γ i + j − 1 4 . (3.5)
By similar reasoning
Z N n a=1 Tr X pa = 1 (4π) N N k 1 ,...,kn=1 det N ×N 1 + (−1) i+ a paδ ika +j 2 · Γ i + a p a δ ika + j − 1 4 . (3.6)
These relatively simple results mask the complexity of the model inside a determinant. To illustrate this complexity, we evaluate the partition function for small values of N :
Z 0,1,... = 1, Γ1 4 4π
,
Γ1 4 Γ3 4 16π 2 , Γ 2 1 4 Γ3 4 − 4 Γ 3 3 4 256π 3 , −Γ 4 1 4 + 16 Γ 2 1 4 Γ 2 3 4 − 48 Γ 4 3 4 2 14 π 4 , −Γ 5 1 4 + 20 Γ 3 1 4 Γ 2 3 4 − 96 Γ1 4 Γ 4 3 4 2 18 π 5 , −Γ 5 1 4 Γ3 4 + 17 Γ 3 1 4 Γ 3 3 4 − 72 Γ1 4 Γ 5 3 4 2 20 π 6 , −5 Γ 6 1 4 Γ3 4 + 105 Γ 4 1 4 Γ 3 3 4 − 684 Γ 2 1 4 Γ 5 3 4 + 1296 Γ 7 3 4 2 26 π 7 , . . . (3.7)
where we use the shorthand Γ x ≡ Γ(x
+ 4 Γ 2 3 4 4 Γ1 4 Γ3 4 , Γ 3 1 4 + 4 Γ1 4 Γ 2 3 4 4 Γ 2 1 4 Γ3 4 − 16 Γ 3 3 4 , 16 Γ1 4 Γ 3 3 4 −Γ 4 1 4 + 16 Γ1 4 Γ3 4 − 48 Γ 4 3 4
, . . . , (3.8) with increasingly complicated expressions for larger N .
Nonetheless, the result (3.7) has some obvious structure. The partition function takes the general form:
Z N = 1 (2π) N N i=0 a i;N Γ i 1 4 Γ N −i 3 4 , (3.9)
where the coefficients a i;N are rational. More generally, for any polynomial operator O It is natural to interpret (3.10) as a sum over sectors,
Z N [O] = 1 (2π) N N i=0 a i;N (O) Γ i 1 4 Γ N −i 3 4 ,(3.Z[O] = i Z i [O]
, in which case the complexity of the real-line quartic model can be partially ascribed to the increasing number of sectors -N + 1 for Z N . One approach to solving the model is to solve each sector individually. After identifying the origin of these sectors, we will show that at least some of them admit exact solutions for all N .
Pure and mixed phases
We now generalize to the potential:
S[X] = Tr X r ,(3.11)
which includes the Gaussian (r = 2), cubic (r = 3), and quartic (r = 4) as special cases. The corresponding N = 1 matrix model admits r − 1 closed contours on which the integral converges, constructed as follows. Let B r,j denote the contour from 0 to ω j r · ∞, where ω r ≡ e 2πi/r . These contours are open, but represent all possible asymptotics for which the integral converges. We form the "Fourier-transformed" contours: where the values at the end of each ray (and the color of the ray) denote its weight within the contour. We include the case a = 0, which is not a closed contour but occurs naturally once we include the u-deformation in §4.
C r,a ≡ r−1 j=0 ω −ja r B r,j .
We observe that ∂C r,a = 0 for a ≡ 0 mod r, hence C r,1 , . . . , C r,r−1 form a basis of closed contours on which the integral converges. This basis, illustrated in Figure 3, is the eigenbasis of the Z r symmetry X → ω r X, which maps C r,a → ω −a r C r,a For N > 1, a general contour can be written as a linear combination of
C (r) N 1 ,...,N r−1 ≡ C N 1 r,1 × . . . × C N r−1 r,r−1 , N = r−1 a=1 N a ,(3.13)
i.e. with N a eigenvalues integrated along the contour C r,a . There are N +r−2 r−2 such contours. We refer to matrix models integrated over the contours C N r,a as "pure phases," and those integrated over C (r) N 1 ,...,N r−1 with N 1 , . . . , N r < N as "mixed phases." The principle advantage of this contour basis is that it simplifies the moments:
Ca x p e −x r dx = δ r|p+1−a Γ p + 1 r ,(3.14)
where δ r|p = 1 when r divides p, and vanishes otherwise. The non-vanishing moments are those for which the integrand together with the contour forms a Z r singlet, and each such moment is a rational prefactor times Γ a r . As a consequence
C N 1 ,...,N r−1 O i e −x r i ∼ a Γ Na a r ,(3.15)
up to a rational prefactor, where O is any polynomial insertion. Comparing with (3.10), we see that the sum over sectors previously identified in the real-line quartic is nothing but a sum over pure and mixed phases! In particular, for r = 4, the contour along the real axis is
R = 1 2 (C 4,1 + C 4,3 ), where R N = 1 2 N N i=0 N i C i 4,1 C N −i 4,3 . (3.16)
Not every phase mixture contributes to the partition function, as the Z r symmetry dictates that many insertions vanish. Suppose that O p is a homogeneous polynomial in X of degree p. A necessary condition for Z N 1 ,...,N r−1 [O p ] to be non-vanishing is for it to be a Z r singlet:
p + N 2 − a aN a ≡ 0 mod r . (3.17)
For instance, the partition function (p = 0) of a pure phases vanishes unless N = kr or N = kr + a. For the C 4,1 , C 4,3 two-phase mixture of the real-line quartic, we obtain the constraint
N 1 ≡ N (N + 1) 2 mod 2 , (3.18)
for contributions to the partition function, which explains the non-vanishing terms in (3.7).
Having identified the natural subsectors of the real-line quartic, the obvious question is whether these subsectors are solvable. Remarkably, as we now argue, the pure phases are exactly solvable for any r, a and N .
Summary of results for pure phases
Before explaining how the pure phases can be solved, we present the solution in brief. We consider the pure-phase eigenvalue model
Z (r,a) N = 1 N ! Cr,a i dx i 2π e −x r i i>j (x i − x j ) 2 ,(3.19)
for integers N 0, r > 0, and 0 a < r, with C r,a defined by (3.12
Z (r,a) N = δ r,a (N ) (2π) N N −1 i=0 Γ i r + 1 Γ i − a r + a r + 1 , (3.20)
where δ r,a (N ) = 0, ±1 is given by
δ r,a (N ) = (−1) N r a(a−1) 2 + N r ã(ã−1) 2 N ≡ 0, a mod r 0 otherwise , (3.21) andã ≡ r − a.
Recall that Schur polynomials are multivariate symmetric polynomials defined as
s λ (x) ≡ det x N +λ j −j i det x N −j i , (3.22)
where λ denotes a partition λ 1 . . . λ N 0.
Theorem 2. The averages of Schur polynomials take the simple form:
s λ (X) = δ r (λ) r |λ|/r x∈λ N + c λ (x) r,0 N + c λ (x) r,a h λ (x) r,0 , (3.23)
where n r,a = n n ≡ a mod r 1 otherwise .
(3.24)
Here we interpret λ as a Young diagram with |λ| boxes x = (i, j), rows of length λ i and columns of length λ i . The contents c λ (x) ≡ j −i and hook-length h λ (x) ≡ λ i +λ j −i−j +1 are the same quantities which appear in the dimension formula for representations of the special linear group:
dim SL(N ) (R λ ) = x∈λ N + c λ (x) h λ (x) ,(3.25)
which is similar to (3.23). Finally, the prefactor δ r (λ) = 0, ±1, the "r-signature" of λ, is given explicitly by
δ r (λ) = (−1) |λ| r x∈λ (−1) c λ (x) r + h λ (x) r
The r-core of λ is trivial 0 otherwise .
(3.26)
Here the r-core of λ-the unique result of stripping all possible rim hooks of length r from λ-generalizes the remainder upon division by r, so that Schur polynomials with nonzero averages correspond to Young diagrams which are "divisible by r," or "r-divisible." A necessary but insufficient condition for r-divisibility is that |λ| ≡ 0 mod r. The relevant properties of r-cores, r-signatures, and r-divisible Young diagrams are reviewed in Appendix A.
We note in passing that the Schur average formula (3.23) obeys several non-trivial relations when treated as a formal analytic function of N for fixed N mod r:
s λ (X) (r,a) N = s λ (X) (r,r−a) N , s λ (X) (r,a) −N = (−1) (r+1) |λ| r s λ (X) (r,r−a) N ,(3.
Solution by orthogonal polynomials
We now prove Theorems 1 and 2, i.e. derive formulas (3.20) and (3.23), using orthogonal polynomials. Suppose that p n (x) = x n + . . . is some polynomial basis. The Vandermonde determinant can be rewritten as det x j−1 i = det p j−1 (x i ) by a triangular change of basis, so that
Z N = det N ×N Z 1 [p i p j ] .
(3.28)
If we choose a polynomial basis for which the matrix Z 1 [p i p j ] is sufficient sparse then the partition function is easily computed. The usual approach is to choose Z 1 [p i p j ] = t i δ ij for some normalization t i , so that Z N = N −1 i=0 t i . This approach is well-suited to the Gaussian model (r = 2 and a = 1), but is impossible in the pure phases for r > 2, because Z N =kr and Z N =kr+a do not vanish, whereas Z N vanishes for other values of N , implying that some of the t i vanish and others are infinite.
Instead, we consider orthogonal polynomials satisfying the orthogonality relation:
Z 1 [p m p n ] = t m δ (r,a) m,n ,(3.29)
where δ (r,a) m,n is the block-diagonal matrix diag(J a , J r−a , J a , J r−a , . . .) with J n the n×n antidiagonal permutation matrix. This condition is chosen so that δ
(x) = x iL ( a r −1) k (x r ) 0 i < a x iL ( a r ) k (x r ) a i < r , (3.30) where L (α) k (x) denotes the generalized Laguerre polynomial L (α) k (x) = k p=0 (−1) p k + α k − p x p p! , (3.31) andL (α) k (x) = (−1) k k!L (α) k (x) = x k + .
. . is monic. The Laguerre polynomials satisfy the orthogonality relation:
∞ 0 x α L (α) m (x)L (α) n (x)e −x dx = Γ(n + α + 1) n! δ mn . (3.32)
Together, (3.30) and (3.32) are sufficient to derive (3.29), where the homogeneity property p n (ω r x) = ω n r p n (x) implies that the integral over C r,a either vanishes or is equivalent to an integral over B 0 = (0, ∞), which can be reduced to (3.32) by a change of variables y = x r . We obtain
t i = 1 2π Γ i r + 1 Γ i − a r + a r + 1 .det(z j − X) = λ λ 1 k (−1) |λ| s λ (X)sλ(z) ,(3.34)
whereλ is the partitionλ i = (N − λ k , . . . , N − λ 1 ). It is straightforward to check using the orthogonality relation (3.29) that:
k j=1 det(z j − X) = 1 det 1 i,j k z j−1 i det p N (z 1 ) . . . p N +k−1 (z 1 ) . . . . . . p N (z k ) . . . p N +k−1 (z k ) . (3.35)
In general, this holds when Z 1 [p i p j ] = 0 for i < N and j N , which follows from (3.29) when Z N = 0, i.e., when N = kr or N = kr + a.
Combing (3.34) with (3.35) and using sλ(z) = det z
N −λ j +j−1 i / det z j−1 i , we obtain det 1 i,j k p N +j−1 (z i ) = λ λ 1 k (−1) |λ| s λ (X) det 1 i,j k z N −λ j +j−1 i . (3.36)
For orthogonal polynomials of the general form p i = j p i;j x j , we have
det 1 i,j k p N +i−1 (z j ) = p 1 ,...,p k k j=1 z p j j det 1 i,j k p N +i−1; p j , (3.37) so that s λ (X) = (−1) |λ| det 1 i,j k p N +i−1; N −λ j +j−1 , k λ 1 . (3.38)
The general result (3.38) can be applied to the case at hand by noting that
p i;j = δ r|(i−j) (−1) i−j r i−j r ! t i t j ,(3.39)
where the Kronecker delta enforces i−j r ∈ Z. We obtain
s λ (X) = (−1) |λ| k j=1 t N +j−1 t N +j−1−λ j det 1 i,j k δ r|(λ j +i−j) (−1) λ j +i−j r ( λ j +i−j r )! ,(3.40)
for k λ 1 .
To evaluate the determinant, we use the results of §A.1. We must have
det 1 i,j k (−1) λ j +i−j (λ j + i − j)! = (−1) |λ| x∈λ 1 h λ (x) , k λ 1 ,(3.41)
to reproduce the Kadell formula for r = 1 [3]. The hook-lengths in the various components of the r-quotient λ/r (µ) are just 1/r times the hook lengths divisible by r in λ, so we obtain
det 1 i,j k δ r|(λ j +i−j) (−1) λ j +i−j r ( λ j +i−j r )! = r |λ|/r δ r (λ) x∈λ 1 h λ (x) r,0 ,(3.42)
using (A.9) and Theorem 9. Using (3.33) and the identity Γ n+I−a r + a r + 1 Γ n−a r + a r + 1
= r −( n+I−a r − n−a r ) I i=1 n + i r,a ,(3.43)
to simplify the product over ts, we obtain the Schur average formula (3.23).
Single-trace correlators
Let L (I,J) denote the L-shaped partition with I +1 rows and J +1 columns, i.e. L Combining this with the Schur average formula (3.23), we compute the expectation values of single-trace operators. Using δ r (L (I,J) ) = δ r|(I+J+1) (−1) I+ I r , we obtain:
Tr X qr = 1 qr q+1 qr−1 J=0 (−1) I r I r,0 ! J r,0 ! J i=−I N + i r,0 N + i r,a ,(3.45)
for q > 0, where I = qr − 1 − J and n r,a ! ≡ n i=1 i r,a . By (3.43), this can be rewritten as
Tr X qr = 1 q qr−1 J=0 (−1) I r Γ N +J r + 1 Γ N +J−a r + a r + 1 Γ N +J r − q + 1 Γ N +J−a r + a r − q + 1 I r ! J r ! . (3.46)
Collecting terms, this takes the form of a sum of hypergeometric series Tr X qr N =kr = af k, a r (q) + (r − a)f k, a r +1 (q) , Tr X qr N =kr+a = af k+1, a r (q) + (r − a)f k, a r +1 (q) ,
(3.47) for 0 < a < r, where f k,x (q) ≡ (−1) q−1 q! q−1 j=0 Γ(k + j + 1)Γ(k + j + x)(1 − q) j Γ(k + j − q + 1)Γ(k + j − q + x)j! ,(3.48)
and (x) n ≡ x(x + 1) . . . (x + n − 1) denotes the rising factorial. Using a pair of resummation identities for 3 F 2 , this can be rewritten as 14 Tr X qr (x)q
f k,x (q) = k (x) q 3 F 2 1 − k, 1 + q, −q x, 2 ; 1 .
Combining (3.47) and (3.49), it is straightforward to compute any single-trace correlation function of interest. For instance,
r Tr X r = N 2 , r 2 Tr X 2r = 2N 3 + aãN r 3 Tr X 3r = 5N 4 + (r 2 + 6aã)N 2 ± aã(a −ã)N , r 4 Tr X 4r = 14N 5 + 10(r 2 + 3aã)N 3 ± 10aã(a −ã)N 2 + 3aã(2r 2 − aã)N , . . . (3.50) where the upper (lower) sign corresponds to N = kr (N = kr + a).
The large N limit
We briefly consider the large N limit of these pure phase eigenvalue models. From (3.50) we see that the contour dependence enters at O(1/N 2 ) relative to the leading large N behavior, and that there are subleading corrections suppressed by odd powers of N . This can be shown more generally by rewriting
f k,x (q) = k q p=0 (2q − p)! p!(q − p)!(q − p + 1)! (x + q − 1) (p) (k − 1) (q−p) ,(3.51)
where x (p) ≡ x(x − 1) . . . (x − p + 1) = Γ(x+1) Γ(x+1−p) denotes the falling factorial. Expanding in k 1 and retaining the first few terms, we obtain
r q Tr X qr = (2q)! q!(q + 1)! N q+1 + (2q − 2)! 12(q − 1)!(q − 2)! (r 2 (q − 2) + 6aã)N q−1 ± (2q − 2)! 12(q − 1)!(q − 3)! aã(a −ã)N q−2 + . . . , (3.52)
where the upper (lower) sign corresponds to N = kr (N = kr + a), as above.
Similar results hold for the free energy. We first rewrite the partition function (3.20) in terms of the Barnes G-function, defined by the Weierstrass product
G(z + 1) = e −ζ (−1)− z+(1+γ)z 2 2 ∞ k=1 1 + z k k e z 2 2k −z ,(3.53)
which can be shown to satisfy 15
G(z + 1) = Γ(z) √ 2π G(z) , G(1) = e −ζ (−1) . (3.54)
where ζ(s) is the Riemann zeta function and the normalization is chosen for future convenience. We define
Z r,a (N ) ≡ G N r + 1 r G N + a r a G N + a r + 1 r−a ,(3.55)
so that
Z (r,a) N =kr = δ r,a (N ) Z r,a (N ) Z r,a (0) , Z (r,a) N =kr+a = δ r,a (N ) Z r,ã (N ) Z r,a (0) . (3.56)
We have the asymptotic expansion
log G(n + 1) = n 2 2 log n − 3 4 n 2 − 1 12 log n + ∞ g=2 B 2g 2g(2g − 2) n 2−2g ,(3.57)
where B 2g are the Bernoulli numbers. Thus,
log Z r,a (N ) = N 2 r + aã 2r − r 6 log N r − 3N 2 2r + aã(a −ã) 6N r − r 4 − 15a 2ã2 120rN 2 + . . . , (3.58)
from which we obtain the large N free energy: 16 Reproducing these results with a large N analysis along the lines of [33][34][35] (see, e.g., [28] and references therein for a more comprehensive review of large N techniques) is an interesting open problem. This calculation is non-trivial for several reasons. Firstly, the large 15 A more common but ultimately less convenient convention is G 2 (z + 1) = (2π) z 2 e ζ (−1) G(z + 1), which satisfies G 2 (z + 1) = Γ(z)G 2 (z) and G 2 (1) = 1. 16 Notice that the free energy satisfies F 17 We drop the prefactor δ r,a (N ) from the large N expansion, as it is periodic in N with period 2r, hence formally non-perturbative in N .
F = F 0 + N 2 r + aã 2r − r 6 log N r − 3N 2 2r ± aã(a −ã) 6N r − r 4 − 15a 2ã2 120rN 2 + . . .
N analysis of [33][34][35] is naturally expressed in a contour basis of Lefschetz thimbles, corresponding to the saddle points of the potential. This basis is degenerate when the potential is monomial for r > 2, as r − 1 critical points coincide at the origin. Secondly, the genus expansion is organized in even powers of N , hence the appearance of O(1/N 3 ) corrections is unexpected in a standard analysis.
These two issues may be linked. To pick out the pure-phase contour C r,a , one can deform the potential by ε Tr X 2 to resolve the r − 1 critical points and then express C r,a in a thimble basis, later taking ε → 0. The change of basis between C r,a and the thimbles gives a linear combination of the saddle points weighted by binomial coefficients (e.g., in the case r = 3). The O(N ) terms in the sum over saddle points can change the large N scaling, and we hypothesize that this gives rise to the unexpected odd powers of N .
In the language of topological recursion (see [36] for a review), the appearance of subleading corrections suppressed by odd powers of N means that these theories are not of topological type ( [36], Definition 3.6). Hence they are not described by the standard large-N tools-spectral curve topological recursion [37,38]-or even its more general "blobbed" version [39]. However, the difference from the topological type case is actually rather mild. The ward identities (2.10) are not broken-they still admit solutions of topological type (in contrast to β-ensembles [40], where for generic β solutions of topological type are forbidden). Rather, the unusual dependence on N enters through initial conditions; for example, for r = 3 the first non-topological correlator is (Tr X) 3 = ±N .
The large-N behavior of these monomial matrix models is interesting for the following reason. If one computes the standard spectral curve (forgetting for now that pure phase correlators are not described by it), one gets y ∼ x r , which is the symplectic dual (x ↔ y) [41] to the spectal curve for the r-Gelfand-Dickey hierarchy (see [42] Theorem 7.3). Since pure phases are very natural from the matrix model point of view one can expect that the relevant generalization of the topological recursion is also natural. Once available, it would immediately provide a generalization of the r-Gelfand-Dickey hierarchy (and, via a lift to cohomology, of Witten's r-spin class [43]). We defer this problem to future work.
A General Orbifold Construction for Matrix Models
The exact solutions found in the previous section arise from a more general construction, which we now explain. This construction is, roughly speaking, an orbifold, where a matrix model with a single-trace potential W (X) is replaced by one with a single-trace potential W (X r ).
General results
Definition 1. For any one-matrix model
Z n,u = 1 n! C n n i=1 dx i 2π x u i i<j (x i − x j ) 2 e − Tr W (X) , (4.1)
the (pure phase) r-fold matrix model is defined as
Z (r,a) N,U = 1 N ! C N r,a N i=1 dx i 2π (x r i ) U r i<j (x i − x j ) 2 e − Tr W (X r ) , (4.2)
for any r ∈ N and choice of contour 0 a < r, with C r,a = r−1 j=0 ω −ja r C 1/r j , where C 1/r is the principal rth root of C and C 1/r j is C 1/r rotated by ω j r .
Here W (X) is any (single-trace) potential and C is any contour, 18 where we isolate a contribution −u log X from the potential for future convenience. 19 We assume that the solution to the parent matrix model is known, and use it to solve the corresponding r-fold model.
Let p (u)
m be a set of orthogonal polynomials for the parent model (4.1):
Z 1,u [p (u) m p (u) n ] = t (u) m δ m,n . (4.3)
Now consider the polynomials P (U ;r,a)
kr+i (x) = x i p ( U +a r −1) k (x r ) , 0 i < a , x i p ( U +a r ) k (x r ) , a i < r .= t ( U +a r −1) k , 0 i < a , t ( U +a r ) k , a i < r . (4.5) where δ (r,a)
m,n is the same as in (3.29). In particular, (4.5) follows from the Z r orbifold projection implied by the contour C r,a together with the change of variables
r C 1/r x a−1 f (x r ) dx = C y a r −1 f (y) dy , (4.6)
where (x r ) 1/r = x for − π r < arg x π r . Using (4.5), we can compute the partition function Z (4.7)
Re-expressing this in terms of the partition function of the parent model,
Z n,u = n−1 i=0 t (u)
i , we find 20 Theorem 3. The partition function of an r-fold matrix model is a product of r copies of the partition function of the parent model:
Z (r,a) N,U = δ r,a (N ) r−1 µ=0 Z nµ,uµ , (4.8) where n µ = N − µ − 1 r + 1 , u µ = U + a r + N − µ − a − 1 r − N − µ − 1 r . (4.9)
Note that µ n µ = N and µ u µ = U .
A similar factorized structure occurs in the Schur average formula. To derive it, we start with the general result (3.38). Writing p Figure 1.
Choosing K such that K + N ≡ 0 (mod r) and applying (A.17), we obtain
s λ (X) (r,a) N,U = δ r (λ) r−1 µ=0 (−1) |λ/r (µ) | det 1 i,j kµ p (uµ) nµ+i−1; nµ+j−(λ/r (µ) ) j −1 . (4.12)
where k µ = K+µ r . From the definition of the r-quotient (A.8), we find
k µ λ 1 + µ r (λ/r (µ) ) 1 , (4.13)
so that s λ/r (µ) (X) nµ,uµ , (4.14)
where n µ and u µ are defined in Theorem 3.
An example application of this theorem is shown in Figure 4.
Example: logarithmic models
As an example of the general construction given in the previous section, we consider the eigenvalue model 16) with B j now the finite segment [0, ω j r ]. This is an r-fold generalization of the β = 1 Selberg integral [1,2] (see also [7]): 21
Z (r,a) N = 1 N ! C N r,a i dx i 2π i<j (x i − x j ) 2 i (x r i ) u (1 − x r i ) v , (4.15) where C r,a = r−1 j=0 ω −ja r B j ,(4.1 N ! 1 0 i dx i i<j (x i − x j ) 2 i x u i (1 − x i ) v = N i=1 Γ(i + u)Γ(i + v)Γ(i) Γ(N + i + u + v) . (4.17)
The model (4.15) is related to the exponential model (3.19) by the scaling limit
lim v→∞ 1 − 1 v x r v = e −x r , (4.18) hence lim v→∞ S 0,v [X/v 1/r ] = Tr X r , where S u,v [X] is the logarithmic potential S u,v [X] = −u Tr ln X r − v Tr ln 1 − X r . (4.19)
This logarithmic generalization is natural, for example, from the perspective of conformal field theories, where free-field representations of correlation functions are typically given by generalized matrix models with logarithmic potentials [26] sometimes referred to as Selberg integrals [44].
Using Theorem 3 and (4.17), we obtain the partition function
Z (r,a) N = δ r,a (N ) (2π) N N −1 I=0 Γ I−a r + a r + u + 1 Γ I r + v + 1 Γ I r + 1 Γ N +I−a r + a r + u + v + 1 , (4.20)
where we use the substitution (4.21) to simplify the product of r Selberg integrals.
I = ir − µ − 1 , N = kr , (i − 1)r + µ , N = kr + a ,
The expectation value of a Schur polynomial with the Selberg measure (4.17) is given by the Kadell formula [3]:
s λ (X) = x∈λ (N + c λ (x))(N + c λ (x) + u) h λ (x)(2N + c λ (x) + u + v)
. (4.22) Using this together with Theorem 4, we can compute the average of a Schur polynomial in the r-fold model (4.15). To simplify the result, we rewrite the product over contents in the r-quotient as a product of columns using x∈λ Γ(α + c λ (x)) = λ 1 j=1 Γ(α+j) Γ(α+j−λ j ) . Rearranging and applying a generalization of (3.43), the result can be expressed as a product over boxes of the parent Young diagram:
s λ (X) = δ r (λ) x∈λ N + c λ (x) r,0 N + c λ (x); ru r,a h λ (x) r,0 2N + c λ (x); ru + rv r,a ,(4.23)
where the deformed bracket n; x r,a generalizes (3.24)
n; x r,a = n + x n ≡ a mod r 1 otherwise .
(4.24)
It is straightforward to check that (4.20) and (4.23) reduce to (3.20) and (3.23) in the appropriate limit.
These logarithmic models not only preserve all the solvability properties discussed so far; they also reveal additional ones not present in the original polynomial models. They enjoy exact formulas for the correlation function of two Schur polynomials (with an appropriate shift in the argument). This remarkable property has already been noted for r = 1 models [44][45][46][47]; in this paper we propose a generalization to r > 1, see §5.1.
Some q analogs
The orbifold-like construction of §4.1 can also be extended to q-analogs of random matrix models. q calculus is based on the replacement
df dx −→ d q f d q x ≡ f (x) − f (qx) x − qx ,(4.25)
for some 0 < q < 1, where the limit q → 1 takes dqf dqx → df dx . The inverse of the q derivative is the Jackson integral
f (x)d q x ≡ (1 − q)x ∞ k=0 q k f (q k x) ,(4.26)
where definite integrals are defined by the fundamental theorem of calculus,
b a f (x)d q x = F (b) − F (a) for F (x) = f (x)d q x.
q calculus satisfies a limited form of the chain rule
d q r (ax r ) = a[r] q x r−1 d q x ,(4.27)
where [n] q denotes the q-number
[n] q ≡ 1 − q n 1 − q , (4.28)
which satisfies lim q→1 [n] q = n. Such calculus appears naturally from several closely related perspectives. Physically, it has most recently attracted attention in the context of five-dimensional gauge theories, where the finite difference parameter q is the exponentiated radius of the fifth dimension [48] in the spirit of Kaluza and Klein. Mathematically, q-numbers appear in enumerative geometry of symplectic resolutions as K-theory weights associated with fixed points of equivariant torus action [49]. From the perspective of integrability theory, q-numbers correspond to trigonometric integrable models, which occupy an intermediate level of complexity between rational (corresponding to usual numbers) and elliptic (corresponding to elliptic numbers, [50]) models.
The reasoning of §4.1 now goes as follows.
Definition 2. For any q-deformed one-matrix model
Z n,u;q = 1 n! C n n i=1 d q x i 2π x u i i<j (x i − x j ) 2 e − Tr W (X;q) ,(4.29)
the (pure phase) r-fold matrix model is defined as
Z (r,a) N,U ;q = 1 N ! C N r,a N i=1 d q x i 2π (x r i ) U r i<j (x i − x j ) 2 e − Tr W (X r ;q r ) ,(4.30)
with C r,a defined as in Definition 1.
This definition is chosen so that the q-deformed measures of the parent and r-fold models are related by a change of variables. We find the r-fold orthogonal polynomials P (U ;r,a)
kr+i (x; q) = x i p ( U +a r −1) k (x r ; q r ) , 0 i < a , x i p ( U +a r ) k (x r ; q r ) , a i < r ,kr+i;q = r [r] q · t ( U +a r −1) k;q r , 0 i < a , t ( U +a r ) k;q r , a i < r .
(4.32)
By the same reasoning as above, we conclude that s λ/r (µ) (X) nµ,uµ;q r , (4.33) as in Theorems 3 and 4.
As an example, we consider the q-Selberg integral
1 N ! 1 0 i d q x i i<j (x i − x j ) 2 i x u i (qx i ; q) v = q (u+1)( N 2 )+2( N 3 ) N i=1 Γ q (i + u)Γ q (i + v)Γ q (i) Γ q (N + i + u + v) .
(4.34) Here
(x; q) ∞ ≡ ∞ i=0 (1 − q i x) , (x; q) n ≡ (x; q) ∞ (xq n ; q) ∞ , (4.35)
is the q-Pochhammer symbol and Γ q (
x) ≡ (1 − q) 1−x (q; q) x−1 is the q-gamma function, which satisfies Γ q (x + 1) = [x] q Γ q (x), Γ q (1) = 1, and lim q→1 Γ q (x) = Γ(x).
The r-fold generalization of the q-Selberg integral is the eigenvalue model:
Z (r,a) N = 1 N ! C N r,a N i=1 d q x i 2π (x r i ) u (q r x r i ; q r ) v i<j (x i − x j ) 2 .
(4.36)
Using (4.33), we obtain
Z (r,a) N = δ r,a (N ) (2π) N (q r ) nr(N,u) r [r] q N × N −1 I=0 Γ q r I−a r + a r + u + 1 Γ q r I r + v + 1 Γ q r I r + 1 Γ q r N +I−a r + a r + u + v + 1 , (4.37)
where n r (N, u) ≡ r−1 µ=0 (u µ + 1) nµ 2 + 2 nµ
3
. 22 Apart from the q-dependent prefactor, this is a straightforward q-analog of (4.20).
Likewise, using the q-Selberg Schur average formula [3] s λ (X) = q n(λ) with n(λ) ≡ i (i − 1)λ i , we obtain
x∈λ [N + c λ (x)] q [N + c λ (x) + u] q [h λ (x)] q [2N + c λ (x) + u + v] q ,(4.s λ (X) (r,a) = δ r (λ) (q r ) nr(λ) x∈λ N + c λ (x) (q) r,0 N + c λ (x); ru (q) r,a h λ (x) (q) r,0 2N + c λ (x); ru + rv (q) r,a ,(4.39)
where n; x
(q)
r,a is given by the obvious q-deformation n + x → [n + x] q of (4.24) and n r (λ) ≡ µ n(λ/r (µ) ). This is a q-analog of (4.23).
The q → ω r limit There is a another class of q-analogs of the r-fold models discussed in this paper. To see this, consider a Jackson integral
C e −W (x;q) d q x ,(4.40)
where for simplicity we take C to be an interval along the positive real axis. Whereas in the limit q → 1 we obtain an ordinary integral along C, in the limit q → ω r , we find
lim q→ωr [r] q b a e −W (x;q) d q x = C r,0 e −Wr(x) dx ,(4.41)
where W r (x) = W (x; ω r ) and C r,0 is the a = 0 r-fold contour of Definition 1. Other r-fold contours can be obtained by inserting appropriate branch cuts (in the form x −a (x r ) a/r ) into the integrand.
Thus, for a given r-fold potential W (x r ), any q-deformed potential W (x; q) satisfying W (x; ω r ) = W (x r ) defines a q-analog which reduces to the r-fold model in the q → ω r limit. For example, in the q-Selberg measure we have
lim q→ωr (qx i ; q) V = (1 − x r i ) V /r ,(4.42)
provided that V ∈ rZ. Thus, the r = 1 q-Selberg integral is in some sense a q-analog of the r-fold logarithmic model discussed in §4.2. Using this connection and analytic continuation off the integers, one obtains an alternate proof of (4.20) and (4.23) from the q-Selberg integral.
The limit q → ω r has an interesting connection to orbifolds (see, e.g., [20]), which plays a role in several possible physical applications for r-fold matrix models, as discussed in §5.
Refactorization of the Vandermonde
The results of §4.1 can be reduced to a combinatoric identity of the integrand, as follows. Define the projection operator
P a f (x) = 1 r r−1 j=0 ω −ja r f (ω j r x) ,(4.43)
and notate P a 1 ,...,a N f (x 1 , . . . , g(x r 1 , . . . , x r N ) for some polynomial g(y 1 , . . . , y N ). Since f is alternating within each sub-block I µ,i for fixed µ, so too is g, hence g contains Vandermonde factors, and (4.45) holds for some polynomial δ = δ(x r 1 , . . . , x r N ). The total degree of δ in the x variables is 47) where N µ denotes the number of variables x I for which a I ≡ µ (mod r). Maximizing (4.47) at fixed N , we find the conditions N µ − N ν ∈ {0, 1} for µ ν, thus the unique maximum satisfies N µ = n µ from (4.9). In this case, one can check that (4.47) vanishes, implying that δ is a constant, whereas for N µ = n µ the total degree of δ is negative, requiring δ = 0.
x N ) = P (x 1 ) a 1 . . . P (x N ) a N f (x 1 , . . . , x N ) .r−1 µ=0 µN µ + N µ (N µ − 1) 2 − N (N − 1) 2 ,(4.
To fix the constant δ in the case where N µ = n µ , note that we can fix a I ≡ I − 1 (mod r) up to a permutation of the labels. Comparing the coefficients of I x I−1 I on both sides, we conclude that δ 0,...,N −1 = 1, whereas σ is the identity permutation. Generalizing by permuting the labels gives δ a 1 ,...,a N = (−1) σ . Since moreover σ is a permutation on [0, N ] iff N µ = n µ , the result (4.45) is proven.
In fact, Theorem 6 encompasses our earlier results Theorems 3-5. For instance, the pure-phase partition function can be derived by rewriting 48) up to symmetrization in the variables x i . The integral over C r,a imposes the projection P a−1,...,a−1 on the integrand, hence P a−N,...,a−1 on the Vandermonde i<j (x i − x j ). By Theorem 6, this splits the Vandermonde into r non-interacting blocks, and we recover Theorem 3. Likewise, using the definition (3.22) we find
1 N ! det N ×N x N −j i 2 ∼ = i x N −i i i<j (x i − x j ) ,(4.1 N ! s λ (X) det N ×N x N −j i 2 ∼ = i x N +λ i −i i i<j (x i − x j ) ,(4.49)
up to symmetrization. As before, the projection implied by the contour splits the Vandermonde into r non-interacting blocks, and we recover Theorem 4.
Theorem 6 can also be used to answer several important questions that we will not discuss in detail in the present paper, but which deserve further attention in future work. Firstly, using this identity we can compute the unnormalized insertions Z 0, a (mod r)). More importantly, Theorem 6 can be used to derive a factorized structure for mixed phases, as follows.
After fixing the S N permutation symmetry as in (4.48) or (4.49), a mixed phase integrated on the contour r−1 a=0 C Na r,a ( a N a = N ) becomes an integral over the symmetrization of the contour a C Na r,a . This symmetrization can be broken up into ordered components N i=1 C r,a i , such that x i is integrated along the contour C r,a i . This is a finer basis of contours than that discussed in §3.2 because in a mixed phase each symmetrized contour has multiple ordered components N i=1 C r,a i . In particular, for p eigenvalue contours there are p N ordered contours but only N +p−1 p−1 symmetrized contours. The larger basis of contours does not lead to further ambiguities in the loop equations (recall §2.2) because the weights of ordered contours related by permutations are constrained to be equal. Nonetheless, the basis of ordered contours is useful because on each ordered contour the eigenvalue interactions (4.48) and (4.49) factor into the product of r subblocks using Theorem 6, much as in Theorems 3 and 4.
This means, for instance, that the partition function Z of the reflection-positive quartic matrix model discussed in §3.1 can be written explicitly as a sum of 2 N terms (since (C 4,1 + C 4,3 )/2 equals the real line), as can Z[s λ (X)]. While the number of terms grows rapidly with N , the growth is much slower than the N ! terms which appear in the determinant (3.5).
Moreover, this approach gives a straightforward solution to any given "symmetrized" mixed phase as a finite sum over "ordered" mixed phases, where the ordered mixed phases are given by r-fold products of the parent theory. Whether this leads to further insights into the mixed phases (and by extension, strongly-interacting reflection-positive models) is a question for the future.
Applications and Future Directions
In this section we discuss various generalizations and applications of our results. Several open questions still remain here and clarifying those questions is a promising direction of future research.
Two-Schur correlators
Perhaps the most puzzling and not fully understood phenomenon, observed in a wide range of matrix models, is a possibility to write an exact formula for a two-Schur correlator : an average of a product of two Schur polynomials. While existence of a formula for the average of a single Schur polynomial is not surprising and relates to, e.g., the determinant structure of Schur polynomials, an equally simple reason for the exact solvability of two-Schur correlators is-to the best of our knowledge-unknown.
In this section, we present such a formula for two-Schur averages in the general case of the r-fold q-deformed logarithmic model. We have tested this formula with numerous computer experiments. 23 Conjecture 1. For a pair of partitions λ, µ let Z λµ be the following product, with the linear factors given by
Z λµ = 1≤i<j≤N λ i − λ j + j − i; 0 r,0 j − i; 0 r,0 1≤i<j≤N µ i − µ j + j − i; 0 r,0 j − i; 0 r,0 1≤i,j≤N 2N + 1 − i − j; u + v r,a 2N + 1 + λ i + µ j − i − j; u + v r,a (i,j)∈λ N + j − i; u r,a N + j − i; u + v r,a (i,j)∈µ N + j − i; v r,0 N + j − i; 0 r,0 ,(5.x; y r,a = Λ q (x+ry)/2 − q −(x+ry)/2 q 1/2 − q −1/2 , (x + a) mod r = 0 , ω x/2 r − ω −x/2 r ω 1/2 r − ω −1/2 r , (x + a) mod r = 0 . (5.2)
Then,
χ λ X χ µ Tr X k → Tr X k + rv δ r|k = Z λµ , if deg Λ Z λµ = 0 , 0, otherwise. (5.3)
Here the notation s λ (Tr X k → Tr X k + rvδ r|k ) means that we first write s λ as a linear combination of multitrace operators and then replace Tr X k → Tr X k + rvδ r|k . Note that this formula does not fully survive the polynomial limit: when x → x/v and v → ∞, the v-shift in the second Schur polynomial implies that the µ-dependence goes away, and the formula reduces to a single Schur correlator. Thus, there is no analogous two-Schur average formula in the monomial matrix models considered in §3.
The r = 1 case of Conjecture 1 was given in [46,47,51] and used to formulate a proof of the AGT conjecture. For v = 0 these conjectures reduce to a theorem proven by Kadell [45]. The space of conjectured solvable two-Schur correlators is diagrammed in Figure 5. Here we include the possibility of an elliptic generalization, see, e.g., [44].
As illustrated in Figure 5, the r-fold q-deformed model discussed in §4.3 generalizes all of the known solvable two-Schur correlators, with the possible exception of an elliptic version of the formula. For r = 1 one recovers the correlators associated to the q-Selberg model from [51], of which the other existing conjectures are special cases. An additional relationship in the diagram arises from the q → ω r limit of the r = 1 q-deformed formula, which gives the r-fold formula without q-deformation, see §4.3. If an elliptic generalization exists for r = 1, we speculate that the r-fold q-deformed formula may be some limit of it.
In the next subsection we outline a possible physical interpretation for the existence of the factorization formula, Conjecture 1.
Five-dimensional partition functions
The AGT conjecture [18,19], by now proven by several different approaches [46,47,[52][53][54][55][56][57][58][59][60][61][62], states a correspondence between conformal blocks of q-deformed Toda CFTs with Walgebra symmetry and partition functions of N = 1 5d gauge theories. Since conformal blocks of (q-deformed) Toda CFTs can be represented in matrix model form, this is really a relation between three quantities, i.e., a "threesome" [63] or "triality" [59]. In this paper we only need the simplest example of an AGT correspondence: equality between a 3-point conformal block of W q,t (sl 2 )-i.e., the q-Virasoro algebra-and the partition function of a 5d T 2 gauge theory. In particular, in the free-field formalism the 3-point conformal block is exactly represented by the q-Selberg integral. This gives a simple 5d argument for why the q-Selberg integral is solvable (i.e., is fully factorized as a product of Gamma-functions with linear arguments), because the partition function of a 5d T 2 gauge theory is fully perturbative, and does not have a non-trivial instanton part.
An important special case that has attracted significant attention is when the q-deformation parameter is an rth root of unity. In this case, as shown in [20,64], the 5d partition function reduces to a 4d partition function on an ALE space, the C 2 /Z r orbifold. At the same time the matrix model (q-Selberg integral) reduces to precisely the integrals that we study in the current paper, see §4. 3. 24 This provides a 5d physical perspective for why the non-q-deformed model with r > 1 is solvable.
At the same time this argument also hints at a physical explanation of why the qdeformed model with r > 1 is solvable. Namely, instead of taking the root of unity limit, one can consider directly the 5-dimensional T 2 theory on an orbifold C 2 /Z r . The resulting theory would depend on both q and r and -because of the relation to orbifold T 2 theory - 24 It should be noted that a different matrix model was derived for the same partition function earlier in [20]. This other matrix model was obtained using the methods of [65] and is not the usual AGT description. The relation between AGT-type matrix models and Sulkowski-type matrix models is not a simple change of variables; rather, it should involve a spectral duality transformation [66].
should be given by a product of Gamma-functions with linear arguments. This could also shed light on why the two Schur average is factorized for these models.
To connect this answer to our integrals, however, there has to be an orbifold version of the 5d AGT correspondence. Orbifold generalizations of AGT have been considered in [21][22][23][24][25], but we are unaware of an exact relationship to the matrix models considered in our paper. Thus, the physical interpretation of the topmost nodes in Figure 5 remains an open problem.
This argument also allows us to make a connection to mathematics in the enumerative geometry of symplectic resolutions. The T 2 theory in five dimensions is known to be equivalent to a U (1) theory with two fundamental hypermultiplets. The partition function of this theory is mathematically a K-theoretic integral (push-forward) of a sheaf on a Hilbert scheme of points on C 2 that corresponds to fundamental matter. The results of the current paper correspond to considering instead a Hilbert scheme of points on C 2 /Z r .
Superconformal indices and surface defects
Another way to explain the solvability of the q-Selberg integral is to note that it is a special case of the elliptic Selberg integral [6,7], which is itself equal to the superconformal index of a 4d N = 1 gauge theory with gauge group Sp(2N ), one chiral multiplet in the antisymmetric tensor representation, and six fundamental chiral multiplets. This gauge theory is s-confining [10][11][12][13], hence it admits an infrared description in terms of chiral multiplets interacting via an irrelevant superpotential. The absence of a gauge group in this description implies that the index factors into a product of elliptic gamma functions, which in turn implies the factorization of the q-Selberg integral.
Recall that the superconformal index is the partition function of the theory on S 3 × S 1 . Replacing S 3 by a Lens space L(r, p) leads to a refined index which has received some attention in the literature [14][15][16]. Since, of course, L(r, n) is a Z r orbifold of S 3 , this suggests that perhaps the r-fold q-Selberg integral considered in §4.3 is a special case of the superconformal index of the same Sp(2N ) gauge theory on L(r, n) × S 1 (for some n, e.g., n = 1). More generally, other r-fold matrix models of the kind considered in §4 may relate to the index of the same CFT on a Lens space as generates the parent matrix model on S 3 .
An intriguing feature of this potential connection to indices is that it could explain the physics behind the factorization of Schur averages. It is natural to conjecture that the insertion of a Schur function (or the appropriate elliptic analog) into a superconformal index is related to the insertion of a surface defect in the partition function of the gauge theory, see for instance [67]. If so, the factorization of Schur averages in the r-fold matrix models considered in this paper could relate to the physics of surface defects on Lens spaces.
Nonetheless, at the time of writing we have been unable to make the connection between r-fold matrix models and Lens space indices explicit, and it remains an interesting open problem for future research. of λ. As we have seen, these play an important role in evaluating determinants of the form (A.1). More generally, the determinant of an arbitrary matrix M IJ satisfying M IJ = 0 for (λ I + J − I) mod p = 0 can be found using the formula for p functions f µ (x, y), where (A.9) is a special case.
A.2 The abacus diagram
The p-core, p-quotient, and p-signature have a graphical interpretation, as follows. Consider a plot consisting of the points
(i − λ i − 1) mod p, λ i − i p i ∈ N , (A.11)
in Z p × Z, which is the same as the plot µ, (λ mod p) µ + (λ/p (µ) ) i − i i ∈ N, 0 µ < p . (A.12) This is known as the "abacus diagram", and can be constructed from the Young diagram associated to λ as follows.
We assign a binary digit 1 (0) to each vertical (horizontal) line segment along the lowerright boundary of the Young diagram. Reading off the resulting sequence from upperright to lower-left and padding the beginning (end) with an infinite number of 0s (1s), we obtain an infinite binary sequence which encodes the Young diagram, see Figure 6(a). This sequence has a natural centerline where it crosses the diagonal of the Young diagram (the point where the number of 1s preceding is equal to the number of 0s following). For each "inversion"-a pair of digits 1, 0, where the 1 precedes the 0-there is a corresponding box in the Young diagram with hook length equal to the distance between the two digits. Swapping to the two digits corresponds to removing the associated rim hook from the Young diagram, see Figure 6(b).
To construct the abacus diagram, we print out the binary sequence left-to-right, topto-bottom, in p columns, arranged so that the centerline falls on a line break. Each 1 in the sequence corresponds to the end of a row in the Young diagram, whereas in the abacus diagram the 1s coincide with the points of the plot (A.11), henceforward referred to as "beads". Conversely, using (A.12) we see that the binary sequence in the column µ, 0 µ < p, corresponds to the p-quotient λ/p (µ) , where the centerline (determined by equating the 1s preceding and the 0s following, as before) is a height (λ mod p) µ above the centerline of the abacus diagram.
The remainder (λ mod p) µ is unaffected by sliding beads vertically in the abacus diagram (adding or removing a rim-hook of length p), hence the p-core is determined uniquely by sliding all the beads downwards as far as they can go. This reproduces the standard definition of the p-core: the Young diagram which results from removing length p rimhooks until none remain. The relation between the Young diagram, the abacus diagram, and the p-quotient and p-core is summarized in Figure 7.
The p-signature can also be given a diagrammatic interpretation, as follows. Reading top to bottom in the kth column, label the jth bead with k + (j − 1)p. Reading left to right, top to bottom, σ(i) is the label associated to the ith bead, and δ p (λ) is the signature of σ. Using this construction it is straightforward to show that the effect of adding a rim-hook spanning the rows i, i + 1, . . . , j is to map σ → σ • (j, j − 1, . . . , i) , (A.13)
in cycle notation, so that the group algebra is related to sliding beads in the abacus diagram. This observation provides another method (easier to apply by hand) for computing the psignature. Removing length p rim-hooks successively until no boxes remain, the p-signature of λ is equal to (−1) k when there are k rim-hooks spanning an even number of rows.
A.3 Basic theorems
We now catalog a few basic properties of p-cores, p-quotients, and p-signatures. For the sake of brevity, we omit most proofs, leaving them as an exercise for the interested reader.
To state these properties concisely, we define the functions H p,i (λ) = |{x ∈ λ|h λ (x) ≡ i (mod p)}| , C p,i (λ) = |{x ∈ λ|c λ (x) ≡ i (mod p)}| , (A.14)
which count the number of boxes in λ with hook-length and contents congruent to i (mod p), respectively.
We have Theorem 7. For any partition λ and p ∈ N:
Figure 1 :
1Evaluating a Schur polynomial average using (1.3) for the pure cubic potential S[X] = Tr(X 3 ) with a = 1 and N ≡ 0 (mod 3). In this example, we find s 3
7
Recall that an operator equation A = B holds if and only if AO = BO for any operator O.
Figure 2 :
2A basis of possible integration contours for a cubic potential. in terms of its N eigenvalues x i , i = 1, . . . , N . To do so, we apply the Faddeev-Popov gauge-fixing procedure to the U (N ) gauge symmetry of the Hermitian matrix model. The result is the eigenvalue model:
I p ≡ Z[(Tr X) p ] for 0 p N will depend on linearly independent combinations of the c i,N −i , 8 so that the data not fixed by the loop equations exactly correspond to 8 Assume the opposite. This implies the existence of a linear combination of contours such that Z[(Tr X) p ] = 0 for 0 p N , hence by the loop equations Z[O] = 0 for any polynomial operator.
cycle defined by the U (N ) orbit of the eigenvalue contours. This cycle can be defined in a U (N )-invariant way by equations of the form [X, X * ] = 0 , f i (Tr X, . . . Tr X N ) = 0 , (2.19)
coefficients a i;N (O), since the eigenvalue integral can be evaluated by expanding the Vandermonde determinant times O into a sum of monomials and applying (3.4).
Figure 3 :
3A natural basis of integration contours for a cubic (r = 3) potential,
27) where N indicates the opposite value of N mod r. The first equation relates the two possible values of N mod r for fixed N , whereas the second equation relates N → −N for fixed N mod r, similar to negative rank duality (see, e.g.,[30, Ch. 13]).Since any symmetric polynomial in x i , i = 1, . . . , N , can be expressed in terms of the Schur polynomials, in principle (3.20) and (3.23) provide a complete solution to the matrix model (3.19) for any N . 12 For instance, single-trace correlation functions are calculated explicitly in §3.5, see (3.47), (3.49), (3.50) and (3.52).
n is nonzero on a subset of the nonzero entries of Z 1 [x m x n ], i.e., those satisfying m + n ≡ a − 1 (mod r). The solution is p rk+i
function is therefore Z N = δ r,a (N ) N −1 i=0 t i , where δ r,a (N ) = det N ×N δ (r,a) m,n is the determinant of upper-left N × N block of δ (r,a) m,n . This matches (3.20). The Schur polynomial average (3.23) can also be derived using orthogonal polynomials, as we now show. 13 Our starting point is the formula [32, p. 67] k j=1
14
Note that the generating function ∞ q=0
0 is an N -independent constant 17 which depends on the normalization of the partition function and the the upper (lower) sign corresponds to N = kr (N = kr + a). Ignoring the logs, the contour dependence again enters at O(1/N 2 ) relative to the leading terms, and there are O(1/N 3 ) corrections to the free energy.
,
similar to(3.27).
20Figure 4 :
4Here we use the substitution I = ri + (N − µ − 1) mod r for i 0 and 0 µ < r. Simpler substitutions are possible, such as I = ri + µ, but this particular form occurs naturally in the Schur average formula. One example of applying Theorem 4, c.f.
Theorem 4 .
4The average of Schur polynomials in an r-fold matrix model is the product of r averages in the parent model:
Theorem 5 .
5The partition function and Schur averages of a q-deformed r-fold matrix
38) 22 +
22One can check that n r (N, u) = r(u + 1) N/r (N mod r)(u + N/r − 1), hence it does not depend separately on a.
Theorem 6 .
6The projection of the Vandermonde factorsP a 1 ,...,a N I>J (x I − x J ) = δ a 1 ,...,i − x r I µ,j ) ,(4.45)where I µ,i denotes the ith value of I for which a I ≡ µ (mod r) and δ a 1 ,...,a N ∈ {0, ±1} is defined as follows: consider the function σ : [1, N ] → N σ(I µ,i ) = 1 + r(i − 1) + µ .(4.46) If σ is a permutation on [1, N ], then δ a 1 ,...,a N = (−1) σ . Otherwise δ a 1 ,...,a N = 0. Proof. Call the polynomial on the left hand side f (x 1 , . . . , x N ). The homogeneity conditions f (. . . , ω r x I , . . .) = ω a I r f (. . . , x I , . . .) imply that f (x 1 , . . . , x N ) = I x a I mod r I
Figure 5 :
5The landscape of two-Schur correlators
n µ + (λ/p (µ) ) i − i, n µ − j , (A.10)
Figure 6 :
6(a) The lower-right boundary of a Young diagram determines a binary sequence, where we indicate the centerline of the sequence (the point for which the number of 1's to the left and 0's to the right are equal) with a bar. (b) For each inversion in the sequence (1 before a 0), there is a corresponding box in the Young diagram (red) and an associated rim hook (grey). Removing the inversion (exchanging the circled digits) corresponds to removing the rim hook.
Figure 7 :
7Translating between (a) a Young diagram, (b) the corresponding abacus diagram, and (c) the p-quotient and p-core, in the case p = 4 for the Young diagram inFigure 6(a).
). As usual, insertions and expectation values are defined as in (2.1-2.3). Theorem 1. The partition function of the monomial matrix model (3.19) is
1 )
123 The necessary MAPLE code may be found at http://math.harvard.edu/~shakirov/.
Specifically, those with gauge group Sp(2N ), one chiral multiplet in the antisymmetric tensor representation, six in the fundamental representation, and no superpotential. These theories confine without chiral symmetry breaking[8,9].6 There is also a possible connection between our results and the (r + 1)-point function of ordinary Louiville theory for a special arrangements of the points.
Technically, we could restrict the operator algebra to polynomial operators, eliminating this problematic operator, but for any fixed value of λ = 0, polynomial operators of finite degree will nonetheless violate reflection-positivity.
2 p p! g −2p .(2.28)11 Resolving these non-perturbative ambiguities in unitary theories is an active research topic, see, e.g.,[29].
2.958675 is trancendental, there are no obvious further simplifications. The same complexity is evident
Our formulas for normalized correlators apply when Z = 0. Unnormalized insertions can still be finite when Z = 0, but besides a brief discussion in §4.4, we leave a thorough treatment of these cases to the future.
We follow a similar approach to[31].
t q can be written in terms of 2 F 1 . However, the resulting expression is no easier to work with than (3.47),(3.49), and this curiosity will play no further role in our discussion.
Note that C 1/r will have additional discontinuities relative to C if C crosses the negative real axis, since the principal rth root is discontinuous there. The pieces of the contour are reconnected in the linear combination C r,a , but the weight of each piece remains discontinuous except in the special case a = 0.19 The presence of the u deformation is linked to the inclusion of the contour a = 0. The additional logarithmic term in the potential adds a saddle point, hence (in the basis of Lefschetz thimbles) an additional integration contour with a boundary at the origin.
One might hope to generalize the β-deformation (x i − x j ) 2 → |x i − x j | 2β to the exactly solvable r > 1 models we study in this paper. However, this is challenging due to the non-analytic integrand that results for non-integral β, and even for integral β we do not find any exact results besides the r = 2, a = 1 case with u = 0, which is merely a special case of the Selberg integral.
We later show that χ is unique, which is not obvious at present.
i,j kµ f µ n µ + i − 1, n µ + j − (λ/p (µ) ) j − 1 .(A.17) for K λ 1 , where k µ = K+µ p and N + K ≡ 0 (mod p).
AcknowledgmentsWe thank G. DunneA Divisibility and Quotients of PartitionsIn this appendix we review the concept of dividing a partition by a natural number as it relates to our work. (see, e.g.,[32, pp. 12-14]for a textbook treatment).A.1 Determinants and p-quotientsTo motivate the concepts of divisibility and quotients for partitions, consider the determinantfor natural numbers p, N ∈ N, a partition λ 1 . . . λ N 0 and any function f (n). Writing the determinant as a sum det M IJ = σ (−1) σ I M Iσ(I) , the permutations which contribute satisfyThis specifies σ up to the ordering of pairs I, I for which I − λ I ≡ I − λ I (mod p). This ordering can be specified by p permutations σ µ , 0 µ < p, constructed so thatwhere I µ,i is the ith value of I for which I − λ I − 1 ≡ µ (mod p). We have explicitlyRecall that a permutation σ S on a set S is finitary (infinitary) if it acts non-trivially on a finite (infinite) subset of S. As an infinitary permutation cannot be written as a finite product of transpositions, it is convenient to define its signature as (−1) σ ≡ 0. With this definition, it is straightforward to check thatSince the permutations which appear in the determinant are finitary, (A.1) vanishes unless δ p (λ) = 0. For reasons which will soon become clear, we call a partition λ with δ p (λ) = 0 "p-divisible." In general, we havewhere σ is finitary iff i µ = 0 for all 0 µ < p. The p integers i µ , constrained by µ i µ = 0, can be thought of as a remainder. Conventionally, this is expressed in terms of the "p-core", λ mod p, which is the smallest partition χ such that iµ .25We use the notation (λ mod p) µ ≡ idefines a partition λ/p (µ) , since (λ/p (µ) ) i+1 (λ/p (µ) ) i and (λ/p (µ) ) i = 0 for I µ,i > N . We findsince i µ = 0 when δ p (λ) = 0.The p partitions λ/p (µ) , 0 µ < p, defined in (A.8) are known as the "p-quotient" Theorem 8. If λ is p-divisible then its p-signature is given by the product over boxesAs above, this formula can be proven by constructing λ from its (empty) p-core and keeping track of both sides, e.g., using (A.13) to track the changes in p-signature.Theorem 9. The p-core, p-quotient, and p-signature transform as follows under a transposition of the Young diagram3. δ p (λ ) = (−1) |λ|+ |λ| p δ p (λ) .The first two parts are obvious from the abacus diagram, whereas the third follows from Theorem 8 as well as16)for λ p-divisible, which is a direct consequence of property 4.iv of Theorem 7.Using Theorem 9, we write down a useful variant of (A.10):
The following are equivalent i. λ is p-divisible, ii. H p,0 (λ) = |λ|/p, iii. 0iv. ∀i ∈ Z, C p,i (λ) = |λ|/pThe following are equivalent i. λ is p-divisible, ii. H p,0 (λ) = |λ|/p, iii. H p,0 (λ) = C p,0 (λ), iv. ∀i ∈ Z, C p,i (λ) = |λ|/p.
If λ is p-divisible, then ∀i ∈ Z, H p,i (λ) + H p,p−i (λ) = 2|λ|. If λ is p-divisible, then ∀i ∈ Z, H p,i (λ) + H p,p−i (λ) = 2|λ|/p.
One way to prove these results is to construct λ from its p-core by sliding beads in the abacus diagram, keeping track of both sides of the relevant (in)equality. One way to prove these results is to construct λ from its p-core by sliding beads in the abacus diagram, keeping track of both sides of the relevant (in)equality.
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"The Effects of Thermonuclear Reaction-Rate Variations on 26 Al Production in Massive Stars: a Sensitivity Study",
"The Effects of Thermonuclear Reaction-Rate Variations on 26 Al Production in Massive Stars: a Sensitivity Study"
]
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"Christian Iliadis [email protected] \nDepartment of Physics and Astronomy\nUniversity of North Carolina at Chapel Hill\nChapel Hill27599-3255NCUSA\n\nTriangle Universities Nuclear Laboratory\n27708-0308DurhamNCUSA\n",
"Art Champagne \nDepartment of Physics and Astronomy\nUniversity of North Carolina at Chapel Hill\nChapel Hill27599-3255NCUSA\n\nTriangle Universities Nuclear Laboratory\n27708-0308DurhamNCUSA\n",
"Alessandro Chieffi [email protected]. \nIstituto Nazionale di Astrofisica -Istituto di Astrofisica Spaziale e Fisica Cosmica\nVia Fosso del Cavaliere, I-00133RomaItaly\n",
"Marco Limongi \nIstituto Nazionale di Astrofisica -Osservatorio Astro-nomico di Roma\nVia Frascati 33, I-00040, Monteporzio CatoneItaly\n"
]
| [
"Department of Physics and Astronomy\nUniversity of North Carolina at Chapel Hill\nChapel Hill27599-3255NCUSA",
"Triangle Universities Nuclear Laboratory\n27708-0308DurhamNCUSA",
"Department of Physics and Astronomy\nUniversity of North Carolina at Chapel Hill\nChapel Hill27599-3255NCUSA",
"Triangle Universities Nuclear Laboratory\n27708-0308DurhamNCUSA",
"Istituto Nazionale di Astrofisica -Istituto di Astrofisica Spaziale e Fisica Cosmica\nVia Fosso del Cavaliere, I-00133RomaItaly",
"Istituto Nazionale di Astrofisica -Osservatorio Astro-nomico di Roma\nVia Frascati 33, I-00040, Monteporzio CatoneItaly"
]
| []
| We investigate the effects of thermonuclear reaction rate variations on 26 Al production in massive stars. The dominant production sites in such events were recently investigated by using stellar model calculations: explosive neon-carbon burning, convective shell carbon burning, and convective core hydrogen burning. Post-processing nucleosynthesis calculations are performed for each of these sites by adopting temperature-density-time profiles from recent stellar evolution models. For each profile, we individually multiplied the rates of all relevant reactions by factors of 10, 2, 0.5 and 0.1, and analyzed the resulting abundance changes of 26 Al. In total, we performed ≈ 900 nuclear reaction network calculations. Our simulations are based on a next-generation nuclear physics library, called STARLIB, which contains a recent evaluation of Monte Carlo reaction rates. Particular attention is paid to quantifying the rate uncertainties of those reactions that most sensitively influence 26 Al production. For stellar modelers our results indicate to what degree predictions of 26 Al nucleosynthesis depend on currently uncertain nuclear physics input, while for nuclear experimentalists our results represent a guide for future measurements. We also investigate equilibration effects of 26 Al. In all previous massive star investigations, either a single species or two species of 26 Al were taken into account, depending on whether thermal equilibrium was achieved or not. These are two extreme assumptions and in a hot stellar plasma the ground and isomeric state may communicate via γ-ray transitions involving higher-lying 26 Al levels.We tabulate the results of our reaction rate sensitivity study for each of the three distinct massive star sites referred to above. It is found that several current reaction rate uncertainties influence the production of 26 Al. Particularly important reactions are 26 Al(n,p) 26 Mg, 25 Mg(α,n) 28 Si, 24 Mg(n,γ) 25 Mg and 23 Na(α,p) 26 Mg. These reactions should be prime targets for future measurements. Overall, we estimate that the nuclear physics uncertainty of the 26 Al yield predicted by the massive star models explored here amounts to about a factor of 3. We also find that taking the equilibration of 26 Al levels explicitly into account in any of the massive star sites investigated here has only minor effects on the predicted 26 Al yields. Furthermore, we provide for the interested reader detailed comments regarding the current status of certain reactions, including 12 C( 12 C,n) 23 Mg, 23 Na(α,p) 26 Mg, 25 Mg(α,n) 28 Si, 26 Al m (p,γ) 27 Si, 26 Al(n,p) 26 Mg and 26 Al(n,α) 23 Na. | 10.1088/0067-0049/193/1/16 | [
"https://arxiv.org/pdf/1101.5553v1.pdf"
]
| 118,507,241 | 1101.5553 | e391336014aaaac3059f3e81694290736f9481d8 |
The Effects of Thermonuclear Reaction-Rate Variations on 26 Al Production in Massive Stars: a Sensitivity Study
28 Jan 2011
Christian Iliadis [email protected]
Department of Physics and Astronomy
University of North Carolina at Chapel Hill
Chapel Hill27599-3255NCUSA
Triangle Universities Nuclear Laboratory
27708-0308DurhamNCUSA
Art Champagne
Department of Physics and Astronomy
University of North Carolina at Chapel Hill
Chapel Hill27599-3255NCUSA
Triangle Universities Nuclear Laboratory
27708-0308DurhamNCUSA
Alessandro Chieffi [email protected].
Istituto Nazionale di Astrofisica -Istituto di Astrofisica Spaziale e Fisica Cosmica
Via Fosso del Cavaliere, I-00133RomaItaly
Marco Limongi
Istituto Nazionale di Astrofisica -Osservatorio Astro-nomico di Roma
Via Frascati 33, I-00040, Monteporzio CatoneItaly
The Effects of Thermonuclear Reaction-Rate Variations on 26 Al Production in Massive Stars: a Sensitivity Study
28 Jan 20111Subject headings: gamma rays: theory -nuclear reactionsnucleosynthesisabundances -stars: evo- lution -supernovae: general
We investigate the effects of thermonuclear reaction rate variations on 26 Al production in massive stars. The dominant production sites in such events were recently investigated by using stellar model calculations: explosive neon-carbon burning, convective shell carbon burning, and convective core hydrogen burning. Post-processing nucleosynthesis calculations are performed for each of these sites by adopting temperature-density-time profiles from recent stellar evolution models. For each profile, we individually multiplied the rates of all relevant reactions by factors of 10, 2, 0.5 and 0.1, and analyzed the resulting abundance changes of 26 Al. In total, we performed ≈ 900 nuclear reaction network calculations. Our simulations are based on a next-generation nuclear physics library, called STARLIB, which contains a recent evaluation of Monte Carlo reaction rates. Particular attention is paid to quantifying the rate uncertainties of those reactions that most sensitively influence 26 Al production. For stellar modelers our results indicate to what degree predictions of 26 Al nucleosynthesis depend on currently uncertain nuclear physics input, while for nuclear experimentalists our results represent a guide for future measurements. We also investigate equilibration effects of 26 Al. In all previous massive star investigations, either a single species or two species of 26 Al were taken into account, depending on whether thermal equilibrium was achieved or not. These are two extreme assumptions and in a hot stellar plasma the ground and isomeric state may communicate via γ-ray transitions involving higher-lying 26 Al levels.We tabulate the results of our reaction rate sensitivity study for each of the three distinct massive star sites referred to above. It is found that several current reaction rate uncertainties influence the production of 26 Al. Particularly important reactions are 26 Al(n,p) 26 Mg, 25 Mg(α,n) 28 Si, 24 Mg(n,γ) 25 Mg and 23 Na(α,p) 26 Mg. These reactions should be prime targets for future measurements. Overall, we estimate that the nuclear physics uncertainty of the 26 Al yield predicted by the massive star models explored here amounts to about a factor of 3. We also find that taking the equilibration of 26 Al levels explicitly into account in any of the massive star sites investigated here has only minor effects on the predicted 26 Al yields. Furthermore, we provide for the interested reader detailed comments regarding the current status of certain reactions, including 12 C( 12 C,n) 23 Mg, 23 Na(α,p) 26 Mg, 25 Mg(α,n) 28 Si, 26 Al m (p,γ) 27 Si, 26 Al(n,p) 26 Mg and 26 Al(n,α) 23 Na.
Introduction
The radioisotope 26 Al is of outstanding importance for γ-ray astronomy and cosmochemistry. It has been discovered in three distinct sites: (i) in the Galactic interstellar medium via detection of its decay emission line at 1809 keV (Mahoney et al. 1982, Diehl et al. 1995; (ii) in meteorites via observed excesses of its radioactive decay (daughter) product 26 Mg (MacPherson, Davies & Zinner 1995), implying an injection of live 26 Al into the early Solar System nebula; and (iii) in presolar dust grains, again via detected 26 Mg excesses (Hoppe et al. 1994;Huss, Hutcheon & Wasserburg 1997), that are uncontaminated by solar system material and thus are of likely stellar origin. Identification of the main sources of 26 Al would have far-reaching implications, ranging from questions related to the circumstances and conditions of the Solar System birth to imposing strong constraints on the chemical evolution of the Galaxy. A number of different sources have been suggested over the years: AGB stars, classical novae, Wolf-Rayet stars, and core collapse supernovae. For reviews, see Prantzos & Diehl (1996) or Diehl & Timmes (1998). However, the origin of 26 Al remains controversial.
The observation of Galactic γ-rays from 26 Al is important since it provides unambiguous direct evidence for the theory of nucleosynthesis in stars. The half-life of 26 Al amounts to 7.17 × 10 5 y and is small compared to the time scale of Galactic chemical evolution (≈10 10 y). Consequently, nucleosynthesis is currently occurring in the interstellar medium and, in particular, 26 Al is synthesized throughout the Galaxy. From the observed γ-ray intensity, depending on the assumption for the density distribution, a present-day 26 Al equilibrium mass of ≈ 2 − 3 M ⊙ in the entire Galaxy has been inferred (Diehl et al. 2006). The observational evidence favors in this case massive stars as a source: first, the all-sky map of the 1809 keV γ-ray line detected by the COMPTEL instrument onboard CGRO showed that 26 Al is confined along the Galactic disk and that the measured intensity is clumpy and asymmetric (Plüschke et al. 2001); second, the comparison between the 26 Al all-sky map from COMPTEL to other all-sky maps for different wavelengths (Knödlseder 1999) revealed that the 1.8 MeV γ-ray emission is cor-related with the Galactic free-free emission, which traces the distribution of ionized gas in the interstellar medium, observed in the microwave domain by the COBE satellite; third, the measurement of the 1809 keV line Doppler shift by the SPI spectrometer onboard INTEGRAL demonstrated that 26 Al co-rotates with the Galaxy and hence supports a Galaxy-wide origin (Diehl et al. 2006). Here we investigate the bulk production of 26 Al in massive stars. A study of 26 Al/ 27 Al ratios observed in meteorites and presolar grains will be subject of a separate study.
Massive stars may produce 26 Al during several different phases of their evolution: (i) during pre-supernova stages in the C/Ne convective shell, where a fraction of the 26 Al survives the subsequent explosion and is ejected into the interstellar medium (Arnett & Wefel 1978); (ii) during core collapse via explosive Ne/C burning (Arnett 1977), where the ejected 26 Al yield may perhaps be modified by the ν-process via neutrino spallation (Woosley et al. 1990); and (iii) in Wolf-Rayet stars, i.e., stars with masses in excess of about 30 M ⊙ , which experience such a strong mass loss that even layers located within the H convective core, hence significantly enriched in 26 Al, are ejected into the interstellar medium (Palacios et al. 2005). These 26 Al production mechanisms (with the exception of the ν-process) were recently analyzed in detail by Limongi & Chieffi (2006) by using extensive stellar model calculations of solar metallicity stars in the mass range of 11M ⊙ ≤ M ≤ 120M ⊙ . In that work they also emphasized the impact of rate uncertainties for selected reactions on the final 26 Al yields. Another discussion along these lines can be found in Woosley & Heger (2007).
In the present work we expand this effort by presenting a comprehensive investigation of the impact of nuclear reaction rate uncertainties on the synthesis of 26 Al. Our method is similar in spirit to earlier nuclear reaction rate sensitivity studies that addressed the nucleosynthesis in classical novae and type I x-ray bursts (Parikh et al. 2008). The general strategy consists of varying the rates of many reactions by different factors (in this work, 10, 2, 0.5 and 0.1) and to analyze the impact of each individual reaction rate change on the final 26 Al yields. Once the yield changes are established for this grid of rate variation factors, more realistic abundance changes based on actual rate uncertainties, if available, are considered. At present it is not feasible to perform this computationally intensive procedure with a self-consistent stellar model. Instead, we extract representative temperaturedensity-time profiles from recent stellar evolution models of massive stars and execute a large number of post-processing reaction network sensitivity calculations using these profiles. Our goal is twofold. On the one hand, we would like to quantify to what degree predictions of 26 Al yields depend on currently uncertain nuclear physics input. On the other hand, by identifying the "most important nuclear reactions", our results represent a guide for future measurements.
There are a number of novel aspects about the present work. First, we employ a new-generation library of nuclear reaction and weak interaction rates, called STARLIB. This library is partially based on a recent evaluation of experimental Monte Carlo reaction rates . Besides recommended reaction rates for a grid of temperature values between 1 MK and 10 GK, the library includes in addition for many reactions the rate uncertainty factor at each temperature. In fact, this work represents the first application of STARLIB. Second, we carefully investigate the equilibration effects of 26 Al. At least two species of 26 Al take part in the nucleosynthesis, the ground state and the isomeric state. In all previous investigations, either a single species or two species of 26 Al were taken into account, depending on whether thermal equilibrium is achieved or not. Obviously, these are two extreme assumptions and in a hot stellar plasma the ground and isomeric state may "communicate" via γ-ray transitions involving higher-lying 26 Al levels.
Our study has some obvious limitations. First, since we perform post-processing calculations, we necessarily focus our investigation on the effects of nuclear reaction rates. In other words, the important effects of convection 1 , mass loss, rotation, 1 It can be shown analytically that the abundance evolution in a convective region in which the turnover time is fast enough to ensure a flat abundance profile of the various nuclear species is equivalent to the evolution of a single mesh in which each local thermonuclear rate, σv ij , is replaced by its mass-weighted average over the convective region, i.e., k σv ij,k dm k /mtot. Hence it is perfectly and so on, are outside the scope of the present work. This also implies that our simulations are unsuitable for defining absolute 26 Al yields. Instead, we claim that our procedure is useful for exploring the effects of 26 Al abundance changes that result from reaction rate variations. Second, we only explore a few temperature-density-time evolutions that are representative of solar metallicity stars. A more comprehensive study covering a broad range of stellar masses and metallicities is also beyond the scope of this work. Third, it is well-known that the radioisotope 60 Fe (half-life of 2.62 × 10 6 y) is likely co-produced with 26 Al in massive stars and that their abundance ratio provides a sensitive constraint on stellar models (see Limongi & Chieffi 2006;Woosley & Heger 2007; and references therein). Indeed, Galactic γ-rays from the decay of 60 Fe have been detected by both RHESSI and the SPI spectrometer onboard INTE-GRAL, and the observed γ-ray line flux ratio for 60 Fe and 26 Al amounts to ≈ 0.1 − 0.2 (Harris et al. 2005). In the present work we only focus on the nucleosynthesis of 26 Al and leave a similar sensitivity study for 60 Fe to future work 2 . Finally, by individually varying each rate and leaving all other rates at their nominal values, we disregard any correlations among different reactions. In our opinion, no single study will cover all of the possible uncertainties, but each approach has advantages and disadvantages. We present only one realization of a sensitivity study, similar to the proceplausible to use a single point evolution for a convective region if the turnover time is faster that the nuclear burning time (e.g., for core H burning). Furthermore, we performed the calculation using un-weighted rates instead of massweighted average rates because we are mostly interested in 26 Al abundance changes. 2 In the recent work of Tur, Heger & Austin (2010), the impact of triple-α and 12 C(α,γ) 16 O rate variations on the 26 Al,44 Ti and 60 Fe yields were investigated using stellar evolution and explosion models. The authors claimed that "...over a range of twice the experimental uncertainty, σ, for each helium-burning rate, the production of 26 Al,60 Fe, and their ratio vary by factors of 5 or more...". By looking in detail at their results, it is clear that the effects on the 60 Fe yield are indeed large. However, it becomes also apparent that the effects on the 26 Al yield are much smaller. For example, their 25M ⊙ model and adopting the initial abundances from Lodders (2003) provides a factor of 1.5 change in 26 Al yield if the triple-α and 12 C(α,γ) 16 O rates are individually varied by their 1σ experimental uncertainties (see their Tab. 3). As will be seen, the rate uncertainty effects explored in the present work result in significantly larger 26 Al yield variations. dure applied in Iliadis et al. (2002) and Parikh et al. (2008). Interestingly, the latter work explored two methods: the one applied here and also a Monte Carlo procedure. It was found by Parikh et al. (2008) that very similar results were obtained with "...minor differences attributed to such correlation effects...". We feel that a Monte Carlo procedure makes most sense if it is performed with reliable experimentally based reaction rate probability densities. However, as will become apparent below, we do not have this information for all of the important reaction rates yet and thus leave such a study to future work when an update of STARLIB becomes available. This paper is organized as follows. Our procedure is explained in more detail in § 2, including a discussion of stellar models, the equilibration of 26 Al, and a description of the library STARLIB. The results of reaction rate sensitivity studies are presented in § 3 for the three predicted main sites of 26 Al synthesis: explosive Ne/C burning, convective C/Ne shell burning, and convective H core burning. A summary and conclusions are given in § 4. More information on reaction and decay rates, together with a discussion of individual reactions, is provided in the Appendix.
General Procedure
Massive star models
The stellar models adopted in the present work are those presented in Limongi & Chieffi (2006). For the sake of completeness, we summarize the basic properties of these models and the main results concerning the production of 26 Al in massive stars. The evolution of each stellar model was computed from the pre-main sequence phase up to the onset of the iron core collapse by using the stellar evolutionary code FRANEC (Frascati RAphson Newton Evolutionary Code, release 5.050218). The kernel of this code has been presented in Limongi & Chieffi (2003) (and references therein). Here we will only mention recent updates. First, the convective mixing and the nuclear burning were coupled together providing a set of diffusion equations that are linearized and solved simultaneously by means of a Newton-Raphson method. This coupling is extremely important in all situations where the nuclear burning timescale of a given nuclide is comparable to the mixing turnover time. Thus the interaction between the local nuclear burning and the convective mixing cannot be disregarded. The nuclear network adopted was the same as in Limongi & Chieffi (2003) and the thermonuclear reaction rates were up-to-date at the time when these models were computed (see Tab. 1 of Limongi & Chieffi 2006). The nuclide 26 Al was treated in a distinct manner, by assuming two separate species (for the ground and isomeric state) for temperatures below T ≈ 1 GK, and a single (thermalized) species above this temperature. Mass loss was included following the prescriptions of Vink et al. (2000) for the blue supergiant phase (T ef f > 12000 K), de Jager et al. (1988) for the red supergiant phase (T ef f < 12000 K), and Nugis & Lamers (2000) for the Wolf-Rayet phase. All of these solar metallicity models had an initial He mass fraction of 0.285 and a global metallicity (by mass) of Z = 0.02. The relative abundances for the various nuclear species were adopted from Anders & Grevesse (1989).
The explosion of the mantle of the star was started artificially, by instantaneously imparting an initial velocity of v 0 to a mass coordinate corresponding to ≈ 1 M ⊙ of the presupernova model. Such a mass coordinate relates to a region located well within the iron core and is chosen in such a way that the initial conditions should not affect the properties of the shock wave too much at a time when it approaches the Fe-Si interface. The formation and propagation of the shock wave, generated in such a way, was calculated by means of a computer code that solves the fully compressible reactive hydrodynamic equations by applying the Piecewise Parabolic Method (PPM) of Colella & Woodward (1984), using a Lagrangian scheme. The chemical evolution of the matter was computed by coupling the same nuclear reaction network that was adopted in the hydrostatic calculations to the system of hydrodynamic equations. The free parameter v 0 was properly adjusted in order to obtain a given final kinetic energy of the ejecta or, equivalently, to eject a given amount of mass above the Fe core. Since 26 Al was synthesized in regions relatively far away from the Fe core (see below), its final yield did not depend on the particular choice of v 0 provided that at least a minimum amount of 56 Ni was ejected.
Based on this set of presupernova models and simulated explosions, it was found that 26 Al is produced in massive stars in three distinct evolutionary stages: core H burning, C convective shell burning just prior to the core collapse, and explosive Ne/C burning. Any 26 Al produced by these massive stars is ejected into the interstellar medium, both by stellar winds and the explosion, in different proportions depending on the initial stellar mass. It was also found that 26 Al was mainly produced by explosive Ne/C burning over most of the initial mass interval between 11 and 120 M ⊙ . Only for the more massive stars, say, M > 60M ⊙ , did the wind component (produced during core H burning) become important (see Tab. 3 and Fig. 2 in Limongi & Chieffi 2006 for details). In the present work, we based our study on 20, 60 and 80 M ⊙ model stars for exploring 26 Al yield sensitivities in explosive Ne/C burning, C convective shell burning and core H burning, respectively. These choices of model stars are motivated by the fact that they provide relatively large 26 Al yields.
Nuclear physics library
The nuclear physics input for the present postprocessing studies is based on a new-generation library, called STARLIB. It originated from a previous version of REACLIB (originally created by F.-K. Thielemann) that one of us modified over the years and was used for all of the reaction network calculations presented in Iliadis 2007. At that point in time several important changes occurred. A recent evaluation of reaction rates for the A=14-40 target range was completed. These 62 experimental rates are based on a Monte Carlo technique, allowing for a rigorous definition of recommended reaction rates and their associated uncertainties. The Monte Carlo procedure also provides, for the first time, for any given temperature the (output) reaction rate probability density function that is based on the (input) probability densities of measured nuclear physics quantities (such as S-factors, resonance energies, resonance strengths, upper limits in spectroscopic factors, etc.). From the cumulative distributions of the rate probability densities, a low rate, median rate and high rate can be defined as the 0.16, 0.50 and 0.84 quantiles, respectively, assuming a coverage probability of 68%. The meaning of these rates is in general different from the commonly reported, but statistically meaningless, literature expressions "lower limit", "nominal value" and "upper limit" of the total reaction rate. It is important to emphasize that the Monte Carlo rates incorporate both statistical and systematic uncertainties, as explained in detail in Longland et al. (2010). Furthermore, it has been shown in Longland et al. (2010) that in the majority of cases the Monte Carlo rate probability density function can be approximated by a lognormal distribution, which is determined by only two parameters: the lognormal location parameter µ and the lognormal spread parameter σ. The former parameter determines the recommended reaction rate via N A σv rec = e µ , while the latter parameter corresponds to the rate factor uncertainty via f.u. = e σ .
The information on the rate probability density was not available previously and opens interesting windows of opportunity for Monte Carlo studies of nucleosynthesis and energy generation in stars. However, it becomes clear from the above discussion that three quantities (T , N A σv rec , lognormal σ) instead of the traditional two (T and N A σv rec ) need to be reported so that the user can calculate the rate probability density for each reaction at each temperature. Therefore, it was decided to convert our 2007 version of the REA-CLIB, which lists the recommended reaction rates as analytical functions of temperature by employing a number of rate fitting parameters, to a tabular format. To be precise, the rate tables are directly derived from the fitting parameters and not from any tabular rates given in the original publications. The new format consists of three columns and lists for each reaction the temperature, the recommended rate and the rate factor uncertainty on a grid of 60 temperatures between 1 MK and 10 GK, allowing for an accurate interpolation between grid points. At this stage the rate factor uncertainty for each reaction is set equal to a nominal value of 10. In a subsequent step, the rates and factor uncertainties of 62 reactions in the A=14-40 region were replaced with their exact Monte Carlo results. In addition, the rates of the following interactions were replaced with more recent information 3 : (i) 10 Big Bang reactions, using the rates of Descouvemont et al. (2004), which were derived from an R-matrix description of the available data; (ii) 30 reactions from the NACRE evaluation of experimental rates , in the mass range of A=1-26; (iii) (n,γ) reactions based on the KADoNiS v0.2 evaluation of experimental rates (Dillmann et al. 2006); (iv) a number of special reactions, such as 14 N(p,γ) 15 O (Bertone et al., in preparation) and 12 C(α,γ) 16 O (Kunz et al. 2002); (v) 550 experimental rates for β-decays and β-delayed particle decays including associated uncertainties, calculated from the half lives and branching ratios compiled in Audi et al. (2003); and (vi) 17 γ-ray transitions rates for 26 Al (see below). For all nuclear reactions mentioned above, the corresponding reverse reaction rates were also calculated and properly accounted for in the library. Furthermore, all experimental reaction rates were corrected for the effects of thermal target excitations using the stellar enhancement factors and partition functions of Rauscher & Thielemann (2000), although it should be noted that these effects are relatively small for the sites of nucleosynthesis discussed here. For all other reactions for which insufficient experimental information is available to compute reliable experimental rates, the results of statistical model calculations (Rauscher & Thielemann 2000) were adopted.
The new library, STARLIB, described above extends in its present version up to antimony (Sb) and is employed for the very first time in the present work. A more detailed account of STAR-LIB will be published elsewhere (Iliadis et al., in preparation). We emphasize that the rate probability density functions contained in STARLIB are not directly used in the present post-processing calculations. However, the tabulated rate factor uncertainties will be useful in later sections for the discussion of reaction rate uncertainties.
Thermal equilibration of 26 Al
A level scheme of 26 Al is shown in Fig. 1. The 5 + ground state, 26 Al g , β-decays with a half-life of f.u. = e σ =
x high /x low , where x i denotes a reaction rate; this expression can be employed to derive approximate rate factor uncertainties, f.u., from published high and low reaction rate boundaries. Furthermore, all of the replaced rates are obtained from the exact rate tables of the original publications, not from any fitting parameters.
T 1/2 = 7.17 × 10 5 y to several excited states (not shown in the figure) in the 26 Mg daughter nucleus that mainly de-excite via γ-ray transitions to the first excited state (at 1809 keV) in 26 Mg. The subsequent decay to the 26 Mg ground state gives rise to the 1809 keV γ-ray line emission observed in the Galactic plane ( § 1). An interesting situation occurs in 26 Al since its first excited state (228 keV; 0 + ), 26 Al m , is an isomer. In other words, the significant angular momentum difference between ground and isomeric state gives rise to a very small γ-ray decay constant (with a mean lifetime on the order of ≈ 10 6 y for a M5 decay). Instead, the isomer β-decays to the ground state of 26 Mg (without emission of a γ-ray) with a half-life of T 1/2 = 6.34 s.
Although the direct γ-ray transition between 26 Al g and 26 Al m is strongly inhibited, they may nevertheless be linked (or "communicate") with each other via thermal excitations involving higher-lying 26 Al levels. Three of these levels are shown in Fig. 1, at excitation energies of 417 keV ( 26 Al a ), 1058 keV ( 26 Al b ) and 2070 keV ( 26 Al c ). For example, the ground and isomeric state are linked via the γ-ray transitions 0 ↔ 417 ↔ 228. It is obvious that at sufficiently high temperatures and long timescales 26 Al g and 26 Al m will achieve thermal equilibrium, implying that their abundance ratio is simply determined by the Boltzmann distribution (i.e., by plasma temperature and energy difference), and that the internal equilibration mechanism is not important. However, it is also clear that, by lowering temperature and time scale, the ground and isomeric state will fall out of thermal equilibrium at some point. The important issue to consider for a reaction network calculation is if and when 26 Al should be treated as a single species ( 26 Al t , denoting thermal equilibrium) or as two distinct species ( 26 Al g and 26 Al m ) that β-decay with their characteristic laboratory half lives. This issue was investigated by Ward & Fowler (1980), who established the generally accepted procedure: above a limiting temperature value of T=0.4 GK, the nuclide 26 Al should be regarded as a single species (with all of its levels in thermal equilibrium), while below this temperature two distinct species (no equilibrium) should be assumed.
Some of the important γ-ray transition rates in 26 Al have not been measured yet (indicated by the 26 Al. Energies and J πvalues are from Endt (1990). The vertical arrows represent γ-ray transitions. In 26 Al, the thick vertical lines denote experimentally measured transitions (Endt 1990), while the decay rates for the thin vertical lines have been estimated using shell model calculations (Coc, Porquet & Nowacki 1999;Runkle, Champagne & Engel 2001). The possible communication of the 26 Al ground state and isomeric state (228 keV) via thermal excitations involving higher-lying levels (417, 1058 and 2070 keV) is apparent. The arrows connecting 26 Al and 26 Mg denote β-decay transitions, according to the shell model calculations of Kajino et al. (1988). The β-decays shown do not necessarily represent direct transitions, but rather indicate if the ground state or first excited state (at 1809 keV) in 26 Mg is predominantly populated by a given transition.
thin vertical arrows in Fig. 1) and Ward & Fowler (1980) had to employ for these rather crude approximations. In more recent work (Coc, Porquet & Nowacki 1999;Runkle, Champagne & Engel 2001) the γ-ray transition rates have been estimated by using shell model calculations and the thermal equilibration of 26 Al was studied in more detail using small reaction networks. These works confirmed the assumption that above T ≈ 0.45 GK the ground and isomeric state are in thermal equilibrium, while below T ≈ 0.15 GK these two states β-decay with their laboratory half lives (i.e., neither equilibrium nor any communication between levels). In the transitional temperature region, T ≈ 0.15 − 0.45 GK, the effective 26 Al decay rate was found to deviate substantially from the results reported by Ward & Fowler (1980) by up to 4 orders-of-magnitude and, furthermore, that 26 Al g and 26 Al m are indirectly linked by γ-ray transitions (i.e., they achieve a quasi-equilibrium). It was pointed out by Runkle, Champagne & Engel (2001) that these findings rest on simplifying assumptions and may not hold if the nuclear reactions that produce or destroy 26 Al are sufficiently fast to disturb the thermal equilibration. See also, Gupta & Meyer (2001).
In the present work we examine carefully the equilibration of 26 Al for each of the three nucleosynthesis sites mentioned above. Two separate post-processing calculations using recommended interaction rates are performed and the resulting 26 Al yields are compared: one assuming either a single or two separate 26 Al species, depending on the temperature regime (according to Ward & Fowler 1980), and one where the communication between ground and isomeric states is explicitly taken into account. For the latter case, no artificial assumptions about the equilibration of 26 Al are made, but additional 26 Al species (i.e., levels at 417, 1058 and 2070 keV; see Fig. 1) need to be taken into account in the reaction network. The γ-ray transition rates linking the various levels in 26 Al, and the most important β-decay transition rates to 26 Mg, are adopted from Runkle, Champagne & Engel (2001). Since these have not been published elsewhere, we list the decay constants versus temperature in Appendix A. Furthermore, when taking more than two 26 Al species into account, the reaction rates for 26 Al g and 26 Al m are separately needed. This represents an additional complication since, as will be seen below, some of these rates are poorly known at present. When performing the two separate post-processing calculations referred to above, it is of utmost importance that an internally consistent set of rates for the production and destruction of 26 Al t , 26 Al g and 26 Al m is employed. Suppose, for example, that the rate of a destruction reaction 26 Al x (a,b) in the first calculation (i.e., assuming a single 26 Al species) is adopted from one specific source, and the rates of the physically related reactions 26 Al g (a,b) and 26 Al m (a,b) in the second calculation (i.e., assuming five different 26 Al species) are adopted from a different source. In such a case, when comparing the results of the two network calculations, one may find significant differences in 26 Al yields. However, these may not be caused at all by the effects of thermal equilibration of 26 Al but may rather reflect spurious results caused by using inconsistent reaction rates. In order to elucidate this issue, we list in Appendix B the rates and sources of all reactions considered in our network that produce and destroy 26 Al.
Procedure and Results
General considerations
The same reaction network is used in the present work for investigating the nucleosynthesis of 26 Al in the predicted main locations of massive stars: explosive Ne/C burning, convective shell C/Ne burning and convective core H burning. The network extends from 1 H to 40 Ca, including 175 proton-and neutron-rich nuclides up to the respective driplines, that are linked by 1648 nuclear interactions ( § 2.2). Initial abundances are listed in Tab. 1 and the temperature-density time evolutions for each of the sites will be discussed in the following subsections. The nucleosynthesis will initially be visualized by considering so-called "abundance flows", which represent the change of abundance per time as a result of an interaction between two nuclides. Since a forward and corresponding reverse reaction occur concurrently, what is of main interest is the net abundance flow (i.e., the difference between forward and reverse flow). Initially, a "standard" network calculation is performed, employing recommended reaction rates. The final abundances, achieved at the end of the standard calculations, are summarized in Tab. 1. Subsequently, a series of network calculations is performed, where the rates of many reactions are varied individually by generic factors of 10, 2, 0.5 and 0.1. Resulting abundance changes of 26 Al are then analyzed in detail. Then we focus our discussion on the actual rate uncertainties in the most relevant temperature region, which differs from site to site. It will become apparent that this temperature region is rather narrow, which significantly reduces any (unknown) systematic effects that are potentially caused by an incorrect temperature dependence of some rates. Finally, the issue of thermal equilibration of 26 Al is investigated.
Our strategy regarding which and how many reaction rates to vary was as follows. We started by considering the net abundance flows (i.e, the difference of total abundance flows between a given forward and corresponding reverse reaction), integrated over the entire duration of a "standard" post-processing calculation. These flows will be displayed graphically below for each of the three burning regimes. All rates of reactions with net flows within 3 orders-of-magnitude of the maximum flow were then selected for the variation procedure. Obviously, the forward and corresponding reverse reaction rate need to be multiplied by the same variation factor. We added to this list all reactions that either destroy or produce 26 Al,27 Al and 25 Mg, if these were not taken into account already. Furthermore, a number of selected other reactions, for example, 12 C( 12 C,n) 23 Mg and 24 Mg(p,γ) 25 Al, were added to the list. We find it unlikely that any other reaction not identified by the above procedure has a major impact on 26 Al nucleosynthesis in massive stars.
A number of important issues need to be considered in detail when performing any reaction rate sensitivity study using post-processing calculations. First, it is assumed that the nuclear reaction rates to be varied do not impact the nuclear energy generation. If a given reaction rate variation changes both the energy generation and the final 26 Al abundance, then this result has no obvious meaning. Clearly, in such cases the rate should be varied using the full, self-consistent stellar model. For example, changing the rate of the 20 Ne(γ,α) 16 O reaction, the process that initiates Ne burning, influences the 26 Al abundance, although the effect is small, as will be seen below. Table 1 Initial and final mass fractions of present post-processing calculations a
Nuclide
Xi a Only mass fractions of stable or long-lived nuclides in access of X ≈ 5 × 10 −7 are listed here. For 26 Al the listed values refer to the ground state. The labels "xNe/C", "C/Ne" and "H" refer to explosive Ne/C burning, convective shell C/Ne burning, and convective core H burning, respectively. b Initial mass fractions at beginning of post-processing calculations (from Limongi & Chieffi 2006). c Final mass fractions at the end of post-processing calculations, obtained from the "standard" calculations (i.e., without any reaction rate variations). Throughout this work, we carefully checked that a rate variation did not impact at the same time the energy generation. Second, it is very important to verify that a given temperature-density-time evolution is followed precisely in a post-processing calculation. The time step is numerically adjusted to track the abundance evolutions above some limiting abundance value. For example, spurious abundance variations may occur if a time step misses the peak temperature even by a few percent. Therefore, we carefully checked that the temperature-density evolution is followed closely (within a fraction of a percent).
b X f c Solar d xNe/C C/Ne H xNe/C C/Ne H 1 H · · · · · · 7.0E-01 · · · · · · 1.4E-06 7.11E-01 2 H · · · · · · 5.0E-05 · · · · · · · · · 2.76E-05 3 He · · · · · · 3.1E-05 · · · · · · · · · 3.40E-05 4 He · · · · · · 2.9E-01 · · · · · · 9.8E-01 2.74E-01 12 C 1.9E-02 1.5E-01 3.2E-03 1.5E-02 1.0E-01 2.6E-04 2.44E-03 13 C · · · · · · 3.8E-05 · · · · · · 7.3E-05 2.96E-05 14 C · · · 1.5E-05 · · · · · · 2.7E-06 · · · · · · 14 N 8.2E-07 1.2E-04 1.2E-03 · · · 2.0E-05 1.3E-02 7.90E-04 15 N · · · · · · 4.6E-06 · · · · · · · · · 3.11E-06 16 O 5.0E-01 7.0E-01 1.0E-02 5.8E-01 6.7E-01 1.0E-04 6.55E-03 17 O · · · 2.9E-05 4.1E-06 · · · 3.7E-06 4.0E-07 2.60E-06 18 O · · · · · · 2.3E-05 · · · · · · · · · 1.
Finally, we need to address the issue of stellar versus laboratory β-decay rates. The stellar plasma affects β-decays in a number of ways. First, at high temperatures thermally excited states in the β-decaying nuclide may undergo transitions to levels in the daughter nuclide. Second, at high (electron) densities the decay constants for electron (or positron) capture will increase. Both effects generally cause a change in the total β-decay rate. Many previous stellar model studies employed the stellar β-decay constants calculated by Fuller, Fowler & Newman (1982) for the mass range of A=21-60, or the more modern results of Oda et al. (1994), that are based on shell model calculations, for the mass range of A=17-39. For the purposes of the present work, the highest temperature and density values are encountered in explosive Ne/C burning, with peak values of T = 2.3 GK and ρ = 3.2 × 10 5 g/cm 3 (see below). By inspecting the tables of Oda et al. (1994) in the region A≤30, we find that for the temperatures and densities of interest here the stellar and laboratory β-decay rates are very similar in magnitude. The only exceptions are the β-decays of the long-lived species 22 Na and 24 Na. However, their destruction via the processes (γ,n), (γ,p) or (p,γ) is much faster compared to the βdecays and, therefore, their stellar β-decay rate is unimportant at high values of T and ρ. In conclusion, it is sufficient to adopt laboratory β-decay rates throughout this work, except for the β-decay of (thermalized) 26 Al t . For the calculation of this decay we only take the ground and isomeric state into account. The decay constants are listed in Appendix A and agree with the more comprehensive results of Oda et al. (1994) for temperatures and densities below 5 GK and 10 6 g/cm 3 , respec-tively.
Explosive Ne/C burning
Standard calculation
At the beginning of the burning, the most abundant nuclides are (in order) 16 O, 20 Ne, 24 Mg,28 Si and 12 C (Tab. 1). The temperature-densitytime profile for simulating explosive Ne/C burning is shown in Fig. 2. It has been extracted from a stellar model calculation of a 20M ⊙ star ( § 2.1). Specifically, we select a mass coordinate of 2.04M ⊙ , corresponding to the zone where the maximum abundance of 26 Al is produced during the explosion. Temperature and density peak at T = 2.3 GK and ρ = 3.2×10 5 g/cm 3 , respectively. The evolution is followed in a post-processing simulation over a total time of t = 12.8 s. At this point the temperature has declined to T = 0.4 GK and no additional 26 Al synthesis is occurring. We assume at this stage thermal equilibrium for 26 Al, i.e., the network contains only a single species, 26 Al t .
The net abundance flows, integrated over a total running time of t = 12.8 s, for the "standard" calculation are displayed in Fig. 3. They provide a first impression regarding the nucleosynthesis and indicate the degree of "nuclear activity". The network consists of all nuclides shown as squares. The strongest net abundance flows, i.e., those within one, two, and three orders of magnitude of the maximum flow, are displayed by the thickest arrows, arrows of intermediate thickness, and the thinnest arrows, respectively. The strongest net flows belong to the reactions 20 Ne(γ,α) 16 O and 20 Ne(α,γ) 24 Mg that drive explosive Ne burning. The released α-particles induce a network of secondary reactions, giving rise to a small, but significant, abundance of light particles. During the explosive burning the mass fractions of protons, α-particles and neutrons reach maximum values of 1.0×10 −8 , 1.1×10 −5 and 3.8×10 −11 , respectively. The main direct process of 26 Al t synthesis, in terms of the net abundance flow, is 25 Mg(p,γ) 26 Al t , with 25 Mg produced by the 24 Mg(n,γ) 25 Mg reaction. The main neutron sources are 25 Mg(α,n) 28 Si and 26 Mg(α,n) 29 Si. On the other hand, 26 Al t is predominantly destroyed via 26 Al t (n,p) 26 Mg and, to a lesser degree, 26 Al t (n,α) 23 Na. In particular, the (Limongi & Chieffi 2006). In the present postprocessing calculation, the evolution was only followed until t = 12.8 s (vertical lines), since for later times T and ρ decline to values where no additional 26 Al synthesis is occurring. The peak temperature and density, near t = 0.6 s, amount to T = 2.3 GK and ρ = 3.2 × 10 5 g/cm 3 , respectively (see text). 26 Al t (p,γ) 27 Si reaction is entirely negligible under explosive burning conditions. These general features have already been discussed by Limongi & Chieffi (2006). The abundance evolutions of the species 26 Al t and 27 Al are shown in Fig. 4. While the abundance of the latter nuclide is approximately constant throughout the calculation, the abundance of the former species increases by more than an order of magnitude during the explosion. The abundance ratio, X( 26 Al t )/X( 27 Al), increases from an initial value of 3.2 × 10 −4 to a final value of 1.2 × 10 −2 (see Tab. 1).
Reaction rate variations
Subsequently, the rates of 70 pairs of forward and reverse reactions were varied. Those reactions whose rate changes have the strongest effect on the final 26 Al yield (i.e., at time t = 12.8 s) are listed in Tab. 2. All other rate changes, as well as those labeled by "..." in the table, produced 26 Al t abundance changes of less than 20%. The reactions are listed in approximate order of importance, as measured by their impact on the final 26 Al abundance. The last two columns display the source of the rate and the reported rate uncertainty at a temperature near the peak of the explosion (i.e., where most of the nucleosynthesis is occurring). Disregarding for a moment the actual rate uncertainties, the six reactions with the strongest impact on 26 Al t nucleosynthesis are: 26 Al t (n,p) 26 Mg, 25 Mg(p,γ) 26 Al t , 25 Mg(α,n) 28 Si, 24 Mg(n,γ) 25 Mg, 20 Ne(α,γ) 24 Mg, and 30 Si(p,γ) 31 P. The first and second reaction destroys and produces 26 Al t , respectively, while the third and fourth reaction destroys and produces, respectively, the 25 Mg seed. The fifth reaction produces 24 Mg, from which 25 Mg is synthesized via neutron capture. The sensitivity of the final 26 Al t abundance to any of these rate changes is thus not surprising.
The manner by which the sixth reaction impacts the synthesis of 26 Al t is interesting. In fact, the sequence 30 Si(p,γ) 31 P(p,α) 28 Si is the main consumer of free protons (together with the proton captures on 26 Mg and 27 Al). When the rate of the 30 Si(p,γ) 31 P reaction is reduced by an order of magnitude, the number of available protons increases near the peak of the explosion and, consequently, more 25 Mg nuclei are converted to 26 Al t . There are 13 more reactions listed in Tab. 2 and their mechanisms by which they impact Explosive Ne/C burning Fig. 3.-Net abundance flows, obtained for a post-processing network calculation of explosive Ne/C burning, integrated over a total running time of t = 12.8 s. The T -ρ profile for this simulation is shown in Fig. 2. The network consists of all nuclides shown as squares. The strongest net abundance flows, i.e., those within one, two, and three orders of magnitude of the maximum flow, are displayed by the thickest arrows, arrows of intermediate thickness, and the thinnest arrows, respectively. Thermal equilibrium for 26 Al has been assumed (i.e., the network contains only a single species, 26 Al t ). Table 2 Factor changes of final 26 Al t abundance resulting from reaction rate variations for explosive Ne/C burning a , assuming thermal equilibrium for 26 26 Al t (n,α) 23 Na 0.55 · · · · · · · · · present 25 Mg(n,γ) 26 Mg 0.75 · · · · · · · · · ka02 28 Si(n,γ) 29 Si 1.4 · · · · · · · · · ka02 29 Si(p,γ) 30 P 1.4 · · · · · · · · · il10 7% 32 S(n,γ) 33 S 1.2 · · · · · · · · · ka02 26 Al t (α,p) 29 Si 0.72 · · · · · · · · · rath 26 Mg(p,γ) 27 Al 1.2 · · · · · · · · · il10 4% a The temperature-density-time profile is extracted from a hydrodynamic model of a 20M ⊙ star of initial solar metallicity, see Limongi & Chieffi (2006). b In total, the rates of 70 different reactions were varied. Listed are only those reactions whose rate changes have the strongest effect on the 26 Al t yield. All other rate changes, as well as those labeled by "...", produced abundance changes of less than 20%. The reactions are listed in approximate order of importance. Thermal equilibrium for 26 Al has been assumed, i.e., the network contains only a single species, 26 Al t . c Reaction rate references: (nacr) Angulo et al. 1999 (NACRE); (ka02) Dillmann et al. (2006) (KADoNiS v0.2); (rath) Rauscher & Thielemann (2000); (il10) Iliadis et al. (2010); (present) hybrid rates, see Appendix C.5 and C.6. d Reaction rate uncertainty near a temperature of 2.3 GK, at the peak of the explosion; no entry implies that the rate uncertainty is difficult to quantify (see text). the final 26 Al t abundance can be easily deduced from arguments similar to those given above. The only reaction that we found to influence somewhat the 26 Al t yield but is not listed in Tab. 2 is 20 Ne(γ,α) 16 O. Varying this rate by a factor of 10 changes the 26 Al t abundance by a factor of ≈2. However, the estimated rate uncertainty of this reaction amounts to only 13% and thus the actual effect is relatively small.
Al(α,p) 30 Si 1.5 · · · · · · 0.72 rath 29 Si(α,n) 32 S 0.65 · · · · · · 1.3 rath 24 Mg(α,γ) 28 Si 0.62 · · · · · · · · · il10 6% 24 Mg(α,p) 27 Al · · · · · · · · · 0.65 il10 6% 27 Al(p,γ) 28 Si 0.60 · · · · · · · · · il10 3% 25 Mg(α,p) 28 Al 0.59 · · · · · · · · · rath
Reaction rate uncertainties
Before proceeding, notice the sources of our reaction rates, listed in column 6 of Tab. 2. Of the 19 reactions listed, the rates of: (i) 8 reactions are available from the Monte Carlo procedure ; § 2.2); (ii) 6 reactions are adopted from the statistical model (Rauscher & Thielemann 2000); (iii) 4 reactions are obtained from KADoNiS v0.2 (Dillmann et al. 2006); and (iv) only one is adopted from NACRE . Note that none of these rates rely anymore on outdated information from Caughlan & Fowler (1988).
We now turn to a discussion of reaction rate uncertainties. These are listed for a temperature of T = 2.5 GK, near the peak of the explosion, in the last column of Tab. 2, when reported in the original source. Rate uncertainty estimates are of obvious importance. Suppose a rate variation of a particular reaction by a factor of 10 changes the 26 Al t abundance by the same factor. Then one may conclude that this particular reaction rate should be known with rather small uncertainty. On the other hand, if a particular rate variation barely affects the abundance of 26 Al t , one may tolerate a much larger uncertainty. In reality, however, the issue is much more complicated and one is usually confronted with the following questions when considering rate uncertainties reported in the literature. What is the (statistical) meaning of a reported rate uncertainty? Is a presumed experimental rate at a given temperature directly based on data, or is it based on a normalization of (theoretical) Hauser-Feshbach rates? Even if a rate is directly based on data, how large is the stellar enhancement factor that must usually be obtained from Hauser-Feshbach models? What may one estimate for a rate uncertainty if no values are reported in the literature? And, even if a given rate is directly based on data and if the stellar enhancement factor is negligible at a given temperature, are there possible systematic errors that were not taken into account in the reported rate uncertainty? All of these issues play an important role and thus reported rate uncertainties are frequently difficult to assess. Below we will give a few examples to emphasize these points.
The first reaction listed in Tab. 2, 25 Mg(α,n) 28 Si, strongly affects the final 26 Al t abundance. Varying the rate by a factor of 10 (2) changes the 26 Al t yield by a factor of 0.1 (0.5). The rate is adopted from the NACRE compilation , and its reported uncertainty of ≈18% near T = 2.5 GK may on first sight indicate a rather reliable rate. However, not enough information is provided in Angulo et al. (1999) to understand how exactly this value of uncertainty has been obtained. Also, beyond T = 2 GK, i.e., the highest temperature for which the rate is directly based on data, the rate was extrapolated with the aid of (theoretical) Hauser-Feshbach model results. Furthermore, even at the lower temperatures, the rate seems to be based on data from an unpublished thesis. Considering these arguments together with the importance of the 25 Mg(α,n) 28 Si reaction, there is no doubt in our minds that this particular reaction should be a target of future experimental work. Consider, on the other hand the fourth reaction listed in Tab. 2, 25 Mg(p,γ) 26 Al t . Varying the rate by a factor of 0.1 (0.5) changes the 26 Al t yield by a factor of 0.14 (0.58). A rate uncertainty of only 4% near T = 2.5 GK has been reported by Iliadis et al. (2010). This value has been obtained from a Monte Carlo procedure, implying a statistically meaningful probability coverage (68%). The rate near the peak of the explosion is directly based on data, i.e., no extrapolation using theoretical reaction models is needed. Furthermore, the experimental rate is normalized to a well-known standard resonance strength (for details, see Iliadis 2007). In conclusion, at present there is less compelling reason for remeasuring this reaction at higher energies compared to the previous case. We emphasize again that each reaction must be treated as a special case and that a reported rate uncertainty needs to be considered carefully. For readers interested in the present status of specific reactions, we provide brief discussions in Appendix C.
The set of rates shown in Tab. 2 that are based on Hauser-Feshbach theory (labelled by "rath") represent a special case. It has been claimed by Rauscher & Thielemann (2000) that "...the accuracy of the rates is estimated to be within a factor of 1.5-2...". Obviously, if too few resonances contribute to the rate at a given temperature, the statistical model will provide a poor description. For this reason, Rauscher & Thielemann (2000) provide a minimum temperature estimate below which the Hauser-Feshbach rates become inaccurate. This minimum temperature value is calculated from a parameterization of nuclear level densities, assuming that at least 10 levels (Rauscher, Thielemann & Kratz 1997) are located in the astrophysically important energy window (e.g., the Gamow peak for charged-particle reactions). Note that for all of the reactions labeled "rath" in Tab. 2 the peak temperature of the explosion (T = 2.3 GK) far exceeds the minimum temperature required for the applicability of the Hauser-Feshbach model according to Rauscher & Thielemann (2000). Unfortunately, the above claims are not supported when comparing the Hauser-Feshbach rates with results that are directly based on experiment. The issue was discussed in Iliadis et al. (2001), who found that for several reactions involving A=20-40 mass targets "... the deviation between theoretical and experimental rates far exceeds the usually quoted factor of 2 reliability of statistical model results ...". Clearly, more work is required to resolve this controversy. At this point it may be argued that all of the reactions labeled by "rath" in Tab. 2 should be targets for future experimental work, including the important destruction reactions 26 Al t (n,p) 26 Mg and 26 Al t (n,α) 23 Na (labeled "present"; see Appendix C.6 and C.5). Neutron capture rates represent another special case. We adopted for these the results presented in the KADoNiS v0.2 evaluation (Dillmann et al. 2006; these rates are labelled by "ka02" in Tab. 2). The most important neutron capture reaction for the purposes of the present work is 24 Mg(n,γ) 25 Mg, as is apparent from the table. In order to obtain a better sense for the uncertainties, we will briefly discuss how the KADoNiS evaluated rates have been obtained and what information is actually incorporated in reaction rate libraries. The arguments below apply equally to the other reactions listed in Tab. 2, i.e., 25 Mg(n,γ) 26 Mg,28 Si(n,γ) 29 Si and 32 S(n,γ) 33 S. The KADoNiS eval-uation tabulates recommended rates for the range of kT = 5 − 100 keV (corresponding to T = 0.06 − 1.2 GK). For the neutron captures on 24,25 Mg, 28 Si and 32 S the rates are obtained from experimental data on resonance properties (with some theoretical corrections for direct neutron capture contributions, if applicable) over the entire tabulated temperature range. According to the KADoNiS evaluation, the "relative uncertainties [of the rates] are similar to those quoted for the 30 keV data" (12% for neutron capture on 24 Mg). The tabulated rates include the stellar enhancement factor ( § 2.2), although these are predicted to be close to unity for the (n,γ) reactions mentioned above. Note that for explosive Ne/C burning the rates are needed at temperatures (T ≈ 2.3 GK) that have not been covered by experiments. Thus it is not obvious how to extrapolate the rates from lower temperatures, where they are based on experimental data, to much higher temperatures. Furthermore, it must be pointed out that the experimental KADoNiS rates are not directly used in reaction rate libraries (including ours). What is usually incorporated for neutron captures are Hauser-Feshbach rates, which are normalized to the experimental rates at a single temperature (kT = 30 keV or T = 0.35 GK). Since the level density for targets in the mass A ≤ 40 range at kT = 30 keV may be too small for the application of statistical models, an additional systematic uncertainty is introduced when extrapolating such normalized rates to higher temperatures. For example, in the case of 24 Mg(n,γ) 25 Mg the (experimental) KADoNiS rate at the upper temperature cutoff (T = 1.2 GK) deviates from the normalized Hauser-Feshbach rate already by ≈40%. Considering the above arguments, we estimate a rate uncertainty of a factor of 2 − 3 for the neutron captures on 25,25 Mg, 28 Si and 32 S near the peak of explosive Ne/C burning. Recall from Tab. 2 that varying the 24 Mg(n,γ) 25 Mg rate by a factor of 2 increases the 26 Al t yield by a factor of 1.6. Clearly, a more reliable experimental rate for 24 Mg(n,γ) 25 Mg at higher temperatures is urgently needed.
Thermal equilibration
We will now consider the issue of thermal equilibration. Recall that we assumed so far a single species of 26 Al, implying thermal equilibrium ( 26 Al t ). We will now relax this assumption and follow the equilibration numerically in the network calculation. To this end, we introduce five different species of 26 Al, as explained in § 2.3. The required γ-and β-decay transitions between and from these levels are discussed in detail in Appendix A. The price we pay is that additional reaction rates, involving 26 Al g and 26 Al m separately, have to be incorporated into the network (see Appendix B). As will be seen, some of these rates are highly uncertain.
As a first step, we performed a standard post-processing network calculation (with recommended rates) using the same temperaturedensity-time evolution as before (Fig. 2). The final 26 Al g abundance, at t = 12.8 s, is found to be identical to our earlier result obtained assuming thermal equilibrium (Tab. 1). Thus the latter assumption seems to be justified. An impression can be gained from Fig. 5, showing the abundance evolution of 26 Al levels. The top part displays the abundances for individual 26 Al species and it is apparent that at any given time the 26 Al g abundance dominates over those of the other species. The bottom part displays the fraction of the total 26 Al abundance that resides in the isomeric state. This curve is directly obtained from the network calculation, but is indistinguishable from the one calculated assuming a Boltzmann distribution (i.e., thermal equilibrium).
Next, the rates of 20 different reactions and transitions, together with their inverse rates, were varied individually by factors of 100, 10, 2, 0.5, 0.1 and 0.01. This list contained all nuclear reactions that produced and destroyed 26 Al g or 26 Al m . It also included those β-and γ-ray decay rates of 26 Al x levels that were estimated using the shell model, as described in § 2.3 and shown in Fig. 1. Experimentally obtained β-and γ-ray decay rates have not been varied since their uncertainties are very small. The final 26 Al g abundance, after each rate variation, was then compared to the standard calculation. The results are listed in Tab. 3. It can be seen that of the 20 interactions only eight, all of them reactions, influence the final 26 Al g yield. In other words, even a variation by a factor of 100 of the shell-model based β-or γ-ray decay rates seems to have no effect on the 26 Al g abundance. The eight reactions displayed in Tab. 3 are listed in approximate order of importance. It must be noted that the impact of these rates, with one exception, on the final abundance of 26 Al g is moderate. For example, consider the second and third reaction, 25 Mg(p,γ) 26 Al g and 25 Mg(p,γ) 26 Al m . Even a rather small rate variation (by a factor of 2) influences the final 26 Al yield (by ≈30%). However, these rates are based on experimental information and their Monte Carlo uncertainties are predicted to amount to only 4-5% near the peak of the explosion.
The one exception is the 26 Al g (n,p) 26 Mg reaction rate. Increasing this rate by a factor of 100 changes the 26 Al g abundance by a factor of 0.017. We adopted for this reaction the same rates as for 26 Al t (n,p) 26 Mg (see Appendix B). Our numerical results indicate that even a factor of 100 variation in the 26 Al g (n,p) 26 Mg rate does not change the thermal equilibrium abundance ratio of 26 Al m and 26 Al g (Fig. 5). In fact, comparison of the first entry of Tab. 3 with the third entry of Tab. 2 immediately reveals that the 26 Al(n,p) 26 Mg reaction impacts the final 26 Al abundance by the same factor changes, no matter if a single (thermalized) or five species of 26 Al are used in the simulation. In conclusion, 26 Al is in thermal equilibrium 4 during explosive Ne/C burning and, consequently, there is no need to introduce the extra complication of five 26 Al levels, and their mutual interactions, into the reaction network.
Convective shell C/Ne burning
Standard calculation
Preliminary studies of the impact of nuclear uncertainties on pre-explosive 26 Al yields can be found in Baldovin, Pignatari & Gallino (2006). These authors performed post-processing studies using a schematic one-zone model consisting of two phases: a constant temperature of T ≈ 1.1 GK until the 12 C mass fraction decreases from an inital value of 0.18 to 0.10 for phase 1, and a constant temperature of T ≈ 1.3 GK until the 12 C mass fraction reaches a value of 0.050 for phase 2. Table 3 Factor changes of final 26 Al g abundance resulting from reaction rate variations for explosive Ne/C burning a , assuming five species of 26 Al
Reaction b
Rate multiplied by 100 10 2 0.5 0.1 0.01 Source c Uncertainty d 26 Al g (n,p) 26 Mg 0.017 0.14 0.57 1.6 2.9 3.8 present 25 Mg(p,γ) 26 26 Al m 1.6 1.6 · · · · · · 0.79 0.79 il10 5% 26 Al g (α,p) 29 Si 0.21 0.71 · · · · · · · · · · · · rath 26 Al g (n,α) 23 Na 0.21 0.54 · · · · · · · · · · · · present 26 Al m (n,p) 26 Mg 0.36 · · · · · · · · · · · · · · · present 26 Al g (p,γ) 27 Si 0.52 · · · · · · · · · · · · · · · il10 7% 26 Al m (n,α) 23 Na 0.79 · · · · · · · · · · · · · · · present a The temperature-density-time profile is extracted from a hydrodynamic model of a 20M ⊙ star of initial solar metallicity, see Limongi & Chieffi (2006). b In total, the rates of 20 different reactions producing or destroying 26 Al g and 26 Al m were varied. Listed are only those reactions whose rate changes have the strongest effect on the 26 Al g yield. All other rate changes, as well as those labeled by "...", produced abundance changes of less than 20%. The reactions are listed in approximate order of importance. No thermal equilibrium for 26 Al has been explicitly assumed, i.e., the network contains five different species ( 26 Al g , 26 Al m , 26 Al a , 26 Al b , 26 Al c ) and takes the interactions between them into account. c Reaction rate references: (il10) Iliadis et al. 2010; (rath) Rauscher & Thielemann 2000;(present) hybrid rate, see Appendix C.5 and C.6. In the latter three cases, we assumed that the rate involving 26 Al g or 26 Al m is the same as the rate for 26 Al t (see comments in Appendix B). d Reaction rate uncertainty near a temperature of 2.3 GK, at the peak of the explosion; no entry implies that the rate uncertainty is difficult to quantify.
In the present work we proceeded as follows. Initially, we extracted the temperature-density-time evolution of the deepest and hottest zone of the convective C/Ne burning shell from a stellar evolution model of a 60 M ⊙ star with initial solar metallicity (Limongi & Chieffi 2006). This profile extended in time from the formation of the shell until a time of t = 3.15 × 10 6 s. Using this T − ρ profile directly in a post-processing study would greatly distort the nucleosynthesis prediction, since the effects of convection are not taken properly into account. On the one hand, convection constantly carries fresh fuel (here 12 C) into the burning region, while, on the other hand, it transports fragile nuclei from the burning region to cooler layers where they survive for a longer period of time. Therefore, in a stellar evolution calculation, convection has the effect of lengthening considerably the duration of nuclear burning. If we would use this profile directly in a post-processing simulation, then the initial 12 C fuel, for example, would be destroyed much faster than in the actual stellar evolution calculation. After some trial attempts, we found that compressing the time axis of the original T − ρ profile by a factor of 60 gives results that are consistent with the stellar evolution calculations. Clearly, this large scaling factor reflects the strong effects of convection during C/Ne shell burning. The results are shown in Fig. 6, displaying the temperature and density dependence on the 12 C mass fraction as solid and dashed lines, respectively. For comparison, the circles indicate the corresponding values from the stellar evolution calculations. The good agreement is encouraging and thus we used the scaled T −ρ profile for our post-processing study. At the beginning of the burning, when X i ( 12 C) = 0.15, temperature and density start at values of T = 1.13 GK and ρ = 6.3 × 10 4 g/cm 3 , respectively. The profile extends over a time period of t = 5.24 × 10 4 s, when X f ( 12 C) = 0.10, and ends with values of T = 1.44 GK and ρ = 1.1 × 10 5 g/cm 3 .
At the beginning of the burning, the most abundant nuclides are (in order) 16 O, 12 C and 20 Ne (Tab. 1). We assume at this stage thermal equilibrium for 26 Al, i.e., the network contains only a single species, 26 Al t . The net abundance flows, integrated over a total running time of t = 5.24×10 4 s, for the standard calculation are displayed in Fig. 7. The strongest net flows belong to the pri-mary carbon burning reactions 12 C( 12 C,α) 20 Ne and 12 C( 12 C,p) 23 Na, and to the secondary reactions 16 O(α,γ) 20 Ne and 23 Na(p,α) 20 Ne that are initiated by the released light particles from the primary reactions. Near the end of convective shell C/Ne burning the mass fractions of protons, α-particles and neutrons reach maximum values of 9.5×10 −14 , 5.0×10 −9 and 2.9×10 −16 , respectively. The main direct process of 26 Al t synthesis, in terms of the net abundance flow, is 25 Mg(p,γ) 26 Al t , with 25 Mg produced by the 22 Ne(α,n) 25 Mg and 24 Mg(n,γ) 25 Mg reactions. The main neutron source is 22 Ne(α,n) 25 Mg. On the other hand, 26 Al t is mainly destroyed via the β-decay 26 Al t → 26 Mg (see column 2 of Tab. 8). In particular, the neutron abundance is too low in the standard calculation for the destruction reactions 26 Al t (n,p) 26 Mg and 26 Al t (n,α) 23 Na to compete successfully with the β-decay of 26 Al t . The abundance evolutions of the species 26 Al t and 27 Al are shown in Fig. 8. While the abundance of the latter nuclide is approximately constant throughout the calculation, the abundance of the former species increases by more than an order of magnitude over the course of the burning. The abundance ratio, X( 26 Al t )/X( 27 Al), increases from an initial value of 6.7 × 10 −5 to a final value of 2.5 × 10 −3 (see Tab. 1).
Reaction rate variations
Subsequently, the rates of 66 pairs of forward and reverse reactions were varied. Those reactions whose rate changes have the strongest effect on the final 26 Al yield (i.e., at the end of the calculation, when X f ( 12 C) = 0.10) are listed in Tab. 4. All other rate changes, as well as those labeled by "..." in the table, produced 26 Al t abundance changes of less than 20%. The reactions are listed in approximate order of importance, as measured by their impact on the final 26 Al abundance. The last two columns display the source of the rate and the reported rate uncertainty at a temperature of ≈1.4 GK near the end of the calculation. Disregarding at first the actual rate uncertainties, the four reactions with the strongest impact on 26 Al t nucleosynthesis are: 25 Mg(p,γ) 26 Al t , 26 Al t (n,p) 26 Mg, 23 Na(p,α) 20 Ne and 23 Na(α,p) 26 Mg. The first reaction produces 26 Al t , while multiplying the rate of the second reaction by a factor of 10 would make it the dominant 26 Al t destruction process, at the Convective shell C/Ne burning Fig. 7.-Net abundance flows, obtained for a post-processing network calculation of convective shell C/Ne burning, integrated over a total running time of t = 5.2 × 10 4 s, when the 12 C mass fraction has decreased to 0.097. The T -ρ profile for this simulation is shown in Fig. 6. The network consists of all nuclides shown as squares. The strongest net abundance flows, i.e., those within one, two, and three orders of magnitude of the maximum flow, are displayed by the thickest arrows, arrows of intermediate thickness, and the thinnest arrows, respectively. Thermal equilibrium for 26 Al has been assumed (i.e., the network contains only a single species, 26 Al t ).
cost of the β-decay of 26 Al t . The third reaction is the main consumer of free protons. When the rate of the 23 Na(p,α) 20 Ne reaction is increased, the number of available protons decreases and, consequently, fewer 25 Mg nuclei can be converted to 26 Al t . The fourth reaction represents the second most important proton-generating process (after the primary 12 C( 12 C,p) 23 Na reaction). When the 23 Na(α,p) 26 Mg reaction rate is increased, more protons are available for producing 26 Al t from 25 Mg.
Other important rate variations that impact the final 26 Al t abundance arise from the reactions 25 Mg(α,n) 28 Si, 24 Mg(n,γ) 25 Mg and 26 Al t (n,α) 23 Na, which have already been discussed in § 3.2.2. In addition, the reactions 16 O(α,γ) 20 Ne, 12 C( 12 C,n) 23 Mg and 26 Mg(α,n) 29 Si play an important role. Decreasing the rate of the first reaction will consume fewer α-particles, thus increasing the production of neutrons (for converting 24 Mg to 25 Mg seed) via 22 Ne(α,n) 25 Mg. Increasing the rate of the second reaction produces an additional burst of neutrons towards the very end of the burning, thereby destroying more 26 Al t nuclei via the (n,p) and (n,α) reactions. For specific comments on the 12 C( 12 C,n) 23 Mg reaction, see Appendix C.1. Similar arguments apply to the 26 Mg(α,n) 29 Si reaction. There are four more reactions listed in Tab. 4, 27 Al(n,γ) 28 Al, 25 Mg(n,γ) 26 Mg, 26 Mg(p,γ) 27 Al and 27 Al(p,α) 24 Mg, which impact the final 26 Al t abundance. The only reactions that we found to influence the 26 Al t yield but are not listed in Tab. 4 are 12 C( 12 C,p) 23 Na and 12 C( 12 C,α) 20 Ne. Varying these rates by a factor of 10 changes the 26 Al t abundance by a factor of ≈2. However, these reactions drive carbon burning and thus strongly influence the nuclear energy generation. Therefore, varying this rate in a post-processing study is not very meaningful. Nevertheless, the effect appears to be relatively small.
Reaction rate uncertainties
Of the 14 reactions listed in Tab. 4, the rates of: (i) 5 reactions are available from the Monte Carlo procedure ; § 2.2); (ii) 3 reactions are adopted from the statistical model (Rauscher & Thielemann 2000); (iii) 3 reactions are obtained from KADoNiS v0.2 (Dillmann et al. 2006); and (iv) 2 reactions are adopted from NACRE ). Only the rates of the 12 C( 12 C,n) 23 Mg reaction are partially based (i.e., the total 12 C+ 12 C rate) on the information provided in Caughlan & Fowler (1988), see Appendix C.1.
Reaction rate uncertainties are listed for a temperature of T = 1.4 GK, near the end of the burning, in the last column of Tab. 4, when reported in the original source. The uncertainties for 23 Na(p,α) 20 Ne, 25 Mg(p,γ) 26 Al t , 16 O(α,γ) 20 Ne, 26 Mg(p,γ) 27 Al and 27 Al(p,α) 24 Mg are relatively small and, therefore, the rate estimates for these reactions seem sufficiently reliable at present. The 26 Mg(α,n) 29 Si and 25 Mg(α,n) 28 Si reactions are listed with rather large rate uncertainties (29% and 59%, respectively, according to ) and, therefore, should be addressed in future work (see also Appendix C.3). No rate uncertainties are given for any of the other reactions listed in the table. These rates are derived, for example, from Hauser-Feshbach theory (Rauscher & Thielemann 2000) or from the KADoNiS v0.2 evaluation (Dillmann et al. 2006) and uncertainties are difficult to quantify, as has already been discussed in § 3.2.3. Clearly, more reliable experimental rates for these reactions are urgently needed. Specific comments on the reactions 26 Al t (n,p) 26 Mg, 26 Al t (n,α) 23 Na, 23 Na(α,p) 26 Mg and 12 C( 12 C,n) 23 Mg can be found in Appendix C.
Thermal equilibration
So far we assumed a single species of 26 Al, implying thermal equilibrium ( 26 Al t ). We will now follow the equilibration numerically in the network calculation. Five different species of 26 Al are incorporated into the network, as explained in § 2.3. The required γ-and β-decay transitions between and from these levels are discussed in detail in Appendix A. The additional reaction rates, involving 26 Al g and 26 Al m separately, are discussed in Appendix B.
As a first step, a standard post-processing network calculation (with recommended rates) is performed using the same temperature-density-time evolution as before (Fig. 2). The final 26 Al g abundance, when the 12 C mass fraction has fallen to a value of X f ( 12 C) = 0.10, is found to be identical to our earlier result obtained assuming thermal equilibrium (Tab. 1). Thus the latter assumption seems to be justified. An impression can be gained Table 4 Factor changes of final 26 Al t abundance resulting from reaction rate variations for convective shell C/Ne burning a , assuming thermal equilibrium for 26 26 Al t (n,α) 23 Na 0.54 0.79 · · · · · · present 16 O(α,γ) 20 Ne · · · 0.83 1.3 1.7 il10 14% 25 Mg(α,n) 28 Si 0.42 · · · · · · · · · nacr 59% 12 C( 12 C,n) 23 Mg 0.46 · · · · · · · · · da77 27 Al(n,γ) 28 Al 1.7 · · · · · · · · · ka02 25 Mg(n,γ) 26 Mg 1.3 · · · · · · · · · ka02 26 Mg(p,γ) 27 Al 0.71 · · · · · · · · · il10 5% 27 Al(p,α) 24 Mg 0.79 · · · · · · · · · il10 7% a The temperature-density-time profile is extracted from a stellar evolution calculation of a 60M ⊙ star with initial solar metallicity, see Limongi & Chieffi (2006). b In total, the rates of 66 different reactions were varied. Listed are only those reactions whose rate changes have the strongest effect on the 26 Al t yield. All other rate changes, as well as those labeled by "...", produced abundance changes of less than 20%. The reactions are listed in approximate order of importance. Thermal equilibrium for 26 Al has been assumed, i.e., the network contains only a single species, 26 Al t . c Reaction rate references: (nacr) Angulo et al. 1999 (NACRE); (ka02) Dillmann et al. (2006) (KADoNiS v0.2); (rath) Rauscher & Thielemann (2000); (il10) Iliadis et al. (2010); (present) hybrid rate, see Appendix C.5 and C.6; (da77) total 12 C+ 12 C rate from Caughlan & Fowler (1988), with neutron branching ratio adopted from Dayras et al. (1977), see Appendix C.1. d Reaction rate uncertainty near a temperature of 1.4 GK, at the end of the calculation; no entry implies that the rate uncertainty is difficult to quantify (see text). from Fig. 9, showing the abundance evolution of 26 Al levels. The top part displays the abundances for individual 26 Al species and it is apparent that at any given time the 26 Al g abundance dominates over those of the other species. The bottom part displays the fraction of the total 26 Al abundance that resides in the isomeric state. This curve is directly obtained from the network calculation, but is indistinguishable from the one calculated assuming a Boltzmann distribution (i.e., thermal equilibrium).
Subsequently, the rates of 20 different interactions, together with their inverse processes, were varied individually by factors of 100, 10, 2, 0.5, 0.1 and 0.01. This list contained all nuclear reactions that produced and destroyed 26 Al g or 26 Al m . It also included those β-and γ-ray decay rates of 26 Al x levels that were estimated using the shell model, as described in § 2.3 and shown in Fig. 1. Experimentally obtained β-and γ-ray decay rates have not been varied since their uncertainties are very small. The final 26 Al g abundance, after each rate variation, was then compared to the standard calculation. The results are listed in Tab. 5. It can be seen that of the 20 interactions only five, all of them reactions, influence the final 26 Al g yield. In other words, even a variation by a factor of 100 of the shell-model based β-and γ-ray decay rates seems to have no effect on the 26 Al g abundance.
The five reactions displayed in Tab. 5 are listed in approximate order of importance. For example, increasing the 25 Mg(p,γ) 26 Al g rate by a factor of 2 will enhance the final 26 Al yield by ≈50%. However, this rate is based on experimental information and their Monte Carlo uncertainty is predicted to amount to only 5% near T = 1.4 GK. As was the case for explosive Ne/C burning ( § 3.2.4), we found that even a factor of 100 variation in the rates of these five reactions has no impact on the thermal equilibrium abundance ratio of 26 Al m and 26 Al g (Fig. 9). Comparison of the factor changes listed in Tab. 5 with those of Tab. 4 reveals that these five reactions impact the final 26 Al abundance by similar amounts, no matter if a single (thermalized) or five species of 26 Al are used in the simulation. In conclusion, 26 Al is in thermal equilibrium during convective shell C/Ne burning and, therefore, there is no need to introduce the extra complication of five 26 Al levels, and their mutual interactions, into the reac-tion network.
Convective core H burning
Standard calculation
We have performed post-processing calculations using the temperature-density profile for convective core H-burning in an 80 M ⊙ star of solar initial composition (Limongi & Chieffi 2006). Following the procedure adopted for convective shell C/Ne burning, we have artificially shortened the burning time in our calculation so that the time evolution of the hydrogen fuel would closely follow that for the stellar evolution calculation. For this profile, compressing the time axis by a factor of 17 gave consistent results, as shown in Fig. 10. Burning starts at X i ( 1 H) = 0.70, T = 0.044 GK and ρ = 2.03 g/cm 3 . The modified profile extends for t = 5.85 × 10 12 s, at which time X f ( 1 H) = 1.4 × 10 −6 , T = 0.088 GK and ρ = 17.9 g/cm 3 . Note, that in this case T and ρ refer to the values at the center of the star.
Our standard calculation assumes that 26 Al g and 26 Al m are distinct species. The net abundance flows are shown in Fig. 11 and not surprisingly, the strongest are within the CNO cycles. Both the ground state of 26 Al and the isomeric level are produced via the 25 Mg(p,γ) 26 Al reaction. For most of the burning period, 26 Al is produced from the reservoir of initial 25 Mg and it is not until the very late stages of burning that 25 Mg is replenished through 24 Mg(p,γ) 25 Al and the subsequent β-decay of 25 Al. The primary destruction route for the isomer is β-decay to 26 Mg, whereas the ground state is destroyed via 26 Al g (p,γ) 27 Si. The abundance evolution of 26 Al g and 27 Al is shown in Fig. 12. The abundance of 27 Al is essentially constant at X( 27 Al) = 6.1 × 10 −5 until late times while 26 Al g grows to a maximum of X( 26 Al g ) = 5.2 × 10 −5 before dropping to X( 26 Al g ) = 4.9 × 10 −5 at the end of burning (see Tab. 1). The ratio 26 Al g / 27 Al shows a similar behavior, reaching a maximum of 0.73 with a final value of 0.45.
Reaction rate variations, thermal equilibration and uncertainties
The rates of 26 pairs of forward and reverse reactions were varied and those reactions whose rate changes have the strongest effect on the final abundance of 26 Al g (i.e., at the end of the calcu- Table 5 Factor changes of final 26 Al g abundance resulting from reaction rate variations for convective shell C/Ne burning a , assuming five species of 26 26 Al m 6.7 3.0 · · · · · · 0.75 0.71 il10 6% 26 Al g (n,α) 23 Na 0.12 0.54 · · · · · · · · · · · · present 26 Al m (n,p) 26 Mg 0.58 · · · · · · · · · · · · · · · present a The temperature-density-time profile is extracted from a stellar evolution calculation of a 60M ⊙ star with initial solar metallicity, see Limongi & Chieffi (2006). b In total, the rates of 20 different reactions producing or destroying 26 Al g and 26 Al m were varied. Listed are only those reactions whose rate changes have the strongest effect on the 26 Al g yield. All other rate changes, as well as those labeled by "...", produced abundance changes of less than 20%. The reactions are listed in approximate order of importance. No thermal equilibrium for 26 Al has been explicitly assumed, i.e., the network contains five different species ( 26 Al g , 26 Al m , 26 Al a , 26 Al b , 26 Al c ) and takes the interactions between them into account. c Reaction rate references: (il10) Iliadis et al. 2010;(present) hybrid rate, see Appendix C.5 and C.6. In the latter two cases, we assumed that the rate involving 26 Al g or 26 Al m is the same as the rate for 26 Al t (see comments in Appendix B). d Reaction rate uncertainty near a temperature of 1.4 GK, at the end of the calculation; no entry implies that the rate uncertainty is difficult to quantify. Fig. 11.-Net abundance flows, obtained for a post-processing network calculation of convective core H burning, integrated over a total running time of t = 5.9 × 10 12 s, when the 1 H mass fraction has decreased to 1.3 × 10 −6 . The T-ρ profile for this simulation is shown in Fig. 10. The network consists of all nuclides shown as squares. The strongest net abundance flows, i.e., those within two, four, and six orders of magnitude of the maximum flow, are displayed by the thickest arrows, arrows of intermediate thickness, and the thinnest arrows, respectively. lation, when X H = 1.3 × 10 −6 ) are listed in Tab. 6. All other rate changes, as well as those labeled by "..." in the table, changed the 26 Al g abundance by less than 20%. The reactions are listed in approximate order of importance, as measured by their impact on the final 26 Al g abundance. The last two columns display the source of the rate and the reported rate uncertainty at a temperature of ≈ 0.09 GK, near the end of the calculation. For these calculations, 26 Al g and 26 Al m are considered to be separate species. It is not surprising that the 25 Mg(p,γ) 26 Al g reaction has the largest impact on the final 26 Al g abundance, but it is interesting that variations in the rate of 25 Mg(p,γ) 26 Al m also affect the 26 Al g abundance. This is because these two reactions are the only significant destruction mechanisms for 25 Mg and thus 25 Mg(p,γ) 26 Al m lowers the amount of 25 Mg available to be converted to 26 Al g . The third reaction, 26 Al g (p,γ) 27 Si, is the major destruction route for 26 Al g . Finally, the 16 O(p,γ) 17 F reaction affects 26 Al production by reducing the abundance of free protons. At no point during the calculation did the temperature reach a point where 26 Al g and 26 Al m could communicate through thermal excitations. This was verified by including the three mediating levels discussed in § 2.3 in a second series of network calculations and no change was seen in the final abundance of 26 Al g .
Core H burning
The rates for all of the reactions listed in Tab. 6 were obtained using the Monte Carlo procedure ; § 2.2) and for the temperatures encountered in convective core H-burning, all of these are based on experimental data. The uncertainties quoted are at the 1-σ level for lognormal probability density functions. Judging from the entries in Tab. 6, none of these reactions will impact the abundance of 26 Al g if their rates are varied within a factor of 2 from their recommended values. Given the quoted uncertainties, a factor of 2 corresponds to confidence intervals of 97.9% for 25 Mg(p,γ) 26 Al g , 98.4% for 25 Mg(p,γ) 26 Al m , 99.3% for 26 Al g (p,γ) 27 Si and ≈ 100% for 16 O(p,γ) 17 F. In other words it is unlikely that the rates for any of these reactions will be a factor of 2 away from the recommended values. This points to the utility of uncertainties with statistical significance. Therefore, we conclude that the rates for the reactions that determine the abundance of 26 Al g during convective core H-burning are known with sufficient precision.
Summary
We presented a comprehensive investigation of the impact of nuclear reaction rate uncertainties on 26 Al production in massive stars. In such stars, 26 Al is likely produced in three distinct sites: (i) during core collapse via explosive Ne/C burning; (ii) during pre-supernova stages in the C/Ne convective shell, where a fraction of the 26 Al survives the subsequent explosion and is ejected into the interstellar medium; and (iii) in Wolf-Rayet stars, that experience such a strong mass loss that even layers located within the H convective core, hence significantly enriched in 26 Al, are ejected into the interstellar medium. These 26 Al production mechanisms were recently analyzed in detail by Limongi & Chieffi (2006). From their stellar evolution models, we extracted representative temperature-density-time profiles and executed a large number of post-processing reaction network sensitivity calculations. The general strategy consisted of varying the rates of many reactions individually by different factors (in this work, 10, 2, 0.5 and 0.1) and to analyze the impact of each individual reaction rate change on the final 26 Al yields. Our results are important for quantifying the influence of current reaction rate uncertainties on predicted 26 Al yields, and for the motivation of future laboratory measurements.
There are a number of novel aspects about the present work. First, we employed a newgeneration library of nuclear reaction and weak interaction rates, called STARLIB. This library contains a recent evaluation of experimental Monte Carlo reaction rates . Besides recommended reaction rates for a grid of temperature values between 1 MK and 10 GK, the library includes in addition for many reactions the rate uncertainty factor at each temperature. This work represents the first application of STAR-LIB. Second, we carefully investigate the equilibration effects of 26 Al. At least two species of 26 Al take part in the nucleosynthesis, the ground state and the isomeric state. In all previous massive star investigations, either a single species or two species of 26 Al were taken into account, depending on whether thermal equilibrium is achieved or Table 6 Factor changes of final 26 Al g abundance resulting from reaction rate variations for convective core H burning a , assuming two species of 26 Al
Reaction b
Rate multiplied by 100 10 2 0.5 0.1 0.01 Source c Uncertainty d 26 Al g (p,γ) 27 Si 0.0035 0.55 · · · · · · · · · · · · Il10 31% 25 Mg(p,γ) 26 Al g 1.20 1.20 · · · · · · 0.33 0.039 Il10 35% 25 Mg(p,γ) 26 Al m 0.049 0.37 · · · · · · 1.20 1.20 Il10 35% 16 O(p,γ) 17 F · · · · · · · · · · · · · · · 1.70 Il10 7%
a The temperature-density-time profile is extracted from a stellar evolution calculation of a 80M ⊙ star with initial solar metallicity, see Limongi & Chieffi (2006). b In total, the rates of 26 different reactions were varied. Listed are only those reactions whose rate changes have the strongest effect on the 26 Al g yield. All other rate changes, as well as those labeled by "...", produced abundance changes of less than 20%. The reactions are listed in approximate order of importance.
c Reaction rate reference: (il10) Iliadis et al. (2010). d Reaction rate uncertainty near a temperature of 0.09 GK, at the end of the calculation. The T -ρ profile for this post-processing network simulation is shown in Fig. 2. The calculation assumes a single species of (thermalized) 26 Al.
not. These are two extreme assumptions and in a hot stellar plasma the ground and isomeric state may "communicate" via γ-ray transitions involving higher-lying 26 Al levels. Some of our results are summarized in Tab. 7, listing those nuclear reactions that significantly impact 26 Al synthesis in massive stars. The reactions are listed in approximate order of importance. The reader should consult Tabs. 2, 4 and 6 for detailed results. Particularly the first five reactions, 26 Al(n,p) 26 Mg,25 Mg(α,n) 28 Si, 24 Mg(n,γ) 25 Mg, 23 Na(α,p) 26 Mg and 26 Al(n,α) 23 Na, should be prime targets for future measurements. The approximate temperature range near which the rate needs to be improved (≈ 2.3 GK for explosive Ne/C burning, ≈ 1.4 GK for convective shell C/Ne burning), as well as the current literature source of a particular rate, is also given in the table. For those five reactions we argued in § 3.2.3, § 3.3.3 and Appendix C that the current rate uncertainties at astrophysically important temperatures amount to about a factor of 2. The sensitivity of 26 Al production to rate variations of these reactions can be estimated from Tabs. 2 and 4: a factor of 2 variation in 26 Al abundance that resides in the isomeric state. The curve is indistinguishable from the one calculated assuming a Boltzmann distribution (i.e., thermal equilibrium). their rates changes the final 26 Al mass fraction by factors of 1.7, 1.9, 1.6, 1.3 and 1.3, respectively. Thus we conclude that the uncertainty of the 26 Al yield predicted by the massive star models explored here amounts to about a factor of 3. This result is obtained on the basis of nuclear physics uncertainties alone and should be considered together with other uncertainties inherent in the stellar models, such as mixing, mass loss and rotation. We do not list any reactions for core H burning in Tab. 6, mainly because in this case reaction rate variations have only a small effect on the 26 Al yield. Here, the most important reaction is 26 Al(p,γ) 27 Si, but even a factor of 10 change in this rate near ≈ 90 MK has only a modest impact on the 26 Al yield (see Tab. 6). Of course, new experimental results for 26 Al(p,γ) 27 Si are useful in any case.
We carefully examined the issue of 26 Al equilibration for each of the three nucleosynthesis sites mentioned above. Two series of post-processing calculations were performed and the resulting 26 Al yields were compared: one assuming either a single or two separate 26 Al species, depending on the temperature regime, and one where the communication between ground and isomeric states was explicitly taken into account. For the latter case, no artificial assumptions about the equilibration of 26 Al are made, but additional 26 Al species (i.e., levels at 417, 1058 and 2070 keV; see Fig. 1) were taken into account in the reaction network. We found that the equilibration of 26 Al levels in any of the massive star sites investigated here has only minor effects on the 26 Al yields. The reason is that in explosive Ne/C burning and convective shell C/Ne burning the temperatures are sufficiently high to ensure thermal equilibration of 26 Al, while in core H burning the temperatures are never high enough to facilitate communication of the ground and isomeric state via thermal excitations. We also verified that current uncertainties in some unmeasured 26 Al γ-ray transition rates do not significantly impact the predicted nucleosynthesis yields.
For the interested reader we provide detailed comments on the status of certain reactions, including 12 C( 12 C,n) 23 Mg,23 Na(α,p) 26 Mg,25 Mg(α,n) 28 Si, 26 Al m (p,γ) 27 Si, 26 Al(n,p) 26 Mg and 26 Al(n,α) 23 Na. For the latter two, particularly important, reactions we provide new rate estimates, which will be presented in more detail in a forthcoming publication (Oginni et al., in print).
This work was supported in part by the U.S. Department of Energy under Contract No. DE-FG02-97ER41041.
Table 7
Summary of nuclear reactions that impact 26 Al production in massive stars a , assuming thermal equilibrium for 26 Al
Reaction
Site b Temperature c Source d 26 Al t (n,p) 26 b Site of 26 Al synthesis in massive star; the labels "xNe/C" and "C/Ne" refer to explosive Ne/C burning and convective shell C/Ne burning, respectively. c Temperature (in units of GK) near which most of 26 Al production occurs in given site. The T -ρ profile for this post-processing network simulation is shown in Fig. 6. The calculation assumes a single species of (thermalized) 26 Al. Time increases from left to right. 26 Al levels via γ-ray transitions is explicitly taken into account. (Top) Abundance evolution of different 26 Al species; (Bottom) Numerically simulated fraction of total 26 Al abundance that resides in the isomeric state. The curve is indistinguishable from the one calculated assuming a Boltzmann distribution (i.e., thermal equilibrium). 10 -3 10 6 10 7 10 8 10 9 10 10 10 11 10 12 10 13
Mass Fraction
Time (s) 27 Al 26 Al g Core H burning Fig. 12.-Abundance evolution (by mass) of 26 Al g and 27 Al during convective core H burning.
The T-ρ profile for this post-processing simulation is shown in Fig. 10.
A. β-AND γ-DECAY RATES OF 26 AL LEVELS
The decay constants for β-and γ-decay of 26 Al levels, in units of s −1 , are listed in Tab. 8 for the temperature range of relevance in the present work. The labels 26 Al g , 26 Al m , 26 Al a , 26 Al b and 26 Al c refer to the levels at E x = 0 keV (5 + ; ground state), 228 keV (0 + ; isomeric state), 417 keV (3 + ), 1058 keV (1 + ) and 2070 keV (2 + ), respectively (see Fig. 1).
The entries in the second column refer to the β-decay of 26 Al t (ground and isomeric state in thermal equilibrium) to the daughter 26 Mg. The decay constant is only listed for temperatures above T = 0.4 GK since for lower temperatures thermal equilibrium is not achieved. The values are calculated from λ( 26 Al t → 26 Mg) = 9.93 × 10 −3 e −2.646/T9 s −1 , where T 9 is the temperature in GK (see Ward & Fowler 1980, Iliadis 2007). This expression, which takes only the ground and isomeric state into account, is valid for temperatures and densities below 5 GK and 10 6 g/cm 3 , respectively. In the temperature and density regimes considered here, the results are in good agreement with the more extensive calculations of Oda et al. (1994). Note that the original REACLIB fit of this particular decay rate is off by ≈20-80% at T = 1 − 3 GK.
The following β-decay constants are not listed in the table since they are constant for the temperature grid shown here:
λ( 26 Al g → 26 Mg)=3.069×10 −14 s −1 λ( 26 Al m → 26 Mg)=1.092×10 −1 s −1 λ( 26 Al a → 26 Mg)=1.283×10 −4 s −1 λ( 26 Al b → 26 Mg)=9.630×10 −2 s −1
The first two values are computed from measured laboratory half-lifes (Audi et al. 2003), while the latter two values are obtained from shell model calculations (Kajino et al. 1988).
The following γ-ray decay constants are not listed in the table since they are nearly constant for the temperature grid shown here: 26 Al a )=1.070×10 13 s −1 λ( 26 Al c → 26 Al b )=3.770×10 13 s −1 All γ-ray decay constants given above and listed in Tab. 8 are calculated from experimental information (Endt 1990), except those connecting the levels 26 0.04 · · · 1.12E-44 5.56E+08 7.11E-25 6.24E-02 · · · · · · · · · · · · · · · 0.05 · · · 3.43E-34 5.56E+08 4.07E-20 6.24E-02 · · · · · · · · · · · · · · · 0.06 · · · 3.45E-27 5.56E+08 6.05E-17 6.24E-02 · · · · · · · · · · · · · · · 0.07 · · · 3.46E-22 5.56E+08 1.11E-14 6.24E-02 · · · · · · 2.21E-38 · · · · · · 0.08 · · · 1.95E-18 5.56E+08 5.58E-13 6.24E-02 4.45E-39 · · · 1.30E-32 · · · · · · 0.09 · · · 1.62E-15 5.56E+08 1.17E-11 6.24E-02 2.86E-33 · · · 3.97E-28 · · · 1.37E-43 0.10 · · · 3.49E-13 5.56E+08 1.33E-10 6.24E-02 1.27E-28 · · · 1.54E-24 · · · 6.36E-38 0.11 · · · 2.83E-11 5.56E+08 9.77E-10 6.24E-02 8.03E-25 · · · 1.33E-21 · · · 2.75E-33 0.12 · · · 1.10E-09 5.56E+08 5.14E-09 6.24E-02 1.18E-21 · · · 3.73E-19 · · · 2.01E-29 0.13 · · · 2.45E-08 5.56E+08 2.09E-08 6.24E-02 5.67E-19 · · · 4.40E-17 · · · 3.72E-26 0.14 · · · 3.50E-07 5.56E+08 6.98E-08 6. Note.- Table 8 is published in the electronic edition of the Astrophysical Journal.
λ( 26 Al b → 26 Al m )=2.780×10 13 s −1 λ( 26 Al c → 26 Al m )=1.500×10 12 s −1 λ( 26 Al b → 26 Al a )=7.240×10 8 s −1 λ( 26 Al c →
Table 9 Stellar reaction rates a N A σv involving 26 Al (in cm 3 mol −1 s −1 ) T (GK) 25 Mg(p,γ) 26 Al t 25 Mg(p,γ) 26 Al g 25 Mg(p,γ) 26 Al m 26 Al t (p,γ) 27 Si 26 Al g (p,γ) 27 Si 26 Al t (n,p) 26 Mg 26 Al t (n,α) 23 Na a The rates for only some reactions are listed; other reactions are discussed in the text. All rates given here (except column 6) account for thermal target excitations.
il10 b il10 b il10 b il10 b il10 b,c present d,
b Experimental Monte Carlo rates of Iliadis et al. (2010). c Same rate is used for 26 Al m (p,γ) 27 Si. d Hybrid rate: at T ≤ 0.2 GK from experiment of ; at T > 0.2 GK from Hauser-Feshbach model of Rauscher & Thielemann (2000).
C. DISCUSSION OF SPECIFIC REACTION RATES
The references of the reaction rates used in the present work are provided in the tables above and the reader is referred to these sources for details. Decays and reactions involving levels of 26 Al are discussed in Appendices A and B, respectively. For some specific reaction rates that are neither evaluated in nor in Angulo et al. (1999), we provide more information below.
C.1. 12 C( 12 C,n) 23 Mg (Q = −2.598 MeV)
We calculated the reaction rate from the total 12 C+ 12 C rate (i.e., summed over all exit channels) and the neutron branching ratio. For T = 1.25 GK, representing the average temperature of convective shell C/Ne burning of the 60 M ⊙ model explored in the present work, the Gamow peak extends over a center-of-mass energy range from 2.6 MeV (i.e., the threshold energy for this endothermic reaction) to about 3.4 MeV. In this energy region the total 12 C+ 12 C cross section has been measured (Costantini et al. 2009, and references therein). For the neutron branching ratio, as a function of temperature in the range of T = 0.5 − 5.0 GK, we adopted the values from Dayras, Switkowski and Woosley (1977). In the latter work, the 12 C( 12 C,n) 23 Mg reaction has been measured down to an energy of 3.54 MeV and, with the aid of statistical model calculations, the results were extrapolated down to the threshold energy in order to extract the neutron branching ratio. The overall reaction rate uncertainty is difficult to quantify at present. Clearly, a new measurement of the 12 C( 12 C,n) 23 Mg reaction at lower energies is desirable.
The 12 C( 12 C,n) 23 Mg reaction rates employed in various rate libraries are inconsistent with each other. An impression can be obtained from Fig. 13. The black line represents the ratio of the rate used in the Basel version of the REACLIB (nucastro.org/reaclib.html#reaclib) to the present rate. The hatched area marks the temperature range of convective shell C/Ne burning explored in the present work. Surprisingly, the ratio amounts to more than an order of magnitude near T = 1.25 GK. The reason may be that the 12 C( 12 C,n) 23 Mg rate in the original REACLIB library was erroneously obtained using a neutron branching ratio of zero for T < 1.75 GK that is listed in Caughlan & Fowler (1988). The blue line in Fig. 13 represents the ratio of the rate used in the MSU version of the REACLIB (groups.nscl.msu.edu/jina/reaclib/db/) to the present rate. The ratio amounts to a factor of 2 near T = 1.25 GK. The reason may perhaps be that their fitted rate deviates from the actual rate. For the interested reader, we provide below a rate fit (in REACLIB format) that reproduces the present and best currently available rate within 10%:
-0.527166E+02 -0.344948E+02 0.140849E+03 -0.878184E+02 0.371377E+01 -0.338673E+00 0.736030E+02 C.2. 23 Na(α,p) 26 Mg (Q = 1.821 MeV)
A direct measurement of this reaction has been reported in Whitmire & Davids (1974). They bombarded a target, fabricated by evaporating NaCl onto a thick Cu backing, with α-particles (E cm = 2.0 − 3.1 MeV) and measured the emitted protons populating the ground and first excited states in 26 Mg. In total, they report the strengths of 39 resonances. There are a number of reasons that warrant a re-measurement of this reaction. Most importantly, the strengths have been determined relative to an absolute strength measurement for the E lab = 3051 keV resonance, assuming that the stoichiometry of their NaCl target amounts to 1:1. This issue has been discussed in detail by Rowland et al. (2002) in connection with the 23 Na(p,α) 20 Ne reaction, where it was shown that during proton bombardment a NaCl target quickly changes its stoichiometry to 5:3, resulting in a significant change in thick-target yield (and in the derived resonance strength). This problem is certainly aggravated when using an α-particle beam incident on a NaCl target. There were other problems in the -Reaction rate ratio for 12 C( 12 C,n) 23 Mg: (black line) ratio of "Basel" rate to present rate; (blue line) ratio of "MSU" rate to present rate. The hatched region marks the temperature range of convective shell C/Ne burning explored in the present work. (blue) Anderson et al. (1983); (black) Wieland (1995). For the first two references, the data were extracted from the published figures and converted from cross sections to S-factors. For the latter reference, the Sfactors are adopted from the NACRE evaluation ). Uncertainty bars have been omitted for reasons of clarity. The hatched horizontal bar marks the region of the Gamow peak near T = 2.3 GK, the peak temperature achieved in explosive Ne/C burning. (2000). The first two rates are based on experimental results, while the latter rate is estimated using the Hauser-Feshbach model. Beyond the vertical line, near T ≈ 0.26 GK, the experimental rate of De represents a lower limit. Note that for this comparison only, the rates represent "laboratory rates", i.e., they do not account for thermal target excitations. ; (blue solid line) Rauscher & Thielemann (2000); squares (Trautvetter et al. 1986). The first (experimental) rate only takes the transition to the first excited state in 26 Mg into account, while the third (experimental) rate represents the combined transitions to the ground and first excited states in 26 Mg. The second rate is estimated using the Hauser-Feshbach theoretical model and includes transitions to all possible final states. Note that for this comparison only, the rates represent "laboratory rates", i.e., they do not account for thermal target excitations.
Fig. 1 .
1-Level scheme of
d
Solar system mass fractions, for comparison (from Lodders 2003).
Fig. 2 .
2-Temperature-density-time evolution for explosive Ne/C burning. The profile was obtained from a hydrodynamic model of a 20M ⊙ star
Fig. 4 .
4-Abundance evolution (by mass) of26 Al t and 27 Al during explosive Ne/C burning.
Fig. 5 .
5-Abundance evolution (by mass) of26 Al during explosive Ne/C burning. The T -ρ profile for this post-processing network simulation is shown inFig. 2. The calculation assumes five species of 26 Al: ground state (g), isomeric state (m), and three excited levels (a, b, c); seeFig. 1. The communication of the different 26 Al levels via γ-ray transitions is explicitly taken into account. (Top) Abundance evolution of different 26 Al species; (Bottom) Numerically simulated fraction of total
Fig. 6 .
6-Temperature-density evolution for convective shell C/Ne burning. The results were obtained from a model of a 60M ⊙ star(Limongi & Chieffi 2006), but the time scale is shortened in the present work (see text). The profile approximates the evolution of the hottest and deepest zone of the C/Ne convective shell. For comparison, the circles indicate the temperature and density values that are directly obtained from the stellar evolution calculations. Time increases from left to right.
d
For reference labels, see Tabs. 2 and 4.
Fig. 8 .
8-Abundance evolution (by mass) of26 Al t and 27 Al during convective shell C/Ne burning.
Fig. 9 .
9-Abundance evolution (by mass) of26 Al during convective shell C/Ne burning. The T -ρ profile for this post-processing network simulation is shown inFig. 6. The calculation assumes five species of 26 Al: ground state (g), isomeric state (m), and three excited levels (a, b, c); seeFig. 1. The communication of the different
Fig. 10 .
10-Temperature-density evolution for convective core H burning. The results were obtained from a model of a 80M ⊙ star(Limongi & Chieffi 2006), but the time scale is shortened in the present work (see text). The circles indicate the temperature and density values that are directly obtained from the above stellar evolution calculations. Time increases from left to right.
Al m ↔ 26 Al a and 26 Al a ↔ 26 Al b , which have been found from shell model calculations (Runkle, Champagne & Engel 2001; see also Fig. 1).
Fig. 13.-Reaction rate ratio for 12 C( 12 C,n) 23 Mg: (black line) ratio of "Basel" rate to present rate; (blue line) ratio of "MSU" rate to present rate. The hatched region marks the temperature range of convective shell C/Ne burning explored in the present work.
Fig. 14 .
14-Experimental astrophysical S-factors for 25 Mg(α,n) 28 Si: (red)Van der Zwan and Geiger (1981);
Fig. 15 .
15-Reaction rates for 26 Al(n,α) 23 Na: (black solid line) De Smet et al. (2007); (dashed line) Koehler et al. (1997); (blue solid line) Rauscher & Thielemann
Fig. 16 .
16-Reaction rates for26 Al(n,p)26 Mg: (dashed line)
Al Reaction b
Rate multiplied by
10
2
0.5
0.1
Source c Uncertainty d
25 Mg(α,n) 28 Si
0.10 0.49 1.8
4.0
nacr
18%
24 Mg(n,γ) 25 Mg 5.2
1.6
0.61 0.24 ka02
26 Al t (n,p) 26 Mg 0.14 0.58 1.6
3.2
present
25 Mg(p,γ) 26 Al t 1.7
1.4
0.58 0.14 il10
4%
30 Si(p,γ) 31 P
0.51 0.77 1.3
2.0
il10
14%
20 Ne(α,γ) 24 Mg 1.8
1.4
0.64 0.28 il10
11%
27
Mg(n,γ) 25 Mg xNe/C; C/Ne ≈ 2.3; ≈ 1.4 ka02 23 Na(α,p)26 Mg C/Ne ≈ 1.4 rath26 Al t (n,α) 23 Na xNe/C; C/Ne ≈ 2.3; ≈ 1.4 present 27 Al(α,p) In approximate order of importance; for full results, see Tabs. 2, 4 and 6.Mg xNe/C; C/Ne
≈ 2.3; ≈ 1.4
present
25 Mg(α,n) 28 Si
xNe/C
≈ 2.3
nacr
24 30 Si
xNe/C
≈ 2.3
rath
29 Si(α,n) 32 S
xNe/C
≈ 2.3
rath
26 Mg(α,n) 29 Si
C/Ne
≈ 1.4
nacr
a
Table 8
8Decay constants of26 Al levels (in s −1 ) T (GK)26 Al t → 26 Mg 26 Al g → 26 Al a 26 Al a → 26 Al g 26 Al m → 26 Al a 26 Al a → 26 Al m 26 Al m → 26 Al b 26 Al m → 26 Al c 26 Al a → 26 Al b 26 Al a → 26 Al c 26 Al b → 26 Al c
Assuming that the reaction rate probability density function can be approximated by a lognormal distribution, it can be shown) that for a coverage probability of 68% the lognormal spread parameter is given by
The reader may suspect circular reasoning in our arguments, in the sense that we used the same rates for the ground and isomeric states as for the thermalized26 Al target, and then conclude that26 Al is in thermal equilibrium. However, this is not the case since our assumption for the nominal rates serves as a starting point only and we fully explore individual rate changes by factors up to 100. For more information, see App. B.
This 2-column preprint was prepared with the AAS L A T E X macros v5.2.
We argued in § 2.3 that it is important to ensure internal consistency of the rates used. For example, for the first three listed reactions ( 25 Mg+p) the rates are based on the same nuclear physics input and are thus consistent. Monte Carlo rates from Iliadis et al. 2010); "rath" (theoretical Hauser-Feshbach rates from Rauscher & Thielemann 2000); "present" (hybrid rate, see below). Similar arguments apply to the following two reactions ( 26 Al x +psecond row lists the source of the rates: "il10" (experimental Monte Carlo rates from Iliadis et al. 2010); "rath" (theoretical Hauser-Feshbach rates from Rauscher & Thielemann 2000); "present" (hybrid rate, see below). We argued in § 2.3 that it is important to ensure internal consistency of the rates used. For example, for the first three listed reactions ( 25 Mg+p) the rates are based on the same nuclear physics input and are thus consistent. Similar arguments apply to the following two reactions ( 26 Al x +p).
Note that these two rates were predicted by Caughlan & Fowler (1988) to be similar within a factor of ≈5. Our assumption should be regarded as a starting point for exploring the effects of 26 Al m (p,γ) 27 Si reaction rate variations. In Angulo, However, the situation for the 26 Al m (p,γ) 27 Si reaction is a different matter. Rates have been estimated in Caughlan & Fowler (1988). In the absence of a better procedure, we approximated the 26 Al m (p,γ) 27 Si rate by the (experimental) ground state rate (column 6 in Tab. For the 26 Al t (n,α) 23 Na reaction we use a hybrid rate, which is based on experimental information fromHowever, the situation for the 26 Al m (p,γ) 27 Si reaction is a different matter. Rates have been estimated in Caughlan & Fowler (1988) and in Angulo et al. (1999), while initial experimental studies are reported in Deibel et al. (2009) and Lotay et al. (2009). There can be no doubt that this rate is highly uncertain at present (see Appendix C.4). In the absence of a better procedure, we approximated the 26 Al m (p,γ) 27 Si rate by the (experimental) ground state rate (column 6 in Tab. 9). Note that these two rates were predicted by Caughlan & Fowler (1988) to be similar within a factor of ≈5. Our assumption should be regarded as a starting point for exploring the effects of 26 Al m (p,γ) 27 Si reaction rate variations. For the 26 Al t (n,α) 23 Na reaction we use a hybrid rate, which is based on experimental information from
The predicted stellar enhancement factors are relatively small (43% at 2.5 GK) and, therefore, we adopt these rates also for the 26 Al g (n,α) 23 Na reaction. Furthermore, in order to ensure internal consistency, we approximated the 26 Al m (n,α) 23 Na rate by the thermalized rate. Note that the thermalized and isomeric state rates were predicted by Caughlan & Fowler (1988) to be similar within a factor of ≈7. Again, our assumptions serve as starting points to explore the effects of 26 Al x (n,α) 23 Na reaction rate variations. A similar procedure has been followed for the 26 Al x (n,p) 26 Mg reaction rates (see column 7 of Tab. 9 and Appendix C.6). The rates of the following reactions are not explicitly listed in Tab. De Smet, T ≤ 0.1 GK, and on Hauser-Feshbach results from Rauscher & Thielemann (2000) at higher temperatures (see Appendix C.5; the rate is listed in the last column of Tab. 9. 922Na(α,γ) 26 Al t , 25 Al(n,γ) 26 Al tDe Smet et al. (2007) at T ≤ 0.1 GK, and on Hauser-Feshbach results from Rauscher & Thielemann (2000) at higher temperatures (see Appendix C.5; the rate is listed in the last column of Tab. 9). The predicted stellar enhancement factors are relatively small (43% at 2.5 GK) and, therefore, we adopt these rates also for the 26 Al g (n,α) 23 Na reaction. Furthermore, in order to ensure internal consistency, we approximated the 26 Al m (n,α) 23 Na rate by the thermalized rate. Note that the thermalized and isomeric state rates were predicted by Caughlan & Fowler (1988) to be similar within a factor of ≈7. Again, our assumptions serve as starting points to explore the effects of 26 Al x (n,α) 23 Na reaction rate variations. A similar procedure has been followed for the 26 Al x (n,p) 26 Mg reaction rates (see column 7 of Tab. 9 and Appendix C.6). The rates of the following reactions are not explicitly listed in Tab. 9: 22 Na(α,γ) 26 Al t , 25 Al(n,γ) 26 Al t ,
Al t. Mg, 2627Al t , 29 P(n,α) 26 Al t , 26 Al t (n,γMg(α,p) 26 Al t , 26 Si(n,p) 26 Al t , 29 P(n,α) 26 Al t , 26 Al t (n,γ) 27
Since the stellar enhancement factors are predicted to be small, we also adopted these results for the respective rates involving 26 Al g . Note that the corresponding reactions involving 26 Al m are absent in the original REACLIB. We disregarded these as well, on the grounds that their net abundance flows (for 26 Al t ) in our network calculations are at least 3 orders of magnitude smaller than the maximum flow. Si. For these we used the Hauser-Feshbach rates of Rauscher & Thielemann. Al, 26 Al t (α,γ) 30 P and 26 Al t29except the rates of 26 Al t (n,γ) 27 Al, which were adopted from the KADoNiS v0.2 evaluation (Dillmann et al. 2006). For all forward reactions discussed above, the corresponding reverse reaction rates are also implementedAl, 26 Al t (α,γ) 30 P and 26 Al t (α,p) 29 Si. For these we used the Hauser-Feshbach rates of Rauscher & Thielemann (2000), except the rates of 26 Al t (n,γ) 27 Al, which were adopted from the KADoNiS v0.2 evaluation (Dillmann et al. 2006). Since the stellar enhancement factors are predicted to be small, we also adopted these results for the respective rates involving 26 Al g . Note that the corresponding reactions involving 26 Al m are absent in the original REACLIB. We disregarded these as well, on the grounds that their net abundance flows (for 26 Al t ) in our network calculations are at least 3 orders of magnitude smaller than the maximum flow. For all forward reactions discussed above, the corresponding reverse reaction rates are also implemented
T > 0.1 GK from Hauser-Feshbach model of Rauscher & Thielemann. e Hybrid rate: at T ≤ 0.1 GK from experiment of De Smet ete Hybrid rate: at T ≤ 0.1 GK from experiment of De Smet et al. (2007); at T > 0.1 GK from Hauser-Feshbach model of Rauscher & Thielemann (2000).
Same rate is used for 26 Al g and 26 Al m. f Same rate is used for 26 Al g and 26 Al m .
(i) protons populating higher-lying final states could only be resolved at the highest measured energies; (ii) the uncertainty in the resonance energies is relatively large, amounting to ≈10 keV; (iii) the assumed uncertainties of 1% for the stopping cross sections of α-particles in Na and Cl seem unreasonably small. Furthermore. Astrophysical Journal. analysis of Whitmire & Davids. 9for temperatures typical of convective shell C/Ne burning (T ≈ 1.25Note.-Table 9 is published in the electronic edition of the Astrophysical Journal. analysis of Whitmire & Davids (1974): (i) protons populating higher-lying final states could only be resolved at the highest measured energies; (ii) the uncertainty in the resonance energies is relatively large, amounting to ≈10 keV; (iii) the assumed uncertainties of 1% for the stopping cross sections of α-particles in Na and Cl seem unreasonably small. Furthermore, for temperatures typical of convective shell C/Ne burning (T ≈ 1.25
GK) the Gamow peak is covering a center-of-mass energy range of E cm = 1.2 − 2.2 MeV, i.e., significantly below the energy range covered by experiment. Considering the substantial uncertainties involved, we prefer to use for the reaction rate the estimate provided by the Hauser-Feshbach model. Rauscher & ThielemannGK) the Gamow peak is covering a center-of-mass energy range of E cm = 1.2 − 2.2 MeV, i.e., significantly below the energy range covered by experiment. Considering the substantial uncertainties involved, we prefer to use for the reaction rate the estimate provided by the Hauser-Feshbach model (Rauscher & Thielemann
Clearly, an improved measurement of this reaction is called for. Clearly, an improved measurement of this reaction is called for.
The reaction rate recommended by the NACRE collaboration (Angulo et al. 1999) is exclusively based on these data sets. Si ; Anderson, 654 MeV) Direct measurements of the 25 Mg(α,n) 28 Si and 25 Mg(α,nγ) 28 Si reactions have been reported in Van der Zwan and Geiger. The NACRE collaboration also reports the experimental S-factors for the three references. However, below an energy of 3Si (Q = 2.654 MeV) Direct measurements of the 25 Mg(α,n) 28 Si and 25 Mg(α,nγ) 28 Si reactions have been reported in Van der Zwan and Geiger (1981), Anderson et al. (1983) and Wieland (1995). The reaction rate recommended by the NACRE collaboration (Angulo et al. 1999) is exclusively based on these data sets. The NACRE collaboration also reports the experimental S-factors for the three references. However, below an energy of 3
in the NACRE evaluation. The Gamow peak region at T = 2.3 GK is shown in Fig. 14 as a hatched horizontal bar. Clearly, the Gamow peak region at this temperature, in fact, up to T = 4 GK, is entirely covered by experimental data. Thus, there is no reason to use Hauser-Feshbach rates and thereby introduce another source of uncertainty. We feel that a proper re-analysis of the existing cross section data will not only improve the rate of the 25 Mg(α,n) 28 Si reaction. Geiger Mev The Data From Van Der Zwan, Anderson, ) have been disregarded by NACRE. As a result, in this energy range, their rate is based exclusively on the unpublished work of Wieland. It is interesting to note that the data of Anderson et al. (1983) as reported by the NACRE collaboration disagree with the corresponding curve shown in Fig. 14 by a factor of ≈2. The reason is presumably that NACRE extracted the (α,n) data (which represent only upper limits, as explicitly stated in Anderson et al. 1983) instead of the (α,nγ) data (which are much less susceptible to background). but will likely provide better estimates of the rate uncertainty. at least up to temperatures of 4 GK. Such an analysis is left for future work. We would also like to reiterateMeV the data from Van der Zwan and Geiger (1981) and from Anderson et al. (1983) have been disregarded by NACRE. As a result, in this energy range, their rate is based exclusively on the unpublished work of Wieland (1995), since it is argued in Angulo et al. (1999) that at lower energy background contributions in the earlier works dominate the neutron yield. This conclusion is only partially correct since, for example, Anderson et al. (1983) have also measured the 25 Mg(α,nγ) 28 Si reaction, which shows much less background compared to the (α,n) reaction. The current situation is shown in Fig. 14. The data from Van der Zwan and Geiger (1981) and from Anderson et al. (1983) have been extracted as cross sections from the original figures and converted to astrophysical S-factors. The data from Wieland (1995) are adopted from Angulo et al. (1999). It is interesting to note that the data of Anderson et al. (1983) as reported by the NACRE collaboration disagree with the corresponding curve shown in Fig. 14 by a factor of ≈2. The reason is presumably that NACRE extracted the (α,n) data (which represent only upper limits, as explicitly stated in Anderson et al. 1983) instead of the (α,nγ) data (which are much less susceptible to background). We conclude from Fig. 14 that the available data are in reasonable agreement. It is also inexplicable why already "...above T 9 = 2, H[auser]F[eshbach] rates are used..." in the NACRE evaluation. The Gamow peak region at T = 2.3 GK is shown in Fig. 14 as a hatched horizontal bar. Clearly, the Gamow peak region at this temperature, in fact, up to T = 4 GK, is entirely covered by experimental data. Thus, there is no reason to use Hauser-Feshbach rates and thereby introduce another source of uncertainty. We feel that a proper re-analysis of the existing cross section data will not only improve the rate of the 25 Mg(α,n) 28 Si reaction, but will likely provide better estimates of the rate uncertainty, at least up to temperatures of 4 GK. Such an analysis is left for future work. We would also like to reiterate
In Angulo et al. (1999), the rate for this reaction was obtained by multiplying the experimental (ground state) rate for 26 Al g (p,γ) 27 Si by the ratio of isomeric and ground state rates. but it is not apparent from their work how the results have been obtained. Caughlan & FowlerThe latter ratio was obtained from the Hauser-Feshbach model. However. it is clear from our comments in § 3.2.3 that the latter reaction model may not be applicable to 26 Al+p. Furthermore, the rate in Angulo et al. (1999) is only listed at temperatures of T = 0.018 − 0.4 GK, but the rate needs to be known at higher temperatures as well in order to study the equilibration of 26 Al levels ( § 2.3Rates for this reaction are listed in Caughlan & Fowler (1988), but it is not apparent from their work how the results have been obtained. Presumably their rates were estimated using statistical model calculations (see comments in Ward & Fowler 1980). In Angulo et al. (1999), the rate for this reaction was obtained by multiplying the experimental (ground state) rate for 26 Al g (p,γ) 27 Si by the ratio of isomeric and ground state rates, N A σv m /N A σv g . The latter ratio was obtained from the Hauser-Feshbach model. However, it is clear from our comments in § 3.2.3 that the latter reaction model may not be applicable to 26 Al+p. Furthermore, the rate in Angulo et al. (1999) is only listed at temperatures of T = 0.018 − 0.4 GK, but the rate needs to be known at higher temperatures as well in order to study the equilibration of 26 Al levels ( § 2.3).
He,t) and ( 3 He,α) reaction studies and the subsequent proton decay to the isomeric state was observed in coincidence, providing values for excitation energies and proton branching ratios. In the latter work, the 12 C( 16 O,n) reaction was used to measure γ-ray transitions in 27 Si, allowing for a determination of excitation energies, J π -values and level lifetimes. Nevertheless, too much experimental information is still lacking (i.e., missing levels, spectroscopic factors, proton partial widths, and resonance strengths) in order to estimate this rate reliably over the temperature range of interest. More measurements are clearly in order. In the absence of a more reliable estimate, we approximated in this work the 26 Al m (p,γ) 27 Si rate by the (experimental) ground state rate (see comments in Appendix B). Deibel, the former work, levels in the 27 Si compound nucleus near the proton threshold were populated in. Our assumption is a starting point for exploring the effects of 26 Al m (p,γ) 27 Si reaction rate variationsRecently, some new experimental information has been reported by Deibel et al. (2009) and Lotay et al. (2009). In the former work, levels in the 27 Si compound nucleus near the proton threshold were populated in ( 3 He,t) and ( 3 He,α) reaction studies and the subsequent proton decay to the isomeric state was observed in coincidence, providing values for excitation energies and proton branching ratios. In the latter work, the 12 C( 16 O,n) reaction was used to measure γ-ray transitions in 27 Si, allowing for a determination of excitation energies, J π -values and level lifetimes. Nevertheless, too much experimental information is still lacking (i.e., missing levels, spectroscopic factors, proton partial widths, and resonance strengths) in order to estimate this rate reliably over the temperature range of interest. More measurements are clearly in order. In the absence of a more reliable estimate, we approximated in this work the 26 Al m (p,γ) 27 Si rate by the (experimental) ground state rate (see comments in Appendix B). Our assumption is a starting point for exploring the effects of 26 Al m (p,γ) 27 Si reaction rate variations.
. Al , 23 Na (Q = 2.966 MeVAl(n,α) 23 Na (Q = 2.966 MeV)
We may draw a number of conclusions from the figure. First, the two experimental rates do not agree, even if the large rate uncertainty (26%) in the earlier work is taken into account (see discussion in De Smet et al. 2007 for the possible source of the discrepancy). Second, taking both the experimental uncertainty of the De Smet rate into account, as well as the fact that their rate represents a lower limit near their hightemperature cutoff, the agreement with the Hauser-Feshbach rate near T ≈ 0.3 GK is reasonable. Recall that for the purposes of the present work the rate is of interest at temperatures between 1.1 GK (convective shell C/Ne burning) and 2.3 GK (explosive Ne/C burning), i.e., in a region that has not been covered by experiments. Current reaction rate uncertainties are difficult to quantify and new measurements are called for. Koehler, A direct measurement of the 26 Al(n,α 0 ) 23 Na reaction (i.e., for population of the 23 Na ground sate) has been reported by. Hilaire & KoningThis rate represents a lower limit above T = 0.26 GK (indicated by the vertical line). The theoretical rate, based on the Hauser-Feshbach model, is adopted from Rauscher & Thielemann 2000 and is displayed as a solid blue line. A similar theoretical rate has been reported by Goriely. In the absence of more reliable results, we use a hybrid rate consisting of the results from De Smet et al. (2007) and from Rauscher & Thielemann (2000) below and above T ≈ 0.1 GK, respectively. The rate is listed in the last column of Tab. 9A direct measurement of the 26 Al(n,α 0 ) 23 Na reaction (i.e., for population of the 23 Na ground sate) has been reported by Koehler et al. (1997), while De Smet et al. (2007) have measured the 26 Al(n,α 0 + α 1 ) 23 Na reaction (i.e., for population of the ground and first excited state in 23 Na). The current situation for the reaction rates is displayed in Fig. 15. The experimental rate for 26 Al(n,α 0 ) 23 Na from Koehler et al. (1997) is extracted from their Fig. 4 and is shown as a dashed line. Note that their rates are claimed to be reliable only for T ≤ 0.08 GK. The more recent experimental rate of De Smet et al. (2007) is shown as a black solid line and was obtained by converting the Maxwellian-averaged cross sections (MACS), shown in their Fig. 8, to reaction rates. This rate represents a lower limit above T = 0.26 GK (indicated by the vertical line). The theoretical rate, based on the Hauser-Feshbach model, is adopted from Rauscher & Thielemann 2000 and is displayed as a solid blue line. A similar theoretical rate has been reported by Goriely, Hilaire & Koning (2008). We may draw a number of conclusions from the figure. First, the two experimental rates do not agree, even if the large rate uncertainty (26%) in the earlier work is taken into account (see discussion in De Smet et al. 2007 for the possible source of the discrepancy). Second, taking both the experimental uncertainty of the De Smet rate into account, as well as the fact that their rate represents a lower limit near their high- temperature cutoff, the agreement with the Hauser-Feshbach rate near T ≈ 0.3 GK is reasonable. Recall that for the purposes of the present work the rate is of interest at temperatures between 1.1 GK (convective shell C/Ne burning) and 2.3 GK (explosive Ne/C burning), i.e., in a region that has not been covered by experiments. Current reaction rate uncertainties are difficult to quantify and new measurements are called for. In the absence of more reliable results, we use a hybrid rate consisting of the results from De Smet et al. (2007) and from Rauscher & Thielemann (2000) below and above T ≈ 0.1 GK, respectively. The rate is listed in the last column of Tab. 9.
. Al , 26Al(n,p) 26
. Mg, Q = 4.787 MeVMg (Q = 4.787 MeV)
Mg reaction for the transitions to both the ground and first excited 26 Mg state was directly measured by. The 26 Al. Trautvetter et al.26at a number of neutron energies (corresponding to T ≈ 0.36−3.6 GKThe 26 Al(n,p) 26 Mg reaction for the transitions to both the ground and first excited 26 Mg state was directly measured by Trautvetter et al. (1986) at a number of neutron energies (corresponding to T ≈ 0.36−3.6 GK).
measured the 26 Al(n,p 1 ) 26 Mg reaction, i.e., for population of the first excited state in. Koehler, Koehler et al. (1997) measured the 26 Al(n,p 1 ) 26 Mg reaction, i.e., for population of the first excited state in
Near the overlap region, T ≈ 0.3 GK, the latter rate exceeds the former rate by a factor of 2 (see discussion in Koehler et al. 1997 for the possible source of the discrepancy). Mg, any case, it is shown in both Trautvetter et al. (1986) and in Skelton. The latter rates are claimed to be reliable only for T ≤ 0.3 GK. Kavanagh & Sargood (1987) that the 26 Al(n,p 0 ) 26 Mg reaction rate (i.e., for population of the ground state in 26 Mg) is predicted to be much smaller than the (n,p 1 ) rate. The current situation for the reaction rates is displayed in Fig. 16. The experimental rate forMg. The latter rates are claimed to be reliable only for T ≤ 0.3 GK. Near the overlap region, T ≈ 0.3 GK, the latter rate exceeds the former rate by a factor of 2 (see discussion in Koehler et al. 1997 for the possible source of the discrepancy). The disagreement cannot be explained by unaccounted transitions, because the Koehler rate exceeds the Trautvetter rate. In any case, it is shown in both Trautvetter et al. (1986) and in Skelton, Kavanagh & Sargood (1987) that the 26 Al(n,p 0 ) 26 Mg reaction rate (i.e., for population of the ground state in 26 Mg) is predicted to be much smaller than the (n,p 1 ) rate. The current situation for the reaction rates is displayed in Fig. 16. The experimental rate for
Considering the experimental uncertainty of the Koehler rate near their high-temperature cutoff (20%), the agreement between experimental and theoretical rates near T ≈ 0.2 GK seems reasonable (deviation of 40%). Therefore, we adopt a hybrid rate consisting of the results from Koehler et al. (1997) and from Rauscher & Thielemann (2000) below and above T ≈ 0.2 GK, respectively. The rate is listed in column 7 of Tab. 9. Al ; Of Koehler, The theoretical 26 Al(n,p) 26 Mg rate, based on the Hauser-Feshbach model (Rauscher & Thielemann 2000). Trautvetter et al.Hilaire & KoningNote that we prefer the more recent rates from Koehler et al. (1997) and Rauscher & Thielemann (2000) over the earlier experimental result of. which is displayed as data points in the figure. Our adopted rate exceeds the prediction of Trautvetter et al.. factor of ≈3 near T = 2.5 GK. It is currently difficult to estimate rate uncertainties and new measurements are urgently neededAl(n,p 1 ) 26 Mg from the more recent work of Koehler et al. (1997) is extracted from their Fig. 5 and is shown as a dashed line. The theoretical 26 Al(n,p) 26 Mg rate, based on the Hauser-Feshbach model (Rauscher & Thielemann 2000), is displayed as a solid blue line. An almost identical theoretical rate has been reported by Goriely, Hilaire & Koning (2008). Considering the experimental uncertainty of the Koehler rate near their high-temperature cutoff (20%), the agreement between experimental and theoretical rates near T ≈ 0.2 GK seems reasonable (deviation of 40%). Therefore, we adopt a hybrid rate consisting of the results from Koehler et al. (1997) and from Rauscher & Thielemann (2000) below and above T ≈ 0.2 GK, respectively. The rate is listed in column 7 of Tab. 9. Note that we prefer the more recent rates from Koehler et al. (1997) and Rauscher & Thielemann (2000) over the earlier experimental result of Trautvetter et al. (1986), which is displayed as data points in the figure. Our adopted rate exceeds the prediction of Trautvetter et al. (1986) by a factor of ≈3 near T = 2.5 GK. It is currently difficult to estimate rate uncertainties and new measurements are urgently needed.
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| []
|
[
"Quantum effects on plasma screening for thermonuclear reactions in laser-generated plasmas",
"Quantum effects on plasma screening for thermonuclear reactions in laser-generated plasmas"
]
| [
"David Elsing \nMax-Planck-Institut für Kernphysik\nSaupfercheckweg 1D-69117HeidelbergGermany\n",
"Adriana Pálffy \nMax-Planck-Institut für Kernphysik\nSaupfercheckweg 1D-69117HeidelbergGermany\n\nDepartment of Physics\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nD-91058ErlangenGermany\n",
"Yuanbin Wu \nMax-Planck-Institut für Kernphysik\nSaupfercheckweg 1D-69117HeidelbergGermany\n"
]
| [
"Max-Planck-Institut für Kernphysik\nSaupfercheckweg 1D-69117HeidelbergGermany",
"Max-Planck-Institut für Kernphysik\nSaupfercheckweg 1D-69117HeidelbergGermany",
"Department of Physics\nFriedrich-Alexander-Universität Erlangen-Nürnberg\nD-91058ErlangenGermany",
"Max-Planck-Institut für Kernphysik\nSaupfercheckweg 1D-69117HeidelbergGermany"
]
| []
| A quantum plasma screening model based on the density matrix formalism is used to investigate theoretically the thermonuclear reactions 13 C(α, n) 16 O and 2 H(d, n) 3 He in laser-generated plasmas over a large range of densities and temperatures. For cold and dense (solid-state density) plasmas, our results show that quantum effects can enhance the plasma screening for thermonuclear reactions up to one order of magnitude compared to the classical case. This result can have impact on nuclear astrophysics predictions, and also may play a role for fusion energy gain prospects. Our simulations allow us to identify the laser-generated plasma experimental setting in which the quantum effects on plasma screening could be confirmed at existing high-intensity laser facilities. | 10.1103/physrevresearch.4.l022004 | [
"https://arxiv.org/pdf/2103.13311v2.pdf"
]
| 232,335,519 | 2103.13311 | 79d398a27affd65ca1e127c17071dc50ad94f37e |
Quantum effects on plasma screening for thermonuclear reactions in laser-generated plasmas
20 Apr 2022
David Elsing
Max-Planck-Institut für Kernphysik
Saupfercheckweg 1D-69117HeidelbergGermany
Adriana Pálffy
Max-Planck-Institut für Kernphysik
Saupfercheckweg 1D-69117HeidelbergGermany
Department of Physics
Friedrich-Alexander-Universität Erlangen-Nürnberg
D-91058ErlangenGermany
Yuanbin Wu
Max-Planck-Institut für Kernphysik
Saupfercheckweg 1D-69117HeidelbergGermany
Quantum effects on plasma screening for thermonuclear reactions in laser-generated plasmas
20 Apr 2022(Dated: April 21, 2022)arXiv:2103.13311v2 [physics.plasm-ph]
A quantum plasma screening model based on the density matrix formalism is used to investigate theoretically the thermonuclear reactions 13 C(α, n) 16 O and 2 H(d, n) 3 He in laser-generated plasmas over a large range of densities and temperatures. For cold and dense (solid-state density) plasmas, our results show that quantum effects can enhance the plasma screening for thermonuclear reactions up to one order of magnitude compared to the classical case. This result can have impact on nuclear astrophysics predictions, and also may play a role for fusion energy gain prospects. Our simulations allow us to identify the laser-generated plasma experimental setting in which the quantum effects on plasma screening could be confirmed at existing high-intensity laser facilities.
Introduction. In plasmas, long-range electric fields are screened by the dynamic flow of moving particles. This charge screening enhances the nuclear reaction cross sections by reducing the Coulomb barrier that reacting ions must overcome [1,2]. Plasma screening is a crucial aspect for thermonuclear reactions in astrophysical plasmas such as star cores where nucleosynthesis occurs [1], but also for industrial fusion energy gain [2], for instance inertial confinement fusion [3][4][5][6][7] that may provide future sources of alternative energy. Theoretical plasma screening models focus mostly on classical approaches [8][9][10][11][12][13], while quantum plasma models only address the weak screening regime and have shown good agreement with the classical weak screening [14][15][16][17].
Hints on the important role of quantum effects come from the atomic counterpart of plasmas screening, the plasma-induced ionization potential depression (IPD). Direct measurements [18] have been shown to conflict the extensively used Stewart-Pyatt IPD model [19] which interpolates between the limits of the Debye-Hückel theory [20,21] and the ion-sphere model [8]. The debate is still ongoing [22][23][24][25] and has shed light on the role of quantum effects [26][27][28]. On the front of plasma screening in thermonuclear reactions, the lack of experimental evidence could not resolve several controversies [1,7,29,30], despite numerous theoretical studies [8][9][10][11][12][13][14][15][16][17][31][32][33][34][35][36][37][38][39]. Fortunately, the development of laser technology in the past decades promises the appropriate conditions for conclusive experiments in the lab. Both X-ray Free Electron Lasers (XFELs) [40][41][42][43] and ultra-strong optical lasers [44][45][46][47][48][49] open so-far unavailable parameter regimes for the study of nuclear physics in laser-generated plasmas [3,7,30,[50][51][52]. In particular, theoretical predictions show that experiments at petawatt optical lasers should allow tests of the widely used Salpeter weak screening model for thermonuclear reactions [53,54].
In this Letter we investigate the role of quantum effects for screening in laser-generated plasmas in the intermediate screening regime of low temperature and high den-sity. This regime has became available experimentally at newly commissioned laser facilities and is relevant for the evolution of low mass stars, brown dwarfs, and pre-mainsequence stars as well as the lithium depletion problem [55][56][57][58][59][60][61][62][63][64][65][66]. We employ the density matrix formalism (DMF) derived in quantum statistical mechanics [14] to include quantum effects in plasma screening and compare our results with classical predictions over a large range of densities and temperatures accessible in laser-generated plasmas. Surprisingly, in the intermediate screening regime the DMF quantum plasma model predicts up to one order of magnitude higher screening factors than the classical plasma models. This enhancement is sufficiently large to be observed experimentally in laser-generated plasma experiments. We investigate three realistic experimental settings at existing petawatt and x-ray laser facilities and determine nuclear reaction rates for currently accessible experimental parameters. Based on our predictions, we identify the most promising experimental scenario and put forward an experimental test of quantum effects in the intermediate screening regime.
As case studies we choose two thermonuclear reactions: (i) 13 C(α, n) 16 O which is one of the important helium burning processes as well as one of the main neutron sources for the s-process [67][68][69][70][71][72], and (ii) 2 H(d, n) 3 He, one of the key reactions in the study of inertial confinement fusion [4,6,7]. The quantum screening results are compared with the Salpeter weak screening [8] and the Mitler formula [9,10], both based on classical plasma models. In addition, for the 13 C(α, n) 16 O reaction in a cold and dense plasma case we also compare our results with interpolation formulae for the Salpeter and Von Horn intermediate screening (SVH) [11] and the GDGC classical plasma model by Graboske et al. [13]. These models use numerical interpolation of classical model results for the intermediate screening regime. Our numerical results show that all models display good agreement for low plasma densities and high temperatures, confirming previous results [14][15][16][17]. However, for low temperature and high density, the DMF quantum plasma model presents a substantial enhancement of the plasma screening. This holds true also for the comparison with results of the Mitler formula [9,10] which is based on the Stewart-Pyatt IPD model and should be valid for the full range of plasma parameters.
Theory. We consider the fusion reaction of two positively charged nuclei with charge numbers Z 1 and Z 2 . Due to screening, the nuclear reaction rate in plasmas can be enhanced by a factor g scr [2] defined as <σv> scr = g scr < σv >, where < σv > is the the averaged reactivity neglecting screening, σ is the nuclear reaction cross section and v the particle relative velocity, respectively. < σv > is given by the averaging of σv over the reactant velocity distribution. In weakly coupled plasmas, i.e., plasmas in which the Coulomb interaction energy between the nucleus and the nearest few electrons and nuclei is small compared to the thermal energy, the classical Salpeter model [8,14,53] gives the plasma screening enhancement factor (in atomic units with the Boltzmann constant k B = 1) g scr = exp [Z 1 Z 2 /(λ D T )], where λ D is the Debye length, and T is the plasma temperature. This holds for low-density and high-temperature plasmas. For dense plasmas, Salpeter applied the ion-sphere approximation to the strong screening [8]. The intermediate regime can be described by numerical interpolations in the SVH approach [11].
Starting from the Stewart-Pyatt IPD model [19], Mitler considered the charge density to be constant for small distances close to the nucleus (ion-sphere model), while applying the Debye-Hückel theory at large distances [9,10]. This lead to an expression valid over the entire range of plasma parameters for the screening enhancement factor [9,10] g scr = exp 8π 2 n 2
e λ 5 D |∆G| /(5T ) ,(1)
where n e is the electron density and
∆G = (ζ Z1 + ζ Z2 + 1) 5 3 − (ζ Z1 + 1) 5 3 − (ζ Z2 + 1) 5 3 + 1, (2) with ζ Z = 3Z/(4πn e λ 3 D )
. We now turn to the DMF quantum plasma model introduced in Ref. [14]. In this model, the electron density is derived via the density matrix equation known in quantum statistical mechanics [14,73],
∂ρ(r ′ , r; β) ∂β = ∇ 2 r ′ /2 + Φ(r ′ ) ρ(r ′ , r; β)(3)
with the initial condition ρ(r ′ , r;
β = 0) = δ (3) (r ′ − r),
where β = 1/T and Φ is the potential around the nuclear charge. The electron density should be normalised by the solution of Eq. (3) for nuclear charge Z = 0, ρ 0 (β) = (2πβ) −3/2 [14]. Using this normalisation, the total electron density becomes ρ e (r) = n e (2πβ) 3/2 ρ(r, r; β). The potential Φ surrounding a nuclear charge Z is then described by the modified Poisson-Boltzmann equation [14]
∇ 2 δΦ = −4π n b i Z i X i A i exp (−βZ i Φ) − ρ e ,(4)
where δΦ = Φ − Z/r, n b is the baryon density, and X i and A i are the mass fraction and the mass number of i-species ion, respectively. Equations (3) and (4) can be solved self-consistently. We note that the spatial dependence of the screening potential has also been considered in the WKB approximation in Refs. [74][75][76], however, for the strong screening regime. In our case, with the solution of Eqs. (3) and (4) for the density distribution and potential, the plasma screening enhancement factor can be obtained as
g scr = exp [−β (F Z1+Z2 − F Z1 − F Z2 )]
where, F Z stands for the free energy obtained in terms of the electrostatic energy [14]. Numerical approach. The numerical approach for solar plasma parameters described in Ref. [14] fails to provide accurate electron densities for small temperatures and high densities. For this parameter regime, the numerical integration of Eq. (3) becomes cumbersome for two reasons: (1) The initial value is a Dirac δ function, which is approximated by a Gaussian with an appropriate width, requiring a small grid spacing near r ′ = r;
U Z via F Z = 1 β β 0 U Z (τ )dτ
(2) The potential Φ(r ′ ) diverges at r ′ = 0 and has to be approximated numerically by using a regularization procedure [14]. The increasing error stemming from the β integration in Eq. (3) becomes quickly noticeable when going to smaller temperatures.
Eq. (3) has cylindrical symmetry and is solved on a non-linear two-dimensional grid in cylindrical coordinates R and z. Since the grid spacing is very small at the points where ∂ρ/∂β is very large, we combine an implicit Crank-Nicolson step size [77] with the Runge-Kutta-Fehlberg method [78]. For large r ′ , Φ is approximated by the weak screening potential of the Mitler model, while the density matrix is set to zero for the finite differences at the boundaries of the numerically considered r ′ region. Between the iterations in solving Eqs. (3) and (4), both the density and the potential are interpolated to the new grid by cubic spline interpolation. With the numerical procedure described above, free energies F Z obtained agree with the ones in Ref. [14] on the level of a few percent.
Numerical results. We calculate the plasma screening enhancement factor g scr for the 13 C(α, n) 16 O reaction occurring in a Helium plasma. Figure 1 presents the calculated screening enhancement factor compared to classical model results as a function of plasma temperature for He number densities 10 24 cm −3 and 10 21 cm −3 (in the inset). The results show that for lower density, the DMF model predicts similar enhancement factors as the weak (Salpeter) screening model, confirming previous predictions [14]. At the same time, the Mitler formula pre-dicts sightly smaller enhancement factors than the weak screening at low temperatures, but the overall differences between models remain small. Thus, no significant quantum effects are expected to affect experiments based on the 13 C(α, n) 16 O reaction aimed at determining g scr in the weak screening regime [53].
For the high density case, the five plasma screening models show good overall agreement only for high temperatures and spread out significantly for temperatures of few hundreds eV. Here, DMF predicts the highest plasma screening enhancement factor among the considered models. The Mitler, SVH and GDGC predictions lie below the Salpeter weak screening, which in turn remains one order of magnitude lower than the DMF results at 200 eV temperature. Figure 2 shows the density dependence of the plasma screening factor at this temperature and reveals that the disagreement between models becomes increasingly visible starting with densities of few times 10 22 cm −3 . We note that the Mitler formula is expected to be valid for the full range of plasma parameters, while the intermediate screening SVH formula is expected to have an error on the level of 30% for the cases of screening enhancement factor around the value of e, the Euler number [11]. In addition, the GDGC intermediate screening formula obtained assuming completely degenerate electrons [12,13], would underestimate the screening effect [14] when applied to our case which is only weakly degenerate.
From Figs. 1 and 2, we can conclude that the quantum plasma effects become significant in the intermediate screening regime where the Salpeter weak screening also deviates significantly from the other classical plasma models. For both classical and quantum models, the low temperature and high density regime highlights the Coulomb effects in the immediate vicinity of the reacting nuclei. It is this physical region where the model assumptions for the Salpeter weak screening and the Mitler formula differ most, and where the quantum effects originate. Thus, low temperature (approx. 200 eV) and increasing density > 10 23 cm −3 are the physical conditions under which the quantum plasma effects become important. This in turn indicates that laser experiments based on gas jet targets cannot distinguish between screening models, as the highest density of gas jets achieved at present is approx. 10 21 cm −3 [79,80]. Solid-state density plasma targets should be used instead.
We now turn to the second investigated thermonuclear reaction 2 H(d, n) 3 He occurring in a deuterium plasma. 2 H(d, n) 3 He is one of the key reactions in the study of the inertial confinement fusion [4,6,7]. Calculated plasma screening enhancement factors g scr using the Salpeter week screening, the Mitler formula and the DMF model are presented in Table I for selected plasma density and temperature values. We observe also in this case the same general trend. While the three models agree very well for k B T = 400 eV and deuterium ion density n i = 10 21
• • • • • • • • [ ] • • • • • • • • FIG. 1.
Plasma screening enhancement factor gscr for the 13 C(α, n) 16 O reaction as a function of the He plasma temperature for weak screening [8] (black solid curve), Mitler formula [9] (brown dashed curve), SVH interpolation [11] (magenta dotted curve), GDGC interpolation [13] (grey dash-dotted curve), and the DMF model (orange filled circle). We consider the plasma density 10 24 cm −3 (10 21 cm −3 for inset). cm −3 , discrepancies occur going to larger densities and lower temperatures. The predicted values for n i = 10 21 cm −3 show that also for the 2 H(d, n) 3 He reaction, the screening effect would be negligible in gas-jet experiments which are bound to low ion density values, as already implied in Ref. [54]. For higher plasma densities, the DMF model provides the largest screening factor, while the Mitler formula gives the smallest one. However, the disagreements are on the order of only 2% even for a plasma temperature of 200 eV and plasma density 10 24 cm −3 . The quantum plasma effects are weaker in the case of 2 H(d, n) 3 He compared with 13 C(α, n) 16 O. This can be explained by the lower charges of the nuclear reactants and therefore the weaker Coulomb fields in the immedi- [81][82][83][84] or by using XFELs [85,86]. Thus, the intermediate regime screening conditions should be accessible experimentally. In the following we address three realistic scenarios that promise to shed light on the screening mechanism and the role of quantum effects. In all cases, determining the screening factor requires three successive experiments: the thermonuclear reaction occurring in the setup (1) in the absence of a plasma target, (2) with a low-density and (3) with a high-density plasma target, the latter two at the same temperature. In parallel, plasma diagnostics on density, temperature, and energy distributions will be necessary in order to allow for data reconstruction. By comparison between the three experimental outcomes and with theory, the value of the plasma screening factor could be deduced [53].
Setup I. At mega-Joule laser facilities such as the National Ignition Facility (NIF) in the US [49], thermonuclear reactions have been observed in solid-state or even higher density plasmas with temperatures from a few hundred eV to a few keV [4,6,7,50,87]. However, the considered 13 C(α, n) 16 O reaction has the disadvantage that experimental cross section data is not available for few hundreds eV energies [88], rendering unreliable extrapolations to a region of very small cross sections necessary. The situation is more promising for the reaction 2 H(d, n) 3 He, for which cross sections have been well measured experimentally down to rather low energy [88]. To give a numerical estimate, we follow the spherical plasma model in Ref. [54]. We assume a laser energy of 1 MJ per pulse, deuterium ion density 10 24 cm −3 , plasma temperature is 200 eV, and the interaction time 1 ns. The calculated 2 H(d, n) 3 He event numbers per pulse are 4.01 · 10 7 s −1 for the DMF model, 3.94 · 10 7 s −1 for the weak screening, and 3.86 · 10 7 s −1 for the Mitler model, respectively. Unfortunately, these values are too close to identify the underlying plasma screening mechanism.
Setup II. At the upcoming Nuclear Pillar of the Extreme Light Infrastructure facility (ELI-NP) [46], an experimental setup involving two laser beams that generate two colliding plasmas has been proposed [30]. A laser pulse interacting with a first solid-state target should produce a rapidly streaming projectile plasma by means of target normal sheath acceleration (TNSA). This particle beam interacts with the (secondary) target plasma generated by the interaction of a second laser pulse on a gas jet target [30]. This setup has been already successfully applied in the context of aneutronic fusion reactions [3]. Theoretical predictions show that at ELI-NP, this experimental design will be suitable for testing weak screening models [53]. For the parameter regime investigated here, a modified setup could be employed by changing the secondary gas jet target to a liquid drop target or a solid target [89] to test also the intermediary screening regime.
For an ELI-NP beam for TNSA with 10 PW power, 25 fs pulse duration, 800 nm wavelength and intensity of 3 · 10 19 W/cm 2 , and a target thickness of 2 µm, our simulations show that an accelerated beam of C 5+ ions could be generated. We choose the parameters such that in the center-of-mass reference frame, the reaction energy is shifted towards 1 MeV, in a region where the 13 C(α, n) 16 O cross sections have been measured experimentally with precision of a few percent [90] and astrophysical Sfactor has a smooth energy dependence. Using a plasma target of temperature 200 eV, thickness 10 µm and density 10 24 cm −3 , the calculated reaction events per laser pulse are 4.4·10 5 for the DMF, 1.2·10 5 for weak screening and 3.4 · 10 4 for the Mitler models, respectively. While the reaction numbers lie apart by seizable factors, the experimental conditions at ELI-NP are likely to introduce temperature and density spatial and temporal gradients or non-thermal ion distributions, which are difficult to model theoretically or check experimentally.
Setup III. A strategy to mitigate the non-ideal plasma effects described above is to replace the second optical laser by an XFEL beam. Due to the high x-ray penetration power, the XFEL-generated plasma is expected to have more uniform conditions [86]. The generation of cold and dense plasmas isochoricaly at XFEL has been demonstrated experimentally [18,85,86]. We therefore address in the following a setup combining optical and x-ray laser beams as the one at the Helmholtz International Beamline for Extreme Fields (HIBEF) [91]. In this scenario, an optical laser would generate via TNSA the rapidly streaming projectile plasma, while the XFEL heats the secondary target to solid-state density [18,85,86]. As suggested in Ref. [92], in order to match XFELs in both power and repetition rate, mJ-class optical lasers are adopted. We consider laser pulses with 100 mJ energy, 100 fs duration, and 800 nm wavelength. The intensity of the optical laser is assumed to be 2 × 10 19 W/cm 2 . The optical laser interacts with a solid 13 C target with a thickness of 2 µm to generate a 13 C ion beam. The ion beam interacts with a 4 He plasma target generated by the XFEL with 4 He ion density of 10 24 cm −3 and 200 eV temperature. Also in this case we choose the parameters such that the center-of-mass reaction energy is shifted towards 1 MeV to avoid a complicated energy dependence of the astrophysical S-factor of the nuclear reaction. Assuming a repetition rate of 10 kHz for the optical laser and XFEL, we obtain the reaction rates 1.31 · 10 7 s −1 for the DMF, 3.65 · 10 6 s −1 for weak screening, and 1.02 · 10 6 s −1 for the Mitler models, respectively. With the advantage of high repetition rates, more uniform plasma conditions, together with well measured nuclear reaction cross sections in relevant energies, and the large discrepancies of the reaction event rate between different screening models, this scenario could give a compelling evidence of the plasma screening model in the 13 C(α, n) 16 O reaction.
Outlook. The enhancement predicted by the quantum plasma screening model occurs in the intermediate screening regime, which could be rescaled to the astrophysical conditions relevant to the evolution of low mass stars, brown dwarfs, and pre-main-sequence stars as well as the lithium depletion problem [55][56][57][58][59][60][61][62][63][64][65][66]. Such enhancement would lead to significantly impact in relevant realistic astrophysical scenarios, and hence would play important roles in the evolution of the above mentioned stars and the understanding of the lithium depletion problem. Furthermore, we may speculate that an experimental confirmation of quantum effects in plasma screening might shed light on the origin of the conflict between direct IPD measurements [18] and the extensively used Stewart-Pyatt IPD model [19]. Once confirmed, the quantum effect on the plasma screening enhancement may also play an important role for fusion energy gain prospects [3,4,6,7,93].
AP gratefully acknowledges support from the Heisenberg Program of the Deutsche Forschungsgemeinschaft (DFG).
FIG. 2 .
2Plasma screening enhancement factor gscr for the 13 C(α, n)16 O reaction as function of helium plasma density at temperature 200 eV for the same models as inFig. 1.
TABLE I .
IPlasma screening enhancement factors gscr for the
reaction 2 H(d, n) 3 He. A deuterium plasma with ion density
ni in units of cm −3 is assumed. The plasma temperature T
is given in units of keV. Salpeter weak screening [8], Mitler
formula [9], and the present DMF model are considered.
ate vicinity of the nuclei. The small quantum effects are
correlated with a better agreement between the Salpeter
and Mitler screening factors, showing that for the 2 H(d,
n) 3 He reaction, the intermediate screening regime would
require even higher density and lower temperature.
Possible experimental verification. Recent works have
shown that solid-state density targets can be heated to
temperatures starting from a few hundred eV up to a
few keV either via isochoric heating
* Present address: Institute of Nanotechnology, KIT Campus North, Hermann-von-Helmholtz-Platz. 176344* Present address: Institute of Nanotechnology, KIT Campus North, Hermann-von-Helmholtz-Platz 1, 76344
Eggenstein-Leopoldshafen, Germany † Corresponding [email protected]. Eggenstein-Leopoldshafen, Germany † Corresponding [email protected]
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